Springer Series in

OPTICAL SCIENCES founded by H.K.V. Lotsch Editor-in-Chief: W.T. Rhodes, Atlanta Editorial Board: A. Adibi, Atlanta T. Asakura, Sapporo T.W. Hänsch, Garching T. Kamiya, Tokyo F. Krausz, Garching B. Monemar, Linköping H. Venghaus, Berlin H. Weber, Berlin H. Weinfurter, München

35

Springer Series in

OPTICAL SCIENCES The Springer Series in Optical Sciences, under the leadership of Editor-in-Chief William T. Rhodes, Georgia Institute of Technology, USA, provides an expanding selection of research monographs in all major areas of optics: lasers and quantum optics, ultrafast phenomena, optical spectroscopy techniques, optoelectronics, quantum information, information optics, applied laser technology, industrial applications, and other topics of contemporary interest. With this broad coverage of topics, the series is of use to all research scientists and engineers who need up-to-date reference books. The editors encourage prospective authors to correspond with them in advance of submitting a manuscript. Submission of manuscripts should be made to the Editor-in-Chief or one of the Editors. See also www.springeronline.com/series/624

Editor-in-Chief William T. Rhodes Georgia Institute of Technology School of Electrical and Computer Engineering Atlanta, GA 30332-0250, USA E-mail: [emailprotected]

Editorial Board Ali Adibi

Bo Monemar

Georgia Institute of Technology School of Electrical and Computer Engineering Atlanta, GA 30332-0250, USA E-mail: [emailprotected]

Department of Physics and Measurement Technology Materials Science Division Linköping University 58183 Linköping, Sweden E-mail: [emailprotected]

Toshimitsu Asakura Hokkai-Gakuen University Faculty of Engineering 1-1, Minami-26, Nishi 11, Chuo-ku Sapporo, Hokkaido 064-0926, Japan E-mail: [emailprotected]

Theodor W. Hänsch Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Straße 1 85748 Garching, Germany E-mail: [emailprotected]

Takeshi Kamiya Ministry of Education, Culture, Sports Science and Technology National Institution for Academic Degrees 3-29-1 Otsuka, Bunkyo-ku Tokyo 112-0012, Japan E-mail: [emailprotected]

Ferenc Krausz Ludwig-Maximilians-Universität München Lehrstuhl für Experimentelle Physik Am Coulombwall 1 85748 Garching, Germany and Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Straße 1 85748 Garching, Germany E-mail: [emailprotected]

Motoichi Ohtsu University fo Tokyo Department of Electronic Engineering 7-3-1 Hongo, Bunkyo-ku Tokyo 113-8959, Japan E-mail: [emailprotected]

Herbert Venghaus Fraunhofer Institut für Nachrichtentechnik Heinrich-Hertz-Institut Einsteinufer 37 10587 Berlin, Germany E-mail: [emailprotected]

Horst Weber Technische Universität Berlin Optisches Institut Straße des 17. Juni 135 10623 Berlin, Germany E-mail: [emailprotected]

Harald Weinfurter Ludwig-Maximilians-Universität München Sektion Physik Schellingstraße 4/III 80799 München, Germany E-mail: harald.weinfurter@physik. uni-muenchen.de

Keigo Iizuka

Engineering Optics Third Edition

With 433 Figures

K EIGO I IZUKA Department of Electrical & Computer Engineering University of Toronto Toronto, ON Canada M5S 1A4 [emailprotected]

ISBN 978-0-387-75723-0

e-ISBN 978-0-387-75724-7

Library of Congress Control Number: 2007937162 c 2008 Springer Science+Business Media, LLC ° All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com

Preface to the Third Edition

On my 1990 sabbatical leave from the University of Toronto, I worked for the NHK (Japan Broadcasting Station) Science and Technology Laboratory. I recall my initial visit to the executive floor of NHK, a spacious room with a commanding view of metropolitan Tokyo. I could not help but notice High Definition Televisions (HDTVs) in every corner of the room. At that time HDTV technology was just breaking into the marketplace, and there was fierce competition for mass-producing HDTVs. When I entered the room the HDTVs were turned on one after the other as my host proclaimed, “This is the future of the television. Isn’t the quality of the picture superb?” I replied, “They are indeed superb, but if the HDTV images were 3D, that would be really out-of-this world.” The host continued, “3D HDTV is our future project.” That was nearly twenty years ago, and much progress has been made in 3D imaging techniques since then and this future is here with us. The amount of information in the literature on this subject is now substantial. 3D imaging techniques are not merely limited to the television industry. The basic techniques are also used in the automobile, electronics, entertainment and medical industries. Inspired by this area of growth, I added Chap. 16 on 3D imaging, which starts with a brief historical introduction of 3D imaging from the ancient Greek era to modern times. The chapter then expands on basic principles of 13 different types of 3D displays. The errata of the second edition of Engineering Optics, which were previously published on my Web site, have been corrected in this edition. A few new problem set-type questions have been added for teaching purposes. I would like to express my sincere gratitude to my colleague and fellow engineer, Ms. Mary Jean Giliberto for her excellent editing assistance. This book and its previous edition would not have been possible without the unconditional caring support of my wife and family. Toronto, October 2007

Keigo Iizuka

v

Preface to the Second Edition

The first edition of this textbook was published only last year, and now, the publisher has decided to issue a paperback edition. This is intended to make the text more affordable to everyone who would like to broaden their knowledge of modern problems in optics. The aim of this book is to provide a basic understanding of the important features of the various topics treated. A detailed study of all the subjects comprising the field of engineering optics would fill several volumes. This book could perhaps be likened to a soup: it is easy to swallow, but sooner or later heartier sustenance is needed. It is my hope that this book will stimulate your appetite and prepare you for the banquet that could be yours. I would like to take this opportunity to thank those readers, especially Mr. Branislav Petrovic, who sent me appreciative letters and helpful comments. These have encouraged me to introduce a few minor changes and improvements in this edition. Toronto, September 1986

Keigo Iizuka

vii

Preface to the First Edition

“Which area do you think I should go into?” or “Which are the areas that have the brightest future?” are questions that are frequently asked by students trying to decide on a field of specialization. My advice has always been to pick any field that combines two or more disciplines such as Nuclear Physics, Biomedical Engineering, Optoelectronics, or even Engineering Optics. With the ever growing complexity of today’s science and technology, many a problem can be tackled only with the cooperative effort of more than one discipline. Engineering Optics deals with the engineering aspects of optics, and its main emphasis is on applying the knowledge of optics to the solution of engineering problems. This book is intended both for the physics student who wants to apply his knowledge of optics to engineering problems and for the engineering student who wants to acquire the basic principles of optics. The material in the book was arranged in an order that would progressively increase the student’s comprehension of the subject. Basic tools and concepts presented in the earlier chapters are then developed more fully and applied in the later chapters. In many instances, the arrangement of the material differs from the true chronological order. The following is intended to provide an overview of the organization of the book. In this book, the theory of the Fourier transforms was used whenever possible because it provides a simple and clear explanation for many phenomena in optics. Complicated mathematics have been completely eliminated. Chapter 1 gives a historical prospective of the field of optics in general. It is amazing that, even though light has always been a source of immense curiosity for ancient peoples, most principles of modern optics had to wait until the late eighteenth century to be conceived, and it was only during the mid-nineteenth century with Maxwell’s equations that modern optics was fully brought to birth. The century following that event has been an exciting time of learning and a tremendous growth which we have been witnessing today. Chapter 2 summarizes the mathematical functions which very often appear in optics and it is intended as a basis for the subsequent chapters.

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Preface to the First Edition

Chapter 3 develops diffraction theory and proves that the far field diffraction pattern is simply the Fourier transform of the source (or aperture) function. This Fourier-transform relationship is the building block of the entire book (Fourier optics). Chapter 4 tests the knowledge obtained in Chaps. 2 and 3. A series of practical examples and their solutions are collected in this chapter. Chapter 5 develops geometrical optics which is the counterpart of Fourier optics appearing in Chap. 3. The power of geometrical optics is convincingly demonstrated when working with inhomogeneous transmission media because, for this type of media, other methods are more complicated. Various practical examples related to fiber optics are presented so that the basic knowledge necessary for fiber optical communication in Chap. 13 is developed. Chapter 6 deals with the Fourier transformable and image formable properties of a lens using Fourier optics. These properties of a lens are abundantly used in optical signal processing appearing in Chap. 11. Chapter 7 explains the principle of the Fast Fourier Transform (FFT). In order to construct a versatile system, the merits of both analog and digital processing have to be cleverly amalgamated. Only through this hybrid approach can systems such as computer holography, computer tomography or a hologram matrix radar become possible. Chapter 8 covers both coherent and white light holography. The fabrication of holograms by computer is also included. While holography is popularly identified with its ability to create three-dimensional images, the usefulness of holography as a measurement technique deserves equal recognition. Thus, holography is used for measuring minute changes and vibrations, as a machining tool, and for profiling the shape of an object. Descriptions of microwave holography are given in Chap. 12 as a separate chapter. Knowledge about the diffraction field and FFT which are found in Chaps. 3 and 6 are used as the basis for many of the discussions on holography. Chapter 9 shows a pictorial cook book for fabricating a hologram. Experience in fabricating a hologram could be a memorable initiation for a student who wishes to be a pioneer in the field of engineering optics. Chapter 10 introduces analysis in the spatial frequency domain. The treatment of optics can be classified into two broad categories: one is the space domain, which has been used up to this chapter, and the other is the spatial frequency domain, which is newly introduced here. These two domains are related by the Fourier-transform relationship. The existence of such dual domains connected by Fourier transforms is also found in electronics and quantum physics. Needless to say, the final results are the same regardless of the choice of the domain of analysis. Examples dealing with the lens in Chap. 6 are used to explain the principle. Chapter 11 covers optical signal processing of various sorts. Knowledge of diffraction, lenses, FFT and holography, covered in Chaps. 3, 6, 7 and 8, respectively, is used extensively in this chapter. In addition to coherent and incoherent optical processing, Chap. 11 also includes a section on tomography. Many examples are

Preface to the First Edition

xi

given in this chapter with the hope that they will stimulate the reader’s imagination to develop new techniques. Chapter 12 is a separate chapter on microwave holography. While Chap. 8 concerns itself primarily with light wave holography, Chap. 12 extends the principles of holography to the microwave region. It should be pointed out that many of the techniques mentioned here are also applicable to acoustic holography. Chapter 13 describes fiber-optical communication systems which combine the technologies of optics and those of communications. The treatment of the optical fiber is based upon the geometrical-optics point of view presented in Chap. 5. Many of the components developed for fiber-optical communication systems find applications in other areas as well. Chapter 14 provides the basics necessary to fully understand integrated optics. Many an integrated optics device uses the fact that an electro- or acousto-optic material changes its refractive index according to the external electric field or mechanical strain. The index of refraction of these materials, however, depends upon the direction of the polarization of the light (anisotropic) and the analysis for the anisotropic material is different from that of isotropic material. This chapter deals with the propagation of light in such media. Chapter 15 deals with integrated optics, which is still such a relatively young field that almost anyone with desire and imagination can contribute. Mrs. Mary Jean Giliberto played an integral role in proof reading and styling the English of the entire book. Only through her devoted painstaking contribution was publication of this book possible. I would like to express my sincere appreciation to her. Mr. Takamitsu Aoki of Sony Corporation gave me his abundant cooperation. He checked all the formulas, and solved all problem sets. He was thus the man behind the scenes who played all of these important roles. I am thankful to Dr. Junichi Nakayama of Kyoto Institute of Technology for helping me to improve various parts of Chap. 10. I am also grateful to Professor Stefan Zukotynski and Mr. Dehuan He of The University of Toronto for their assistance. The author also wishes to thank Professor T. Tamir of The Polytechnic Institute of New York, Brooklyn and Dr. H. K. V. Lotsch of Springer Verlag for critical reading and correcting the manuscript. Mr. R. Michels of Springer-Verlag deserves praise for his painstaking efforts to convert the manuscript into book form. Megumi Iizuka helped to compile the Subject Index. Toronto, August 1985

Keigo Iizuka

Contents

Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1

History of Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Mysterious Rock Crystal Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Ideas Generated by Greek Philosophers . . . . . . . . . . . . . . . . . . . . . . . 1.3 A Morning Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Renaissance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Lengthy Path to Snell’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 A Time Bomb to Modern Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Newton’s Rings and Newton’s Corpuscular Theory . . . . . . . . . . . . . 1.8 Downfall of the Corpuscle and Rise of the Wave . . . . . . . . . . . . . . . 1.9 Building Blocks of Modern Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Quanta and Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Reconciliation Between Waves and Particles . . . . . . . . . . . . . . . . . . . 1.12 Ever Growing Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 6 7 10 11 12 16 17 20 23 24 24

2

Mathematics Used for Expressing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Spherical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Cylindrical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Interference of Two Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Spatial Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Relationship Between Engineering Optics and Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Special Functions Used in Engineering Optics and Their Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27 28 31 33 34 37

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Contents

2.7.1 The Triangle Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 The Sign Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 The Step Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.4 The Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.5 The Comb Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Fourier Transform in Cylindrical Coordinates . . . . . . . . . . . . . . . . . . 2.8.1 Hankel Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Examples Involving Hankel Transforms . . . . . . . . . . . . . . . 2.9 A Hand-Rotating Argument of the Fourier Transform . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38 38 39 40 41 43 44 47 50 50

3

Basic Theory of Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Kirchhoff’s Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fresnel-Kirchhoff Diffraction Formula . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Fresnel-Kirchhoff’s Approximate Formula . . . . . . . . . . . . . . . . . . . . 3.4 Approximation in the Fraunhofer Region . . . . . . . . . . . . . . . . . . . . . . 3.5 Calculation of the Fresnel Approximation . . . . . . . . . . . . . . . . . . . . . 3.6 One-Dimensional Diffraction Formula . . . . . . . . . . . . . . . . . . . . . . . . 3.7 The Fresnel Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 57 60 63 63 66 68 71

4

Practical Examples of Diffraction Theory . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Diffraction Problems in a Rectangular Coordinate System . . . . . . . 4.2 Edge Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Diffraction from a Periodic Array of Slits . . . . . . . . . . . . . . . . . . . . . 4.4 Video Disk System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Reflection Grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Principle of the Video Disk System . . . . . . . . . . . . . . . . . . . 4.5 Diffraction Pattern of a Circular Aperture . . . . . . . . . . . . . . . . . . . . . 4.6 One-Dimensional Fresnel Zone Plate . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Two-Dimensional Fresnel Zone Plate . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 78 82 85 85 87 89 91 95 98

5

Geometrical Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1 Expressions Frequently Used for Describing the Path of Light . . . . 101 5.1.1 Tangent Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1.2 Curvature of a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.1.3 Derivative in an Arbitrary Direction and Derivative Normal to a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.2 Solution of the Wave Equation in Inhomogeneous Media by the Geometrical-Optics Approximation . . . . . . . . . . . . . . . . . . . . 107 5.3 Path of Light in an Inhomogeneous Medium . . . . . . . . . . . . . . . . . . . 111 5.4 Relationship Between Inhomogeneity and Radius of Curvature of the Optical Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.5 Path of Light in a Spherically Symmetric Medium . . . . . . . . . . . . . . 117

Contents

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5.6 5.7

Path of Light in a Cylindrically Symmetric Medium . . . . . . . . . . . . 122 Selfoc Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.7.1 Meridional Ray in Selfoc Fiber . . . . . . . . . . . . . . . . . . . . . . 126 5.7.2 Skew Ray in Selfoc Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.8 Quantized Propagation Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.8.1 Quantized Propagation Constant in a Slab Guide . . . . . . . 129 5.8.2 Quantized Propagation Constant in Optical Fiber . . . . . . . 131 5.9 Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6

Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.1 Design of Plano-Convex Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.2 Consideration of a Lens from the Viewpoint of Wave Optics . . . . . 141 6.3 Fourier Transform by a Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.3.1 Input on the Lens Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.3.2 Input at the Front Focal Plane . . . . . . . . . . . . . . . . . . . . . . . 143 6.3.3 Input Behind the Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.3.4 Fourier Transform by a Group of Lenses . . . . . . . . . . . . . . 147 6.3.5 Effect of Lateral Translation of the Input Image on the Fourier-Transform Image . . . . . . . . . . . . . . . . . . . . . 148 6.4 Image Forming Capability of a Lens from the Viewpoint of Wave Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.5 Effects of the Finite Size of the Lens . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.5.1 Influence of the Finite Size of the Lens on the Quality of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.5.2 Influence of the Finite Size of the Lens on the Image Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7

The Fast Fourier Transform (FFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7.1 What is the Fast Fourier Transform? . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7.2 FFT by the Method of Decimation in Frequency . . . . . . . . . . . . . . . . 164 7.3 FFT by the Method of Decimation in Time . . . . . . . . . . . . . . . . . . . . 172 7.4 Values of W k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8

Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 8.1 Pictorial Illustration of the Principle of Holography . . . . . . . . . . . . . 181 8.2 Analytical Description of the Principle of Holography . . . . . . . . . . . 183 8.3 Relationship Between the Incident Angle of the Reconstructing Beam and the Brightness of the Reconstructed Image . . . . . . . . . . . 188 8.4 Wave Front Classification of Holograms . . . . . . . . . . . . . . . . . . . . . . 190 8.4.1 Fresnel Hologram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 8.4.2 Fourier Transform Hologram . . . . . . . . . . . . . . . . . . . . . . . . 190 8.4.3 Image Hologram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 8.4.4 Lensless Fourier Transform Hologram . . . . . . . . . . . . . . . . 191

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8.5 8.6 8.7 8.8

Holograms Fabricated by a Computer . . . . . . . . . . . . . . . . . . . . . . . . 192 White-Light Hologram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Speckle Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Applications of Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.8.1 Photographs with Enhanced Depth of Field . . . . . . . . . . . . 205 8.8.2 High-Density Recording . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.8.3 Optical Memory for a Computer . . . . . . . . . . . . . . . . . . . . . 205 8.8.4 Holographic Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.8.5 Laser Machining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.8.6 Observation of Deformation by Means of an Interferometric Hologram . . . . . . . . . . . . . 210 8.8.7 Detection of the Difference Between Two Pictures . . . . . . 212 8.8.8 Observation of a Vibrating Object . . . . . . . . . . . . . . . . . . . . 213 8.8.9 Generation of Contour Lines of an Object . . . . . . . . . . . . . 214 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

9

Laboratory Procedures for Fabricating Holograms . . . . . . . . . . . . . . . . 217 9.1 Isolating the Work Area from Environmental Noise . . . . . . . . . . . . . 217 9.2 Necessary Optical Elements for Fabricating Holograms . . . . . . . . . 218 9.2.1 Optical Bench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 9.2.2 Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 9.2.3 Beam Director . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 9.2.4 Spatial Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 9.2.5 Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 9.2.6 Photographic-Plate Holder . . . . . . . . . . . . . . . . . . . . . . . . . . 221 9.2.7 Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 9.3 Photographic Illustration of the Experimental Procedures for Hologram Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 9.4 Exposure Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 9.5 Dark-Room Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 9.5.1 Developing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 9.5.2 Stop Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 9.5.3 Fixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 9.5.4 Water Rinsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 9.5.5 Drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 9.5.6 Bleaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 9.6 Viewing the Hologram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

10

Analysis of the Optical System in the Spatial Frequency Domain . . . . 233 10.1 Transfer Function for Coherent Light . . . . . . . . . . . . . . . . . . . . . . . . . 233 10.1.1 Impulse Response Function . . . . . . . . . . . . . . . . . . . . . . . . . 233 10.1.2 Coherent Transfer Function (CTF) . . . . . . . . . . . . . . . . . . . 235 10.2 Spatial Coherence and Temporal Coherence . . . . . . . . . . . . . . . . . . . 237 10.3 Differences Between the Uses of Coherent and Incoherent Light . . 239 10.4 Transfer Function for Incoherent Light . . . . . . . . . . . . . . . . . . . . . . . . 241

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10.5 Modulation Transfer Function (MTF) . . . . . . . . . . . . . . . . . . . . . . . . . 246 10.6 Relationship Between MTF and OTF . . . . . . . . . . . . . . . . . . . . . . . . . 246 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 11

Optical Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 11.1 Characteristics of a Photographic Film . . . . . . . . . . . . . . . . . . . . . . . . 251 11.2 Basic Operations of Computation by Light . . . . . . . . . . . . . . . . . . . . 253 11.2.1 Operation of Addition and Subtraction . . . . . . . . . . . . . . . . 253 11.2.2 Operation of Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 254 11.2.3 Operation of Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 11.2.4 Operation of Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 11.2.5 Operation of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . 257 11.3 Optical Signal Processing Using Coherent Light . . . . . . . . . . . . . . . . 259 11.3.1 Decoding by Fourier Transform . . . . . . . . . . . . . . . . . . . . . 259 11.3.2 Inverse Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 11.3.3 Wiener Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 11.3.4 A Filter for Recovering the Image from a Periodically Sampled Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 11.3.5 Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 11.4 Convolution Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 11.5 Optical Signal Processing Using Incoherent Light . . . . . . . . . . . . . . 273 11.5.1 The Multiple Pinhole Camera . . . . . . . . . . . . . . . . . . . . . . . 274 11.5.2 Time Modulated Multiple Pinhole Camera . . . . . . . . . . . . 276 11.5.3 Low-Pass Filter Made of Randomly Distributed Small Pupils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 11.6 Incoherent Light Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 11.7 Logarithmic Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 11.8 Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 11.8.1 Planigraphic Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 285 11.8.2 Computed Tomography (CT) . . . . . . . . . . . . . . . . . . . . . . . . 287 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

12

Applications of Microwave Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 12.1 Recording Microwave Field Intensity Distributions . . . . . . . . . . . . . 305 12.1.1 Scanning Probe Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 12.1.2 Method Based on Changes in Color Induced by Microwave Heating . . . . . . . . . . . . . . . . . . . . . . 307 12.1.3 Method by Thermal Vision . . . . . . . . . . . . . . . . . . . . . . . . . 308 12.1.4 Method by Measuring Surface Expansion . . . . . . . . . . . . . 309 12.2 Microwave Holography Applied to Diagnostics and Antenna Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 12.2.1 “Seeing Through” by Means of Microwave Holography . 311 12.2.2 Visualization of the Microwave Phenomena . . . . . . . . . . . 313 12.2.3 Subtractive Microwave Holography . . . . . . . . . . . . . . . . . . 314

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Contents

12.2.4 12.2.5

Holographic Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 A Method of Obtaining the Far-Field Pattern from the Near Field Pattern . . . . . . . . . . . . . . . . . . . . . . . . . 316 12.3 Side Looking Synthetic Aperture Radar . . . . . . . . . . . . . . . . . . . . . . . 318 12.3.1 Mathematical Analysis of Side Looking Synthetic Aperture Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 12.4 HISS Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 12.4.1 Hologram Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 13

Fiber Optical Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 13.1 Advantages of Optical Fiber Systems . . . . . . . . . . . . . . . . . . . . . . . . . 334 13.1.1 Large Information Transmission Capability . . . . . . . . . . . . 334 13.1.2 Low Transmission Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 13.1.3 Non-Metallic Cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 13.2 Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 13.3 Dispersion of the Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 13.4 Fiber Transmission Loss Characteristics . . . . . . . . . . . . . . . . . . . . . . 339 13.5 Types of Fiber Used for Fiber Optical Communication . . . . . . . . . . 342 13.6 Receivers for Fiber Optical Communications . . . . . . . . . . . . . . . . . . 343 13.6.1 PIN Photodiode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 13.6.2 Avalanche Photodiode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 13.6.3 Comparison Between PIN Photodiode and APD . . . . . . . . 347 13.7 Transmitters for Fiber Optical Communications . . . . . . . . . . . . . . . . 348 13.7.1 Light Emitting Diode (LED) . . . . . . . . . . . . . . . . . . . . . . . . 348 13.7.2 Laser Diode (LD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 13.7.3 Laser Cavity and Laser Action . . . . . . . . . . . . . . . . . . . . . . 352 13.7.4 Temperature Dependence of the Laser Diode (LD) . . . . . . 357 13.7.5 Comparison Between LED and LD . . . . . . . . . . . . . . . . . . . 357 13.8 Connectors, Splices, and Couplers . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 13.8.1 Optical Fiber Connector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 13.8.2 Splicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 13.8.3 Fiber Optic Couplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 13.9 Wavelength Division Multiplexing (WDM) . . . . . . . . . . . . . . . . . . . . 362 13.10 Optical Attenuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 13.11 Design Procedure for Fiber Optical Communication Systems . . . . . 365 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

14

Electro and Accousto Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 14.1 Propagation of Light in a Uniaxial Crystal . . . . . . . . . . . . . . . . . . . . . 371 14.2 Field in an Electrooptic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 14.2.1 Examples for Calculating the Field in an Electrooptic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 14.2.2 Applications of the Electrooptic Bulk Effect . . . . . . . . . . . 383 14.3 Elastooptic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 14.3.1 Elastooptic Effect in an Isotropic Medium . . . . . . . . . . . . . 386 14.3.2 Elastooptic Effect in an Anisotropic Medium . . . . . . . . . . 388

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14.4

Miscellaneous Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 14.4.1 Optical Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 14.4.2 Faraday Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 14.4.3 Other Magnetooptic Effects . . . . . . . . . . . . . . . . . . . . . . . . . 396 14.4.4 Franz-Keldysh Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

15

Integrated Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 15.1 Analysis of the Slab Optical Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 15.1.1 Differential Equations of Wave Optics . . . . . . . . . . . . . . . . 400 15.1.2 General Solution for the TE Modes . . . . . . . . . . . . . . . . . . . 401 15.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 15.1.4 TM Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 15.1.5 Treatment by Geometrical Optics . . . . . . . . . . . . . . . . . . . . 407 15.1.6 Comparison Between the Results by Geometrical Optics and by Wave Optics . . . . . . . . . . . . . . . . . . . . . . . . . 409 15.2 Coupled-Mode Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 15.3 Basic Devices in Integrated Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 15.3.1 Directional Coupler Switch . . . . . . . . . . . . . . . . . . . . . . . . . 416 15.3.2 Reversed ∆β Directional Coupler . . . . . . . . . . . . . . . . . . . . 419 15.3.3 Tunable Directional Coupler Filter . . . . . . . . . . . . . . . . . . . 423 15.3.4 Y Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 15.3.5 Mach-Zehnder Interferometric Modulator . . . . . . . . . . . . . 425 15.3.6 Waveguide Modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 15.3.7 Acoustooptic Modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 15.4 Bistable Optical Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 15.4.1 Optically Switchable Directional Coupler . . . . . . . . . . . . . 433 15.4.2 Optical Triode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 15.4.3 Optical AND and OR Gates . . . . . . . . . . . . . . . . . . . . . . . . . 437 15.4.4 Other Types of Bistable Optical Devices . . . . . . . . . . . . . . 438 15.4.5 Self-Focusing Action and non-linear optics . . . . . . . . . . . . 440 15.5 Consideration of Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 15.6 Integrated Optical Lenses and the Spectrum Analyzer . . . . . . . . . . . 444 15.6.1 Mode Index Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 15.6.2 Geodesic Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 15.6.3 Fresnel Zone Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 15.6.4 Integrated Optical Spectrum Analyzer . . . . . . . . . . . . . . . . 448 15.7 Methods of Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 15.7.1 Fabrication of Optical Guides . . . . . . . . . . . . . . . . . . . . . . . 449 15.7.2 Fabrication of Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 15.7.3 Summary of Strip Guides . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 15.7.4 Summary of Geometries of Electrodes . . . . . . . . . . . . . . . . 454 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

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16

3D Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 16.1 Historical Development of 3D Displays . . . . . . . . . . . . . . . . . . . . . . . 459 16.2 Physiological Factors Contributing to 3D Vision . . . . . . . . . . . . . . . 464 16.3 How Parallax and Convergence Generate a 3D Effect . . . . . . . . . . . 466 16.3.1 Projection Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 16.3.2 Interception Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 16.4 Details of the Means Used for Realizing the 3D Effect . . . . . . . . . . 471 16.4.1 Methods based on Polarized Light . . . . . . . . . . . . . . . . . . . 471 16.4.2 3D Movies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 16.4.3 Wheatstone’s Stereoscope . . . . . . . . . . . . . . . . . . . . . . . . . . 473 16.4.4 Brewster’s Stereoscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 16.4.5 Anaglyph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 16.4.6 Time Sharing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 16.4.7 Head Mounted Display (HMD) . . . . . . . . . . . . . . . . . . . . . . 479 16.4.8 Volumetric Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 16.4.9 Varifocal Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 16.4.10 Parallax Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 16.4.11 Horse Blinder Barrier Method . . . . . . . . . . . . . . . . . . . . . . . 487 16.4.12 Lenticular Sheet Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 16.4.13 Integral Photography (IP) . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 16.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

Chapter 1

History of Optics

1.1 The Mysterious Rock Crystal Lens It was as early as 4,000 B.C. that the Sumerians cultivated a high level civilization in a region of Mesopotamia which at present belongs to Iraq. The name Mesopotamia, meaning “between the rivers”, [1.1] indicates that this area was of great fertility enjoying natural advantages from the Tigris and Euphrates rivers. They invented and developed the cuneiform script which is today considered to have been man’s first usable writing. Cuneiforms were written by pressing a stylus of bone or hard reed into a tablet of soft river mud. The tip was sharpened into a thin wedge and thus the cuneiform letters are made up of such wedge shaped strokes as shown in Fig. 1.1 [1.2]. These tablets were hardened by baking in the sun. Literally tens of thousands of the tablets were excavated in good condition and deciphered by curious archaeologists. The inscriptions on the tablets have revealed education, religion, philosophy, romance, agriculture, legal procedure, pharmacology, taxation, and so on. It is astonishing to read all about them, and to discover that they describe a highly sophisticated society more than five thousand years ago [1.3]. What is amazing about the tablets is that some of the cuneiform inscriptions are really tiny (less than a few millimeters in height) and cannot be easily read without some sort of a magnifying glass. Even more mysterious is how these inscriptions were made. A rock-crystal, which seemed to be cut and polished to the shape of a plano-convex lens, was excavated in the location of the tablets by an English archaeologist, Sir Austen Layard, in 1885 [1.4]. He could not agree with the opinion that the rock crystal was just another ornament. Judging from the contents of the tablet inscriptions, it is not hard to imagine that the Sumerians had already learned how to use the rock crystal as a magnifying glass. If indeed it were used as a lens, its focal length is about 11 cm and a magnification of two would be achieved with some distortion. Another interesting theory is that these cuneiforms were fabricated by a group of nearsighted craftmen [1.5]. A myopic (nearsighted) eye can project a bigger image

1

2

1 History of Optics

Fig. 1.1 Cuneiform tablets excavated from Mesopotamia. The contents are devoted to the flood episode. (By courtesy of A. W. Sj¨orberg, University Museum, University of Pennsylvania)

Fig. 1.2 a, b. One candle power at the beginning of Greek empire (a) evolved to 15 candle power (b) at the end of the empire

onto the retina than an emmetropic (normal) eye. If the distance b between the lens and the retina is fixed, the magnification m of the image projected onto the retina is m=

b − 1, f

(1.1)

where f is the focal length of the eye lens. The image is therefore larger for shorter values of f . In some cases, the magnifying power of the myopic eye becomes 1.7 times of that of the emmetropic eye, which is almost as big a magnification as the rock crystal “lens” could provide. However, it is still a mystery today how these tiny tablet inscriptions were fabricated and read. Two other significant developments in optics during this period, besides the questionable rock crystal lens, were the use of a lamp and a hand-held mirror. Palaeolithic wall paintings are often found in caves of almost total darkness. Such lamps, as shown in Fig. 1.2, were used by the artists and have been excavated

1.2 Ideas Generated by Greek Philosophers

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from the area. Animal fat and grease were used as their fuel. The lamps were even equipped with wicks. The paintings themselves demonstrated a sophisticated knowledge about colour. Other optical instruments of early discovery are metallic mirrors, which have been found inside an Egyptian mummy-case. By 2,000 B.C., the Egyptians had already mastered a technique of fabricating a metallic mirror. Except for the elaborate handle designs, the shape is similar to what might be found in the stores today.

1.2 Ideas Generated by Greek Philosophers Greece started to shape up as a country around 750 B.C. as a collection of many small kingdoms. Greek colonies expanded all around the coast lines of the Mediterranean Sea and the Black Sea including Italy, Syria, Egypt, Persia and even northeast India [1.6]. A renowned Greek scientist, Thales (640–546 B.C.), was invited by Egyptian priests to measure the height of a pyramid [1.4]. For this big job, all that this brilliant man needed was a stick. He measured the height of the pyramid by using the proportion of the height of the stick to its shadow, as shown in Fig. 1.3. He also predicted the total solar eclipse that took place on May 28, 585 B.C. The greatest mathematician and physicist of all time, Euclid (315–250 B.C.), and his student Archimedes (287–212 B.C.) and many others were from the University of Alexandria in Egypt which was then a Greek colony. The “Elements of Geometry” written by Euclid has survived as a text book for at least twenty centuries. The progress of geometry has been inseparable from that of optics. Democritus (460–370 B.C.), Plato (428–347 B.C.) and Euclid all shared similar ideas about vision. Their hypothesis was that the eyes emanate vision rays or eye rays and the returned rays create vision. Under this hypothesis, vision operated on a principle similar to that of modern day radar or sonar. As a result of the concept of the eye ray, arrows indicating the direction of a ray always pointed away from the eye. Not only the direction of the arrows, but also the designation of the incident and

Fig. 1.3 Measurement of the height of a pyramid by the shadow of a stick

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Fig. 1.4 Reversed direction of arrows as well as reversed designation of the incident and reflected angles

Fig. 1.5 Democritus (460–370 BC) said “a replica of the object is incident upon the eyes and is imprinted by the moisture of the eye”

reflected angles were reversed from what these are today, as illustrated in Fig. 1.4. As a matter of fact, it took fourteen hundred years before the direction of the arrows was reversed by Alhazen (around 965–1039) of Arabia in 1026. Democritus further elaborated this theory of vision. He maintained that extremely small particles chip off from the object and those chipped particles form a replica which is imprinted on the eyes by the moisture in the eyes. His proof was that, on the surface of the viewer’s eye, one can see a small replica of the object going into the viewer’s eyes as illustrated in Fig. 1.5. The basis of a corpuscular theory of light had already started percolating at this early age. Aristotle (384–322 B.C.), who was one of Plato’s pupils, showed dissatisfaction about the emission theory saying, “why then are we not able to see in the dark?” [1.4, 7, 8]. Hero(n) (50?) tried to explain the straight transmission of a ray by postulating that the ray always takes the shortest distance. An idea similar to Fermat’s principle was already conceived in those early days. The left and right reversal of the image of a vertical mirror such as shown in Fig. 1.4, or the upside-down image of a horizontal mirror, such as shown in Fig. 1.6,

1.2 Ideas Generated by Greek Philosophers

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Fig. 1.6 Inversion of mirror images

Fig. 1.7 Corner mirror which forms “correct image” was invented by Hero (5 0 ?). Direction of the arrows is still wrong

aroused the curiosity of the Greek philosophers. Plato attempted the explanation but without success. Hero even constructed a corner mirror which consisted of two mirrors perpendicular to each other. One can see the right side as right, and the left side as left, by looking into the corner mirror, as illustrated in Fig. 1.7. Around the second century A.D., more quantitative experiments had started. Claudius Ptolemy (100–160) [1.4, 7] made a series of experiments to study refraction using the boundary between air and water. He generated a table showing the relationship between the incident and refracted angles, and obtained an empirical formula of refraction

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Fig. 1.8 Ptolemy’s explanation of Ctesibiu’s coin and cup experiment

r = u i − k i 2,

(1.2)

where u and k are constants. This formula fitted the experiments quite nicely for small incidence angles. Again, man had to wait for another fifteen hundred years before Snell’s law was formulated in 1620. Ptolemy attempted to explain Ctesibiu’s coin-in-a-cup experiment. Ctesibiu’s experiment was first performed at the University of Alexandria around 50 B.C. As shown in Fig. 1.8, a coin placed at the point c at the bottom of an empty opaque cup is not visible to an eye positioned at e, but the coin becomes visible as soon as the cup is filled with water. Ptolemy recognized that the light refracts at the interface between air and water at 0 and changes its course toward the coin and the coin becomes visible to the eye. Ptolemy even determined the apparent position of the coin at the intercept c of the extension of e0 with the vertical line hc. The exact position is somewhat to the right of c [1.9]. After fierce fighting, the Romans invaded Greece. Greek gods had to bow their heads to Roman gods and eventually to the Christian god. The fall of the Greek Empire in the fifth century A.D. marked the beginning of the Dark Ages. Progress in the science of light was turned off. It was not until the Renaissance (14th–16th centuries) that science was roused from its slumber.

1.3 A Morning Star A star scientist who twinkled against the darkness of the Middle-Ages especially in the field of optics, was Abu Ali Al-Hasen ibn al-Hasan ibn Al-Haytham, in short Alhazen (around 965–1039) of the Arabian Empire [1.4, 7, 8, 10]. His contribution to the field of optics was very broad and prolific. By measuring the time elapsed from the disappearance of the setting sun from the horizon to total darkness, i.e., by measuring the duration of the twilight which is

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caused by scattering from particles in the atmosphere, he determined the height of the atmosphere to be 80 km [1.4]. This value is shorter than 320 km which is now accepted as the height of the atmosphere, but nevertheless, his observation showed deep insight. He investigated reflection from non-planar surfaces (concave, convex, spherical, cylindrical, etc.) and finally formulated Alhazen’s law of reflection, i.e. the incident ray, the normal of the point of reflection, and the reflected ray, all three lie in the same plane. He also explained the mechanism of refraction taking place at the boundary of two different media by resolving the motion into two components: parallel and normal to the surface. While the component parallel to the surface is unaffected, that normal to the surface is impeded by the denser medium, and as a result, the resultant direction of the motion inside the second medium is different from that of the incident motion. Unfortunately, with this explanation, the refracted ray changes its direction away from the normal at the boundary when light is transmitted into a denser medium, contrary to reality. Alhazen discarded the idea of the eye ray supported by the Greek philosophers. He could not agree with the idea of an eye ray searching for an object. He recognized that the degree of darkness and colour of an object changes in accordance with that of the illuminating light, and if eye rays were responsible for vision, then vision should not be influenced by external conditions. Alhazen finally reversed the direction of the arrows on the rays showing the geometry of vision. Alhazen tried to understand the mechanism of vision from comprehensive anatomical, physical and mathematical viewpoints; in today’s terms, his approach was interdisciplinary. From the fact that eyes start hurting if they are exposed to direct sunlight, and from the fact that the effect of the sun is still seen even after the eyes are shut, he concluded that the light ray must be causing some reaction right inside the human eye. Alhazen’s ocular anatomy was quite accurate. Based upon his detailed anatomy, he constructed a physical model of an eye to actually identify what is going on inside the eye. In this model, several candles were placed in front of a wall with a small hole at the center. The upside down images of the candles were projected onto a screen placed on the other side of the hole. He verified that the position of the image was on a line diagonally across the hole by obstructing each candle one at a time, as shown in Fig. 1.9. He had to give an explanation why man can see images right side up, while those of his physical model were upside down. He was forced to say that the image was sensed by the first surface of the crystalline lens even though his anatomy diagram shows the retina and its nerve structure. This experiment is called camera obscura [1.8].

1.4 Renaissance Renaissance means “rebirth” and it was during this period in history that many disciplines, including optics, experienced a revival. The Renaissance began in Italy in the fourteenth century and quickly spread to other countries.

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Fig. 1.9 Alhazen’s camera obscura

A great Italian architect, sculptor and scientist, Leonardo da Vinci (1452–1519), followed up Alhazen’s experiments and developed the pinhole camera. It was only natural that da Vinci, who was one of the most accomplished artists of all times, indulged in the study of colour. He made an analogy between acoustic and light waves. He explained the difference in colour of smoke rising from a cigar and smoke exhaled from the mouth. The former is lighter, and the frequency of oscillation is higher, so that its colour is blue; while the latter is moistened and heavier, and the frequency of oscillation is lower, and hence its colour is brown. This statement implies that he believed that light is a vibrating particle and that its colour is determined by the frequency of vibration. Around da Vinci’s time, the printing press was invented by Johannes Gutenberg and financed by Johann Fust both in Germany. This invention accelerated the spread of knowledge. The astronomer Galileo Galilei (1565–1642), working on a rumor that a Dutchman had succeeded in constructing a tool for close viewing with significant magnification (the Dutchman could have been either Hans Lippershey [1.4] or De Waard [1.8] of Middleburg), managed to construct such a telescope, as shown in Fig. 1.10, for astronomical observation. Using the telescope of his own make, Galilei indulged in the observation of stars. During consecutive night observations from January 7 to January 15, 1610, he discovered that the movement of Jupiter relative to four adjacent small stars was contrary to the movement of the sky. He was puzzled but continued observation led him to conclude that it was not Jupiter that was moving in the opposite direction but rather it was the four small stars that were circling around Jupiter. When Galilei first claimed the successful use of the telescope, academic circles were skeptical to use the telescope for astronomy. These circles believed that whatever was observed by using such a telescope would be distorted from reality because of its colour coma and aberration.

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Fig. 1.10 Telescopes made by Galileo Galilei (1564–1642). (a) Telescope of wood covered with paper, 1.36 meters long; with a biconvex lens of 26 mm aperture and 1.33 m focal length; planoconcave eye-piece; magnification, 14 times. – (b) Telescope of wood, covered with leather with gold decorations: 0.92 m long; with a biconvex objective of 16 mm useful aperture and 96 m focal length; biconcave eye-piece (a later addition); magnification, 20 times

Galilei kept correspondence with Johann Kepler (1571–1630) of Germany who was rather skeptical about the telescope because he himself tried to build a telescope without much success. Galilei presented the telescope he built to Kepler. Kepler was amazed by the power of the telescope and praised Galilei without qualification in his Latin treatise published in September 1610. Although Kepler is most well known for his laws of planetary motion, he was also interested in various other fields. Kepler turned his attention to earlier results that Ptolemy and Alhazen worked out about refraction. During experiments, he discovered that there is an angle of total reflection. He also explored the theory of vision beyond Alhazen and identified the retina as the photo-sensitive surface. It thus took six centuries for the light to penetrate from the surface of the crystalline lens to the retina. Kepler was not worried about the upside down vision because he reached the conclusion that the inversion takes place physiologically. Ren´e Descartes (1596–1650) proposed the idea of a luminiferous aether to describe the nature of light for the first time [1.11]. The aether is a very tenuous fluid-like medium which can pass through any transparent medium without being obstructed by that medium. The reason why it is not obstructed was explained by using the analogy of the so called Descartes’ Vat, such as shown in Fig. 1.11. The liquid which is poured from the top can go through between the grapes and can come out of the vat at the bottom. Later, however, Descartes abandoned this idea and proposed a new postulate, namely, that light is not like fluid, but rather a stream of globules. The laws of reflection and refraction were then explained by using the kinematics of a globule.

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Fig. 1.11 Descartes’ Vat

1.5 The Lengthy Path to Snell’s Law The one law which governs all refraction problems was finally formulated around 1620 by the Dutch professor Willebrod Snell (1591–1626) at Leyden. Starting with Ptolemy in ancient Greece, the problems of refraction had been relayed to Alhazen, and then to Kepler and finally to Snell’s inspiration; this process covers fifteen hundred years of evolving ideas. Snell’s law is indeed one of the most basic and important laws of optics to-date. The cosecant rather than sine of the angles was used when Snell introduced the law. Descartes attempted to explain Snell’s law by using the model that light is a stream of small globules. Like Alhazen, he decomposed the motion of a globule into tangential and normal components. However, Descartes, was different from Alhazen in treating the normal component of the motion. He realized that in order to be consistent with Snell’s law, the normal component had to be larger so that the light would bend toward the normal. He therefore used the analogy that a ball rolling over a soft carpet is slower than the same ball rolled over a hard bare table, thus the motion in the optically dense medium is larger than that in a less-dense medium. Descartes used sine functions in his illustration and brought the results in today’s form of Snell’s law. He also proposed a hypothesis concerning the dispersion of white light into a spectrum of colours after refraction from glass. He postulated that the globules of light rotate due to impact at the interface, and consequently the light starts displaying colour. This hypothesis was, however, disproved by Robert Hooke (1635–1703) who arranged glass slabs such that the light refracts twice in opposite directions, in which case the colours should disappear according to Descartes’ theory, but they did not. Returning to the refraction problem, Pierre Fermat (1601–1665) of France did not agree with Descartes’ explanation that the normal movement of the light globules in water is faster than that in air. Instead, Fermat focused his attention on Hero’s idea of the shortest path length, which fitted very nicely with the observation of the path

1.6 A Time Bomb to Modern Optics

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of reflected light. However, Hero’s idea fails when one tries to use it to explain the path of refraction. Fermat postulated the concept of the least time rather than the least length for the light to reach from one point in the first medium to another point in the second medium. He argued that, in the case of the boundary between air and water, the speed in water is slower than that in air, so that the light takes a path closer to the normal in the water so as to minimize the transit time. Fermat, being a great mathematician, could determine the exact path which makes the total time the least. It is noteworthy that, even though Descartes’ globules approach made a wrong assumption of a faster speed of light in water than air, he reached a conclusion that is consistent with Snell’s law.

1.6 A Time Bomb to Modern Optics One of the most important books in the history of optics, “Physico – Mathesis de Lumine coloribus et iride, aliisque annexis libri”, was written by a professor of mathematics at the University of Bologna, Italy, Father Francesco Maria Grimaldi (1618–1663) and was published in 1665 after his death. The book openly attacked most of the philosophers saying “. . .let us be honest, we do not really know anything about the nature of light and it is dishonest to use big words which are meaningless” [1.8]. Needless to say, his frank opinions did not go over well in academic circles, and he lost some friends. He developed experiments and an analysis based upon his own fresh viewpoint, at times totally disregarding earlier discoveries. His book planted many a seed which were later cultivated to full bloom by Huygens, Newton, Young and Fresnel. His major contributions were in the field of diffraction, interference, reflection and the colour of light. Probably the most important contribution of Grimaldi was the discovery of the diffration of light. Even such minute obscure phenomena did not escape his eyes. His sharp observations can never be overpraised. Since he used white light as a source, he had to use light coming out of an extremely small hole in order to maintain a quasi-coherency across the source. This means that he had to make his observations with a very faint light source. The small hole also helped to separate the direct propagation from that of the diffracted rays. Using this source, the shadow of an obstacle was projected onto a screen. He discovered that the light was bending into the region of the geometrical shadow, so that the edge of the shadow was always blurred. In addition to this blurriness of the edge of the shadow, he recognized a bluish colour in the inner side of the edge, and a reddish colour in the outer side of the edge. He named this mysterious phenomenon diffusion of light. Could he ever have dreamed that this faint phenomenon would form the basis of optical computation or holography of today? He tried to explain the phenomenon by proposing an analogy with the spreading of fire by sparks. Wherever a spark is launched on the grass, a new circle of fire starts, and each point of fire again generates new sparks. Such sparks can even turn the corner of a building. Diffraction

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Fig. 1.12 Grimaldi’s explanation of refraction

of light takes place in exactly the same manner. Even though he made this analogy, he was not satisfied with it because a secondary wave is generated wherever the light reaches, and soon all area will be filled with light. Despite this drawback of the analogy, Grimaldi’s explanation has a lot in common with Huygens’ principle. Grimaldi shared Descartes’ view that all materials have numerous tiny pores and the degree of transparency is determined by the density of the pores. A transparent material has a higher density and an opaque material has a lower density of pores. Based upon this hypothesis, refraction and reflection between air and water were explained by Grimaldi as follows. Since the density of pores is smaller in water than that in air, the light beam has to become wider in the water, as shown in Fig. 1.12, so that the flow of the light beam will be continuous. Thus the transmitted beam bends toward the normal at the boundary. Using the same hypothesis, he tried to explain reflection when a light beam is incident upon a glass slab held in air. When the light beam hits the surface of the glass whose density of pores is less than that of air, only a portion of the light beam can go into the glass and the rest has to reflect back into air. This was a convincing argument, but he could not explain why the light transmitted into the glass reflects again when it goes out into air from glass. His experiments were wide in scope and subtle in arrangement. He succeeded in detecting interference fringes even with such a quasi-coherent source as the pinhole source. He illuminated two closely spaced pinholes with a pinhole source and, in the pattern projected onto a screen, he discovered that some areas of the projected pattern were even darker than when one of the holes was plugged. His observation was basically the same as the observation of Thomas Young (1773–1829) in an experiment performed one and a half centuries later.

1.7 Newton’s Rings and Newton’s Corpuscular Theory Sir Isaac Newton (1642–1726) was born a premature baby in a farmer’s family in Lincolnshire, England. In his boyhood, he was so weak physically that he used to be bothered and beaten up by his school mates. It was a spirit of revenge that motivated little Isaac to beat these boys back in school marks.

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When he entered the University of Cambridge in 1665, the Great Plague spread in London, and the University was closed. He went back to his home in Lincolnshire for eighteen months. It was during these months that Newton conceived many a great idea in mathematics, mechanics and optics, but it was very late in life at the age of 62 when he finally published “Opticks” which summarized the endeavours of his lifetime. Profound ideas expressed in an economy of words were the trademark of his polished writing style. He was renowned for his exceptional skill and insight in experiments. Newton had been in favour of the corpuscular theory of light, but he conceded that the corpuscular theory alone does not solve all the problems in optics. His major objection to the undulatory theory or wave theory is that waves usually spread in every direction, like a water wave in a pond, and this seemed inconsistent with certain beam-like properties of light propagation over long distances. The light beam was considered as a jet stream of a mixture of corpuscles of various masses. He explained that refraction is caused by the attractive force between the corpuscles and the refracting medium. For denser media, the attractive force to the corpuscles is stronger, when the ray enters into a denser medium from a less dense medium, the ray is bent toward the normal because of the increase in the velocity in that direction. A pitfall of this theory is that the resultant velocity inside a denser medium is faster than that in vacuum. Quantitative studies on the dispersion phenomena were conducted using two prisms arranged as in Fig. 1.13. A sun beam, considered to be a heterogeneous mixture of corpuscles of different masses, was incident upon the first prism. The second prism was placed with its axis parallel to that of the first one. A mask M2 with a small hole in the center was placed in between the two prisms. The spread of colour patterns from prism P1 was first projected onto the mask M2 . The ray of one particular colour could be sampled by properly orienting the small hole of M2 . The sampled colour ray went through the second prism P2 and reached the screen. The difference in the amount of change in velocity between the light and heavy corpuscles was used to explain the dispersion phenomenon. While the output from the first prism was spread out, that of the second prism was not. This is because only corpuscles of the same mass had been sampled out by the hole M2 . Thus, the monochromatic light could be dispersed no more. Newton repeated a similar

Fig. 1.13 Newton’s two prism experiment

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Fig. 1.14 Newton’s cross prism experiment

experiment with a few more prisms after prism P2 to definitely confirm that the beam splits no further. The fact that monochromatic light cannot be further split is the fundamental principle of present-day spectroscopy. The angle of refraction from the second prism P2 was measured with respect to the colour of light by successively moving the hole on M2 . He also examined the case when the second refraction is in a plane different from that of the first refraction; accordingly, he arranged the second prism in a cross position, as shown in Fig. 1.14. The first prism spread the spectrum vertically, and the second spread it side-ways at different angles according to the colour of light. A slant spectrum was observed on the screen. Even the brilliant Newton, however, made a serious mistake in concluding that the degree of dispersion is proportional to the refractive index. It is most natural that he reached this conclusion because he maintained the theory that the dispersion is due to the differential attractive forces. He did not have a chance to try many media to find that the degree of dispersion varies even among media of the same refractive index. This is known as “Newton’s error”. He went as far as to say that one can never build a practical telescope using lenses because of the chromatic aberration. This error, however, led to a happy end. He wound up building an astronomical telescope with parabolic mirrors such as shown in Fig. 1.15. He also attempted to construct a microscope with reflectors but he did not succeed. Next to be mentioned are the so-called Newton’s rings. He placed a plano-convex lens over a sheet glass, as shown in Fig. 1.16, and found a beautiful series of concentric rings in colour. His usual careful in-depth observation led him to discover such fine details about the rings as: (i) The spacings between the rings are not uniform but become condensed as they go√ away from the center. To be more exact, the radius of the nth order ring grows with n. (ii) The locations of the dark lines occur where the gap distance between the lens and plate is an integral multiple of a basic dimension. (iii) Within any given order of rings, the ratio of the gap distance where the red ring is seen and the gap distance where violet ring is seen is always 14 to 9. (iv) The rings are observable from the bottom as well as from the top and they are complimentary to each other.

1.7 Newton’s Rings and Newton’s Corpuscular Theory

Fig. 1.15 Newton’s reflector telescope

Fig. 1.16 Newton’s rings

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For Newton, who had tried to maintain the corpuscular theory, it was most difficult and challenging to explain those rings. He introduced a “fit” theory as an explanation. Wherever reflection or refraction takes place, there is a short transient path. Because the gap is so small in the Newton ring case, before the first transient dies off the second starts; thus the transient is accentuated and the so called “fit” phenomenon takes place. The colourful rings are generated where the “fit” of easy reflection is located and the dark rings, where the “fit” of easy transmission is located. It is interesting to note that there is some similarity between the “fit” to explain the fringe pattern by Newton and the transition of an orbital electron to explain the emission of light by Niels Bohr (1885–1962).

1.8 Downfall of the Corpuscle and Rise of the Wave Newton had acquired such an unbeatable reputation that any new theory that contradicted his corpuscular theory had difficulty gaining recognition. Christiaan Huygens (1629–1695) of The Hague, Holland, who was a good friend of Newton’s, was the first to battle against the corpuscular theory. Huygens postulated that light is the propagation of vibrations like those of a sound wave. He could not support the idea that matterlike corpuscles physically moved from point A to point B. Figure 1.17 illustrates Huygens’ principle. Every point on the wave front becomes the origin of a secondary spherical wave which, in turn, becomes a new wave front. Light is incident from air to a denser medium having a boundary O–O . A dotted line A–O is drawn perpendicular to the direction of incidence. Circles are drawn with their centers along O–O and tangent to A–O . The

Fig. 1.17 Huygens’ principle

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dotted line A–O would be the wave front if air were to replace the denser medium. Since the velocity in the denser medium is reduced by 1/n, circles reduced by 1/n are drawn concentric with the first circles. The envelope A –O of the reduced circles is the wave front of the refracted wave. As a matter of fact, the envelope A –O is the wave front of the reflected wave. Unlike Newton’s or Descartes’ theory, the velocity in the denser medium is smaller, and an improvement in the theory is seen. Huygens was also successful in treating the double refraction of Iceland spar (Calcite CaCO3 ) by using an ellipsoidal wave front rather than a spherical wave front. Figure 1.17 could also be used for explaining total reflection. A great scholar who succeeded Huygens and Newton was Thomas Young (1773– 1829) of England. It is not surprising that Young, who was originally trained as a medical doctor, had a special interest in the study of vision. He introduced the concept of three basic colours and concluded that there are three separate sensors in the retina sensitive to each colour. This concept of basic colours is still supported to date. Young’s greatest contribution to optics was a successful explanation of interference phenomena. For the explanation of Newton’s rings, he made an analogy with the sound wave. He considered that the path difference is the source of the dark and bright rings. Referring to Fig. 1.16, the path difference of a ray reflecting from the convex surface of the lens and that reflecting from the bottom flat plate is responsible for the generation of the ring pattern. He further explained that, if the path difference is a multiple of a certain length, the two rays strengthen each other but, if the path difference is the intermediate of these lengths, the two rays cancel each other. He also noted that this length varies with the colour of light. Young conducted an experiment on interference that now bears his name. He arranged a pinhole such that a sunbeam passing through the pinhole illuminated two pinholes located close together. The interference fringe pattern from the two pinholes was projected onto a screen located several meters away. From measurements of the spacings between the maxima of the fringe pattern associated with each colour, the wavelengths of each colour light were determined. His measurement accuracy was very high and the wavelengths of 0.7 µm for red and 0.42 µm for violet were obtained.

1.9 Building Blocks of Modern Optics Augustin Jean Fresnel (1788–1827) started out his career as a civil engineer and was engaged in the construction of roads in southern France while maintaining his interest in optics. Fresnel spoke openly against Napoleon, and as a consequence, he had to resign his government position. This freed him to devote himself entirely to optics and he asked Dominique Franc¸ois Jean Arago (1786–1853) at the Paris observatory for advice. It was suggested to Fresnel that he read such materials as written by Grimaldi, Newton and Young. Fresnel, however, could not follow Arago’s advice because he could read neither English nor Latin [1.8].

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Fresnel reexamined the distribution of light across an aperture. He measured the aperture distribution at various distances from the aperture to see how the positions of the maxima moved with an increase in the distance from the aperture. He discovered that the size of the pattern did not increase linearly with distance as Newton’s corpuscular theory predicts. The positions of the maxima followed a hyperbola suggesting that the fringe pattern is due to the interference of two waves originating from the edge of the aperture and the incident light. Fresnel earned Arago’s confidence as the result of a skillful experiment on the fringe pattern made by a piece of thread. He noticed that, when light is incident onto a piece of thread, fringe patterns are observed on both sides of the thread, but as soon as the light on one of the sides is blocked by a sheet of paper, the fringes on the shadow side disappear. From this, he proved that the fringes are due to the interference of the two waves coming from both sides of the thread. Arago worriedly reported to Fresnel that even a transparent sheet of glass that does not block the light creates the same result. With confidence, Fresnel replied that a thick glass destroys the homogeneity of the wave front and with a thin sheet of glass the fringes will come back even though they may be shifted and distorted. Arago verified this prediction by experiments and Fresnel thus made another point. Fresnel participated in a scientific-paper contest sponsored by the French Academie des Sciences of which Arago was the chairman of the selection committee [1.8]. Fresnel realized that Huygens’ principle could not explain the fact that, in the diffraction pattern, the added light can be even darker than the individual light. Essentially what Fresnel did was to develop a mathematical formulation of Huygens’ principle but with a newly introduced concept of the phase of the contributing field element. With phase taken into consideration, the same field can contribute either constructively or destructively depending upon the distance. His theory agreed superbly with the experiments. His paper was nominated as the first prize in the contest. Sim´eon Poisson (1781–1840), who was one of the examiners of the contest calculated the diffraction pattern of a disk using Fresnel’s formula and reached the conclusion that the field at the center of the shadow is the brightest and he refuted Fresnel’s theory. Arago performed the experiment by himself and verified that Fresnel’s theory was indeed correct [1.12]. Again Fresnel was saved by Arago. Poisson himself acquired his share of fame in the event. This spot has become known as Poisson’s spot. Thus, Fresnel established an unshakeable position in the field of optics. Fresnel also challenged the experiment with double refraction of Iceland spar (Calcite CaCO3 ). He discovered that the ordinary ray and the extraordinary ray can never form a fringe pattern. With his unparalleled ability of induction, this experimental observation led him to discard the idea of longitudinal vibration of the light wave. Because longitudinal vibration has only one direction of vibration, the effect of the direction of vibration cannot be used to explain the observation on the double refraction. He proposed that the vibration of the light is transverse, and that the directions of the vibration (or polarization) of the ordinary and extraordinary rays are perpendicular to each other. Two vibrations in perpendicular directions cannot cancel one another and thus a fringe pattern can never be formed by these two rays.

1.9 Building Blocks of Modern Optics

19

Fresnel kept in close contact with Young and the two advanced the wave theory to a great extent and together they laid a firm foundation for optics in the nineteenth century. Other noteworthy contributors to optics before the end of the nineteenth century were Armand H. Louis Fizeau (1819–1896) and Leon Foucault (1819– 1868), both of whom were Arago’s students and determined the speed of light by two different methods. James Clerk Maxwell (1831–1879), who became the first director of the Cavendish Laboratory at Cambridge, opened up a new era in history by developing the electromagnetic theory of light. He established elegant mathematical expressions describing the relationship between the electric and magnetic fields and postulated that light is an electromagnetic wave which propagates in space by a repeated alternating conversion between the electric and magnetic energies. His postulate was experimentally verified by a German scientist, Heinrich Hertz (1857–1894), eight years after his death, and Maxwell’s equations became the grand foundation of electromagnetic theory. American scientists Albert Abraham Michelson (1852–1931) and Edward Williams Morley (1838–1923) collaborated and performed a sophisticated measurement of the speed of light to challenge the existence of the aether first proposed by Descartes. They believed that if there is an aether, there must also be an aether wind on the earth because the movement of the earth is as high as 30 km/s. If indeed there is such wind, it should influence the velocity of light. They put the so-called Michelson’s interferometer in a big mercury tub so that it could be freely rotated, as shown in Fig. 1.18. The interference between the two beams, S − M0 − M1 − M0 − T and S − M0 − M2 − M0 − T , taking mutually perpendicular paths was observed with one of the beam axes parallel to the direction of the orbital motion of the earth. Then, the whole table was carefully rotated by 180◦ to observe any changes in the fringe pattern from before. The conclusion of the experiment was that there is no difference and thus there is no aether wind. This result eventually led to the theory of relativity by Einstein.

Fig. 1.18 Michelson-Morley experiment to examine the drag by aether wind

20

1 History of Optics

1.10 Quanta and Photons In 1900, at the dawn of the twentieth century, Max Karl Ernst Ludwig Planck (1858– 1947) made the first big step toward the concept of a quantum of energy. Radiation from a cavity made of a metal block with a hole was intensively investigated by such physicists as Stefan, Boltzmann, Wien and Jean. Radiant emission as a function of frequency was examined at various temperatures of the cavity. Planck introduced a bold postulate that the radiant emission is due to the contributions of tiny electromagnetic oscillators on the walls and these oscillators can take only discrete energy levels, and the radiation takes place whenever there is an oscillator that jumps from a higher level to a lower level. The discrete energy levels are integral multiples of h v, h being Planck’s constant and v the light frequency. For this contribution, Planck was awarded the Nobel prize in 1918. Philipp Eduard Anton von Lenard (1862–1947), a collaborator of Hertz, made experiments on the photoelectric effect by using a vacuum tube such as that shown in Fig. 1.19. When light is incident onto the cathode, electrons on the surface of the cathode are ejected from the surface with kinetic energy m υ 2 /2 by absorbing energy from the light. When the electrons reach the anode, the current is registered by the galvanometer. The amount of kinetic energy can be determined from the negative potential Vs needed to stop the anode current. When the condition 1 (1.3) mυ 2 2 is satisfied, the emitted electrons will be sent back to the cathode before reaching the anode. Lenard discovered that the stopping potential Vs could not be changed no matter how the intensity of the incident light was raised. An increase in the intensity e Vs =

Fig. 1.19 Photoelectric effect experiment: E = 12 mv 2 = eVs E = hv − W0

1.10 Quanta and Photons

21

Fig. 1.20 a, b Experimental results of the photoelectric effect. (a) Photoelectric current vs applied potentials with wavelength and intensity of the incident light as parameters. (b) Energy needed to stop photoelectric emission vs. frequency

of the light resulted only in a proportionate increase in the anode current. Only a change in the frequency of the incident light could change the value of the stopping potential, as seen from the plot in Fig. 1.20 a. Another observation was that there was practically no time delay between the illumination and the change in the anode current. In 1905, Albert Einstein (1879–1955) published a hypothesis to explain the rather mysterious photoelectric phenomena by the equation E = hv − W0 .

(1.4)

Einstein supported Planck’s idea of photo quanta. Radiation or absorption takes place only in quanta of h ν. The amount of energy that can participate in the photoelectric phenomena is in quanta of h ν. In (1.4), the energy of each photon is hν, the kinetic energy of an electron is E, and W0 is the work function which is the energy needed to break an electron loose from the surface of the cathode. Painstaking experimental results accumulated for ten years by Robert Andrews Millikan (1868–1953) validated Einstein’s photoelectric formula (1.4). The plot of the stopping potential with respect to the frequency of the incident light, such as shown in Fig. 1.20b, matched (1.4) quite well. From such a plot, Planck’s constant h and the work function W0 can be determined respectively from the intercept and the slope of the plot. The value of h which Planck obtained earlier from the radiation experiment agreed amazingly well with Millikan’s value. For this work, Millikan was awarded the Nobel prize in 1923. Einstein’s hypothesis denies that light interacts with the cathode as a wave by the following two arguments: i) If it interacts as a wave, the kinetic energy of the emitted electrons should have increased with the intensity of light, but it was observed that the stopping potential was independent of the intensity of light.

22

1 History of Optics

The cut-off condition was determined solely by wavelength. If it were a wave, the cut-off should be determined by the amplitude because the incident energy of the wave is determined by its amplitude and not by its wavelength. ii) If light is a wave, there is a limit to how precisely the energy can be localized. It was calculated that it would take over 10 hours of accumulation of energy of the incident light “wave” to reach the kinetic energy of the emitted electron, because of the unavoidable spread of the incident light energy. But if light were a particle, the energy can be confined to the size of the particle, and almost instant response becomes possible. Einstein added a second hypothesis that the photon, being a particle, should have a momentum p in the direction of propagation where hv . (1.5) c Arthur Holly Compton (1892–1962) arranged an experimental set-up such as that shown in Fig. 1.21 to prove Einstein’s hypothesis. He detected a small difference ∆λ in wavelengths between the incident and reflected x-rays upon reflection by a graphite crystal. He recognized that the difference in wavelength ∆λ (the wavelength of the x-ray was measured by Bragg reflection from a crystal) varies with the angle of incidence ψ as p=

Fig. 1.21 a, b. Compton’s experiment. (a) Experimental arrangement. (b) Change of wavelength with angle

1.11 Reconciliation Between Waves and Particles

∆λ = 0.024(1 − cos ψ).

23

(1.6)

In wave theory, the frequency of the reflected wave is the same as that of the incident wave so that this phenomena cannot be explained by the wave theory. Only after accepting Einstein’s hypothesis that the x-ray is a particle just like the electron, could Compton interpret the difference in the wavelengths. From the conservation of momentum and energy among the colliding particles, he derived an expression that very closely agrees with the empirical formula (1.6).

1.11 Reconciliation Between Waves and Particles Maxwell’s equations, which worked so beautifully with wave phenomena like diffraction and refraction, were not capable of dealing with such phenomena as photoelectric emission or Compton’s effect. Thus, Einstein’s hypothesis of duality of light gained support. Prince Louis-Victor de Broglie (1892–) extended this hypothesis even further by postulating that any particle, not just a photon, displays wave phenomena (de Broglie wave). This hypothesis was experimentally verified by Clinton Joseph Davisson (1881–1958) and Lester Germer (1896–) who succeeded in 1927 in detecting a Bragg type of reflection of electrons from a nickel crystal. The idea of the de Broglie wave was further extended to a wave function by Erwin Schr¨odinger (1887–1961) in 1926. Schr¨odinger postulated that the de Broglie wave satisfies the wave equation just like the electromagnetic wave and he named its solution a wave function ψ. The wave function is a complex valued function but only |ψ|2 has a physical meaning and represents the probability of a particle being at a given location. In other words, |ψ|2 does not have energy but guides the movement of the particle which carries the energy. Interference of light takes place due to the relative phase of the ψ functions. According to Richard P. Feynman (1918–) for example, the interference pattern between the light beams from the same source but taking two separate routes to the destination, like those in Young’s experiment, is represented by (1.7) |ψ1 + ψ2 |2 where ψ1 and ψ2 are wave functions associated with the two separate routes. The relative phase between ψ1 and ψ2 generates interference. It is important that both ψ1 and ψ2 satisfy Schr¨odinger’s equation and their phase depends upon the path and the path has to conform with Fermat’s principle. It was only through inspiration and perspiration of such Nobel laureates as Paul Adrien Maurice Dirac (in 1933), Wolfgang Pauli (in 1945), Max Born (in 1954) and Sin Itiro Tomonaga, Julian Schwinger, Richard P. Feynman (all three in 1965), and others, that today’s theories which reconcile waves and particles have been established.

24

1 History of Optics

1.12 Ever Growing Optics Dennis Gabor (1900–1979) invented the hologram during his attempt to improve the resolution of an electron microscope image in 1948. The development of the technique of holography was accelerated by the invention of the laser in 1960. In 1962, E. N. Leith and J. Upatnieks developed a two-beam hologram which significantly improved the image quality and widened the scope of the engineering application of holography. Nicolaas Bloembergen (1920–) introduced the concept of the three-level solidstate maser in 1956, and succeeded in building one by using chromium-doped potassium cobalt cyanide K3 (Co1−x Crx )(CN)6 . Charles Townes (1915–), who was then at Columbia University, built a similar maser but with ruby. On the occasion when both of them were presented the 1959 Morris Leibmann award, Townes’ wife received a medallion made from the ruby used for his maser. When Bloembergen’s wife admired the medallion and hinted for a similar momento of his maser, he replied, “Well dear, my maser works with cyanide!” [1.13]. The maser principle was extended from microwaves to light, and the first laser was constructed by Theodore Maiman in 1960. The availability of a high-intensity light source opened up brand new fields of nonlinear optics, second-harmonic generation, and high-resolution nonlinear laser spectroscopy for which contribution Bloembergen was awarded the Nobel prize in 1981. Charles Kuen Kao (1933–) predicted the possibility of using an optical fiber as a transmission medium in 1966, and the first relatively low-loss optical fiber of 20 dB/km was fabricated by Corning Glass in 1969. Integrated optics has been making rapid progress. This area covers such optical components as lenses, mirrors, wavelength division multiplexers, laser diode arrays, receiver arrays, spectrum analyzers and even a sophisticated signal processor, all of which can now be fabricated on glass, piezo-electric crystals, or on semi-conductor wafers smaller than a microscope deck glass. Due to the extremely rapid development of present-day technology, its degree of complexity has been ever increasing. “Interdisciplinary” is the key word in today’s technology. Optics has become exceptionally close to electronics. The rendez-vous of optics and electronics has given birth to many hybrid devices such as the solidstate laser, acoustooptic devices, magnetooptic devices, electrooptic devices, etc. All of these consist of interactions among photons, electrons and phonons in a host material. However, what has been discovered so far is only the tip of an iceberg floating in an ocean of human imagination. What a thrilling experience it will be to observe these developments!

Problems 1.1. Describe a series of historical battles between the particle theory and wave theory of the light. 1.2. What is the historical evolution of the law of refraction of light?

Chapter 2

Mathematics Used for Expressing Waves

Spherical waves, cylindrical waves and plane waves are the most basic wave forms. In general, more complicated wave forms can be expressed by the superposition of these three waves. In the present chapter, these waves are described phenomenologically while leaving the more rigorous expressions for later chapters. The symbols which are frequently used in the following chapters are summarized.

2.1 Spherical Waves The wave front of a point source is a spherical wave [2.1]. Since any source can be considered to be made up of many point sources, the expression for the point source plays an important role in optics. Let a point source be located at the origin and let it oscillate with respect to time as e−jωt . The nature of the propagation of the light emanating from the point source is now examined. It takes r/υ seconds for the wave front emanating from the source to reach a sphere of radius r, υ being the speed of propagation. The phase at the sphere is the same as that of the source r/υ seconds ago. The field on the sphere is given by E(t, r ) = E(r ) e−jω(t−r/υ) E(r ) e−jωt+jkr k=

where

(2.1)

2π ω = . υ λ

Next, the value of E(r ) is determined. The point source is an omnidirectional emitter and the light energy is constantly flowing out without collecting anywhere. Hence the total energy W0 passing through any arbitrary sphere centered at the source per unit time is constant regardless of 25

26

2 Mathematics Used for Expressing Waves

Fig. 2.1 Energy distribution emanating from a point source with a spherical wave front

its radius. The energy density of the electromagnetic wave is ε|E|2 , ε being the dielectric constant of the medium. This, however, is true only when the wave front is planar. If the sphere’s radius is large compared to the dimensions of the selected surface area, the spherical surface can be approximated by a plane, as shown in Fig. 2.1. The contributions from the various elements of the surface area over the sphere are summed to give 4πr 2 ε|E(r )|2 υ = W0 , E(r ) =

with

E0 , r

(2.2) (2.3)

where E 0 is the amplitude of the field at unit distance away from the source. (It is difficult to define the field at r = 0). Equation (2.3) is inserted into (2.1) to obtain the expression of the spherical wave E(t, r ) =

E 0 −j(ωt−kr ) . e r

(2.4)

In engineering optics, a planar surface such as a photographic plate or a screen is often used to make an observation. The question then arises as to what is the light distribution that appears on this planar surface when illuminated by a point source. Referring to Fig. 2.2, the field produced by the point source S located at (x0 , y0 , 0) is observed at the point P(xi , yi , z i ) on the screen. The distance from S to P is r = Z i2 + (xi − x0 )2 + (yi − y0 )2 = Zi 1 +

(xi − x0 )2 + (yi − y0 )2 . z i2

Applying the binomial expansion to r gives r = Zi +

1 (xi − x0 )2 + (yi − y0 )2 + ··· 2 zi

(2.5)

2.2 Cylindrical Waves

27

Fig. 2.2 Geometry used to calculate the field at the observation point P due to a point source S

where the distance is assumed to be far enough to satisfy (xi − x0 )2 + (yi − y0 )2 1. z i2 Inserting (2.5) into (2.4) yields (xi − x0 )2 + (yi − y0 )2 E0 E(t, xi , yi , z i ) = exp jk z i − jωt . zi 2z i

(2.6)

Equation (2.6) is an expression for the field originating from a point source, as observed on a screen located at a distance r from the source. The denominator r was approximated by z i .

2.2 Cylindrical Waves The field from a line source will be considered. As shown in Fig. 2.3, an infinitely long line source is placed along the y axis and the light wave expands cylindrically from it. The total energy passing through an arbitrary cylindrical surface per unit time is equal to the total energy w0l radiated per unit time by the line source: 2πrlε|E|2 υ = w0l,

(2.7)

where l is the length of the source, and w0 is the radiated light energy per unit time per unit length of the source. The spatial distribution of the cylindrical wave is therefore E0 (2.8) E(r ) = √ , r where E 0 is the amplitude of the field at unit distance away from the source. As in the case for the spherical wave, (2.8) is true only when the cylinder radius is large

28

2 Mathematics Used for Expressing Waves

Fig. 2.3 Energy distribution emanating from a line source

enough for the incremental area on the cylindrical surface to be considered planar. The expression for the line source, including the time dependence, is E0 E(t, r ) = √ e−j(ωt−kr ) . r

(2.9)

The amplitude of a spherical wave decays as r −1 whereas that of a cylindrical wave decays as r −1/2 with distance from the source.

2.3 Plane Waves This section is devoted to the plane wave. As shown in Fig. 2.4, a plane wave propagates at an angle θ with respect to the x axis. The unit vector kˆ denotes the direction of propagation. For simplicity, the z dimension is ignored. The point of observation, P, is specified by the coordinates (x, y). The position vector r joins P to the origin and is written as r = iˆ x + jˆ y, (2.10) where iˆ and jˆ are unit vectors in the x and y directions. Referring to Fig. 2.4, another point P0 is determined by the intersection of the ˆ P and P0 have the line through kˆ and the line passing through P perpendicular to k. same phase. If the phase at the origin is φ0 , the phase at P0 is φ=

2π OP0 + φ0 λ

(2.11)

where λ is the wavelength, and OP0 is the distance from O to P0 . The line segment OP0 is actually the projection of r onto kˆ and can be expressed as the scalar product ˆ Thus, of r and k. 2π ˆ φ= (2.12) k · r + φ0 . λ

2.3 Plane Waves

29

Fig. 2.4 Position vector and the wave vector of a plane wave propagating in the kˆ direction

Defining a new vector as k=

2π ˆ k, λ

(2.13)

Eq. (2.12) becomes φ = k · r + φ0 .

(2.14)

The vector k is called the wave vector and can be decomposed into components as follows: ˆ cos θ + ˆj |k| sin θ. k = i|k| (2.15) From (2.13, 15), the wave vector is written as k = iˆ

2π 2π + ˆj . λ/ cos θ λ/ sin θ

From Fig. 2.4, λx = λ/ cos θ and λ y = λ/ sin θ are the wavelengths along the x and y axis respectively, so that 2π 2π + ˆj . k = ˆik x + ˆj k y = iˆ λx λy 2 2π 2π 2π 2 |k| = + . = λ λx λy

(2.16)

Making use of the definitions (2.12), the plane wave in Fig. 2.4 can be expressed as E(x, y) = E 0 exp [ j(k x x + k y y − ωt + φ0 )], or upon generalizing to three dimensions E(x, y, z) = E 0 (x, y, z) exp [ j(k · r − ωt + φ0 )].

(2.17)

30

2 Mathematics Used for Expressing Waves

Here E 0 (x, y, z) is a vector whose magnitude gives the amplitude of the plane wave and whose direction determines the polarization. The mathematical analysis becomes quite complex when the direction of polarization is taken into consideration. A simpler approach, usually adopted for solving engineering problems, is to consider only one of the vector components. This is usually done by replacing the vector quantity E(x, y, z) with a scalar quantity E(x, y, z). Such a wave is called a “vector wave” or “scalar wave” depending on whether the vector or scalar treatment is applied. Figure 2.4 is a time-frozen illustration of a plane wave. The sense of the propagation is determined by the combination of the signs of k · r and ωt in (2.17) for the plane wave. This will be explained more fully by looking at a particular example. Consider a plane wave, which is polarized in the y direction and propagates in the x direction, as given by

(2.18) E = Re E y e j(kx−ωt) = E y cos (kx − ωt). The solid line in Fig. 2.5 shows the amplitude distribution at the instant when t = 0. The first peak (point Q) of the wave form E corresponds to the case where the argument of the cosine term in (2.18) is zero. After a short time (t = ∆t) the amplitude distribution of (2.18) becomes similar to the dashed curve in Fig. 2.5. In the time ∆t, the point Q has moved in the positive direction parallel to the x-axis. Since the point Q represents the amplitude of the cosine term when the argument is zero, the x coordinate of Q moves from 0 to ∆x = (ω/k)∆t, as t changes from 0 to ∆t. On the other hand, when t changes from 0 to ∆t in an expression such as E = E y cos(kx + ωt)

(2.19)

the position of Q moves from x = 0 to ∆x = −(ω/k)∆t which indicates a backward moving wave. Similarly, E = E y cos(−kx + ωt) represents a forward moving wave and E = E y cos(−kx − ωt) denotes a backward moving wave. In short, when the signs of kx and ωt are different, the wave is propagating in the forward (positive) direction; when kx and ωt have the same sign, the wave is propagating in the backward (negative) direction.

Fig. 2.5 Determination of the directional sense (positive or negative) of a wave

2.4 Interference of Two Waves

31

Fig. 2.6 Summary of the results of Exercise 2.1

As mentioned earlier, during the time interval ∆t, the wave moves a distance ∆x = (ω/k)∆t so that the propagation speed of the wave front (phase velocity) is υ=

ω ∆x = . ∆t k

(2.20)

Exercise 2.1. A plane wave is expressed as √ √ E = (−2 iˆ + 2 3 ˆj ) exp [ j( 3x + y + 6 × 108 t)]. Find; 1) the direction of polarization, 2) the direction of propagation, 3) the phase velocity, 4) the amplitude, 5) the frequency, and 6) the wavelength. Solution

√ 1) − iˆ +√ 3 ˆj 2) k = 3 iˆ +√ˆj . This wave is propagating in the −k direction (210◦ ) 3) |k| = k = 3 + 1 = 2 υ = ωk = 3 × 108 m/s √ 4) |E| = 22 + 22 × 3 = 4 v/m ω = π3 × 108 Hz 5) f = 2π υ 6) λ = f = π m.

A summary of the results is shown in Fig. 2.6. It was shown that there are two ways of expressing waves propagating in the same direction. To avoid this ambiguity, we adopt the convention of always using a negative sign on the time function. Consequently, a plane wave propagating in the positive x direction will be written as exp [ j(kx − ωt)] and not as exp [ j(ωt − kx)].

2.4 Interference of Two Waves When considering the interference of two waves [2.1], the amplitudes and frequencies are assumed to be identical. In particular, consider the two waves

32

2 Mathematics Used for Expressing Waves

Fig. 2.7 Components of wave vectors k1 and k2

E 1 = E 0 exp [ j(k1 · r + φ1 − ωt)] E 2 = E 0 exp [ j(k2 · r + φ2 − ωt)].

(2.21)

First, as shown in Fig. 2.7, the vectors k1 , k2 are decomposed into components in the same, and opposite directions. The line PQ connects the tips of k1 and k2 . The vectors k1 and k2 are decomposed into k1 and k2 , which are parallel to OC, and k1 and k2 , which are parallel to CP and CQ, with k1 = k1 + k1 , k1 = k2 = k ,

k2 = k2 + k2 ,

k1 = −k2 = k2 .

(2.22)

The resultant field E = E 1 + E 2 is obtained by inserting (2.22) into (2.21).

E = 2E 0 cos(k · r + ∆φ) e j(k ·r+φ−ωt) , phase amplitude φ1 + φ 2 φ1 − φ2 φ= , ∆φ = . 2 2

(2.23)

(2.24)

The first factor in (2.23) has no dependence on time. Consequently, along the locus represented by π (2.25) k · r + ∆φ = (2n + 1) , 2 the field intensity is always zero and does not change with respect to time. These loci, shown in Fig. 2.8 a, generate a stationary pattern in space known as an interference fringe. The presence of interference fringes is related only to the vectors k and −k , which are components pointed in opposite directions, and is not dependent on k at all. When the frequencies of the two waves are identical and when no components occur in opposite directions, no interference fringe pattern appears. The spacing between the adjacent null lines of the fringes in k is obtained from (2.25) by subtracting the case for n from that of n + 1 and is π/|k | and the period is twice this value. It is also interesting to note that the fringe pattern can be shifted by changing the value of ∆φ. The second factor in (2.23) is equivalent to the expression for a plane wave propagating in the k direction.

2.5 Spatial Frequency

33

Fig. 2.8 a, b. Interference pattern produced by the vector components k . (a) Geometry, (b) analogy to a corrugated sheet

To better understand the travelling-wave and standing-wave aspects of (2.23), an analogy can be drawn between the interference of the two waves and a corrugated plastic sheet. If one aligns k in the direction cutting the ridges and grooves of the corrugation, the variation of the amplitude of the standing wave is represented by the corrugation, while the direction along the grooves aligns the direction k of the propagation of the travelling wave (Fig. 2.8 b).

2.5 Spatial Frequency The word “frequency” is normally used to mean “temporal frequency”, i.e., frequency usually denotes a rate of repetition of wave forms in unit time. In a similar manner, “spatial frequency” is defined as the rate of repetition of a particular pattern in unit distance. The spatial frequency is indispensable in quantitatively describing the resolution power of lenses and films and in signal processing analysis. The units of spatial frequency are lines/mm, lines/cm, or lines/inch. In some cases, such as when measuring human-eye responses, the units of spatial frequency are given in terms of solid angles, e.g., lines/minute of arc or lines/degree of arc. Since most wave form patterns in this text are treated as planar, the spatial frequencies of interest are those pertaining to the planar coordinates, usually x and y. Figure 2.9 illustrates the definition of these spatial frequencies, namely spatial frequency f x in x =

1 1 = OA λx

spatial frequency f y in y =

1 1 . = OB λy

(2.26)

34

2 Mathematics Used for Expressing Waves

Fig. 2.9 Definition of spatial frequency

The spatial frequency f = 1/λ in the OH direction is obtained from λ = λx cos ∠HOA λ = λ y sin ∠HOA whereby, 1 f = = x

1 λx

2 +

1 λy

2 =

( f x )2 + ( f y )2 .

(2.27)

2.6 The Relationship Between Engineering Optics and Fourier Transforms In this section, it will be shown that there is a Fourier transform relationship between the light distribution of a source and the light distribution appearing on a screen illuminated by the source [2.2]. Consider the configuration shown in Fig. 2.10. A one-dimensional slit of length a is placed along the x0 axis. The slit is illuminated from behind by a nonuniform light source. The amplitude distribution along the slit is E(x0 ). The slit can be approximated as a composite of point sources. From (2.6), the amplitude E(xi , 0) of the light observed along the xi axis is, to within a constant multiplying factor K , given by xi2 K exp jk z i + E(xi , 0) = zi 2z i a/2 × −a/2

xi E(x0 ) exp −j2π x0 d x0 , λz i

(2.28)

where x02 /λz i 1 was assumed. It will be shown in the next chapter that K = −j/λ. By carefully examining (2.28), one finds that this equation is in the same form as a Fourier transform, where the Fourier transform is defined as

2.6 The Relationship Between Engineering Optics and Fourier Transforms

35

Fig. 2.10 Geometry used to calculate the field at the observation point P from a one dimensional slit

∞ G( f ) = F {g(x0 )} =

g(x0 ) exp(−j2π f x0 ) d x0 .

(2.29)

−∞

By letting

xi , g(x0 ) = E(x0 ) λz i Eq. (2.28) can be rewritten with the help of (2.29) as xi2 K E(xi , 0) = exp jk z i + F {E (x0 )} , zi 2z i f =

(2.30)

f = xi /λz i.

(2.31)

In short, the amplitude distribution of light falling on a screen is provided by the Fourier transform of the source light distribution, where f in the transform is replaced by xi /λz i . In other words, by Fourier transforming the source function, the distribution of light illuminating a screen is obtained.

36

2 Mathematics Used for Expressing Waves

This fact forms the basis for solving problems of illumination and light scattering, and also plays a key role in optical signal processing. The branch of optics where the Fourier transform is frequently used is generally known as Fourier optics. The usefulness of the Fourier transform is not limited to the optical region alone, but is applicable to lower frequency phenomena as well. For example, the radiation pattern of an antenna is determined by taking the Fourier transform of the current distribution on the antenna. Furthermore, the Fourier transform concept has been an important principle in radio astronomy. By knowing the distribution of the radiated field reaching the earth, the shape of the radiating source can be found. This is the reverse process of what was discussed in this chapter. The shape of the source is obtained by inverse Fourier transforming the observed radiated field. The difficulty with radio astronomy, however, is that the phase information about the radio field is not readily available to analyze. Returning to (2.31), the problem of evaluating the Fourier transform in this equation is now considered. Most transforms are well tabulated in tables, but as an exercise the transform will be calculated for a simple example. Let the distribution of the light along the slit be uniform and unity, and let the slit have unit length as well. The function describing the source is then E(x0 ) = (x0 ),

(2.32)

where

1 |x| < = 1/2, 0 elsewhere.

(x) =

(2.33)

The value of the function is therefore unity in the range |x| < = 1/2 and zero elsewhere. This function, which has been given the symbol , is known as either the rectangle function or the gating function. Its graph is shown in Fig. 2.11 a. The rectangle function in its more general form is written as x −b a and represents a function that takes on the value of unity in the interval of length a centered at x = b, and is zero elsewhere. Equation (2.33) corresponds to the special case where a = 1 and b = 0. The Fourier transform of (x) is ∞ F { (x)} =

−j2π f x

(x) e

1/2 dx =

∞

=

e jπ f

e−j2π f x d x

−1/2

− e−jπ f j2π f

=

sin π f . πf

(2.34)

Equation (2.34) is a sine curve that decreases as 1/π f . This function is called the sinc function, so that

2.7 Special Functions Used in Engineering Optics and Their Fourier Transforms

37

Fig. 2.11 (a) Rectangle function (x) and (b) its Fourier transform

Fig. 2.12 a,b. The field distribution on an observation screen produced by a one dimensional slit. (a) Amplitude distribution, (b) intensity distribution

F { (x)} = sinc( f ) =

sin π f . πf

(2.35)

The graph of (2.35) is shown in Fig. 2.11 b. Inserting (2.35) into (2.31), the final expression for E(xi , yi ) is obtained as xi2 + yi2 K xi exp jk z i + E(xi , yi ) = , (2.36) sinc zi 2Z i λz i The amplitude distribution on the screen is shown in Fig. 2.12 a and the intensity distribution in Fig. 2.12 b.

2.7 Special Functions Used in Engineering Optics and Their Fourier Transforms The gating function described in the previous section is one of several functions that appear often in engineering optics, and consequently, have been given special symbols to represent them [2.2, 3]. This section defines these functions and presents their Fourier transforms.

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2 Mathematics Used for Expressing Waves

Fig. 2.13 (a) Triangle function Λ(x) and (b) its Fourier transform sinc2 f

Fig. 2.14 a, b Curves of (a) sign function sgn x and (b) its Fourier transform 1/jπ f

2.7.1 The Triangle Function The triangle function is shaped like a triangle, as its name indicates. Figure 2.13 a illustrates the wave form. The function can be obtained by the convolution of (x) with itself Λ(x) = (x) ∗ (x). (2.37) Using the fact that F { (x) ∗ (x)} = F { (x)} F { (x)} , the Fourier transform of (2.37) is found to be (Fig. 2.13 b) F {Λ(x)} = sinc2 f.

(2.38)

2.7.2 The Sign Function The function defined to be +1 in the positive region of x and −1 in the negative region of x is called the sign function (Fig. 2.14 a). It is written as sgn x: ⎧ ⎨−1 x < 0, 0 x = 0, sgn x (2.39) ⎩ +1 x > 0. The Fourier transform of the sign function is obtained as follows:

2.7 Special Functions Used in Engineering Optics and Their Fourier Transforms

0 F {sgn x}

−e−j2π f x d x +

−∞

∞

39

e−j2π f x d x.

(2.40)

The integral in (2.40) contains the improper integral ∞ e−j2π f x = ?. −j2π f 0

The exponential in the above expression is an oscillating function and such a function is not defined at ∞. However, the integral may still be evaluated by using appropriate limiting operations, as explained below. Adding ‘αx’ terms to the exponents in (2.40) and taking the limit as α approaches zero from the right gives 0 F {sgn x} = − −∞

−j2π f x+αx

lim e

α→0+

∞ dx +

lim e−j2π f x−αx d x.

α→0+ 0

Interchanging the operation of the limit with that of the integral results in 0 ∞ −e−j2π f x+αx e−j2π f x−αx + lim . F {sng x} = lim α→0+ −j2π f − α −j2π f + α α→0+ −∞

For positive α and negative x, the expression exp (αx) exp (− j2π f x) is the product of an exponentially decaying function and a bounded oscillating function. Thus, at x = −∞, exp (αx) is zero making exp (αx) exp (− j2π f x) zero. Likewise, for positive α and x, the expression exp (−αx) exp (− j2π f x) is zero at x = ∞. Equation (2.40) then reduces to 1 1 1 − 0 + lim 0 − . = lim − α→0+ α→0+ −j2π f + α −j2π f x − α jπ f In conclusion, the Fourier transform of the sign function is (Fig. 2.14 b) F {sgnx} =

1 . jπ f

(2.41)

2.7.3 The Step Function The function whose value is zero when x defined to be the step function ⎧ ⎪ ⎨0 H(x) = 12 ⎪ ⎩ 1

is negative and unity when x is positive is for for for

x < 0, x = 0, x > 0.

(2.42)

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2 Mathematics Used for Expressing Waves

Fig. 2.15 a, b. Curves of (a) the step function H(x) and (b) its Fourier transform 1 2 [δ(x) + 1/jπ f ]

Fig. 2.16 (a) Analogy showing how the δ(x) function has constant area. (b) δ(x) and an arbitrary function f (x)

The step function can be rewritten by using the sgn function as H(x) =

1 (1 + sgnx) . 2

The Fourier transform of H(x) is obtained using (2.41). 1 1 F {H(x)} = δ( f ) + , 2 jπ f

(2.43)

where the symbol δ represents the delta function to be explained in the next section. Figure 2.15 illustrates the step function and its Fourier transform.

2.7.4 The Delta Function The delta function δ(x) possesses the following two properties [2.4]: 1) The value of the function is zero except at x = 0. The value at x = 0 is infinity. 2) The area enclosed by the curve and the x axis is unity. A simple description of the delta function can be made using an analogy. As shown by the picture in Fig. 2.16 a, no matter how high a piece a dough is pulled up, the volume is constant (in the case of the delta function, area is constant), and the width decreases indefinitely with an increase in height. The delta function δ(x) satisfies ε δ(x) d x = 1. −ε

(2.44)

2.7 Special Functions Used in Engineering Optics and Their Fourier Transforms

41

It is more common to use the delta function in an integral form than by itself, as follows. ε ε f (x) δ(x) d x = f (0) δ(x) d x = f (0). (2.45) −ε

−ε

Because the value of δ(x) is zero everywhere except in a small region of |x| < ε, the integration need only extend from −ε to +ε. If the function f (x) is slowly varying in the interval |x| < ε, then f (x) can be considered constant over this interval, i.e., f (x) is approximately f (0) in the interval |x| < ε, as illustrated in Fig. 2.16 b. Equation (2.45) was obtained under this assumption. Equation (2.45) thus expresses an important property of the delta function and is sometimes used as a definition of the delta function. Another important property of the delta function is δ(ax) =

1 δ(x). |a|

(2.46)

Equation (2.46) is readily obtained from (2.45). Let a be a positive number. Using δ(ax) in place of δ(x) in (2.45) and then making the substitution ax = y gives ε −ε

1 f (x) δ(ax)d x = a

aε f −az

y a

δ(y) dy =

1 f (0). a

(2.47)

Comparing both sides of (2.45, 47), one concludes δ(ax) =

1 δ(x). a

(2.48)

1 δ(x) a

(2.49)

Similarly, it can be shown that δ(−ax) =

where a again is a positive number. Equation (2.46) is a combination of (2.48, 49), and holds true for positive and negative a. The Fourier transform of the delta function is ∞ δ(x) e−j2π f x d x = e j2π f ·0 = 1, and thus −∞

F {δ(x)} = 1.

(2.50)

2.7.5 The Comb Function The comb function, shown in Fig. 2.17, is an infinite train of delta functions. The spacing between successive delta functions is unity. The comb function is given the symbol III and is expressed mathematically as

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2 Mathematics Used for Expressing Waves

Fig. 2.17 (a) Shah function generator, (b) shah function III (x)

III(x)

n=∞

δ(x − n).

(2.51)

n=−∞

Since the symbol of III is a cyrillic character and is pronounced shah, the comb function is sometimes called the shah function. The comb function with an interval of a can be generated from (2.46, 51). III

x a

=

∞

δ

x

n=−∞

a

∞ δ(x − an). −n =a

(2.52)

n=−∞

There is a scale factor a appearing on the right hand side of (2.52) which can loosely be interpreted as an increase in the ‘height’ of each delta function with an increase in the interval a thus keeping the total sum constant. Equation (2.51) can be used to generate a periodic function from a function with an arbitrary shape. Since g(x) ∗ δ(x − n) = g(x − n) is true, a function generated by repeating g(x) at an interval a is h(x) =

x 1 g(x) ∗ III . a a

(2.53)

Figure 2.18 illustrates this operation. Next, the Fourier transform of the comb function is obtained. The comb function III(x) is a periodic function with an interval of unity and can be expanded into a Fourier series, as indicated in Fig. 2.19. The definition of the Fourier series is f (x) =

∞

an e j2π n(x/T )

where

(2.54)

n=−∞

1 an = T

T /2 −T /2

f (x)e−j2πn(x/T ) d x.

(2.55)

2.8 Fourier Transform in Cylindrical Coordinates

43

Fig. 2.18 Convolution of an arbitrary function g (x) with the shah function resulting in a periodic function of g (x)

Fig. 2.19 Fourier expansion of the shah function

Inserting f (x) = III(x) and T = 1 into (2.55), an becomes unity, and the Fourier expansion of III(x) is found to be ∞

III(x) =

e j2π nx .

(2.56)

n=−∞

Equation (2.56) is an alternate way of expressing the comb function. The Fourier transform of (2.56) is F {III(x)} =

∞

δ( f − n) = III( f ),

(2.57)

n=−∞

where the relationship

∞

e j2π( f −n) dx = δ( f − n)

−∞

and then (2.46) were used. The result is that the Fourier transform of a comb function is again a comb function. This is a very convenient property widely used in analysis.

2.8 Fourier Transform in Cylindrical Coordinates In engineering optics, one deals with many phenomena having circular distributions. The cross-section of a laser beam or circular lenses are two such examples.

44

2 Mathematics Used for Expressing Waves

In these cases, it is more convenient to perform the Fourier transform in cylindrical coordinates [2.5].

2.8.1 Hankel Transform The two-dimensional Fourier transform in rectangular coordinates is ∞ ∞ G( f x , f y ) =

g(x, y) exp [−j2π( f x x + f y y)] d x d y.

(2.58)

−∞ −∞

The expression for this equation in cylindrical coordinates is sought. Figure 2.20 shows the relationship between rectangular and cylindrical coordinates, namely, x = r cos θ y = r sin θ d x d y = r dr dθ

(2.59)

and in the Fourier transform plane f x = cos φ f y = sin φ d f x d f y = d dφ

(2.60)

Inserting (2.59, 60) into (2.58), the result is ∞ 2π G(, φ) = o

g(r, θ) e−j2πr cos(θ−φ)r dr dθ,

(2.61)

Fig. 2.20 a, b. Fourier transform in cylindrical coordinates. (a) Spatial domain, (b) spatial frequency domain

2.8 Fourier Transform in Cylindrical Coordinates

45

where g(r cos θ, r sin θ ) was written as g(r, θ). As long as g(r, θ) is a single valued function, g(r, θ) equals g(r, θ + 2nπ ). Therefore, g(r, θ) is a periodic function of θ with a period of 2π and can be expanded into a Fourier series. The coefficients of the Fourier series are a function of r only, and referring to (2.54, 55), ∞

g(r, θ) =

gn (r ) e jnθ ,

where

(2.62)

n=−∞

1 gn (r ) = 2π

2π

g (r, θ ) e−jnθ dθ.

(2.63)

Equations (2.62, 63) are inserted into (2.61) to give G(, φ) =

∞ ∞ n=−∞

2π r gn (r ) dr

e−j2π r cos(θ−φ)+jnθ dθ.

(2.64)

Using the Bessel function expression in integral form 1 Jn (z) = 2π

2π +a

e j(nβ−z sin β) dβ,

(2.65)

a

Eq. (2.64) can be simplified. In order to apply (2.65), result (2.64) must be rewritten. Let cos(θ − φ) be replaced by sin β, where1 θ −φ =β +

3π . 2

(2.66)

Inserting (2.66) into the second integral in (2.64) gives 2π

−j2πr cos(θ−φ)+jnθ

e

3 dθ = exp jn π + jnφ 2

= (− j) e

n jnφ

2π −φ−3π/2

e j(nβ−2π r sin β) dβ

−φ−3π/2

2π Jn (2πr ).

(2.67)

Inserting (2.67) into (2.64) gives

1

Another substitution

π −β 2 also seems to satisfy the condition. If this substitution is used, the exponent becomes −jnβ + jn(φ + π/2) and the sign of jnβ becomes negative so that (2.65) cannot be immediately applied. θ −φ =

46

2 Mathematics Used for Expressing Waves

G(, φ) =

∞

∞ (− j) e n

jnφ

r gn (r )Jn (2π r ) dr, where

2π

n=−∞

gn (r ) =

1 2π

2π

(2.68)

g(r, θ)e− jnθ dθ.

(2.69)

Similarly, the inverse Fourier transform is ∞

g(r, θ) =

∞ ( j) e n

jnθ

n=−∞

1 G n (ρ) = 2π

∞

ρG n (ρ)Jn (2πr ) d.

2π 0

(2.70)

G(ρ, φ) e− jnφ dφ

In the special case where no variation occurs in the θ direction, i.e., when g(r, θ) is cylindrically symmetric, the coefficients gn (r ) can be written as g0 (r ) for n = 0, (2.71) gn (r ) = 0 for n = 0, and (2.68, 70) simplify to ∞ G 0 () = 2π

r g0 (r )J0 (2πr ) dr,

(2.72)

G 0 ()J0 (2πr )d.

(2.73)

∞ g0 (r ) = 2π 0

Equations (2.68–73) are Fourier transforms in cylindrical coordinates. Essentially, they are no different than their rectangular counterparts. Answers to a problem will be identical whether rectangular or cylindrical transforms are used. However, depending on the symmetry of the problem, one type of transform may be easier to calculate than the other. The integrals in (2.68, 70) are called the Hankel transform of the nth order and its inverse transform, respectively. The zero-th order of the Hankel transform (2.72, 73) is called the Fourier Bessel transform and its operation is symbolized by B{}. Now that the Fourier Bessel transform has been studied, another special function is introduced that can be dealt with very easily by using the Fourier Bessel transform. A function having unit value inside a circle of radius 1, and zero value outside the circle, as shown in Fig. 2.21 a, is called the circle function. The symbol ‘circ’ is used to express this function

2.8 Fourier Transform in Cylindrical Coordinates

47

Fig. 2.21 (a) Circle function circ(r ) and (b) its Fourier transform J1 (2π)/

circ(r ) =

1 for r < =1 0 for r > 1.

(2.74)

To obtain the Fourier transform of (2.74), result (2.72) can be used since circ(r ) has circular symmetry. Therefore, the Fourier Bessel transform of circ(r ) is 1 G() = B {circ(r )} = 2π

r J0 (2πr ) dr.

(2.75)

2π J1 (2π ) τ J0 (τ ) dτ = .

(2.76)

Introducing a change of variables, 2πr = τ, the transform becomes 1 G() = 2 π 2

where, the formula for the Bessel integral z zJ0 (αz) dz = 0

z J1 (αz) α

(2.77)

was used. Figure 2.21 b illustrates the shape of (2.76).

2.8.2 Examples Involving Hankel Transforms A few examples of problems requiring the use of Hankel transforms will be presented.

48

2 Mathematics Used for Expressing Waves

Fig. 2.22 Diagram showing geometry of Exercise 2.2 (mirror placed at an angle γ to the plane perpendicular to the optical axis)

Exercise 2.2. As shown in Fig. 2.22, a mirror is tilted at an angle γ about the x-axis. Let the light reflected from the mirror be expressed by 2 (2.78) g(x, y) = e j2ky tan γ circ ( x 2+y ). The projection of the mirror’s light distribution on the x y plane is a circle of unit radius. a) Express g(x, y) in cylindrical coordinates. b) Expand the above result for g(r, θ) in a Fourier series with respect to θ. c) Find the Hankel transform of g(r, θ). Solution (a) From Fig. 2.22, y can be expressed in terms of r and θ as y = r sin θ so that 2ky tan γ = kβr sin θ, where β = 2 tan γ .

(2.79)

Inserting (2.79) into (2.78) gives g(r, θ) = e jkβr sin θ circ(r ).

(2.80)

(b) By using (2.63), one obtains circ(r ) gn (r ) = 2π

2π

e j(kβr sin θ−nθ) dθ.

(2.81)

In order to use (2.65), the substitution θ = −θ must be made gn (r ) =

circ(r ) 2π

0 −2π

e j(nθ −kβr sin θ ) dθ .

(2.82)

2.8 Fourier Transform in Cylindrical Coordinates

49

Setting a = −2π in (2.65) results in gn (r ) = circ(r )Jn (kβr ).

(2.83)

Inserting this into (2.62) gives ∞

g(r, θ) =

circ(r )Jn (kβr )e jnθ .

(2.84)

n=−∞

(c) Inserting the above result into (2.68) gives ∞

G(, φ) = 2π

1 (− j) e

r Jn (kβr ) Jn (2π r ) dr.

n jnφ

n=−∞

(2.85)

Applying the following Bessel integral formula, 1 Jn (βx)Jn (αx) x d x 0

=

1 [αJn (β)Jn+1 (α) − βJn (α)Jn+1 (β)], α2 − β 2

(2.86)

Equation (2.85) becomes G(, φ) =

∞ n=−∞

2π

(−j)n e jnφ [kβJn (2π ) Jn+1 (kβ) (kβ)2 − (2π )2

− 2π Jn (kβ)Jn+1 (2π )].

(2.87)

This example was chosen from the mode analysis at a slant laser mirror. Exercise 2.3. Express the plane wave E = exp ( jky), propagating in the positive y direction, in cylindrical coordinates and expand it into a Fourier series in θ. Solution The results of Exercise 2.2 are used to obtain E = e jky = e jkr sin θ .

(2.88)

Using (2.80, 84) with kr instead of kβr gives e jkr sin θ =

∞

Jn (kr )e jnθ .

(2.89)

n=−∞

A plane wave exp (−jky) propagating in the negative y direction can be obtained in a similar way ∞ e−jkr sin θ = Jn (kr ) e−jnθ . (2.90) n=−∞

Equations (2.89, 90) can be thought of as a group of waves which are propagating in the ±θ direction and varying as exp (±jnθ ).

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2 Mathematics Used for Expressing Waves

2.9 A Hand-Rotating Argument of the Fourier Transform Consider the meaning of the Fourier transform ∞ G( f ) =

g(x) e−j2π f x d x.

−∞

Suppose g(x) contains many frequency components and is expressed as g(x) = G 0 + G 1 e j2π f 1 x + G2 e j2π f 2 x + G 3 e j2π f 3 x . . . . The phasor of each term rotates with x. Now g(x) is multiplied by the factor exp(−j2π f x). Only the term whose phasor rotates at the same speed as f , but in the opposite direction, stops rotation, whereas all other terms still rotate with x.

A hand rotating argument of the Fourier transform

When the integration with respect to x is performed, all the terms whose phasor still rotates and whose real and imaginary parts oscillate with respect to the zero axis become zero, whereas the integration of the terms whose phasor has stopped rotating does not have a zero value. Thus, the Fourier transform operation can be considered as a means of picking a term whose phasor rotates at a specific speed.

Problems 2.1. A vector wave is expressed as jˆ ˆ iˆ 9 E = − − + k e j2π [10(x+y+z)−3×10 t] . 2 2 By using MKS units, find: 1) the direction of polarization, 2) the direction of propagation, 3) the velocity,

Problems

51

4) the amplitude, 5) the frequency, 6) the wavelength. 2.2. Two simultaneous measurements upon an unknown sinusoidal wave were made along the x and y axes. The phase distribution along the x axis was φx = 10x √ radians and that along the y axis was φ y = 310y radians. What is the direction of incidence of this unknown wave and what is its wavelength? 2.3. Prove the following relationships a) F ddx g(x) = j2π f G( f ), where F {g(x)} = G( f ). b)

d d x [ f (x) ∗ g(x)]

=

d dx

f (x) ∗ g(x) = f (x) ∗ ddx g(x) .

c) III(−x) = III(x). d) III(x − 12 ) = III(x + 12 ). e) π δ(sin π x) = III(x). f) III

x 2

= III(x) + III(x)e jπ x .

Fig. 2.23 a, b. Masks with transmission distributions t (x)

Fig. 2.24 Semi-circular mask (see Problem 2.7)

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2 Mathematics Used for Expressing Waves

! ! ! Nπx ! g) III(x) = lim ! sin sin π x !. N →∞

2.4. Find the formulae which describe the two transmittance distributions shown in Figure 2.23 and calculate their Fourier transforms. 2.5. Prove that the expression for the delta function in cylindrical coordinates is ∞ δ() = 2π

r J0 (2π r ) dr. 0

2.6. Find the Fourier transform G(, ϕ) of the transmittance function g(r, θ) = circ (r ) cos pθ. where p is an integer. Hint Jn (z) = (−1)n Jn (z) The g(r, θ) has the pth angular symmetry. As you see from your result the same pth symmetry exists in G(, ϕ). This is the basic principle of the crystrography by which the symmetry in the crystal lattice is found from the symmetry of its X ray diffraction pattern. 2.7. Let the transmittance function of a semi-circular window, as shown in Fig. 2.24, be t (r, θ). Find its Fourier transform T (, ϕ).

Chapter 3

Basic Theory of Diffraction

In order to explain the diffraction phenomenon of light sneaking into shaded regions, C. Huygens (1678) claimed, “In the process of propagation of the wave, new wave fronts are emanated from every point of the old wave front”. This principle, however, had a serious drawback in that the wave front, as drawn, generated an unwanted wave front in the backward direction in addition to the wave front in the forward direction. This drawback was overcome by A. J. Fresnel (1818) and later by G. Kirchhoff (1882) by taking the periodic nature of the wave into consideration.

3.1 Kirchhoff’s Integral Theorem Let the light wave be expressed by E(x, y, z). As an electromagnetic wave, E satisfies the wave equation (3.1) ∇ 2 E + k 2 E = 0. A simplified theory [3.1–5] will be adopted here which focuses on only one of the vector components. This component will be represented by the scalar quantity υ. In a region where there is no source, υ satisfies the wave equation ∇ 2 υ + k 2 υ = 0.

(3.2)

A spherical wave in free space can be expressed in terms of the solution of (3.2) in spherical coordinates. Since the only variation of υ is in the r direction, ∇ 2 υ becomes 1 d 2 (r υ) , ∇ 2υ = r dr 2 so that (3.2) reduces to d 2 (r υ) + k 2 (r υ) = 0. dr 2

(3.3)

53

54

3 Basic Theory of Diffraction

The general solution of (3.3) is r υ = Ae jkr + B e−jkr , υ=A

e jkr

or

(3.4)

e−jkr

+B . (3.5) r r Since this book uses the exp (−jωt) convention, the first term of (3.5) represents a wave diverging from the origin and the second, a wave converging toward the origin. Next, an approximate expression is derived for the field when the radiation medium is not free space. According to Green’s theorem the following relationship holds for given functions υ and u inside a volume V enclosed completely by the surface S, i.e. 2 2 (3.6) I = (υ∇ u − u∇ υ) d V = (υ∇u − u∇υ) · nˆ d S. V

S

where nˆ is the outward normal to the surface S. Green’s theorem is used for converting a volume integral into a surface integral. The only condition imposed on (3.6) is that the volume V , as shown in Fig. 3.1, be formed in such a way as to exclude the points P1 , P2 , P3 , . . . , Pn where the second derivatives of υ and u are discontinuous. Other than this condition, there are no restrictions on the choice of the functions υ and u. Since the choice of υ and u is arbitrary so long as the above-mentioned condition is satisfied, υ will be taken as (3.7) υ = e jkr /r, which is the solution of (3.2), and u as the solution of ∇ 2 u + k 2 u = −g.

(3.8)

Equation (3.8) is the wave equation inside a domain with a radiating source like the one in Fig. 3.2. The integral I was obtained by inserting (3.2, 8) into the left-hand side of (3.6) so that

Fig. 3.1 A volume V completely enclosed by a surface S. Note that the volume should not include any singularities such as P1 , P2 , P3 , . . . Pn

3.1 Kirchhoff’s Integral Theorem

55

Fig. 3.2 A volume V containing a source g . Note that the observation point is excluded from the volume but the source g is not

I =−

g

e jkr d V. r

(3.9)

V

Since the first and second derivatives of υ are both discontinuous at the origin r = 0, the volume must not include this point. Let the spherical wave g be expanding from the observation point P0 . A small sphere centered at P0 with radius ε is excluded from the volume V . The removal at this sphere creates a new surface S1 and the total surface of the volume becomes S1 + S2 , where S2 is the external surface as shown in Fig. 3.2. Inserting (3.7, 9) into (3.6) gives jkr e jkr e jkr e g dV + ∇u − u∇ · nˆ d S r r r V

S1

+

e jkr ∇u − u∇ r

e jkr r

· nˆ d S = 0.

(3.10)

S2

As the first step toward evaluating (3.10), the integration over the surface S1 will be examined. Expressing the integrand in spherical coordinates gives d e jkr01 1 e jkr01 e jkr01 rˆ 01 · nˆ (3.11) · nˆ = rˆ 01 · nˆ = jk − ∇ r01 dr01 r01 r01 r01 where rˆ 01 is a unit vector pointing radially outward from the center P0 of the sphere, to the surface S1 of integration. Since the unit vectors rˆ 01 and nˆ are parallel but in the opposite directions on the spherical surface with radius ε, the value of rˆ 01 · nˆ is −1. Hence the value of (3.11) on the surface of the sphere S1 is 1 e jkε . (3.12) − jk − ε ε

56

3 Basic Theory of Diffraction

Inserting this result into the second term on the left-hand side of (3.10) and taking the limit as ε → 0, and assuming that the integrand on the surface of this small sphere is constant, yields jkε jkε e 1 e (3.13) (∇u) · nˆ + jk − u(P0 ) = −4π u(P0 ), lim 4π ε2 ε→0 ε ε ε where u(0) is the value of u at the observation point P0 . Inserting (3.13) into (3.10) and replacing u(0) by u p , produces the result jkr01 1 e jkr01 1 e jkr01 e up = g dV + ∇u − u∇ · nˆ d S. (3.14) 4π r01 4π r01 r01 V

S2

This formula gives the field at the point of observation. The amplitude of the illumination is, of course, influenced by the shape of the source, or the location of the observation, but should not be influenced by the choice of the volume of integration. A domain V such as shown in Fig. 3.3, which excludes the light source, is equally applicable and the result for u p is identical to that obtained using domain V . A rigorous proof of this statement is left as an exercise, but a general outline on how to proceed is given here. In the case of V , the value of e jkr01 1 g dV 4π r01 is zero, but new surfaces S3 , S4 and S5 are created in the process of digging a hole to exclude the source, as illustrated in Fig. 3.3. The surface integral over S3 makes up for the loss of e jkr01 1 dV g 4π r01 and the value of u p does not change. The surface integral over S4 +S5 is zero because the integral over S4 exactly cancels that over S5 due to the fact that nˆ 4 = −nˆ 5 . The

Fig. 3.3 A domain V which is similar to V in Fig. 3.2 but with the source excluded. The resulting u p using this geometry is identical to that calculated using the geometry in Fig. 3.2

3.2 Fresnel-Kirchhoff Diffraction Formula

57

value of u p in this case is the surface integral over S = S2 + S3 , excluding a small spherical surface at the point of observation P0 . jkr01 1 e jkr01 e ∇u − u∇ · nˆ d S. (3.15) up = 4π r01 r01 S

Either (3.14) or (3.15) is called the integral theorem of Kirchhoff. Using this theorem, the amplitude of the light at an arbitrary observation point can be obtained by knowing the field distribution of light on the surface enclosing the observation point. The next section rewrites the Kirchhoff formula in a form better suited for solving diffraction problems, and we derive a result which is known as the Fresnel-Kirchhoff diffraction formula.

3.2 Fresnel-Kirchhoff Diffraction Formula The Kirchhoff diffraction theorem will be used to find the diffraction pattern of an aperture when illuminated by a point source and projected onto a screen [3.2, 3]. Consider a domain of integration enclosed by a masking screen Sc , a surface SA bridging the aperture, and a semi-sphere SR centered at the observation point P0 with radius R, as shown in Fig. 3.4. The integral on the surface of the masking screen Sc is zero since the light amplitude is zero here. The integral of (3.15) becomes jkr01 1 e jkr01 e ∇u − u∇ · nˆ d S. (3.16) up = 4π r01 r01 SA +SR

It can also be shown that, under certain conditions, the integral of (3.16) over SR is zero. The integral being examined is

Fig. 3.4 The shape of the domain of integration conveniently chosen for calculating the diffraction pattern of an aperture using the Kirchhoff diffraction theorem

58

3 Basic Theory of Diffraction

1 e jk R e jk R ∇u − u jk − rˆ 03 · nˆ d S, R R R

SR

where rˆ 03 is a unit vector pointing from P0 to point P3 on SR . When R is very large, the integral over SR can be approximated as 4π

e jk R [(∇u) · nˆ − jku] R 2 dΩ, R

(3.17)

where Ω is the solid angle from P0 to SR . Since the directions of rˆ 03 and nˆ are identical, rˆ 03 · nˆ is unity. It is not evident immediately that the value of (3.17) is zero, but if the condition lim R[(∇u) · nˆ − jku] = 0

R→∞

(3.18)

is satisfied, the integral indeed vanishes. This condition is called the Sommerfeld radiation condition. The boundary condition at an infinite distance is rather delicate. The condition that u = 0 at R = a with a → ∞ is not enough because there are two cases which satisfy this condition. Case (i) corresponds to a single smoothly decaying outgoing wave with no reflected wave (incoming wave). This outgoing wave dies down with an increase in distance. Case (ii) can occur when the field is set equal to zero by placing a totally reflecting screen at R = a such that the resultant field of the outgoing and reflected waves satisfies the condition of u = 0 at R = a. One should note that in case (ii), both u 1 = R −1 exp ( jk R) and u 2 = R −1 exp (−jk R) exist and both u 1 and u 2 approach zero as R is increased so that the simple boundary condition of u = 0 at R = ∞ is applicable. At present, the field at an infinite domain without reflection, i.e. case (i) is under consideration. Equation (3.18) is useful to guarantee case (i) since only the outgoing wave can satisfy (3.18). The first factor of (3.18) specifies the conservation of energy and the second factor specifies the direction of propagation. The solution that makes the second factor zero is A exp ( jk R), which is the outgoing wave. [When j in (3.18) is replaced by −j, the corresponding solution becomes B exp (−jk R) and such a condition is called the Sommerfeld absorption condition]. If the expression for u is simple, it is easy to discern the outgoing wave from the incoming wave, but quite often the expression is so complicated that the radiation condition has to be utilized. After having established that the integral over SR is zero, Fig. 3.4 will be used to find the value of the integral over SA . Taking u as the amplitude of a spherical wave emanating from the source P2 , the radius r21 is the distance from P2 to a point P1 on the aperture. The distribution on the aperture is therefore u=A

e jkr21 , r21

where A is a constant representing the strength of the source.

3.2 Fresnel-Kirchhoff Diffraction Formula

59

The direction of the vector ∇u is such that the change in u with respect to a change in location is maximum, so that ∇u is in the same direction as rˆ 21 . Hence, ∇u becomes d 1 e jkr21 e jkr21 and (3.19) = rˆ 21 A jk − ∇u = A rˆ 21 dr21 r21 r21 r21 e jkr01 1 e jkr01 ∇ , (3.20) = rˆ 01 jk − r01 r01 r01 where r01 is the distance from the observation point P0 to the point P1 on the aperture. When r01 and r21 are much longer than a wavelength, the second term inside the brackets on the right hand-side of (3.19, 20) can be considered very small and therefore ignored when compared to the first term inside the brackets. Inserting all the above results into (3.16) one obtains exp [ jk(r01 + r21 )] jA ˆ d S. (ˆr 21 · nˆ − rˆ 01 · n) (3.21) up = 2λ r01r21 SA

ˆ and letting the angle Letting the angle between unit vectors rˆ 21 and nˆ be (ˆr 21 , n) ˆ we have between rˆ 01 and nˆ be (ˆr 01 , n), exp [ jk(r01 + r21 )] A ˆ − cos (ˆr 21 , n)] ˆ d S. up = [cos (ˆr 01 , n) (3.22) j2λ r01r21 SA

Equation (3.22) is called the Fresnel-Kirchhoff diffraction formula. Among the factors in the integrand of (3.22) is the obliquity factor ˆ − cos (ˆr 21 , n)], ˆ [cos (ˆr 01 , n)

(3.23)

which relates to the incident and transmitting angles. For the special case in which the light source is centrally located with respect to the aperture, the obliquity factor becomes (1 + cos χ ), χ being the angle between rˆ 01 and the normal to the sphere SA centered at the light source P2 , as shown in Fig. 3.5. One then obtains exp [ jk(r01 + r21 )] A (1 + cos χ ) d S. (3.24) up = j2λ r01 r21 SA

When both r01 and r21 are nearly perpendicular to the mask screen, the obliquity factor becomes approximately 2 and (3.22, 24) become 1 e jk r01 e jkr21 u SA · d S where u SA = A . (3.25) up = jλ r01 r21 SA

Hence, if the amplitude distribution u SA of the light across the aperture is known, the field u p at the point of observation can be obtained.

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3 Basic Theory of Diffraction

Fig. 3.5 Special case of the source P2 being placed near the center of the aperture, where SA is a spherical surface centered around the source. The obliquity factor is (1 + cos χ)

Equation (3.25) can be interpreted as a mathematical formulation of Huygens’ principle. The integral can be thought of as a summation of contributions from innumerable small spherical sources of amplitude (u SA d S) lined up along the aperture SA . According to (3.24), the obliquity factor is zero when the point P0 of observation is brought to the left of the aperture and χ = 180◦ , which means there is no wave going back to the source. This successfully solved the difficulty associated with Huygens’ principle!

3.3 Fresnel-Kirchhoff’s Approximate Formula Despite the seeming simplicity and power of (3.25), there are only a limited number of cases for which the integration can be expressed in a closed form. Even when one can perform the integral, it may be too complicated to be practical. In this section a further approximation is imposed on (3.25). In developing the theory of the Fresnel-Kirchhoff diffraction formula, it was convenient to use P0 to denote the point of observation, since the integrations were performed with P0 as the origin. However, in most engineering problems, the origin is taken as either the location of the source or the location of the source aperture. In order to accommodate the usual engineering convention, a re-labelling is adopted, as shown in Fig. 3.6. Here, the origin P0 is located in the plane of the aperture, and the coordinates of any given point in this plane are (x0 , y0 , 0). The observation point P is located in the plane of observation, and the coordinates of a point in this plane are (xi , yi , z i ). The distance r01 appearing in (3.25) will simply be referred to as r . Using the rectangular coordinate system shown in Fig. 3.6, the diffraction pattern u(xi , yi ) of the input g(x0 , y0 ) is considered. Expressing (3.25) in rectangular coordinates, one obtains

3.3 Fresnel-Kirchhoff’s Approximate Formula

61

Fig. 3.6 Geometry for calculating the diffraction pattern {u(xi , yi , z i )} on the screen produced by the source {g (x0 , y0 , 0)}

Fig. 3.7 Diagram showing the relative positions of the Fresnel (near field) and Fraunhofer (far field) regions

1 u(xi , yi ) = jλ r=

∞ g(x0 , y0 ) −∞

e jkr d x0 dy0 r

z i2 + (xi − x0 )2 + (yi − y0 )2 .

where

(3.26)

Equation (3.26) has a squared term inside the square root, which makes integration difficult. In a region where (x0 − x)2 + (y0 − yi )2 is much smaller than z i2 , the binomial expansion is applicable and

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3 Basic Theory of Diffraction

r=

z i2 + (xi − x0 )2 + (yi − y0 )2 = z i 1 +

(xi − x0 )2 + (yi − y0 )2 z i2

(xi − x0 )2 + (yi − y0 )2 [(xi − x0 )2 + (yi − y0 )2 ]2 ∼ − ···· = zi + 2z i 8z i3

(3.27)

Thus, (xi − x0 )2 + (yi − y0 )2 2z i

xi2 + yi2 xi x0 + yi y0 x02 + y02 [(xi − x0 )2 + (yi − y0 )2 ]2 + r = zi + − − . 2z i zi 2z i 8 z i3 zi +

Fraunhofer approximation

(3.28)

Fresnel approximation The region where only the first 3 terms are included is called the farfield region or the Fraunhofer region. The region where the first 4 terms are included is called the near field region or the Fresnel region. It should be remembered that in the region very close to the aperture neither of these approximations are valid because use of the binominal expansion is no longer justified. It is the value of the 4th term of (3.28) that determines whether the Fresnel or Fraunhofer approximation should be used. Generally speaking, it is determined according to whether the value of k(x02 +y02 )/2z i is larger than π/2 (Fresnel) or smaller than π/2 (Fraunhofer). For instance, in the case of a square aperture with side dimension D, the value of k (x02 + y02 )/2z i becomes π/2 radian when z i = D 2 /λ. This value of z i is often used as a criterion for distinguishing the Fresnel region from the Fraunhofer region. It should be realized that the distance z i required to reach the Fraunhofer region is usually very long. For example, taking D = 6 mm, and λ = 0.6 × 10−3 mm, the value of z i = D 2 /λ is 60 m. It, however, is possible to observe the Fraunhofer pattern within the Fresnel region by use of a converging lens which has the effect of cancelling the 4th term of (3.28). The thickness of a converging lens decreases, as the rim is approached, and the phase of its transmission coefficient is of the form x 2 + y02 . (3.29) −k 0 2f The phase of the transmitted light, therefore, is the sum of the phases in (3.28, 29). When z i = f , the 4th term of (3.28) is cancelled by (3.29) and the far-field pattern can be observed in the near field. This is the basic principle behind performing a Fourier transform by using a lens, and will be discussed in more detail in Chap. 6.

3.5 Calculation of the Fresnel Approximation

63

3.4 Approximation in the Fraunhofer Region As mentioned earlier, the Fraunhofer approximation makes use of only the first three terms of (3.28). When these three terms are inserted into (3.26), the field distribution on the observation screen is xi2 + yi2 1 exp jk z i + u(xi , yi ) = jλz i 2 zi

∞ ×

g(x0 , y0 ) exp −j2π −∞

xi x0 yi y0 + λ zi λ zi

d x0 dy0 ,

(3.30)

where the approximation r ∼ = z i was used in the denominator of (3.30). Since (3.30) is of the form of a two-dimensional Fourier transform, the diffraction pattern can be expressed in Fourier transform notation as follows G( f x , f y ) = F {g(x0 , y0 )}, xi2 + yi2 1 u(xi , yi ) = exp jk z i + [G( f x , f y )] jλz i 2 zi

(3.31)

with f x = xi /λz i and f v = yi /λ z i . Equation (3.31) is the Fraunhofer approximation to the Fresnel-Kirchhoff diffraction formula in rectangular coordinates. An alternative expression for the diffraction pattern involves the use of the angles φi and θi shown in Fig. 3.6, where φi is the angle between the z-axis and a line connecting the origin to the point (xi , 0, z i ), and θi is the angle between the z-axis and a line connecting the origin to the point (0, yi , z i ). Since sin φi xi /z i and sin θi yi /z i , the values of f x , f y in (3.31) can be rewritten as fx =

sin φi , λ

fy =

sin θi . λ

(3.32)

3.5 Calculation of the Fresnel Approximation The Fresnel approximation is obtained by inserting the first four terms of (3.28) into (3.26). Two types of expressions for the Fresnel approximation can be obtained depending on whether or not (xi − x0 )2 + (yi − y0 )2 is expanded; one is in the form of a convolution and the other is in the form of a Fourier transform. If (xi − x0 )2 + (yi − y0 )2 is inserted into (3.26) without expansion, 1 j k zi (xi − x0 )2 + (yi − y0 )2 g(x0 , y0 ) exp jk e d x0 dy0 u(xi , yi ) = jλz i 2 zi (3.33)

64

3 Basic Theory of Diffraction

is obtained. By recognizing that (3.33) is a convolution, the diffraction pattern can be written using the convolution symbol as follows u(xi , yi ) = g(xi , yi ) ∗ f zi (xi , yi ), where

(3.34)

xi2 + yi2 1 exp jk z i + . f zi (xi , yi ) = jλz i 2 zi

Here f zi (xi , yi ) is of the same form as the approximation obtained by binomially expanding r in the expression exp (jkr )/r for a point source located at the origin. For this reason, f zi (xi , yi ) is called the point-source transfer function. If (xi − x0 )2 + (yi − y0 )2 of (3.28) is expanded and used in (3.26), one gets xi2 + yi2 1 u(xi , yi ) = exp jk z i + (3.35) jλz i 2 zi ∞ x02 + y02 x0 xi y0 yi g(x0 , y0 ) exp jk − j2π + d x0 dy0 . × 2z i λ zi λz i −∞

Since (3.35) resembles the Fourier-transform formula, it can be rewritten as " jk(x02 + y02 ) xi2 + yi2 1 exp jk z i + F g(x0 , y0 ) exp u(xi , yi ) = jλz i 2z i 2 zi jk(xi2 + yi2 = exp (3.36) F {g(x0 , y0 ) f zi (x0 , y0 )} 2 zi with f x = xi /λz i and f z = yi /λz i . One should get the same answer regardless of whether (3.34) or (3.36) is used, but there is often a difference in the ease of computation. Generally speaking, it is more convenient to use (3.34) when u(xi , yi ) has to be further Fourier transformed because the Fourier transform of the factor jk(xi2 + yi2 exp (3.37) 2 zi appearing in (3.34) is already known. The Fourier transform of (3.37) is quite easily obtained by using the following relationship for the Fourier transform with respect to x0 alone. " x02 2 (3.38) F exp jπ = z i λ e jπ/4 e−jπ λzi f x . λ zi Since an analogous relation holds for y0 , the Fourier transform of the point-source transfer function is F { f zi (x0 , y0 )} = e jkzi exp [−jπ λz i ( f x2 + f y2 )].

(3.39)

3.5 Calculation of the Fresnel Approximation

65

Fig. 3.8 Geometry used to calculate the diffraction pattern due to a point source using the Fresnel approximation (Exercise 3.1)

Exercise 3.1. As shown in Fig. 3.8, a point source is located at the origin. Find its Fresnel diffraction pattern on the screen, and compare it with its Fraunhofer diffraction pattern. Solution The source function g(x0 , y0 ) is g(x0 , y0 ) = δ(x0 ) δ(y0 ).

(3.40)

Both (3.34) and (3.36) will be used in the Fresnel region to demonstrate that they give the same results. First, applying (3.34), xi2 + yi2 1 exp jk z i + u(xi , yi ) = [δ(xi ) δ(yi )] ∗ jλz i 2z i 1 exp = jλz i

x 2 + yi2 jk z i + i 2z i

(3.41)

is obtained. Next using (3.36), one finds xi2 + yi2 1 exp jk z i + u(xi , yi ) = jλz i 2z i × F δ(x0 ) δ(y0 ) exp

x 2 + y02 jk 0 2z i

" (3.42)

with f x = xi /λz i and f y = yi /λz i . Since the Fourier transform appearing in (3.42) is unity, the diffraction pattern simplifies to xi2 + yi2 1 exp jk z i + , (3.43) u(xi , yi ) = jλz i 2 zi which is exactly the same as (3.41).

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3 Basic Theory of Diffraction

When the screen is in the Fraunhofer region, (3.31) is used xi2 + yi2 1 u (xi , yi ) = exp jk z i + F {δ (x0 ) δ (y0 )} f z =xi /λzi , f y =yi /λzi jλz i 2z i 1 = exp jλz i

x 2 + yi2 jk z i + i 2z i

.

(3.44)

The results of the Fresnel and Fraunhofer approximations are identical. This is because the size of the source is infinitesimally small and even a very close zone is already a far field. The result is identical to the point-source transfer function, as expected.

3.6 One-Dimensional Diffraction Formula As shown in Fig. 3.9, when the light source or aperture varies only in the x0 direction, the aperture function can be written g(x0 , y0 ) = g(x0 ), and hence the Fresnel-Kirchhoff approximation, (3.26), becomes ⎞ ⎛ ∞ ∞ jkr e 1 u (xi , yi ) = g (x0 ) ⎝ dy0 ⎠ d x0 jλ r −∞

Fig. 3.9 Geometry used for calculating the one-dimensional diffraction pattern

−∞

(3.45)

(3.46)

3.6 One-Dimensional Diffraction Formula

where r=

67

z i2 + (xi − x0 )2 + (yi − y0 )2 .

Of special interest is the integration over y0 in (3.46) which will be referred to as I , where ∞ jkr e (3.47) dy0 . I = r −∞

As shown in Fig. 3.9, the projection of r onto the plane y0 = 0 is denoted by , where = z i2 + (xi − x0 )2 . (3.48) In terms of , r is rewritten as r = ρ 2 + (yi − y0 )2 .

(3.49)

A change of variable is introduced as follows y0 − yi = sinh t, dy0 = cosh t. dt

(3.50) (3.51)

Combining (3.49, 50), r is expressed as a function of t r = cosh t.

(3.52)

Equations (3.51, 52) are inserted into (3.47) to obtain ∞ I =

(1)

e jkcosh t dt = jπ H0 (k)

(3.53)

−∞ (1)

where H0 (k) is the zeroth-order Hankel function of the first kind. The formula (1) H0 (a)

1 = jπ

∞ e jacosht dt

(3.54)

−∞

was used. Inserting (3.53) into (3.46) gives u (xi ) = =

π λ

∞

(1)

g(x0 )H0 (k) d x0 , where

(3.55)

−∞

z i2 + (xi − x0 )2 .

This is the one-dimensional Fresnel Kirchhoff formula.

(3.56)

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3 Basic Theory of Diffraction

Under certain conditions, the integration in (3.55) can be simplified. When k 1 the Hankel function can be approximated as λ j(k−π/4) 1 (1) ∼ , (3.57) e H0 (k) = π and when z i2 (xi − x0 )2 , can be written as = zi +

(3.58)

(xi − x0 )2 . 2z i

(3.59)

Inserting (3.57, 59) into (3.55) gives exp [ j(kz i − π/4)] u (xi ) = √ zi λ

∞ −∞

jk(xi − x0 )2 g(x0 ) exp 2z i

d x0 .

(3.60)

Equation (3.60) is the formula for the one-dimensional Fresnel approximation. When λz i x02 , the formula for the one-dimensional Fraunhofer approximation is obtained. xi2 1 π exp j kz i + k − (3.61) u(xi ) = √ [F {g(x0 )}] f =xi /λzi . 2z i 4 zi λ

3.7 The Fresnel Integral In this section, the Fresnel diffraction pattern of a square aperture such as shown in Fig. 3.10 will be calculated using (3.36) [3.2, 5]. Let the size of the aperture be 2a by 2a, and the amplitude of the incident wave be unity. The aperture is expressed in terms of the rectangular function as

Fig. 3.10 Diffraction pattern due to a square aperture

3.7 The Fresnel Integral

69

x 0

y 0

. (3.62) 2a 2a By definition, the aperture function is zero outside the limits (−a, a) and is unity inside. Equation (3.62) is inserted into (3.36) to yield xi2 + yi2 1 exp jk z i + u(x 1 , yi ) = jλz i 2z i π 2 2 × exp j (x + y0 ) − j2π f x x0 + f y y0 d x0 dy0 λz i 0 −a xi2 + yi2 1 2 2 = exp jk z i + − jπ λz i f x + f y jλz i 2z i

a

a × −a

' ( π exp j [( f x λz i − x0 )2 + ( f y λz i − y0 )2 ] d x0 dy0. (3.63) λz i

Changing variables 2 ( f x λz i − x0 )2 = ξ 2 , λz i Eq. (3.63) becomes √

1 u (xi , yi ) = e jkzi 2j

2 f y λz i − y0 λz i √

2/λz i(a− f z λz i )

e

jπ ξ 2 /2

2

= η2,

2/λz i(a− f y λz i )

dξ

√

e jπ η

2 /2

dη (3.64)

√

− 2/λz i (a+ f x λz i )

− 2/λz i (a+ f y λz i )

with f x = xi /λz i and f y = yi /λz i . If the limits of integration in (3.64) were −∞, +∞, the value of the integral would be √ ( 2e jπ/4 )2 However, with finite limits, the computation is more complicated. To assist with the integration in (3.64) a new function F(α), known as the Fresnel integral, is introduced, namely α 2 (3.65) F(α) = e jπ x /2 d x. 0

Before using the Fresnel integral, some of its properties will be investigated. The Fresnel integral can be separated into real and imaginary parts as α C(α) =

cos

π 2

x 2 d x,

(3.66)

α S(α) =

sin 0

π 2

x 2 d x.

(3.67)

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3 Basic Theory of Diffraction

Fig. 3.11 Cornu’s spiral (plot of the Fresnel integral in the complex plane)

The former is called the Fresnel cosine integral and the latter is the Fresnel sine integral. Equation (3.65) is now rewritten as F(α) = C(α) + jS(α).

(3.68)

Figure 3.11 is a plot of (3.68) by taking α as a parameter. As α is increased from zero to plus infinity the curve starts from the origin and curls up around the point (0.5, 0.5). As α is decreased from zero to minus infinity it curls up around the point (−0.5, −0.5). The curve F(α) is called Cornu’s spiral. The points of convergence of the spiral are C(∞) = S(∞) = −C(−∞) = −S(−∞).

(3.69)

In the C(α) – S(α) complex plane, the value of α2 e jπ x

2 /2

d x = [C(α2 ) + jS(α2 )] − [C(α1 ) + j S(α1 )]

α1

can be expressed as the length of a vector whose end points are the points α = α1 and α = α2 . Returning to the problem of the Fresnel diffraction pattern of a square aperture, (3.64) with the insertion of the Fresnel integral F(α) and f x = xi /λz i , f y = yi /λz i becomes 2 2 1 jkzi (a − xi ) − F − (a + X i ) u (xi , yi ) = e F 2j λz i λz i 2 2 × F (3.70) (a − yi ) − F − (a + yi ) . λz i λz i Equation (3.70) can be calculated using mathematical tables of F(α), but a qualitative result may be obtained using Cornu’s spiral. Since the factors involving xi

Problems

71

Fig. 3.12 a, b Relationship between the phasor on Cornu’s spiral and the magnitude of the diffracted field. (a) Phasors representing the magnitude of the light. (b) The magnitude of the field vs position (the Fresnel diffraction pattern)

in (3.70) are essentially the same as those involving yi , only the xi terms will be considered. The value inside the brackets of (3.70) is the difference between the two values of the Fresnel integral. This can be expressed in the C(α) – S(α) complex plane as a line starting from the point represented by the second term and ending at the point represented by the first term. First, the value at xi = 0 is obtained. At this point, the values of the first and second terms are situated diametrically across the origin in the complex plane and is represented by vector A in Fig. 3.12. Next, the value at xi = a/2, is obtained. The first term moves toward the origin on the Cornu’s spiral and the second term moves away from the origin. It can be expressed by vector B. Next, the value at xi = a is obtained. The first term is now at the origin and the second term moves further away from the origin. It can be represented by vector C. Finally, the value at xi > a, corresponding to a point inside the geometricaloptics shadow region, is considered. The first term enters the third quadrant and the second term moves even further away from the origin. It is represented by vector D. The length of D monotonically decreases with an increase in xi and approaches zero. In summary as the point xi moves away from the center of the aperture, the ends of the vector move in the direction of the arrows in Fig. 3.12 a. The results of plotting the length of the vectors versus xi gives the Fresnel diffraction pattern shown in Fig. 3.12 b.

Problems 3.1. Show that, in the process of obtaining the Kirchhoff integral u p , the same results are obtained by taking the domain including the source, as shown in Fig. 3.2, and by taking the domain excluding the source, as shown in Fig. 3.3. Show that the result is

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3 Basic Theory of Diffraction

Fig. 3.13 Illustration of Babinet’s principle

solely dependent on the relative positions of the point of observation and the source and is independent of the shape of the domain. 3.2. Babinet’s principle states that the sum of the diffraction pattern u 1 of a mask with an aperture, as shown in Fig. 3.13 a, and the diffraction pattern u 2 of its compliment, as shown in Fig. 3.13 b, is the same as the field distribution without anything. If either one of the patterns is known, the other can be determined immediately by this principle. Prove Babinet’s principle using the Fresnel-Kirchhoff diffraction formula. 3.3. When there is no variation in the y direction so that the input function can be expressed as g(x0 , y0 ) = g(x0 ), prove that (3.61) can be derived by applying the Fraunhofer approximation in the x0 direction and the Fresnel approximation in the y0 direction to (3.33). 3.4. Prove that the angle between the tangent of Cornu’s spiral taken at α and the C axis is π α2 /2. Using this result, obtain the values of α that give the extrema of C(α) and S(α). 3.5. Prove that Cornu’s spiral has point symmetry with respect to the origin.

Chapter 4

Practical Examples of Diffraction Theory

The best way to understand the principles associated with a given discipline is through examples which apply these principles. This chapter devotes itself to illustrating methods for solving specific problems, and also serves as a collection of practical examples involving diffraction phenomena.

4.1 Diffraction Problems in a Rectangular Coordinate System Exercise 4.1. Obtain the diffraction pattern of a rectangular aperture with dimensions a × b when projected onto the screen located at z = z i in the Fraunhofer region, as shown in Fig. 4.1. Solution The input function g(x0 , y0 ) is given by x y 0 0 II . g (x0 , y0 ) = II a b

(4.1)

From (2.35) and (3.31), the diffraction pattern is xi2 + yi2 ab yi xi exp jk z i + sinc b . (4.2) u (xi , yi ) = sinc a jλz i 2z i λz i λz i If the first factor of (4.2) is suppressed, the distributions of the amplitude and intensity along the xi axis are represented by the curves in Fig. 4.2 a and b. Similar curves can be drawn along the yi axis, while keeping in mind that the width of the main beam of the diffraction pattern is inversely proportional to the dimension of the side of the aperture. The narrower beam is in the direction of the wider side of the rectangular aperture. A rule of thumb is that the beam width of an antenna with an aperture of 60 λ is 1◦ .

73

74

4 Practical Examples of Diffraction Theory

Fig. 4.1 Geometry of diffraction from a rectangular aperture

Fig. 4.2 a, b. Diffraction pattern of a rectangular aperture. (a) Amplitude. (b) Intensity

Exercise 4.2. Consider a light source with dimensions 2a × 2a. An obstructing square with dimensions a × a is placed with its center at (ξ, η) inside the light source, as shown in Fig. 4.3 a. Compare the Fraunhofer diffraction patterns with and without the obstructing square. Obstruction caused by an antenna feed shown in Fig. 4.3 b is a practical example. Solution The input function g(x0 , y0 ) representing the light source is obtained by subtracting the smaller square from the larger square. x y x0 − ξ y0 − η 0 0 II − II II . (4.3) g (x0 , y0 ) = II 2a 2a a a The Fourier transform G( f x , f y ) of (4.3) is G( f x , f y ) = 4a 2 sinc (2a f x ) sinc (2a f y ) − a 2 sinc (a f x ) sinc (a f y ) e−j2π( f x ξ + f y η ).

(4.4)

Using (3.31), the final result is xi2 + yi2 1 exp jk z i + u(xi , yi ) = {G( f x , f y )} jλz i 2z i

(4.5)

with f x = xi /λz i and f y = yi /λz i . The curve determined by (4.5) when ξ = η = 0 is drawn with respect to f x in a solid line in Fig. 4.4. The curve in the absence of

4.1 Diffraction Problems in a Rectangular Coordinate System

75

Fig. 4.3 a, b. Square light source obstructed by a square. (a) Geometry. (b) Antenna with center feed (EM Systems, Inc.)

Fig. 4.4 Diffraction patterns with and without the obstructing square

the obstructing square is drawn by a dotted line. It can be seen from the figure that the existence of the mask does not cause too much change in the diffraction pattern. Exercise 4.3. Find the far-field diffraction pattern of a triangular aperture. The edges of the triangle are expressed by x0 = a, y0 = x0 , y0 = −x0 (Fig. 4.5). The observation screen is placed at z = z i . Solution It should be noticed that the limits of the integration with respect to y0 are not constant but vary with x0 . Hence, integration with respect to y0 in a thin strip

76

4 Practical Examples of Diffraction Theory

Fig. 4.5 Aperture in a triangular shape

between x0 and x0 + d x0 ought to be performed first, followed by integration of x0 from 0 to a. Integrating with respect to y0 gives a G( f x , f y ) =

x0 d x0 −x0

a =

e−j2π( f x x0 + f y y0 ) dy0

−j2π f x x0

e 0

j = 2π f y

a

+j2π f y x0

e−j2π f y x0 −e −j2π f y

d x0

(e−j2π( f x + f y )x0 − e−j2π( f x − f y )x0 ) d x0.

(4.6)

Integrating with respect to x0 and rearranging terms yields the result ja {e−jπ( f x + f y )a sinc [( f x + f y ) a]−e−jπ( f x − f y )a sinc [( f x − f y ) a]}. 2π f y (4.7) Since (4.7) for the far-field diffraction pattern of a triangular aperture is somewhat more complex than previous examples using square apertures, a closer examination of the final result is worth making. First, the distribution along the f x = xi /λz i axis is considered. If the value of f y = 0 is directly inserted into (4.7), G( f x , 0) takes on the indefinite 0/0 form. It is therefore necessary to evaluate the limit of G( f x , f y ) as f y approaches zero. By defining ∆ as G( f x , f y ) =

∆ = πa f y πa f x , Eq. (4.7) can be approximated as G( f x , ∆) = a 2

sin ∆ ∆

e−jπ f x a sinc ( f x a).

(4.8)

Taking the limit as ∆ → 0 gives G( f x ) = lim G( f x , ∆) = a 2 e−jπ f x a sinc ( f x a). ∆→0

(4.9)

4.1 Diffraction Problems in a Rectangular Coordinate System

77

Fig. 4.6 Amplitude and phase distribution along f x for the diffraction pattern of a triangular aperture

The curve determined by (4.9) is shown in Fig. 4.6. It is interesting to note that even though the phase distribution is antisymmetric with respect to the f y axis, the amplitude and intensity distributions are both symmetric. It is quite surprising to see such symmetry exist considering that the shape of the aperture is far from symmetric. The expression for the distribution along the f y axis is obtained from (4.7) and is G( f y ) = G(0, f y ) = a 2 [sinc ( f y a)]2 .

(4.10)

The decay of G( f y ) with respect to f y is much faster than that of G( f x ) with respect to f x . Next, the distribution of the intensity along a line 45◦ with respect to the f x axis, namely, along the line f y = f x , is examined. In order to make the interpretation of the expression simpler, the coordinates will be rotated by 45 degrees. The variables in (4.7) are converted according to √ √ 2X = f y + f x , 2 Y = f y − fx (4.11) to obtain ja G(X, Y ) = √ 2 π(X + Y ) √

× [e−j

2 πa X

√ √ √ sinc( 2a X ) − e j 2 πaY sinc( 2 aY )].

(4.12)

The distribution along the X axis is √ √ ja [e−j 2 πa X sinc ( 2 a X ) − 1]. G(X, 0) = √ 2πX

(4.13)

The expression for G(0, Y ) is exactly the same as G(X, 0) with Y substituted for X . Figure 4.7 a shows a photograph of the diffraction pattern of the aperture. The photograph clearly shows that the dominant features of the diffraction pattern are aligned along directions which are perpendicular to the aperture edges. The longest trail of diffracted light lies along the direction perpendicular to the longest aperture edge. It is generally true that the diffraction is predominantly determined by the shape of the edges. A rule of thumb is that the trails of diffracted light are in the

78

4 Practical Examples of Diffraction Theory

Fig. 4.7 a, b. Aperture diffraction. (a) Photograph of the diffraction pattern of a right-angled triangular aperture. (b) Diffraction due to iris

directions perpendicular to the tangent of the aperture edges. This rule of thumb is handy when an analytical approach is difficult. When gazing directly at the sun, bright star like streaks are seen. These are considered due to the diffraction from the edges of the iris.

4.2 Edge Diffraction As demonstrated by the above example, the edge plays an important role in the diffraction pattern. This section is devoted to a detailed study of the diffraction pattern of a semi-infinite plane aperture having one infinitely long straight edge. This aperture is ideally suited to studying the effect of the edge alone. Exercise 4.4. Find the diffraction pattern of the aperture which is open in the entire right half plane, as shown in Fig. 4.8. Solution The step function provides a good description of such an aperture. The source function g(x0 , y0 ) is expressed by g (x0 , y0 ) = H (x0 ).

(4.14)

The Fourier transform of (4.14) with respect to x0 was given by (2.43) and that with respect to y0 is δ( f y ). Using (3.31), the diffraction pattern is given by xi2 1 1 1 u(xi , yi ) = δ( f y ) exp jkz i + jk + δ( f x ) (4.15) jλz i 2z i j2π f x 1 with f x = xi /λz i and f y = yi /λz i . The curve of (4.15) excluding the first factor is shown in Fig. 4.9, which shows that light is diffracted not only into the region f x > 0 (where the direct ray is incident) but also into the region f x < 0 (where the direct ray is blocked by the aperture).

4.2 Edge Diffraction

79

Fig. 4.8 Semi-infinite aperture

Fig. 4.9 Diffraction pattern of the semiinfinite aperture with respect to f x

The diffraction pattern of the semi-infinite aperture is now compared with that of an aperture of infinite dimension. The value of the diffraction pattern of the infinite aperture, which corresponds to the bracketed portion of (4.15), is δ( f x ) δ( f y ). In the case of the semi-infinite aperture, obviously one half of the incident light energy is transmitted through the aperture. Examining the bracket of (4.15) reveals that one quarter of the incident light energy, or one half of the energy transmitted through the aperture, is not diffracted and goes straight through while the rest of the energy is diffracted equally on each side of the edge. It is a somewhat puzzling fact that the diffraction of an infinite aperture is a point expressed by δ( f x ) δ( f y ) and that of a semi-infinite aperture is of delta function form only in the f y direction. Does this really mean that the far-field diffraction pattern of the infinite aperture (no edge) becomes a point? The answer is “yes” and “no”. According to (3.28), in order for the region of observation to be considered far field, the distance to the plane of observation has to be much larger than the dimension of the source. Due to the fact that the dimensions of the infinite aperture are indeed infinite, it is theoretically impossible to find the region of observation that satisfies the far-field condition of (3.28) and therefore no legitimate far-field region exists. There is yet another more enlightening interpretation. In reality, it is impossible to make a source of truly infinite size. A beam that travels straight should have the same cross-section no matter how far the plane of the observation is moved back. However, due to diffraction, the beam deviates from the center line, and this deviation distance increases continuously as the screen is moved indefinitely further

80

4 Practical Examples of Diffraction Theory

away. If the finite cross-section of the incident beam is compared with the deviation distance, there must be a distant place where the cross-section of light can be considered as a point (delta function) compared with the increasingly large deviation of the diffracted beam away from the center line. Another characteristic of the far-field pattern of the edge is that its amplitude decreases monotonically with f x and there is no interference phenomenon. This is because the non-diffracted beam contributes to the δ function at the origin, but does not contribute in any other region. Now that the diffraction patterns of the “no edge” and “one edge” aperture have been explored, the next logical step is to look at the diffraction produced by an aperture containing two edges. The aperture considered is a slit constructed by two infinitely long edges. The semi-infinite plane displaced by a/2 to the left, as shown in Fig. 4.10 a and that displaced the same amount to the right, as shown in Fig. 4.10 b, are combined to make up the slit shown in Fig. 4.10 c. The Fourier transform of the functions shown in Fig. 4.10 a and b can be obtained by applying the shift theorem to the result of Exercise 4.4. The source function of Fig. 4.10 a is expressed by Ha (x0 ) =

* 1) a sgn x0 + +1 2 2

(4.16)

and its Fourier transform is 1 1 jπa f x Ha ( f x ) = e + δ( f x ) . 2 jπ f x

(4.17)

Similarly, the source function of Fig. 4.10 b is expressed by Hb (x0 ) =

* 1) a −sgn x0 − +1 2 2

(4.18)

Fig. 4.10 a–c. Composition of a slit. (a) Half plane shifted by a/2 to the left. (b) Half plane shifted by a/2 to the right. (c) Slit made up of two half planes

4.2 Edge Diffraction

81

Fig. 4.11 Expression of a slit using step functions

and its Fourier transform is Hb ( f x ) =

1 1 −jπa f x e + δ( f x ) . − 2 jπ f x

The source function displayed in Fig. 4.10 c has been already obtained as x 0 Hc (x0 ) = II a

(4.19)

(4.20)

and its Fourier transform is H ( f x ) = a sinc (a f x ). By comparing these three figures (Figs. 4.10 a–c) it is easy to see that proper manipulation of (a) and (b) generates (c). The mathematical expression for (c) is obtained with the aid of Fig. 4.11, i.e. a a − H x0 − Hc (x0 ) = H x0 + 2 2 * 1) * 1) a a = sgn x0 + +1 − sgn x0 − +1 . 2 2 2 2 Its Fourier transform is e−jπ f x a e jπ f x a sin π f x a − = α sinc ( f x a) =a Hc ( f x ) = j2π f x j2π f x π fx a

(4.21)

which is the same result as above. Caution It should be noted that, as shown in Fig. 4.12, the expression for (c) is not the sum of the expressions for (a) and (b), but rather the difference of the two. The law of superposition is not applicable here. Only the values of field can be superposed but not the geometry, medium or boundary conditions. Cases where superposition may be applied occur in calculating the field due to multiple sources. In these cases, only one source is assumed at a time and the field due to each such individual source is calculated separately inside the same geometry. The fields due to all of the individual sources are then superposed to obtain the final result. Another point to be aware of is the existence of multiple reflections between the two edges of a slit. The wave diffracted by the left edge illuminates the right edge

82

4 Practical Examples of Diffraction Theory

Fig. 4.12 Mere addition of two half plane functions. Note that it does not represent a slit. The law of superposition is applied to the field but not to the boundary conditions

whose diffraction field again illuminates the left edge. Due to this mutual coupling between the edges, the distribution of the actual field in the close vicinity of the slit is very complicated. It should be noted that Kirchhoff’s integral approximation uses the original field which was present before the edges were introduced to this location to calculate the integral, thus excludes the effect of this mutual coupling between the edges.

4.3 Diffraction from a Periodic Array of Slits By using the comb function, periodic functions of various shapes can be easily generated [4.1–3]. Exercise 4.5. Consider a mask which is made up of periodic slits, as shown in Fig. 4.13. The width of each slit is a and its period is b. The total length of the array is c. Find the Fraunhofer diffraction pattern of the mask. Solution The input source function g(x0 ) is g(x0 ) =

x * x 1 ) x0 0 0 II ∗ III · II . b a b c

(4.22)

The factor inside the bracket is the expression for a periodic slit function of infinite length. The last factor truncates this function to a length c. The Fourier transform of (4.22) is G( f ) = ac [(sinc (a f ) · III (b f )] ∗ sinc (c f ). (4.23) A separate plot for each factor of (4.23) is shown in Fig. 4.14. The factor in the bracket represents a chain of delta functions whose amplitude is modulated by sinc (a f ), as shown by the left graph. The period of the delta functions is inversely related to the period of the slits and is given by 1/b. In other words, the delta function period 1/b becomes shorter as the period of the slits becomes longer. It is also noticed that the zero-crossing point of sinc (a f ) is inversely proportional to the width of the slit. The curve on the right of Fig. 4.14 is sinc (c f ) and its zero-crossing point is likewise inversely proportional to the total length of the array c. By convolving the left with the right graph, the final result shown in Fig. 4.15 is obtained. It is interesting to note that, as far as the source dimensions are concerned, c is the largest, b is the next largest and a is the smallest, but as far as the dimensions in

4.3 Diffraction from a Periodic Array of Slits

83

Fig. 4.13 One-dimensional periodic source

Fig. 4.14 Convolution of (sinc (a f )) ∗ III(b/ f ) with sinc (c f )

Fig. 4.15 Graph of ac[sinc (a f )III(b f )] ∗ sinc (c f )

the Fourier transform are concerned, the smallest is the width 1/c of sinc (c f ), the next is the period 1/b of the comb function III (b f ), and the largest is the width 1/a of sinc (a f ). The Fourier widths are therefore in reverse order of the geometrical widths. As seen from Fig. 4.15, when c is large, the order of the calculation of (4.23) can be altered as

84

4 Practical Examples of Diffraction Theory

G( f ) = ac sinc (a f ) [I I I (b f ) ∗ sinc (c f )] . element factor

(4.24)

array factor

Equation (4.24) is now separated into two factors, one of which is associated with the characteristics of the slit element itself, and the other is associated with the arrangement of the slit elements (period and the overall length of the array). Because the element factor and the array factor can be separated, the procedure for designing an antenna array can be greatly simplified. Changing the shape of the slit results in a change only in the shape of the dotted line in Fig. 4.15 and all other details such as the period and the width of each peak will not be affected. Therefore, the choice of the element antenna and that of the period can be treated separately. Generalizing, it is possible to express the radiation pattern of an antenna array as (pattern) = (element factor) × (array factor).

(4.25)

Exercise 4.6. Consider an N element linear array, as shown in Fig. 4.16. The width of each element antenna is a. Find the spacing b between the elements such that the height of the first side lobe in the radiation pattern equals 2/π times that of the main lobe. Assume that the number of antenna elements N is very large. The radiation pattern considered is in the x z plane. Solution Exercise 4.5 dealt with an array having an element width of a, a spacing between elements of b, and a total length of c. The results of Exercise 4.5 are shown in Fig. 4.15 and can be applied to the current problem. With this in mind, Fig. 4.17 was drawn as an expanded view of the central portion of Fig. 4.15. The tip of the side lobe follows the envelope of E 0 sinc (a f ) when 1/b is varied. This is indicated by the dashed line in Fig. 4.17. Therefore the answer is obtained by finding f such that sinc (a f ) is 2/π , i.e. 2 sin aπ f = . (4.26) aπ f π The solution of (4.26) is f = 0.5/a, and the value of 1/b is set equal to this value of f , namely b = 2a. (4.27)

Fig. 4.16 Horn antenna array

4.4 Video Disk System

85

Fig. 4.17 Method of designing the antenna array

The answer is therefore that the period should be set to twice the width of the aperture of the element antenna.

4.4 Video Disk System An example of the reflection grating is given, followed by an explanation of a video disk system [4.1, 4].

4.4.1 Reflection Grating Exercise 4.7. A reflection grating is fabricated by periodically depressing a reflective metal surface. Obtain the far-field pattern of a reflection grating with dimensions as shown in Fig. 4.18. The width of the upper reflecting surface is a and that of the depressed surface is a . The distance from the lower to the higher surface is h and the light is incident normal to the surfaces. Assume the number of periods is infinite. Solution First the radiation pattern of one period is considered. Referring to Fig. 4.18, it is seen that the ray path reflected from the bottom surface is longer by AB + BD than that reflected from the top surface. The path difference l is l = AB + BC + CD = h +

b sin θ + h cos θ. 2

(4.28)

Hence, the radiation pattern G e ( f ) of one period can be obtained by summing the diffraction patterns of the top and bottom surfaces while taking their relative phase differences into consideration: G e ( f ) = a sinc (a f ) + a e jφ sinc (a f )

(4.29)

86

4 Practical Examples of Diffraction Theory

Fig. 4.18 Diffraction by a reflection grating

where the value of φ is from (4.28): b 2π h (1 + cos θ ) + sin θ . φ= λ 2

(4.30)

The grating array factor G a ( f ) with period b is III(b f ); hence the radiation pattern G( f ) of the entire reflection grating is G( f ) = III (b f ) [a sinc (a f ) + a e jφ sinc (a f )].

(4.31)

For the special case of a = a and θ 1, the intensity distribution of the grating is φ 2 I ( f ) = 4a 2 III (2a f ) sinc (a f ) cos (4.32) 2 where

4π h. λ The intensity distribution with h = λ/4 is shown in Fig. 4.19. Notice that the spectral intensity is significantly different from the case of a flat plate. The fact that the diffraction pattern can be controlled by varying geometry is used to store information in the video disk. φ = 2πa f +

Alternate derivation of G e ( f ). Imagine a fictitious aperture A parallel and close to the reflecting surface in Fig. 4.18. The field distribution of the reflected wave across this aperture is obtained with the help of Fig. 4.18 as x x0 + b/2 0 +Π e2jkh , g (x0 ) = Π a a where the point D is taken as the origin of the coordinates, and 2 kh accounts for the phase delay of the bottom surface. The element pattern is given by its Fourier transform G e ( f ), with f = sin θ/λ,

4.4 Video Disk System

87

Fig. 4.19 Diffraction pattern of a reflection grating with b = 2a for θ 1

G e ( f ) = a sinc (a f ) + a e jφF sinc (a f ) 2π b 2h + sin θ . with φF = λ 2 Thus for small θ, the results of this fictitious aperture approach are equivalent to (4.29, 30), which had been derived by tracing the phase delay along the ray path.

4.4.2 Principle of the Video Disk System The video disk is a memory device designed to store pre-recorded television programs using a spiral string of pits on the surface of a spinning disk. Figure 4.20 shows the play back mechanism of the video disk. The light is focused onto the reflecting surface and the intensity of the reflected beam is modulated according to the pitch length of the pits. The reflected light is then detected by the photodiode which sends the signal to a television set. Figure 4.21 a shows a photograph of the video disk and Fig. 4.21 b shows an electron microscope photograph of a string of pits on the video disk. Information is encoded in the length and the spacing of the pits along the track [4.5]. Figure 4.22 illustrates the mode of encoding the pits. The video frequency modulated signal, such as that shown in (a), and the audio frequency, such as that shown in (b), are simultaneously recorded. The dc value of the video signal is changed according to the amplitude of the audio signal. The zero crossing points of the subsequent signal are used to determine the length of the pits. Therefore, the frequency of the video signal is represented by the spacing between the pits, and the audio signal is represented by the length of the pits. The main features of the video disk system are: 1) There is no contact between the disk and read out system, thus, there is no wearing of the disk. 2) The disks are suitable for mass production.

88

4 Practical Examples of Diffraction Theory

Fig. 4.20 Principle of a video disk system

Fig. 4.21 a, b. Video disk. (a) Photograph of the video disk. (b) Electron microscope photograph of the disk surface (By courtesy of Philips)

4.5 Diffraction Pattern of a Circular Aperture

89

Fig. 4.22 a–c. Pits on the surface of the disk formed from the zero crossing points. (a) Frequency modulated video signal. (b) Audio signal. (c) Pits formed from zero crossing points

3) When the protective plastic layer (Fig. 4.20) is made thick, the quality of the signal is hardly affected by dust, finger prints or scratches on the disk surface. 4) The density of the recording is high. For example, the typical spacing between adjacent tracks is 1.6 µm. On a 30 cm diameter disk this results in a linear track length of 34 km or a storage capacity of 54,000 still pictures.

4.5 Diffraction Pattern of a Circular Aperture Exercise 4.8. Find the Fraunhofer diffraction pattern of a circular aperture of unit radius, as shown in Fig. 4.23. Solution First (3.31) is converted into cylindrical coordinates and then the Fourier Bessel transform is performed. l2 1 exp jk z i + (4.33) u(l) = B {g(r )}|=l/λzi or sin θ/λ . jλz i 2z i In (4.33), r refers to the radius coordinate in the plane of the aperture, and l refers to the radius coordinate in the plane of the diffraction pattern. Since the input function is

90

4 Practical Examples of Diffraction Theory

Fig. 4.23 Circular aperture

Fig. 4.24 Diffraction pattern of a circular aperture

g (r ) = circ (r ), Eq. (2.76) may be used to obtain ! 1 l2 J1 (2π ) !! u(l) = exp jk z i + . ! jλz i 2z i =l/λz i

(4.34)

(4.35)

The diffraction pattern is circularly symmetric and the amplitude distribution in the radial direction is shown in Fig. 4.24. The values of υ which make the value of J1 (π υ) zero are υ = 1.22, 2.23, 3.24, . . ., and the corresponding values of the l’s are l = 0.61 λz i , 1.12 λz i , 1.62λz i , . . .. The diameter to the first zero of the diffraction pattern is 1.22λz i . This value is compared with that of a square aperture with dimensions 2 × 2. The distance between the first zeros of the diffraction pattern of the square is λz i and is slightly narrower than that of the circular aperture (1.22 λz i ). Exercise 4.9. In many optical systems, the cross-sectional amplitude distribution of a light beam is a Gaussian function of the form exp (−a 2 r 2 ). Prove that if the distribution in one plane is Gaussian, then the distribution in a plane at an arbitrary distance away from the first plane is also Gaussian. Solution Let the first plane be z = 0 and the second plane be z = z i . First, the Fourier Bessel transform G() of the Gaussian distribution is calculated so that (4.33) may be used. From (2.72), the Fourier Bessel transform is

4.6 One-Dimensional Fresnel Zone Plate

91

∞ G() = 2π

r e−a

2r 2

J0 (2πr ) dr.

(4.36)

1 −b2 /4a 2 e , 2a 2

(4.37)

Using Bessel’s integral formula ∞

e−a

2x2

xJ0 (bx) d x =

Eq. (4.36) becomes

π −π 2 2 /a 2 e a2 and substituting = l/(λz i ), the distribution u(l) of the light at z = z i is π l2 u(l) = 2 exp jk z i + · exp [−(π/(aλz i ))2l 2 ]. 2z i ja λz i G() =

(4.38)

(4.39)

Thus it has been proven that the amplitude distribution at z = z i is again Gaussian. As a matter of fact, there are only a very limited number of functions whose Fourier Bessel transform can be found in a closed form. The Gaussian distribution function is one of these special functions. Another function of this kind is the delta function.

4.6 One-Dimensional Fresnel Zone Plate When a plane wave is incident upon a mask whose transmission is distributed according to Fig. 4.25 a, the transmitted light is focussed to a point. Such a mask behaves like a lens and is called a Fresnel zone plate [4.3]. The Fresnel zone plate is one of the most powerful means of forming images in the spectral regions of infra-red light, x-rays, and gamma rays where glass lenses are of no use. In Exercise 4.5, it was shown that the equal spacing of the periodic slits produced peaks of the diffraction pattern along the xi axis, but not along the z axis. It will be demonstrated, however, that a quadratic spacing produces peaks along the z axis, but not along the xi axis. Since the transmission of the zone plate shown in Fig. 4.25 a is periodic with x02 , a conversion of variables 2 2 (4.40) x X= λp 0 is made. Equation (4.40) makes it possible to expand the transmission function into a Fourier series. The expansion coefficients an are

92

4 Practical Examples of Diffraction Theory

Fig. 4.25 √ a, b. Transmission distribution of a one dimensional zone plate expressed (a) as a function of 2/λp x0 and (b) as a function of X = (2/λp) x02

1 an = 4 1 4

1 −1

2

t (X ) e−j2πn(X/4) d X

−2

1 π π 1 cos n X d X + j sin n X d X, 2 4 2 −1

where the limits of the integral were changed because t (x) = 0 for 1 < |x| < 2. The integral in the second term is zero because its integrand is an odd function, so that an =

sin n π2 . nπ

(4.41)

The transmittance function t (X ) then can be expressed by t (X ) =

∞ sin n π2 jn(π/2)X . e nπ n=−∞

(4.42)

Changing n to −n, and reversing the order of the summation, and using (4.40) to transform back to the variable x0 , gives t (x0 ) =

∞ sin n π2 −jn(π/λp)x 2 0. e nπ n=−∞

(4.43)

This is a series expression for the transmittance distribution of a one-dimensional Fresnel zone plate. Next, the diffraction pattern of the Fresnel zone plate when illuminated by a parallel beam is examined. The parallel beam is launched along the z-axis and the

4.6 One-Dimensional Fresnel Zone Plate

93

Fig. 4.26 Distribution of the light transmitted through a Fresnel zone plate

Fresnel zone plate is placed in the plane of z = 0, as shown in Fig. 4.26. The distribution function u(xi , yi ) of the diffracted light is obtained by inserting (4.43) into (3.60) and manipulating the equation in a manner similar to that used to arrive at (3.60). The result is ∞ sin n π xi2 1 π 2 exp j kz i + k − u(xi , z i ) = √ 2z i 4 nπ zi λ n=−∞ ∞ × −∞

! ! π 1 n exp j − , x02 − j2π f x0 d x0 !! λ zi p f =xi /λz i

(4.44)

where the Fresnel zone plate was assumed to extend to infinity. Equation (4.44) is a summation of a series with respect to n. The term with n satisfying 1 n − =0 zi p

(4.45)

becomes predominantly large, in which case the integral takes on the value xi (4.46) δ λz i and the intensity peaks up. For a Fresnel zone plate of finite dimension a, t (x0 ) is replaced by t (x0 ) II (x0 /a) and the value corresponding to (4.46) becomes a sinc (axi /λz i ).

94

4 Practical Examples of Diffraction Theory

The term with n = 1 corresponds to a converged beam focused at z i = p and p is called the principal focal length. The focused point is indicated by P in Fig. 4.26. Besides the point P, the light is focused at the points z = pn where pn = − p, − p/3, − p/5 . . . p/n, 0, p/n . . . p/5, p/3, p. The reason that the peaks with even n are absent is that sin(nπ/2) in (4.44) becomes zero with even values of n. The lower curve in Fig. 4.26 shows the distribution of these amplitude peaks. The curves on the right-hand side of Fig. 4.26 show the relative contribution of each value of n when the screen is placed at the principal focal plane. The curves represent the relative intensities and it should be noted that the resultant intensity distribution is not the mere sum of these intensities, but rather each amplitude should be added taking its phase into consideration, and the intensity should be calculated from this sum. As the position of the screen is moved to another focal point along the z axis, the intensity associated with that particular focal length becomes predominantly high, compared to the intensities associated with other focal lengths. The distribution in Fig. 4.26 is for the case when n = 1. If the screen is moved to z i = p/3, then the intensity associated with n = 3 becomes high compared to all others. Now that some understanding of the action of the Fresnel zone plate has been gained, it would be instructive to perform the integration in (4.44). The integral is obtained with the aid of (3.38) ∞ sin n π xi2 1 2 − π/4 u(xi p) = √ exp j kp + k 2p nπ pλ n=−∞ " + 2 xi pλ × exp jπ 1/4 − (4.47) 1−n (1 − n)λp + ∞ xi2 pλ 1 jkp sin n π2 exp jk (n−1) =√ e (4.48) nπ 1−n pλ 2 p n=−∞ n

Recall that a cylindrical wavefront measured along a line located a distance d away is 2 (4.49) Ae jk(xi /2d) . Notice that the nth terms of (4.48) have the same form as (4.49), except that the light source is located at a distance (n − 1) p/n from the screen rather than d. When the Fresnel zone plate is used to converge a parallel beam, not only the contribution from the term with n = 1 in (4.48) but also the contributions of all other terms like n = 0, ±3, ±5, ± (2n − 1) are present. As a result, the converged beam has background illumination and is not as sharp as that made by a convex glass lens. Notice that both positive and negative n terms are included. While positive n corresponds to convex lenses, negative n corresponds to concave lenses, and thus the Fresnel zone plate possesses the properties of both kinds of lenses. Another important point to remember is that the focal length of the Fresnel zone plate varies with the wavelength of light.

4.7 Two-Dimensional Fresnel Zone Plate

95

4.7 Two-Dimensional Fresnel Zone Plate A Fresnel zone plate can be made two-dimensional, as shown in Fig. 4.27, by changing the array of strips into an array of concentric rings. The radii Rn of the concentric rings are the same as the one-dimensional strip spacing of the zone plate in the previous section, i.e., + λp (2n − 1). (4.50) Rn = 2 The intensity distribution in the principal focal plane can be obtained in a manner similar to that of the one-dimensional Fresnel zone plate. The transmittance distribution t (r ) of a two-dimensional Fresnel zone plate is t (r ) =

∞ sin n π2 −jn(π/λp)r 2 . e nπ n=−∞

(4.51)

When the Fresnel zone plate is placed in the plane z = 0 and illuminated by a beam propagating parallel to the z axis, the diffraction pattern on the screen at z = z i in cylindrical coordinates is 1 l2 exp jk z i + u(l, z i ) = jλz i 2z i ∞ ! r2 × 2π r P(r )t (r ) exp jk (4.52) J0 (2π r ) dr !=l/λzi 2z i 0

where P(r ) is the pupil function of the aperture. P(r ) is unity inside the aperture and zero outside the aperture. Inserting (4.51) into (4.52) gives ∞ sin n π2 1 l2 exp jk z i + 2 u(l, z i ) = jλz i 2z i n n=−∞ ∞ × 0

! π n 1 r P(r ) exp j − r 2 J0 (2π r ) dr !=l/λzi λ zi p

Fig. 4.27 Two dimensional Fresnel zone plate

(4.53)

96

4 Practical Examples of Diffraction Theory

The term that satisfies the condition 1 n − =0 (4.54) zi p dominates, as in the case of one-dimensional zone plate, and its distribution can be approximated by ! sin n π2 1 l2 exp jk pn + (4.55) B{P(r )} !=l/λpn . u(l, pn ) = jλpn 2 pn nπ From (4.55) it is seen that the distribution of light in the focal plane is a Fourier Bessel transform of the pupil function associated with the two-dimensional Fresnel zone plate. Up to this point, the distribution of the transmittance across the two-dimensional zone plate was made up of zones of either full transmission or no transmission in a uniform pattern. The transmittance distribution of the zone plate can be further modulated by an arbitrary function φ(r ). The Fresnel zone plate whose transmittance is further modulated in this way is called a Modulated Zone Plate, abbreviated MZP. The light distribution in the back focal plane of the MZP is the Hankel transform of φ(r ). By choosing an appropriate function, the distribution of the light at the back focal plane can be formed to a desired shape. By combining the MZP with a high-intensity laser, a cutting tool can be made, which cuts sheet metal into a shape determined by the shape of the focused beam. The MZP is usually made out of thin chromium film. Fig. 4.28 shows two arrays of holes made by MZP [4.6]. As mentioned in Chap. 2, the Hankel transform in cylindrical coordinates is identical to that of the Fourier transform in rectangular coordinates. Therefore, the transform of φ(r ) may be evaluated using whichever transform is the easier. When using rectangular coordinates for the Fourier transform, (4.55) becomes

Fig. 4.28 The photographs of the machined holes obtained with a laser beam focused by means of MZP [4.6]. (a) An array of holes made on a chrome film deposited on a sheet glass. (b) Two arrays of blind holes made on a polished surface of steel

4.7 Two-Dimensional Fresnel Zone Plate

97

Fig. 4.29 a, b. Gamma ray camera. (a) Shadow of zone plate cast by isotope tracer. (b) Reconstruction of the image by coherent light

" sin n π2 xi2 + yi2 1 u(xi , yi , pn ) = exp jk pn + F {φ(x, y)} jλpn 2 pn nπ

(4.56)

with f x = xi /λpn and f y = yi /λpn . Even though the Fresnel zone plate has such drawbacks as a focal length variation with wavelength and the presence of background light along with the main focused beam, it has the invaluable advantage that it can be used with types of electromagnetic radiation other than visible light. As an example of the application of the Fresnel zone plate to radiation outside the visible light spectrum, the gamma-ray camera is considered. The gamma-ray camera uses the Fresnel zone plate to form the image of an isotope administered to a patient for diagnostic purposes. As shown in Fig. 4.29 a, the gamma rays from an isotope in a patient strike a Fresnel zone plate made out of concentric lead rings. The pattern of the shadow cast by the zone plate is recorded either by a photographic plate or by a sodium iodine scintillator connected to a processing computer. The shadow recorded by the photographic plate is reduced in size and illuminated by coherent light. As shown Fig. 4.29 b, the pattern of the superposed zone plate on the photographic plate converges the light to their respective principal focal points and thus reconstructs the entire shape of the patient’s organ. It should be noted that it is not the diffraction pattern of the gamma rays that forms the image, but rather the

98

4 Practical Examples of Diffraction Theory

diffraction pattern of the light which illuminates the reduced pattern made by the gamma rays. The difference between an x-ray and a gamma-ray camera is that the purpose of the former camera is to take a picture of the structure of the organ whereas that of the latter is to examine the function of a particular organ. A comparison of two gammaray images taken at a certain time interval shows a change in the distribution of the labelled compound in the patient with respect to time. This change can be used to assess the functioning of that particular organ.

Problems 4.1. A grating whose surface is shaped like a saw tooth as shown in Fig. 4.30, is called an echelette grating. Find the optimum height h that maximizes the peak intensity of the diffracted light. The wavelength is 0.6 µm(1 µm is 10−3 mm) and the pitch of the tooth is 0.7 µm. 4.2. Figure 4.31 shows a light deflector based on the principle of the photo-elastic effect. Liquid in the deflector is periodically compressed by an acoustic wave generated by the transducer at the bottom, thus creating a periodic variation in the index of refraction. Such a structure can be considered as a kind of diffraction grating. The direction of the deflection of the beam is controlled by the frequency. It is assumed that the phase modulation of the transmitted wave is expressed by φ(x) = φ0 + φm sin

2π x, λa

(4.57)

when the direction of the incident beam is perpendicular to the wall of the deflector. Prove that when the angle of incidence of the beam to the wall is θ0 , the angle θ of the peak of the deflected beam satisfies sin θ = sin θ0 − n

λ , λa

(4.58)

where λa is the wavelength of the acoustic wave, λ is the wavelength of light, and n is an integer. In Fig. 4.31 all the angles are taken as positive when they are measured clockwise from the outward normal of either wall.

Fig. 4.30 Echelette grating

Problems

99

Fig. 4.31 Debye-Sears spatial light deflector

Fig. 4.32 Modulated zone plate (MZP) for machining tool

4.3. Referring to Problem 4.2, when the angles are adjusted to satisfy θ0 = −θ, (4.58) becomes 2 sin θB = n(λ/λa ), which is the formula for Bragg’s condition. Explain why the intensity of the peak deflected in this particular direction is substantially larger than all others. 4.4. A parallel beam is incident upon the modulated zone plate (MZP) shown in Fig. 4.32. Find an expression for the pattern at the back focal plane of this MZP. (A combination of a high intensity laser and such an MZP is used as a cutting tool for thin steel plates.) 4.5. In the text, the transmittance t (x0 ) of the one-dimensional zone plate was made up of strips that were either completely opaque or completely transparent. Show that the semi-transparent mask with the transmittance x02 (4.59) t (x0 ) = 1 + cos k 2f also possesses focusing action, and determine its focal length.

Chapter 5

Geometrical Optics

The rigorous way of treating light is to accept its electromagnetic wave nature and solve Maxwell’s equations. However the number of configurations for which exact solutions can be found is very limited and most practical cases require approximations. Based on the specific method of approximation, optics has been broadly divided into two categories, these being geometrical optics (ray optics) treated in this chapter and wave optics (physical optics) already treated in Chapter 3. The approximation used in geometrical optics puts the emphasis on finding the light path; it is especially useful for tracing the path of propagation in inhomogeneous media or in designing optical instruments. The approximation used in physical optics, on the other hand, puts the emphasis on analyzing interference and diffraction and gives a more accurate determination of light distributions.

5.1 Expressions Frequently Used for Describing the Path of Light In this section, mathematical expressions often used for describing the path of light are summarized [5.1, 2]. The mathematical foundation necessary to clearly understand geometrical optics will also be established.

5.1.1 Tangent Lines The tangent line is often used to describe the direction of curvilinear propagation. With the curve l shown in Fig. 5.1, an expression for the tangent drawn at P is obtained. An arbitrarily fixed point 0 is taken as the reference point. The vector made up by connecting 0 and P is designated by R, and is called the position vector. Another point P on the curve close to P is selected, and a straight line (chord) from 101

102

5 Geometrical Optics

Fig. 5.1 Tangent to a curve

P to P is drawn. The position vector of P is R + ∆R, ∆R being the vectorial expression of the chord. As P approaches P, the chord tends to reach a definite limiting orientation. This limiting orientation is the direction of the tangent to the curve at P. In this definition, the direction of the vector ∆R indeed coincides with the tangent but its magnitude vanishes, thus making it an inconvenient expression. In order to get around this, the ratio ∆R/∆s is used, ∆s being the length measured along the curve from P to P. Now, as P approaches P both the magnitude |∆R| and ∆s decrease at the same time. The magnitude of |∆R|/∆s reaches the limiting value of unity. Thus ∆R/∆s represents the unit vector sˆ of the tangent. In the following we will interchangeably call sˆ the direction of the light ray, or the unit tangent vector, sˆ =

dR . ds

(5.1)

Using rectangular coordinates, the position vector R is written in terms of its components as R = i x + j y + kz. Substituting this expression for R into (5.1), the unit tangent vector sˆ for rectangular coordinates is dy dz dx + j +k . (5.2) sˆ = i ds ds ds Depending on the geometry, some problems are more easily solved using cylindrical rather than rectangular coordinates. Hence, an expression for the unit tangent vector sˆ in cylindrical coordinates is derived. Figure 5.2 shows the projection of a curve l onto the x − y plane. The position vector of a point P on the curve l is the sum of the position vector of the point P , which is the projection of P onto the x − y plane, and the z component vector of the curve l, i.e., ˆ R = rˆ r + kz.

(5.3)

5.1 Expressions Frequently Used for Describing the Path of Light

103

Fig. 5.2 Tangent in polar coordinate system

The position vector for the projection is expressed as rˆ r , where r is the distance from 0 to P , and rˆ is a unit vector pointing in the direction from 0 to P . The unit vector rˆ is observed to point in the direction of constant φ, where φ is the angle between the line OP and the x axis. Before inserting (5.3) into (5.1) the unit vector φˆ is introduced as being a unit vector in the direction of constant radius. These unit vectors can be decomposed into components parallel to the rectangular coordinates as rˆ = i cos φ + j sin φ,

φˆ = −i sin φ + j cos φ

(5.4)

Inserting (5.3) into (5.1) gives sˆ = rˆ

d rˆ dz dr + r¸ + kˆ . ds ds ds

(5.5)

It is important to realize that both rˆ as well as r change as point P moves. From (5.4), the derivative of rˆ in (5.5) is dφ dφ d rˆ = (−i sin φ + j cos φ) = φˆ . ds ds ds Thus, the tangent expressed in cylindrical coordinates becomes sˆ = rˆ

dφ dz dr + φˆ r + kˆ . ds ds ds

(5.6)

(5.7)

104

5 Geometrical Optics

5.1.2 Curvature of a Curve Light changes its direction of propagation when it encounters an inhomogeneity in the medium. The curvature of the path is used to quantify this change of direction. This curvature is defined as the ratio of the change in the direction of propagation to the length measured along the curved path, i.e., the curvature is d sˆ /ds. The radius of curvature of a curve is ! ! 1 !! d sˆ !! = , (5.8) ! ds ! or from (5.1),

! ! 1 !! d 2 R !! =! !. ! ds 2 !

(5.9)

Thus, the inverse of the radius of curvature is the magnitude of the first derivative of the tangent vector, or the second derivative of the position vector. Next, the relationship, ˆ N d sˆ = (5.10) ds known as the Frenet-Serret formula, will be proven with the aid of Fig. 5.3. The ˆ appearing in (5.10) is the unit normal vector to the curve. A circle which vector N shares a common tangent with the curve at a point P is drawn. The radius of such a circle is designated by . The path length ds common with the circle can be represented by ds = dφ. (5.11) The unit tangent vector sˆ does not change its magnitude but changes its direction over the path. The direction changes from sˆ to sˆ , as illustrated in Fig. 5.3. Since |ˆs| = 1, it follows from the figure that |d sˆ | = dφ.

(5.12)

The scalar product of sˆ · sˆ is sˆ · sˆ = 1 = (ˆs + d sˆ ) · (ˆs + d sˆ ) = |ˆs|2 + 2ˆs · d sˆ + |d sˆ |2 = 1 + 2ˆs · d sˆ + |d sˆ |2 .

Fig. 5.3 Geometry used to prove Frenet-Serret formula

5.1 Expressions Frequently Used for Describing the Path of Light

105

When |d sˆ |2 is much smaller than the other terms, sˆ · d sˆ = 0. This means that d sˆ is perpendicular to sˆ and is normal to the curve. In Fig. 5.3, the change from sˆ to sˆ is in a downward direction corresponding to a section of the curve which is concave down. The direction of the normal N is also downward in this case. The direction of N is the same as that of the change in sˆ . Thus d sˆ is a vector whose magnitude is dφ, from (5.12), and whose direction is N, ˆ dφ. d sˆ = N (5.13) Finally, combining (5.11, 13), the Frenet-Serret formula is derived, namely, d sˆ ˆ 1. =N ds This formula will be used in describing the optical path in an inhomogeneous medium.

5.1.3 Derivative in an Arbitrary Direction and Derivative Normal to a Surface While most of the groundwork for a discussion of geometrical optics has now been laid, there remains yet one more important concept to be introduced before proceeding. This concept involves the rate of change of a multivariable function along a specified direction. In dealing with multivariable functions, the partial derivative is commonly used to describe the rate of change of the function along a particular coordinate axis direction. However, there is no reason for being restricted to the coordinate axis directions, and a more general expression for the derivative along an arbitrary direction is sought. Only three variable functions are considered here, although generalizations to a greater number of variables can easily be made. Let us begin with a specific example. The electrostatic potential u observed at a distance r from a point charge Q is u = Q/r . The expression for an equipotential surface of a specific value, let us say u = 2, is x 2 + y 2 + z 2 = (Q/2)2 .

(5.14)

This equipotential surface is a sphere, as shown in Fig. 5.4. All other equipotential surfaces can be drawn in a similar manner. In general, L(x, y, z) = C

(5.15)

represents a surface which is called the level surface associated with the value C.

106

5 Geometrical Optics

Fig. 5.4 Explanation of the direction of differentiation

Referring to Fig. 5.4, the change in L with a movement of the point P of observation from one equipotential surface to the other will be investigated. Let the length of the movement be ∆l. The direction of its movement is arbitrary. The movement ∆l consists of a movement of ∆x in thex direction, and ∆y in the y direction, and ∆z in the z direction, such that ∆l = ∆x 2 + ∆y 2 + ∆z 2 . The increment ∆L associated with the movement from P(x, y, z) to P (x + ∆x, y + ∆y, z + ∆z) is obtained using the multivariable Taylor’s formula, ∆L =

∂L ∂L ∆L ∆x + ∆y + ∆z + infinitesimals of higher order. ∂x ∂y ∂z

(5.16)

The directional derivative ∇ s L is ∆L ∂ L ∆x ∂ L ∆y ∂ L ∆z = + + ∆l ∂ x ∆l ∂ y ∆l ∂z ∆l + infinitesimals of higher order.

∇ s L = lim

∆l→0

(5.17)

As ∆l decreases, the infinitesimals of higher order in (5.17) tend to zero. Note that ∆x/∆l, ∆y/∆l, ∆z/∆l are the direction cosines cos α, cos β, cos γ of the movement ∆l, as shown in Fig. 5.4. Using the direction cosines, ∇ s L is written as ∇s L =

dL ∂L ∂L ∂L = cos α + cos β + cos γ . dl ∂x ∂y ∂z

(5.18)

5.2 Solution of the Wave Equation in Inhomogeneous Media

107

ˆ The directional derivative ∇ s L is of the form of a scalar product of ∇L and M, ∂L ∂L ∂L + j +k ∂x ∂y ∂z

(5.19)

ˆ = i cos α + j cos β + k cos γ M

(5.20)

∇L = i

ˆ is the unit vector in the direction of the movement of the point P. Thus, where M ˆ ∇ s L = (∇L) · M

or

ˆ ∇L = (∇L) · Mdl = (∇L) · dl

(5.21) (5.22)

where dl is the vector form of dl. For a given L, the direction of ∇L is determined by (5.19) but the value of ∇ s L varies with the choice of the direction of the movement. According to (5.22), the change in L becomes a maximum when the movement is selected in the same direction as that of ∇L. Conversely, it can be said that the direction of ∇L is the direction that gives the maximum change in L for a given length of movement. The direction which gives the maximum change in L with the minimum length of movement is the direction of the normal to the equi-level surface because it connects the two adjacent equi-level surfaces with the shortest distance. The unit vector representing the normal N to the equi-level surface is therefore N=

∇L . |∇L|

(5.23)

In optics, (5.23) is a particularly useful formula because it determines the optical path from the equi-phase surface. With these preparations completed, geometrical optics will be introduced in the next sections.

5.2 Solution of the Wave Equation in Inhomogeneous Media by the Geometrical-Optics Approximation A vector E(x, y, z) representing a light wave needs to-satisfy the wave equation (∇ 2 + ω2 µε) E(x, y, z) = 0

(5.24)

where ω is the radian frequency of the light, µ the permeability of the medium, and ε the dielectric constant of the medium. In a rectangular coordinate system, the equations for E x , E y , and E z are identical if the medium is isotropic1 , so that solving for one of them is sufficient to obtain a general solution. The light wave vector may then be replaced by a scalar function u(x, y, z) and the wave equation becomes 1

A medium is said to be isotropic if the physical properties at each point in the medium are independent of the direction of measurement.

108

5 Geometrical Optics

(∇ 2 + [kn(x, y, z)]2 ) u(x, y, z) = 0,

(5.25)

where ω2 µε = k 2 n 2 (x, y, z), k is the free space propagation constant, and n is refractive index of the medium. When the medium is homogeneous and n(x, y, z) is constant, the solutions are well known trigonometric functions. However, for an inhomogeneous medium in which n(x, y, z) is a function of location, the rigorous solution becomes complicated and some kind of approximation usually has to be employed. The method of approximation treated here is the geometrical-optics approximation [5.3–5] which is valid for large k. The solution of (5.25) is assumed to be of the form u(x, y, z) = A(x, y, z) ej[k L(x,y,z)−ωt] .

(5.26)

Whether or not (5.26) is correct can be verified by substituting it back into the differential equation. The functions A(x, y, z) and L(x, y, z) are unknown and have to be determined in such a way that (5.26) satisfies (5.25). Computing the derivative of (5.26) with respect to x gives ∂A ∂L ∂u = + jk A e j(k L−ωt) . ∂x ∂x ∂x Equation (5.25) is obtained by differentiating once more and adding a term n 2 k 2 u. Similar operations are performed with respect to y and z, giving n2 k 2 u +

" 2 ∂2 A ∂ 2u ∂2 L ∂ A ∂L j(k L−ωt) k 2 n 2 − ∂ L A + = e + jk A + j2k · =0 ∂x ∂x ∂x ∂x2 ∂x2 ∂x2 ↓

↓

↓

↓

↓

n 2 k 2 u + ∇ 2 u = ej(k L−ωt) {k 2 [n 2 − |∇L|2 ] A + ∇ 2 A + jk A∇ 2 L + j2k(∇ A) · (∇L)} = 0,

(5.27) |∇L|2

where means the sum of the squares of the i, j , k components of ∇L. Multiplying (??) by 1/k 2 gives (n 2 − |∇L|2 )A +

1 2 j ∇ A + [ A∇ 2 L + 2(∇ A) · (∇L)] = 0. 2 k k

(5.28)

If the wavelength of light is much shorter than the dimensions of the associated structure, then the second and third terms are very small compared to the first term and can be ignored. Equation (5.28) then becomes |∇L|2 = n 2 .

(5.29)

Expressed in rectangular coordinates, (5.29) is written as

∂L ∂x

2 +

∂L ∂y

2 +

∂L ∂z

2 = n2.

(5.30)

The function u(x, y, z) can be obtained by inserting the solution of (5.30) into (5.26). In this way, (5.25) does not have to be solved directly. It should be noted,

5.2 Solution of the Wave Equation in Inhomogeneous Media

109

however, that the approximations applied in obtaining (5.29) implied not only that k was very large, but also that the values of ∇ 2 L , ∇ 2 A, and ∇ A · ∇L were very small. In other words, the variation of both L and A has to be very small. Referring to (5.22), the variation ∆L due to a movement in an arbitrary direction dl is (∇L) · dl and therefore (5.31) L = (∇L) · dl. arbitrary direction

If the movement is restricted to the ∇L direction which is the direction of the normal to the equi-level surface of L, then (5.29) gives n ds . (5.32) L = |∇L| ds = along normalto equi-level surface

along normal to equi-level surface

At first glance, it appears as if the differential equation (5.29) has been solved directly by the integration of (5.32). However, it is not quite so simple, because, in order to perform the integration, the direction of the normal has to be known, and in order to know the direction of the normal, L has to be known, and L is the quantity one is trying to determine by the integration of (5.32) in the first place. There is no way to avoid solving the differential equation (5.29). Equation (5.29) is an important equation and is called the eikonal equation. The wave front L itself is called the eikonal or optical path. The origin of the word “eikonal” is Greek, meaning “image”. As seen from (5.26), the eikonal is a quantity which specifies the phase front. The movement of a particular phase surface, say the zero phase surface, is considered. From (5.26), such a phase front is on a surface represented by L(x, y, z) =

ω t. k

As time elapses, the zero radian phase front advances, as shown in Fig. 5.5. Recall from (5.23) that the direction of the normal to this equi-phase surface is ∇L/|∇L|. The direction of the normal is called the “wave normal” and coincides with the direction of propagation if the refractive index is isotropic, and is the same in all directions. Since the velocity of light in a medium with refractive index n is v = c/n, the time T that the light takes to travel between the two points P1 and P2 in this medium is P2 T = P1

ds = υ

P2 P1

n L ds = c c

where

(5.33)

110

5 Geometrical Optics

Fig. 5.5 Growth of an equi-phase front with the progress of time

P2 L=

n ds.

(5.34)

P1

The path of integration in (5.34) coincides with the optical path which is the direction of the normal to the equi-phase surface, so that (5.32) can be used. One interpretation of L = cT is that the eikonal is the distance that the light would travel in vacuum during the time that the light travels between the two specified points in the medium. Another interpretation is that the ratio of the eikonal to the speed of light c represents the time it takes for the light ray to travel from P1 to P2 . The value of A(x, y, z) has still to be determined. Since the first term of (5.28) is zero from (5.29), and since the term with 1/k 2 can be assumed negligibly small, (5.28) becomes (5.35) A∇ 2 L + 2(∇ A) · (∇L) = 0. In general, it is difficult to solve (5.35) for A except for special cases, one of which is considered in Problem 5.1. The importance of geometrical optics lies in the solution for L rather than for A. Exercise 5.1. A glass slab is the fundamental building block of many kinds of integrated micro-optic circuits. Consider a glass slab whose index of refraction is variable in the x direction but is constant in both the y and z directions. The index of refraction is denoted by n(x). Find a solution L(x, y, z) of the eikonal equation, and find the geometrical optics solution u(x, y, z) of the light propagating in this medium. Solution L(x, y, z) is assumed to be separable so that it can be expressed by the sum L(x, y, z) = f (x) + g(y) + h(z)

(5.36)

where f (x) is a function solely of x, g(y) is a function solely of y and h(z) is a function solely of z. Whether or not a solution of this form is correct can be verified by inserting it back into (5.29). The inserted result is {[ f (x)2 ] − [n(x)2 ]} + [g (y)2 ] + [h (z)]2 = 0.

(5.37)

5.3 Path of Light in an Inhomogeneous Medium

111

The first term is a function of x only, and the second and the third terms are functions only of y and z, respectively. In order that (5.37) be satisfied at any point in space, in other words, for any possible combination of the values of x, y and z, each term has to be constant, ⎫ f (x)2 − [n(x)]2 = a 2 ⎪ ⎪ ⎬ [g (y)]2 = b2 (5.38) [h (z)]2 = c2 ⎪ ⎪ ⎭ a 2 + b2 + c2 = 0. Integrating (5.38) gives f (x) = ±

x

[n(x)]2 − (b2 + c2 ) d x

(5.39)

m

g(y) = ±by + m 1

(5.40)

h(z) = ±cz + m 2 .

(5.41)

Inserting all of these terms into (5.36), L(x, y, z) becomes L(x, y, z) = ±

x

[n(x)]2 − (b2 + c2 ) d x ± by ± cz.

(5.42)

m

The constants m 1 and m 2 in (5.40, 41) are absorbed in the lower limit of the integral of the first term of (5.42). The values of b and c are determined by the boundary conditions, such as launching position and angle. The plus or minus signs are determined by the direction of propagation. Taking the direction of propagation to be positive for all three x, y and z directions, and inserting (5.42) into (5.26) finally gives ⎫ ⎧ ⎡ x ⎤ ⎬ ⎨ n 2 (x) − (b2 − c2 ) d x + by + cz ⎦ − jωt . u(x, y, z) = A exp jk ⎣ ⎭ ⎩ m

(5.43) The solution for A is left as a problem.

5.3 Path of Light in an Inhomogeneous Medium The direction of light propagation in an isotropic medium is the direction of the wave normal [5.3, 4]. The unit vector N of the wave normal is from (5.23, 29) N=

∇L . n

(5.44)

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5 Geometrical Optics

The wave normal is in the same direction as that of the unit tangent vector sˆ to the light path. Combining (5.1 and 44) gives dR ∇L = . ds n

(5.45)

Equation (5.45) is a differential equation that is used for determining the light path. In a rectangular coordinate system, (5.45) is n

∂L dx = , ˙ ∂x ds

n

dy ∂L = , ds ∂y

n

dz ∂L = . ds ∂z

(5.46)

Using (5.46), a study will be made of the optical path of light launched into a medium whose refractive index varies only in the x direction. The results of Exercise 5.1 are applicable to this situation. Inserting (5.42) into (5.46) gives n(x)

dx = [n(x)]2 − (b2 + c2 ) ds

(5.47)

n(x)

dy =b ds

(5.48)

n(x)

dz = c. ds

(5.49)

From (5.48, 49), one obtains b dy = dz c and therefore

b z + d. (5.50) c There exists a unique plane perpendicular to the y-z plane defined by (5.50). A ray entering a medium characterized by an x dependent refractive index will always remain in this plane regardless of launching conditions. The projection of the launched ray to the y-z plane is, of course, a straight line. The constants are determined by the launching point and angle. For example, let us assume that the launching point is (0, 0, C0 ), and that the ray is launched at an angle φ0 with respect to the y axis and θ0 with respect to the x axis. The expression for the projected line becomes (5.51) z = y tan φ0 + C0 . y=

The expressions for y and z are obtained from (5.47–49), b dx y= 2 [n(x)] − (b2 + c2 ) c dx . z= 2 [n(x)] − (b2 + c2 )

(5.52)

(5.53)

5.3 Path of Light in an Inhomogeneous Medium

113

Referring to Fig. 5.6, dy and dz can be written as ds sin θ cos φ0 = dy,

ds sin θ sin φ0 = dz

(5.54)

where θ is the angle between ds and the x axis at any point on the path. Combining (5.54) with (5.48, 49) gives n(x) sin θ cos φ0 = b,

n(x) sin θ sin φ0 = c.

Eliminating φ0 from the expressions in (5.55) gives n(x) sin θ = b2 + c2 .

(5.55)

(5.56)

This leads to the important conclusion, that n(x) sin θ = constant,

(5.57)

which holds true throughout the trajectory. Equation (5.57) is Snell’s law for a onedimensional stratified medium. At the launching point x = 0, the index of refraction is n 0 and the launching angle θ0 satisfies n 0 sin θ0 = b2 + c2 . (5.58) Inserting (5.55, 58) into (5.52, 53) yields x

y= 0

n 0 sin θ0 cos φ0

dx

and

(5.59)

n 0 sin θ0 sin φ0 d x. [n(x)]2 − n 20 sin2 θ0

(5.60)

[n(x)]2 − n 20 sin2 θ0 x

z − C0 = 0

The values of the lower limits of (5.59, 60) are selected such that y = 0 and z = C0 at x = 0. If the z axis which starts off at (0, 0, C0 ) at angle φ0 , as shown in Fig. 5.6, is taken as a new axis, then (5.60) can be written as

x

z = 0

n 0 sin θ0

d x.

(5.61)

[n(x)]2 − n 20 sin2 θ0

It is worthwhile to examine (5.61) closely. When the quantity inside of the square root becomes negative, the value of z becomes imaginary and the light does not propagate. The light will reflect at the point where [n(x)]2 − n 20 sin2 θ0 becomes negative, i.e., (5.62) n(x) = n 0 sin θ0 . For the case of a refractive index in which the value of n(x) is decreasing monotonically with x, the light will not propagate beyond x = x0 , where

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5 Geometrical Optics

Fig. 5.6 Path of light in a medium whose refractive index is varied only in the x direction

Fig. 5.7 Path of light at various launching angles into a medium whose refractive index is monotonically decreasing with increasing x

n(x0 ) = n 0 sin θ0 , and total reflection takes place at this point. The location of total reflection is a function of the launching angle θ0 , so that the smaller θ0 is, the further away the point of reflection is. If the launching angle is decreased to such an extent that n 0 sin θ0 is smaller than any value of n(x) within the medium, then the light penetrates through the medium. This behavior is illustrated in Fig. 5.7. In a medium with refractive index always larger than that at the launching point, namely n(x) > n 0 , no total reflection takes place regardless of the distribution of refractive index.

5.3 Path of Light in an Inhomogeneous Medium

115

For the special case of a refractive index which is a maximum on the x-axis and then decreases both in the positive and negative x directions, total reflection takes place at both upper and lower limits, so that, under the proper launching conditions, the light is confined to propagate within these limits. This kind of medium has practical significance for guiding light and is used in fiber optical communication. Propagation in this particular type of medium will be discussed in more detail next. A typical distribution which is frequently employed in practical devices is the parabolic distribution (5.63) n 2 = n 2c (1 − α2 x 2 ). A glass slab with such a variation of refractive index is frequently referred to by the tradename of Selfoc. The light path is derived by inserting (5.63) directly into (5.61) ad x (5.64) z = n 2c − a 2 − α2 n 2c x 2 where a = n 0 sin θ0 . When the launching point is at the origin and the launching direction is in the y = 0 plane, as indicated in Fig. 5.8, then a = n c sin θ0 and (5.64) simplifies to x ds sin θ0 z= α cos θ0 2 0 − x2 α which upon integrating becomes z=

sin θ0 sin−1 α

α cos θ0

x.

In optical communications, the angle γ0 = 90◦ − θ0 is conventionally used instead of θ0 (Fig. 5.8). Expressed in terms of γ0 , and solving for x, the result becomes sin γ0 α x= sin z. (5.65) α cos γ0 The optical path in such a medium is sinusoidal with an oscillation amplitude equal to (sin γ0 )/α and a one-quarter period equal to π(cos γ0 )/2α, as shown in Fig. 5.8. As the incident angle γ0 increases, the amplitude grows and the period shortens.

Fig. 5.8 Propagation path of light inside a one dimensional Selfoc slab

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5 Geometrical Optics

Now, compare the ray having a larger incident angle with that having a smaller incident angle. Even though the ray with a larger incident angle travels along a longer path, it travels mostly in an outer region where the refractive index is lower and the velocity is higher. On the other hand, the ray with a smaller incident angle travels along a shorter path but in an inner region where the refractive index is higher and the velocity is lower. As a result, the dispersion or the difference in the travel time of the two rays is small. Hence, the dispersion with the Selfoc slab is smaller than that of a step index slab (a uniform core glass sandwiched by a uniform cladding glass of lower refractive index). Similarly, the dispersion of a Selfoc fiber is less than that of a step index fiber. Therefore the Selfoc fiber has the merit of having a larger information carrying capability than does a step index fiber. Dispersion will be discussed further in Sect. 5.9.

5.4 Relationship Between Inhomogeneity and Radius of Curvature of the Optical Path Whenever light encounters an inhomogeneity of the refractive index, the direction of propagation changes. In this section, the relationship between the inhomogeneity of the refractive index and the radius of curvature associated with the change in direction of the optical path is studied [5.6]. Equation (5.32) can be written in terms of the derivative in the direction of the light path which is along the normal to the equi-level surface as dL = n. ds

(5.66)

Taking the gradient of both sides of (5.66) gives d ∇L = ∇n ds

(5.67)

where the order of applying the gradient and the derivative on the left-hand side was exchanged. Combining (5.45) with (5.67) yields dR d n = ∇n, (5.68) ds ds called the Euler-Lagrange formula. This formula specifies the relationship between the inhomogeneity of the medium and the change in the optical path. It gives the change in the optical path directly without going through L. The more quantitative evaluation of the change in the light path can be found by using the Frenet-Serret formula. Performing the differentiation of (5.68) d sˆ dn d R +n = ∇n ds ds ds

5.5 Path of Light in a Spherically Symmetric Medium

117

and using (5.10), ∇n is expressed as dn ˆ 1 = ∇n. sˆ + n N ds ˆ and making use of Taking the scalar product of both sides of the equation with N, ˆ is perpendicular to sˆ , yields the fact that N n

1 ˆ · ∇n. =N

(5.69)

Now, the meaning of (5.69) is examined. Since was defined as a positive quantity in Sect. 5.1 and since n is always positive, the left-hand side of (5.69) is always ˆ ·∇n must also be a positive quantity. This means that the light ray positive, so that N always bends toward the direction of the maximum incremental increase in n. This relationship is often helpful for finding the direction of the changes of the optical ˆ points down in the path in a complicated optical system. For example, in Fig. 5.8, N ˆ always points region of x > 0 and points up in the region of x < 0. The vector N toward the direction of ∇n. Next, it will be proven that when light is incident upon a medium whose refractive index distribution is spherically symmetric, the light path is always confined to a plane containing the launching point and the center of the symmetry. Take the point of symmetry as being the origin of the coordinate system. Consider a plane containing both the position vector R originating from the point of symmetry and the direction sˆ of the incident ray at the point of entry. This plane naturally contains the origin. The normal vector K to this plane is expressed by K = R × (n sˆ ). As soon as the ray sˆ starts to deviate from this plane, the normal vector defined above starts to change its direction. Conversely, as long as d K /ds = 0, the ray stays in this initial plane. Now, using (5.1), gives dR dR d dR dK = × n +R× n . ds ds ds ds ds The first term is obviously zero and the second term can be rewritten using (5.68) as dK = R × (∇n). ds If the refractive index distribution is spherically symmetric, ∇n points in the radial direction, hence d K /ds = 0. Thus, the ray is always confined in the initial plane, and the situation depicted in Fig. 5.9 is impossible.

5.5 Path of Light in a Spherically Symmetric Medium The light path in a medium whose refractive index distribution is spherically symmetric is examined further [5.3, 4]. The expression for ∇L in spherical coordinates is

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5 Geometrical Optics

Fig. 5.9 Such an optical path is impossible if the medium is spherically symmetric

∇L = rˆ

∂L 1 ∂L 1 ∂L + φˆ + θˆ . ∂r r sin φ ∂φ r ∂θ

(5.70)

As proven at the end of Sect. 5.4, the path of light is always in a plane containing the origin, regardless of launching angle, if the refractive index is spherically symmetric. If the z axis is taken in this plane, then the path direction (or ∇L) does not have a φˆ component. Thus, if the medium is spherically symmetric, the eikonal equation becomes 2 1 ∂L 2 2 ∂L + = [n(r )]2 . (5.71) |∇L| ∂r r ∂θ Applying the method of separation of variables, a solution of the eikonal equation of the form L(r, θ) = R(r ) + Θ(θ ) (5.72) is assumed. Inserting (5.72) into (5.71) gives r 2 {[R (r )]2 − [n(r )]2 } + [Θ (θ )]2 = 0.

(5.73)

The first term of (5.73) is a function of r only and the second is a function of θ only. In order that (5.73) be satisfied for any combination of r and θ, the first and second terms each have to be constant, i.e., [Θ (θ )]2 = a 2

(5.74)

r 2 {[R (r )]2 − [n(r )]2 } = −a 2 .

(5.75)

The solution of (5.74) is Θ(θ ) = ±aθ + m 1 and that of (5.75) is R(r ) = ±

r m1

+ [n(r )]2 −

a 2 r

dr + m 2 .

(5.76)

5.5 Path of Light in a Spherically Symmetric Medium

119

Inserting both results back into (5.72) yields + r

L(r, θ) = a θ +

[n(r )]2 −

a 2

m

r

dr ,

(5.77)

where m 1 and m 2 are included in the lower limit of the integral, and where the propagation is assumed to be in the direction of positive r and θ. Since the eikonal L(r, θ) has been found, the light path can be calculated by inserting L(r, θ) into (5.45). However (5.45) must first be expressed in polar coordinates. Using relationships similar to those that led to (5.7), d R/ds is written as R = r rˆ

(5.78)

dr dr ˆ dθ . = rˆ + θr ds ds ds

(5.79)

Thus, the components of (5.45) become n

dr ∂L = ds ∂r

nr

(5.80)

1 ∂L dθ = . ds r ∂θ

(5.81)

Inserting (5.77) into (5.80, 81) gives dr n = ds nr

+ [n(r )]2 −

a 2 r

1 dθ = a. ds r

Combining (5.82) and (5.83), one obtains r adr θ= m r [r n(r )]2 − a 2

(5.82) (5.83)

(5.84)

where integration constants are included in the lower limit of the integral. Examination of Fig. 5.10 shows that, at an arbitrary point (r, θ), we have r

dθ = sin γ ds

(5.85)

where γ is the angle which the ray makes with the radial vector. Inserting (5.85) into (5.83) gives r n(r ) sin γ = a. (5.86) It is worth mentioning that, based on (5.86), points with the same r have the same angle γ or (180◦ − γ ), as shown in Fig. 5.11a. [Note that sin γ = sin(180◦ − γ ).]

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5 Geometrical Optics

Fig. 5.10 Initial launching condition into a spherically symmetric medium

Fig. 5.11 a, b. Light path in a spherically symmetric medium. In (a) the values of γ at the same r are identical. (b) shows the value of rm n(rm ) for a given launching condition

The constant a is now determined. At the launching point (r0 , θ0 ) with γ = γ0 , (5.86) becomes (5.87) r0 n(r0 ) sin γ0 = a. It is also interesting to examine what happens at γ = 90 ◦ and r = rm , as indicated in Fig. 5.11 b. Equations (5.86, 87) become rm n(rm ) = r0 n(r0 ) sin γ0 .

(5.88)

This equation means, for a given launching condition, the value of rm n(rm ) is immediately known. Exercise 5.2. Find the light path in a medium with a spherically symmetric refractive index given by 1 (5.89) n(r ) = √ . r The launching position and the angle are (r0 , θ0 ) and γ0 respectively.

5.5 Path of Light in a Spherically Symmetric Medium

121

Solution Inserting (5.89) into (5.84), θ becomes adr . (5.90) θ= √ r r − a2 2 2 The square root term is real only for r > = a , so that r = a is a point of total reflection, and there is no propagation in the region of r < a 2 . From a table of integrals [Ref. 5.7, p. 40], 2 −r 2a θ = cos−1 + α, r which can be rewritten as r = 2a 2 − r cos(θ − α).

(5.91)

From the initial condition that the ray starts at r = r0 , θ = θ0 , the value of α can be determined, 2 2a −1 . α = θ0 − cos−1 r0 The value of a, from (5.87), is a=

√ r0 sin γ0 .

Equation (5.91) is the equation of a parabola. The point of total reflection is (a 2 , α). The focus is at the origin. If new axes X and Z are chosen which are rotated by α from the original axes, then the directrix, which is a line used as a reference to generate the locus of the parabola, is at Z = 2a 2 , as shown in Fig. 5.12. The distance from P(r, θ) to H in Fig. 5.12 is r = r . Another way of identifying (5.91) as a parabola is to convert it into rectangular coordinates using Z = r cos(θ − α), and X = −r sin(θ − α). In these coordinates (5.91) becomes X 2 = 4a 2 (a 2 − Z ).

Fig. 5.12 Light path in √ a medium with n(r ) = 1/ r

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5 Geometrical Optics

5.6 Path of Light in a Cylindrically Symmetric Medium Many optical components have cylindrical symmetry with respect to the optical axis [5.8–12]. It is, therefore, important to be able to treat the formulae in cylindrical coordinates. First, the eikonal equation is solved in cylindrical coordinates using the method of separation of variables. The expression for ∇L in cylindrical coordinates is ∇L = rˆ

1 ∂L ∂L ∂L + φˆ + kˆ . ∂r r ∂φ ∂z

(5.92)

The eikonal equation in cylindrical coordinates is

∂L ∂r

2 +

1 ∂L r ∂φ

2 +

∂L ∂z

2 = n2.

(5.93)

Let the solution of (5.93) be L(r, φ, z) = R(r ) + Φ(φ) + Z(z)

(5.94)

where R(r ), Φ(φ) and Z(z) are functions solely of r, φ, and z, respectively. Inserting (5.94) into (5.93) gives 2

(R ) +

1 Φ r

2

+ (Z )2 = n 2 .

(5.95)

If n is assumed to be cylindrically symmetric, the solution is simplified considerably, 2 " Φ 2 2 + (Z )2 = 0. (5.96) R − [n(r )] + r a2 −a 2

The first term is a function of the independent variables r and φ only, and the second term is a function of the independent variable z only. Since for all possible combinations of r, φ, and z, the sum of the two terms has to be constant, each term itself has to be constant. Thus, (5.96) becomes (Z )2 = a 2 R 2 − [n(r )]2 +

Φ 2 r

(5.97) = −a 2 .

(5.98)

Equation (5.98) is further separated in terms of the functions R and , r 2 {R 2 − [n(r )]2 + a 2 } + Φ 2 = 0. −c2

c2

(5.99)

5.6 Path of Light in a Cylindrically Symmetric Medium

123

Similarly, one obtains Φ 2 = c2 R 2 = [n(r )]2 −

(5.100) c2 r2

− a2.

(5.101)

Finally, (5.94, 97, 100, 101) are combined to form the eikonal in a cylindrically symmetric medium r c2 L(r, φ, z) = [n(r )]2 − 2 − a 2 dr + cφ + az (5.102) r m

where all integration constants are absorbed in the lower limit of the integral. Next, the expression for the path of light is derived. Using (5.7, 92) in (5.45) and breaking the equation into its components gives ∂L dr = rˆ component ds ∂r 1 ∂L ˆ dφ = φ component nr ds r ∂φ ∂L ˆ dz k component = n ds ∂z n

(5.103) (5.104) (5.105)

These differential equations are used to derive the path of light from the eikonal in cylindrical coordinates. In particular, when the medium is cylindrically symmetric, the value of L has already been obtained by (5.102) and the above formulae become dr c2 n(r ) (5.106) = [n(r )]2 − 2 − a 2 ds r dφ 1 n(r )r = c (5.107) ds r dz n(r ) = a. (5.108) ds Division of (5.107) and (5.108) by (5.106), followed by integration, gives c dr φ= + 2 r 2 [n(r )]2 − rc2 + a 2 a dr z= + 2 [n(r )]2 − rc2 + a 2 .

(5.109)

(5.110)

Equation (5.109, 110) are general solutions for the optical path in a medium whose refractive index distribution is cylindrically symmetric. The constants a and c are to

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5 Geometrical Optics

Fig. 5.13 Geometry of the optical path in a cylindrically symmetric medium (direction cosines)

be determined from the initial conditions. Referring to Fig. 5.13, dz/ds in (5.108) is the direction cosine cos γ of the path with respect to the kˆ direction and r dφ/ds in (5.107) is the direction cosine cos δ of the path with respect to the φˆ direction. Let the launching condition be r = r0 φ = φ0 z=0 n(r0 ) = n 0 .

(5.111)

Let the launching direction cosines be cos γ0 and cos δ0 , then a = n 0 cos γ0 c = n 0 r0 cos δ0 .

(5.112) (5.113)

The integral in (5.109, 110) simplifies considerably when c = 0, i.e. when δ0 = 90◦ , or in other words, when the launching path is in a plane containing the z axis. The integrand of (5.109) is then zero and the path stays in a plane containing the z axis. Such a ray is called a meridional ray. The integral (5.110) for this case becomes identical to (5.53) of the one-dimensional case. The ray that does not stay in this plane, due to non-zero c, rotates around the z axis and is called a skew ray. As long as the medium is perfectly cylindrically symmetric, only the initial launching condition determines whether the path will be meridional or skew. However, if the medium contains some inhomogeneities, the meridional ray will become a skew ray as soon as the ray moves out of the meridional plane. It is interesting to recognize that any ray launched at the center of the medium is always meridional because c in (5.107) is zero. The general behavior of the light path is discussed by considering the sign of the quantity inside of the square root in (5.109, 110). Let η1 and η2 be functions of r defined as follows, η1 = n 2 (r ) η2 =

c2 + a2. r2

(5.114) (5.115)

Note that η1 is dependent only on the refractive index distribution, and that the value of η2 is dependent only on the launching conditions. Consider the special case

5.7 Selfoc Fiber

125

Fig. 5.14 Determination of the region of propagation of a meridional ray and a skew ray in a Selfoc fiber from the square root in (5.110)

for which the curve of η1 is bell shaped (Fig. 5.14). For a skew ray, the function η2 is monotonically decreasing with respect to r , and light can propagate only in the region rmin < r < rmax , where rmax and rmin are the intersections of η1 and η2 . Outside this region, e.g. along the z axis, the quantity inside the square root in (5.109, 110) becomes negative, and light cannot propagate. As the curve η2 is raised, the region of propagation is narrowed. In the limit that η1 and η2 just touch each other, propagation is possible only on this particular radius, and the projection of the path becomes a circle. For a meridional ray, c is zero and η2 = a 2 , so that the region of propagation is r < rc , which includes propagation along the z axis.

5.7 Selfoc Fiber In Sect. 5.3, the one dimensional Selfoc in slab shape was discussed. In this section, the Selfoc in fiber form will be investigated [5.8, 12]. The distribution of the refractive index of the Selfoc fiber is n 2 = n 2c (1 − α2r 2 ).

(5.116)

With this distribution, light is confined inside the fiber and propagates for a long distance with very little loss, so that the Selfoc fiber plays an important role in fiber-

126

5 Geometrical Optics

optical communication. In the following, the optical path in the Selfoc fiber will be analyzed separately for meridional ray propagation and skew ray propagation.

5.7.1 Meridional Ray in Selfoc Fiber The expression for the meridional ray in the Selfoc fiber is derived. Referring to the equations in Sect. 5.6, it follows that for the case of the meridional ray, c = 0 and the integrand of (5.109) is zero. Equation (5.110) with (5.116) becomes a dr . (5.117) z= 2 n c − a 2 − α2 n 2c r 2 Performing the integration in (5.117) gives n 2c − a 2 α nc sin (z − z 0 ). r= α nc a

(5.118)

This ray path is sinusoidal with an oscillation amplitude equal to n 2c − a 2 /αn c and a period equal to 2πa/αn c . When the launching point is at the center of the fiber and r0 = z 0 = 0, n 0 becomes n c and the expression with the help of (5.112) simplifies to sin γ0 α r= sin z. (5.119) α cos γ0 Figure 5.15 shows the path of light. Comparing (5.119) with (5.65) or even comparing (5.117) with (5.64), one finds that as far as the meridional ray is concerned, the path in the fiber is identical to that in the slab. The result of the one-dimensional treatment is often applicable to the meridional ray in the fiber. The physical reason for this is that, as far as the small section of circumference that reflects the

Fig. 5.15 a, b. Path of the meridional ray in a Selfoc fiber; (a) side view, (b) end view. As seen in the end view, the plane of incidence is perpendicular to the circumference, just as the ray in the slab is perpendicular to the boundary

5.7 Selfoc Fiber

127

meridional ray is concerned, the plane of incidence is always perpendicular to this circumference (Fig. 5.15b), just as the plane of incidence in the slab is perpendicular to the boundaries.

5.7.2 Skew Ray in Selfoc Fiber When c = 0, the integrations of (5.109, 110) are more complex. A short manipulation after inserting (5.116) into (5.110) gives dt a where (5.120) z= 2 2n c α q − (t − p)2 t = r2 p=

n 2c

− a2

2n 2c α2

,

q 2 = p2 −

c

(5.121)

2

nc α

.

The result of the integration in (5.120) is 2n c α (z − m) . t = p + q sin a

(5.122)

(5.123)

Having evaluated (5.110), the next step is to evaluate (5.109) and obtain an expression for φ. By comparing (5.109) with (5.110), one finds that the numerator differs in the constants c and a, and the denominator of (5.109) contains an r 2 factor which the denominator of (5.110) does not. Making use of the similarities in (5.109, 110), the former can be written as dt c . (5.124) φ= 2n c α t q 2 − (t − p)2 The equation below, taken from a table of indefinite integrals [Ref. 5.7, p. 47], is used to evaluate (5.124), acx + bc 1 ds = sin−1 2 , (5.125) √ √ 2 2 2 2 2 2 c x + ab (c x + ab) b − c x bc a −c where, a, b, and c are arbitrary constants not to be confused with a and c in previous equations. Making the following substitutions for x, a, b, and c in (5.125), x =t−p a = p/q

b=q c = 1,

the result of the integration in (5.124) becomes 1 px + q 2 c −1 sin + φ0 . φ= 2n c α p 2 − q 2 q(x + p)

(5.126)

128

5 Geometrical Optics

Writing t − p in place of x, and making use of (5.121, 122), (5.126) becomes c 2 1 = p − q sin 2(φ − φ0 ). (5.127) r 2 nc α If p is replaced by p [cos2 (φ − φ0 − π/4) + sin2 (φ − φ0 − π/4)] and the relationship sin 2(φ − φ0 ) = cos 2(φ − φ0 − π/4) = cos2 (φ − φ0 − π/4) − sin2 (φ − φ0 − π/4) is used, (5.127) can be further rewritten as 1=

r 2 cos2 (φ − φ0 − π/4) r 2 sin2 (φ − φ0 − π/4) + or 2 2 c √

n c α p−q

1=

X2 A2

+

(5.128)

c √ n c α p+q

Y2 . B2

(5.129)

√ Equation(5.128) is theequationof anellipsewithits major axis A = c(n c α p − q)−1 √ and minor axis B = c(n c α p + q)−1 tilted by φ0 + π/4 radians. Inserting all the constants of (5.122) into (5.128), the lengths of the major and minor axes become + 1 2 2 A= √ (n c − a ) + (n 2c − a 2 )2 − 4n 2c α2 c2 (5.130) 2n c α + 1 (n 2c − a 2 ) − (n 2c − a 2 )2 − 4n 2c α2 c2 . (5.131) B=√ 2n c α Thus, the projection of the optical path is found to be an ellipse (Fig. 5.16). It can easily be proven that the solutions rmin and rmax obtained from η1 = η2 in (5.114, 115) are identical to the lengths of the major and minor axes expressed by (5.130, 131).

Fig. 5.16 Skew ray in a Selfoc fiber

5.8 Quantized Propagation Constant

129

In short, the skew ray takes the path of an elliptic helix, as shown in Fig. 5.16. Since the cross section is an ellipse, the radial distance from the z axis in the cross sectional plane modulates as the ray advances. One complete turn around the circumference passes two maxima; hence, the pitch of the helix is found from (5.123) to be 2πa (5.132) zp = nc α which is identical to the pitch of the meridional ray. This is because the skew ray travels longer but stays in the region of lower refractive index. A Selfoc micro lens is a cylindrical glass rod with a parabolic refractive index distribution. The diameter of the Selfoc micro lens, however, is several millimeters and is much larger than that of the Selfoc fiber. The Selfoc micro lens is used to transfer the image on the front surface to the back surface. The reason this device can transfer the image is due to the fact that the pitch is independent of c and all rays having different angle of δ0 have the same pitch so that the image is not blurred. When the innermost square root of (5.130, 131) is zero, the cross section is a circle. It is also seen from these equations that A/B increases with a decrease in c, and the skew ray starts to convert into a meridional ray.

5.8 Quantized Propagation Constant 5.8.1 Quantized Propagation Constant in a Slab Guide It will be demonstrated that only those rays that are incident at certain discrete angles can propagate into the fiber [5.9]. In order to simplify matters, a rectangular slab guide such as the one shown in Fig. 5.17 is taken as a first example. This slab guide (step-index guide) is made of three layers. The inner layer (core glass) has a refractive index n 1 , which is larger than the refractive index n 2 of the two outer layers (cladding glass). If the angle of incidence θ is small enough, total reflection

Fig. 5.17 Explanation of the quantization in the angle of incidence θ

130

5 Geometrical Optics

takes place at R2 and R3 on the boundaries and the incident ray R1 S will take the zigzag path shown in Fig. 5.17, eventually reaching the point T. The equi-phase fronts of the zigzag ray are indicated by the dotted lines drawn perpendicular to the optical path, assuming a plane wave. In addition to this zigzag ray, there is another wave front indicated by the dashed line that propagates directly from S to T. The field at T is the sum of these two wave fronts. The two wave fronts add constructively at T and are accentuated when the incident angle θ is at discrete angles θ N such that the two wave fronts are in phase at T. The field at T is attenuated when the incident angle θ is not at such angles. Here, just one section ST of the zigzag path was considered but the wave fronts originating in the sections preceding this section also contribute to the field at T. The phases of the wave fronts from all of these other sections varies over the entire range of zero to 2π radians, and the resultant field becomes zero, unless the incident angle θ is at such θ N specified above [5.13]. These discrete angles θ N will be calculated by using the geometry in Fig. 5.17. The total length of the zigzag path ST is equal to the length of the straight line SP, and the phase of the zigzag wave is n 1 kSP. The phase of the directly propagated wave at T is the same as that at Q, so that the phase of this wave at T is n 1 kSQ. The difference between these phases is n 1 k(SP − SQ) = n 1 kQP. The length QP from the geometry of Fig. 5.17 is 4d sin θ. Thus θ can take on only such discrete values as specified by 4n 1 kd sin θ N = 2N π

(5.133)

where N is an integer. Next, (5.133) is modified slightly. Since n 1 k sin θ N is the propagation constant βx in the vertical x direction, (5.133) can be rewritten as ψx = 2dβx = N π

(5.134)

which means that the phase difference ψx between the upper and lower boundaries of the core glass can take only discrete values of an integral multiple of π radians. The z component of the propagation constant is n 1 k cos θ N and is normally designated by β N . From (5.133), the propagation constant is expressed as Nπ 2 . (5.135) βN = n1k 1 − 2n 1 kd Thus, it is seen that the values of β N are discrete. The so called Goos-H¨anchen phase shift [5.14], which appears at each reflection on the boundary due to a small spread in incident angle, has been ignored here. The reason that the quantization phenomenon is not readily observed with the thick slab geometry is that the difference θ N − θ N −1 in (5.133) is too minute for quantization to be observed. Although, the above treatment applies to a slab waveguide, the same results hold true for a meridional ray in an optical fiber because, in both cases, the planes containing the rays are perpendicular to the boundary surfaces (Fig. 5.15 b).

5.8 Quantized Propagation Constant

131

5.8.2 Quantized Propagation Constant in Optical Fiber For the case of a skew ray, the explanation of the quantization is more complicated than that of the meridional ray. Quantizations in both the φ and r directions have to be considered. The phase factor ψ of a skew ray is obtained from (5.26, 102) as n 2 − (c/r 2 ) − a 2 dr + kcφ + kaz. (5.136) ψ =k The propagation constants υ, ω and β in the r, φ, and z directions, respectively, are obtained from ψ as follows: υ = ∂ψ/∂r,

ω = r −1 ∂ψ/∂φ,

β = ∂ψ/∂z.

Applying these differentiations to ψ in (5.136) gives υ = k n 2 (c/r 2 ) − a 2 c ω=k r β = ka.

(5.137) (5.138) (5.139)

The quantization in the φ direction will be considered first. Figure 5.18 is a sketch of the path and cross-sectional field distributions of the wave front of a skew ray. The cross-sectional distribution of the ray has tails. The tail of the cross-sectional distribution of the first turn of the spiral path is overlapping with that of the second turn. If the phases of the overlapping wave front are random, then the larger the number of contributing wave fronts is, the smaller the field intensity becomes. However, when their phase differences become an integral multiple of 2π radians, the field grows with the number of participating rays. This condition is satisfied when 2π ωr dφ = 2νπ 0

Fig. 5.18 Interference between the light rays

(5.140)

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5 Geometrical Optics

Fig. 5.19 A wavy line is drawn on a side of a piece of noodle, and then wrapped around a piece of pencil. The line can represent the path of a skew ray

where ν is an integer. Inserting (5.138) into (5.140) gives ν = kc.

(5.141)

Thus, the value of k c is quantized and the condition of quantization is (5.141). Next, quantization in the r direction is considered. The drawing in Fig. 5.19 is a comprehension aid. A piece of soft noodle is wrapped around a piece of pencil. Before wrapping the noodle, a wavy line is drawn on the side of the noodle. If the shape and the period of the wavy line is properly selected, the line becomes an elliptic helix of a skew ray when it is wrapped around the pencil. Before wrapping, the pattern of the wavy lines resembles the one-dimensional zigzag light path shown in Fig. 5.17. As mentioned earlier, the condition of quantization in the one-dimensional zigzag path was that the phase difference between the upper and lower boundaries has to be a multiple of π radians. This condition that the phase difference between the inner and outer boundaries be a multiple of π radians approximately applies to the quantization in the r dimension of the cylindrical guide. Thus, this condition gives rmin υdr

µπ =

(5.142)

rmax

where µ is an integer. In order to make it easier to perform the integral, (5.142) is rewritten. The ratio of (5.106) to (5.108), and (5.137) gives dr υ = , dz ka

5.8 Quantized Propagation Constant

133

and with (5.139), 1 µπ = β

z 2 υ 2 dz.

(5.143)

z1

From (5.137, 139, 141), the condition for quantization in the r direction becomes 1 µπ = β

z 2 k 2 n 2 (r ) −

ν 2 r

− β 2 dz.

(5.144)

z1

Thus, the path of a skew ray is discrete and is controlled by the two integers (µ, ν). The integration of (5.144) will be performed for the case of a Selfoc fiber, and the quantized propagation constant β will be derived. First, the integration constant m in (5.123) is determined such that at z = z 1 , the maximum value of r appears. The maximum value of r appears when 2n c α(z 1 − m)/a = π/2. Putting m back into (5.123) gives (5.145) r 2 = p + q cos [2n c α(z − z 1 )/a] so that, the position z 2 of the adjacent minimum is z2 = z1 +

πa . 2n c α

(5.146)

Inserting (5.116, 122, 145, 146) into (5.144) and using the integration formula dx q + p cos bx 1 cos−1 = (5.147) 2 2 p + q cos bx p + q cos bx b p −q gives the final result

βµ,ν = n c k 1 − 2α

2µ + ν . nc k

(5.148)

When the end of an optical fiber is excited by a light source, only those rays propagate that are incident at discrete angles for which the propagation constant in the z direction satisfies (5.148). Each ray that can propagate is designated by the integers µ and ν, and is called the (µ, ν) mode (of propagation). Next, the average of the radius of the skew path is considered. From (5.145), the 2 is average radius rave 2 = p. (5.149) rave Combining (5.122, 139, 148) gives 2 rave =

2µ + ν . αn c k

(5.150)

The conclusion is that the higher the mode order is, i.e. the larger 2µ + ν is, the further the ray will travel into the outer regions of the optical fiber.

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5 Geometrical Optics

Fig. 5.20 Cross-sectional distribution of the intensity for various mode numbers

In all the above discussions, the plus sign has been employed in (5.102), but the minus sign is equally legitimate. The minus sign indicates a spiraling of the skew ray in the opposite direction. Actually, two skew rays spiraling in opposite directions are present in the fiber. The two waves propagating in opposite directions create a standing-wave pattern. These standing-wave patterns are nothing but the so-called mode patterns. Some of these mode patterns are shown in Fig. 5.20. Recall that in the case of a standing-wave pattern made by the interference of two plane waves travelling in opposite directions, the intensity goes through two extrema for a distance that corresponds to the advance of the phase by 2π radians. For one complete turn in circumference, the phase goes through 2π ν radians as indicated by (5.140) and hence the intensity passes through 2ν extrema. According to (5.142) the phase advances by µπ radians in the radial direction, so that the intensity goes through µ extrema across the radius. Next, the behavior of β, which is the propagation constant in the z direction, is examined more closely. Using (5.148), β is plotted with respect to kn c with µ and ν as parameters in Fig. 5.21. If βµ,ν is imaginary instead of real (propagating), the wave decays rapidly in the z direction. Whether or not the ray propagates is determined by whether the quantity inside the square root in (5.148) is positive or negative. The condition 2µ + ν =0 (5.151) 1 − 2α nc k is called the cut-off condition. Among the modes, the (0, 0) mode is very special. Note that, from either (5.151) or Fig. 5.21, the (0, 0) mode has its cut-off at kn c = 0 and can propagate throughout the frequency band. The cutoff for the next higher mode (0, 1) is kn c = 2α, so that the condition that only the (0, 0) mode is excited is that kn c < 2α. The optical fiber which is designed to propagate only one mode is called a single-mode fiber. Some important characteristics of single-mode fibers are mentioned in the next section.

5.9 Group Velocity The speed of propagation of the light energy is called the group velocity υg of the light [5.9]. Its inverse τ = υg−1 is the time it takes for the light energy to travel a unit

5.9 Group Velocity

135

Fig. 5.21 k − β diagram of the Selfoc fiber

distance and is called the group delay. The group velocity may be obtained from the propagation constant β using the relationship υg =

dω dk =c . dβ dβ

(5.152)

Alternatively, the time needed for the light to travel along the helical path in Fig. 5.16 may be calculated using (5.33) (Problem 5.8). Here, the former method is used. Differentiating (5.148) with respect to k gives dβµ,ν n 2c k 2µ + ν = . (5.153) 1−α dk βµ,ν nc k Equation (5.148) is again used to find an expression for (2µ + ν) in terms of βµ,ν . This expression for (2µ + ν) is substituted into (5.153) and the result is used in (5.152) to obtain c 2(βµ,ν /n c k) . (5.154) υg = n c 1 + (βµ,ν /n c k)2 The lower-order modes have a higher group velocity than the higher-order modes. When light enters into a multimode fiber, several fiber modes may be excited, depending on the launch conditions, and each of these excited modes will support the propagation of light. For an input signal which is composed of a series of light pulses, the difference in the time of arrival at the receiver for the various fiber modes causes a spreading of the individual light pulses. The difference in arrival time between the rays of the various modes limits the time interval between the pulses, and hence, the rate of pulses that can be sent. Following similar reasoning, if the

136

5 Geometrical Optics

input light signal is amplitude modulated, the difference in group velocity causes a distortion of the received waveform. The spreading of a pulse or distortion of an amplitude modulated light signal due to the different velocities between the modes is called mode dispersion. Obviously an isotropic2 single mode fiber has the merit of being free from mode dispersion. An additional advantage of a single-mode fiber is that β is a linear function of frequency (kn c ), as seen from Fig. 5.21, so that the distortion due to the difference in the velocity of transmission among the different frequency components of the signal is eliminated. Because of these special characteristics, a single-mode fiber plays an important role in high density information transmission in the field of fiberoptical communication, as will be discussed in Chapter 12. This section concludes by giving an alternate expression to (5.154) for the group velocity. As discussed previously, β is the propagation constant in the z direction and n c k is the propagation constant along the helical path, so that (β/(n c k)) is cos γ in Fig. 5.13. Equation (5.154) can be written as υg = τ −1 =

c 2 cos γ . n c 1 + cos2 γ

(5.155)

Problems 5.1. By solving the differential equation (5.35), find the value of A for the onedimensional problem; n = n(x), ∂ A/∂ y = ∂ A/∂z = 0. 5.2. Prove that the optical path is circular if light is launched in a direction parallel to the z axis in a medium such as that shown in Fig. 5.22. The refractive index varies only in the x-direction and is expressed by 2

A single-mode fiber actually allows the propagation of two orthogonal polarization modes. If the fiber is isotropic, these two polarization modes propagate with the same velocity. However, in real fibers, there is usually some anisotropy, and the two polarization modes propagate at slightly different velocities. This effect should be taken into account when using polarization sensitive optical devices, or when designing long-haul, high-bandwidth fiber-optical communications links. There are two methods of controlling polarization mode dispersion. In one approach, a large controlled anisotropy is purposely introduced, by, for example, stressing the fiber as it is drawn, so that this controlled anisotropy is much larger in magnitude than any random anisotropy. If polarization of the incident light is aligned with the direction of the stress, the polarization is stabilized. In the other approach, the geometry of the fiber is modified in such a way that one of the two polarization modes is cutoff. An example of such a geometry is a fiber in which the refractive index of two diametrically opposite sections of the cladding is increased. It is important to realize that, in the first approach, both polarization modes will propagate if they are excited. This means that for polarization stabilized operation, the incident light must be launched correctly so that only one polarization mode is excited. In the second approach, an improper launching polarization increases the losses in the cutoff polarization mode, but the output state of polarization does not change.

Problems

137

Fig. 5.22 The light path in a medium with n = b/(a ± x) is an arc of a circle

Fig. 5.23 Direction of the optical path expressed in direction cosines

b a±x where a > 0, b > 0 and x < a. The launching point is (x0 , 0, 0). n± =

(5.156)

5.3. In Exercise 5.2, an expression was derived for the optical √ path in a spherically symmetric medium having the refractive index n = 1/ r . Calculate the optical path for the refractive index n = 1/r . 5.4. Using the relationship between the direction cosines shown in Fig. 5.23, cos2 α + cos2 δ + cos γ 2 = 1 derive (5.106) directly from (5.107, 108). 5.5. A skew ray is launched in the Selfoc fiber with n 2 (r ) = n 2c (1 − α2r 2 ). (a) What is the initial condition that the cross section of the helical path becomes circular? Express the condition in terms of incident angles γ0 and δ0 , the index of refraction n 0 at the point of incidence and distribution coefficient α. (b) What is the radius of such a helical path?

138

5 Geometrical Optics

5.6. Find the length of the major axis A and that of the minor axis B of a skew ray in the Selfoc fiber by a method other than mentioned in the text. (Hint: For r = rmax and r = rmin , dr/ds = 0). 5.7. What is the relationship between φ and z of a skew ray? 5.8. Prove that (5.118) for a meridional ray is a special case of (5.145) for a skew ray. 5.9. Derive the group velocity of a skew ray inside a Selfoc fiber by calculating the time it takes light to travel along the helical path using (5.33).

Chapter 6

Lenses

Lenses are among the most used components in optical systems. The functions that lenses perform include the convergence or divergence of light beams, and the formation of virtual or real images. An additional interesting property of lenses is that they can be used for optically performing the two-dimensional Fourier transform. In this chapter, a special emphasis is placed on the detailed explanation of the Fourier transformable properties of a lens. A special feature of the optical Fourier transform is that the location of the Fourier transformed image is not influenced by that of the input image. This feature is widely used in optical signal processing. The effect of the finite size of a lens on the quality of the output image is also investigated.

6.1 Design of Plano-Convex Lens The contour of a plano-convex lens is made in such a way that light from a point source becomes a parallel beam after passing through the lens. In other words, it is designed so that the optical paths F–P–Q and F–L–R in Fig. 6.1 are identical. Let H be the point of projection from P to the optical axis. Since PQ = HR, the optical paths will be identical if FP equals FH. The optical paths are expressed as follows Optical path of FP = Optical path of FH = f + n ( cos θ − f ) where n is the index of refraction of the glass, and f is the focal length FL. When the optical path of FP is set equal to that of FH, the formula for the contour of the plano-convex lens is found to be =

(n − 1) f . n cos θ − 1

(6.1)

139

140

6 Lenses

Fig. 6.1 Design of the curvature of a plano convex lens

Fig. 6.2 Illustration of the cause of the intensity non-uniformity existing in a parallel beam emerging from a plano convex lens

Next, (6.1) is converted into rectangular coordinates. For now, consider only the y = 0 plane. Taking the origin as the focus of the lens, the coordinates of the point P are ⎫ √ ⎪ = x 2 + z2 ⎬ (6.2) z cos θ = √ .⎪ ⎭ x 2 + z2 Equation (6.2) is inserted into (6.1), and a little algebraic manipulation gives the result x2 (z − c)2 − = 1 where a2 b2 + n−1 f , b= f, a= n+1 n+1

c=

n f. n+1

(6.3)

It can be seen from (6.3) that the contour of the lens is a hyperbola. Even though light from a point source located at the focus becomes a parallel beam after passing through the lens, the intensity distribution across this beam is not uniform. In Fig. 6.2 the solid angle ∠AFB is made equal to solid angle ∠A FB .

6.2 Consideration of a Lens from the Viewpoint of Wave Optics

141

The light beam inside the latter angle is spread more and is therefore weaker than that of the former. This inhomogeneity becomes a problem for large lens diameters.

6.2 Consideration of a Lens from the Viewpoint of Wave Optics In the previous section, the lens was considered from the viewpoint of geometrical optics while in this section the viewpoint of wave optics is adopted. The phase distribution of the transmitted light beam through a lens is examined. As shown in Fig. 6.3, the incident beam is incident parallel to the lens axis z. When the lens is thin, (6.3) can be simplified. The larger b in (6.3) is, the smaller the variation of z with respect to x becomes, which leads to the thin lens condition x 2 1. (6.4) b When (6.4) is satisfied, an approximate formula for (6.3) is obtained by using the binomial expansion a z∼ (6.5) = a + c + 2 x 2. 2b If the plane surface of the lens is at z = a0 , the thickness tn of the glass at x is a (6.6) tn = a 0 − a + c + 2 x 2 , 2b and the phase of the plane wave passed through the lens at x is φ(x) = k[(a0 − a − c) − tn ] + nktn . Inserting (6.6) into (6.7) gives a(n − 1) 2 x . φ(x) = k n(a0 − a − c) − 2b2

Fig. 6.3 Phase distribution of a beam passed through a plano-convex lens

(6.7)

142

6 Lenses

Again, inserting the values for a, b, c from (6.3), the phase becomes x2 where 2f φ0 = kn (a0 − f ).

φ(x) = φ0 − k

(6.8)

Since an analogous relationship holds in the y direction, the phase can be written as x 2 + y2 r2 , (6.9) = φ0 − k φ(x, y) = φ0 − k 2f 2f where r is the distance from the lens axis to the point (x, y). In conclusion, the lens creates a phase distribution whereby the phase advances with the square of the radius from the optical axis, and the rate of phase advance is larger for smaller f .

6.3 Fourier Transform by a Lens Lenses play an important role in optical signal processing, and in particular, Fourier transformability is one of the most frequently used properties of lenses other than image formation. The results of the Fourier transform depend critically on the relative positions of the input object, the lens and the output screen. In the following subsections, every combination will be described [6.1–3].

6.3.1 Input on the Lens Surface As shown in Fig. 6.4, the input transparency is pressed against the lens. Let the input image be represented by the function g (x0 , y0 ). When a parallel beam of unit amplitude is incident along the optical axis of the lens, the distribution of light just behind the lens is x02 + y02 + jφ0 g (x0 , yo ) exp −jk 2f

Fig. 6.4 A pattern projected onto a screen through a convex lens when the input transparency g (x0 , y0 ) is pressed against the lens and the screen is placed at z = z i

6.3 Fourier Transform by a Lens

143

where the diameter of the lens is assumed infinite. The φ0 is insignificant for most of the analysis below; it will be assumed as zero. After propagation to the screen at z = z i located in the Fresnel region, the pattern according to (3.36) is xi2 + yi2 1 exp jk z i + E(xi , yi , z i ) = jλz i 2z i " x02 + y02 x02 + y02 × F g (x0 , yo ) exp −jk +jk (6.10) 2f 2z i with f x = xi /λz i and f y = yi /λz i . It is unfortunate that the same symbol f is used for both the lens focal length and for the Fourier transform variable. As a guideline for distinguishing between the two, note that the Fourier transform variable always appears with a lettered subscript, whereas the focal length is written subscriptless or with a numbered subscript. For the special case in which the screen is placed at the focal plane z i = f , (6.10) becomes xi2 + yi2 1 xi yi exp jk f + , (6.11) G E(xi , yi , f ) = jλ f 2f λf λf where G( f x , f y ) = F {g (x0 , y0 )}. This is a very important result. Except for the factor xi2 + yi2 , exp jk f + 2f the pattern on the screen placed at the focal plane is the Fourier transform G(xi /λz i , yi /λz i ) of the input function g (x0 , y0 ). This Fourier transform property of a lens is used often in optical signal processing [6.3, 4]. The phase factor xi2 + yi2 exp jk 2f can be eliminated by moving the input function to the front focal plane. This will be dealt in the next subsection.

6.3.2 Input at the Front Focal Plane As shown in Fig. 6.5, the input function is now located a distance d1 in front of the lens, and the screen is at the back focal plane. It will be easier to separate the propagation distance into two: propagation from the object to the lens and propagation from the lens to the screen.

144

6 Lenses

Fig. 6.5 Fourier transform by a convex lens; the case when the input transparency g (x0 , y0 ) is located a distance d1 in front of the lens and the screen is at the back focal plane

The field distribution g (x, y) of the input g(x0 , y0 ) which has propagated to the front surface of the lens is, from (3.34), g (x, y) = g(x, y) ∗ f d 1 where fd 1

(6.12)

1 x 2 + y2 = exp jk d1 + . jλd1 2d1

Again, it is unfortunate that the letter f appears in the symbol f d1 which represents the point-source transfer function. The symbol f for both the point-source transfer function and the Fourier transform variable appears with a lettered subscript, however, the former can be distinguished by the fact that its subscript refers to distance along the z direction. Note also that F { f d1 } = Fd1 ( f x , f y ) = exp [jkd1 − jπλd1 ( f x2 + f y2 )].

(6.13)

The pattern at the back focal plane can be obtained by replacing g(x0 , y0 ) in (6.11) with g (x, y). xi2 + yi2 1 xi yi exp jk f + G , (6.14) E(xi , yi , f ) = jλ f 2f λf λf where

G ( f x , f y ) = F {g (x, y)} = G( f x , f y ) · Fd1 ( f x , f y ).

6.3 Fourier Transform by a Lens

145

Fig. 6.6 Fourier transform by a convex lens; the case when the input transparency g (x0 , y0 ) is placed in the front focal plane. When the input transparency is placed at the front focal plane the exact Fourier transform G(xi /λ f, yi /λ f ) is observed at the back focal plane

Inserting (6.13) into (6.14) gives xi2 + yi2 1 xi yi E(xi , yi , f ) = exp jk d1 + f + G , jλ f 2f λf λf 2 " yi 1 j k( f +d1 ) xi 2 × exp −jπ λd1 + e = λf λf jλ f xi2 + yi2 xi yi d1 × exp jk G , . (6.15) 1− 2f f λf λf When the input is placed at the front focal plane d1 = f , the second exponential factor becomes unity and ej2k f xi yi G , . (6.16) E(xi , yi , f ) = jλ f λf λf Thus, the field distribution G(xi /λ f, yi /λ f ) on the screen becomes exactly the Fourier transform of the input function g(x0 , y0 ) if the input function is moved to the front focal plane and the screen is located at the back focal plane. The difference between this case and the previous case is that (6.16) does not contain the factor exp [ jk(xi2 + y 2j )/2 f ]. Figure 6.6 summarizes the results of this subsection.

6.3.3 Input Behind the Lens As shown in Fig. 6.7, the input is now located behind the lens. The positions of the lens and the screen are fixed, and only the input image is moved. Let the input image be placed at d1 behind the lens. Again, the region of the propagation is separated into two: the region between the lens and the input image and that between the input image and the screen.

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6 Lenses

Fig. 6.7 Fourier transform by a convex lens; in the case when the input transparency g (x0 , y0 ) is placed behind the lens

By using (3.34), the light distribution g (x0 , y0 ) illuminating the input image is given by 2 + y2 x 0 (6.17) ∗ f d1 (x0 , y0 ) g (x0 , y0 ) = exp −jk 0 2f where

1 x 2 + y2 f d1 (x, y) = exp jk d1 + jλd1 2d1 F { f d1 (x, y)} = Fd1 ( f x , f y ) = exp [jkd1 − jπλd1 ( f x2 + f y2 )] " x 2 + y2 F exp −jk = −jλ f exp [jπλ f ( f x2 + f y2 )]. 2 f

The expression for g (x0 , y0 ) can be simplified by the successive operations of Fourier transform and inverse Fourier transform, g (x0 , y0 ) = F −1 F {g (x0 , y0 )} = −jλ f e jkd1 F −1 {exp [−jπ λ(d1 − f ) ( f x2 + f y2 )]} therefore

2 + y2 x f 0 exp jk d1 + 0 . g (x0 , y0 ) = f − d1 2(d1 − f )

The field distribution immediately behind the lens is g (x0 , y0 ) g(x0 , y0 ). The image E(xi , yi , f ) that will be produced after this image propagates the distance d2 can be obtained from (3.36)

6.3 Fourier Transform by a Lens

147

E(xi , yi , f ) =

x 2 + yi2 1 f exp jk f + i jλd2 2d2 f − d1 ' 1 × F g (x0 , y0 ) exp jk 2 (d1 − f ) ( 1 (x02 + y02 ) f x = x /λd , . + i 2 2d2

(6.18)

f y = yi /λd2

From Fig. 6.7 d2 = f − d1 , and (6.18) becomes xi2 + yi2 1 yi xi f exp jk f + , G . (6.19) E(xi , yi , f ) = jλ d2 2d2 d2 λd2 λd2 As seen from (6.19), the size of the output image increases with an increase in λd2 and becomes the largest when d2 = f . In short, the size of the image is very easily controlled by moving the input image. With the arrangements discussed in previous subsections, unless one uses lenses of different focal lengths, the size of the output image could not be changed. The ease of adjustment obtained with the present arrangement is a handy advantage. It should be noted that this case contains a phase factor similar to those found in (6.11, 15) and that the intensity of the image varies as ( f /d22 )2 .

6.3.4 Fourier Transform by a Group of Lenses So far, the Fourier transform was performed using a single lens. In this subsection it will be demonstrated that the Fourier transform of the image can be obtained as long as the input image is illuminated by a converging beam, and provided that the screen is placed at the converging point of the beam. Assume that the light beam converges to the point P after passing through a series of lenses, as shown in Fig. 6.8. Since the wave front of a converging beam is spherical, the wave incident on the input image plane can be expressed by x 02 + y02 . E(x0 , y0 ) = A exp −jk 2d2 The negative sign in the exponent denotes a converging beam. This spherical wave is transmitted through the input image and further propagates over the distance d2 to reach the screen. The light distribution on the screen is

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6 Lenses

Fig. 6.8 Fourier transform performed by a converging beam of light. The Fourier transform is observed in a plane containing the point P of convergence. It does not matter how the convergent beam is generated

' xi2 + yi2 E(xi , yi , d2 ) = A exp jk d2 + F g (x0 , y0 ) 2d2 " x02 + y02 x02 + y02 × exp −jk , exp j k 2d2 2d2 f x = xi /λd2 = A exp jk

d2 +

xi2 + yi2 2d2

G

f v = yi /λd2

yi xi , λd2 λd2

,

(6.20)

which is proportional to the Fourier transform of the input image. This is a rather roundabout explanation, but if one considers the group of lenses as one compound convex lens, the analysis would be identical to that of the previous subsection.

6.3.5 Effect of Lateral Translation of the Input Image on the Fourier-Transform Image Figure 6.9a shows the light path when the input object is placed in the center. The parallel incident rays converge to the center of the back focal plane. The beam scattered by the input object has a similar tendency to converge to the center, but it has some spread in the field distribution. This field spread is nothing but the Fourier transform of the input object. Figure 6.9b shows the case when the input object is lowered from the center. The scattered beam starts from a lower location in the input plane, but by the converging power of the lens, the scattered field converges again to the center of the back focal plane. The property that the Fourier transform always appears near the optical axis regardless of the location of the input object, is, as mentioned earlier, one of the most attractive features of Fourier transforming by a lens. This is especially true when the location of the input object is unpredictable. It should, however, be realized that there is a tilt in the Fourier transform image by θ = sin−1 (x1 / f ) when the input image is shifted by x1 (Problem 6.2).

6.4 Image Forming Capability of a Lens from the Viewpoint of Wave Optics

149

Fig. 6.9 a, b. Effect of the lateral translation on the Fourier transform. (a) The object is in the center; (b) the object is laterally translated. Note that Fourier transform always stays on the optical axis

6.4 Image Forming Capability of a Lens from the Viewpoint of Wave Optics It is well known that the condition for forming a real image is 1 1 1 + = , d1 d2 f

(6.21)

where d1 is the distance from the input to the lens, and d2 is the distance from the lens to the image (Fig. 6.10). Using a slightly different approach, this formula will

150

6 Lenses

Fig. 6.10 The image forming capability of a lens is examined from the viewpoint of wave optics

be derived from the viewpoint of wave optics. The propagation distance is separated into two regions as before: the region from the input to the back focal plane of the lens and that from the back focal plane to the screen. As for the propagation from the input to the back focal plane, it was treated already in Sect. 6.3.2 and the result was (6.15). It is rewritten here for convenience, i.e., x 2 + y2 x y d1 1 jk( f +d1 ) exp jk e G , 1− E(x, y, f ) = jλ f 2f f λf λf where the plane at z = f is taken as the x y plane. The above distribution further propagates over the distance d2 − f . From (3.36), the field distribution is xi2 + yi2 1 E(xi , yi · d2 ) = exp jk d1 + d2 + jλ(d2 − f ) 2 (d2 − f ) 1 × F jλ f ×G

x 2 + y2 exp jk 2 f

x y , λf λf

d1 1− f

x 2 + y2 exp jk 2 (d2 − f )

" f x = xi /λ(d2 − f ), f v = yi /λ(d2 − f )

. (6.22)

The exponential term inside the curly brackets of the Fourier transform is treated separately by denoting it by φ d1 1 x 2 + y2 1 1− + φ = jk 2 f f d2 − f x 2 + y2 1 1 d1 d2 1 = jk + − . (6.23) 2 f (d2 − f ) d1 d2 f

6.4 Image Forming Capability of a Lens from the Viewpoint of Wave Optics

151

The value of (6.23) becomes zero when the imaging condition (6.21) is satisfied, and (6.22) becomes xi2 + yi2 −f exp jk d1 + d2 + E(xi , yi · d2 ) = d2 − f 2 (d2 − f ) ×g

f xi f yi − , − d2 − f d2 − f

.

(6.24)

The relationship f d1 = d2 − f d2 obtained from (6.21) is inserted in (6.24) giving d1 d1 xi2 + yi2 E(xi , yi , d2 ) = − exp jk d1 + d2 + d2 d2 2 f ×g

−

xi yi , − d2 /d1 d2 /d1

.

(6.25)

Equation (6.25) means that an image which is d2 /d1 times the original image is formed at the location set by (6.21). The negative signs in xi , yi indicate that the image is inverted. It is noteworthy that both the Fourier transform and the image are simultaneously obtained with only one lens. It would appear that only one lens is needed to perform the same function as the two lens arrangement shown in Fig. 6.11. However, the difference is that, in the case of the single lens, the phase factor is present in both the Fourier transform and the image as seen in (6.11, 15, 19, 25). For a single lens, the field distribution gives a correct representation of the intensity, but not of the phase. However, if two lenses are arranged in the manner shown in Fig. 6.11, both the correct amplitude and the correct phase are retained.

Fig. 6.11 Formation of the image using two successive Fourier transforms. The output image obtained in this manner is identical with the input image not only in magnitude but also in phase

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6 Lenses

6.5 Effects of the Finite Size of the Lens So far, the finite physical extent of the lens has been ignored and all the light scattered from the object was assumed to be intercepted by the lens. In this section, we discuss how the finiteness of the lens affects the quality of the image. Figure 6.12 illustrates the spatial distribution of the signal scattered from the object relative to the lens. The lens is placed in the far-field region of the scattered field and the onedimensional case is considered for simplicity. The input is a transparency g(x) illuminated by a parallel beam from behind. The image of the input is to be formed by a lens with diameter D. The diffracted wave from the input transparency radiates in all directions, but the portion which contributes to the formation of the image is limited by the diameter of the lens. The lens intercepts only the light which falls inside the angle subtended by the lens at the input plane. The input g(x) can be expanded into spatial frequency components by the Fourier transform. At first, only the component with the spatial frequency f 0 is considered. By using (3.31), the diffraction pattern of this component is given by sin θ − f0 , (6.26) U (θ ) = K 0 F {e j2π f 0 x } f = sin θ/λ = K 0 δ λ where K 0 is the phase factor in (3.31). The direction of the diffracted wave is θ = sin−1 ( f 0 λ). The higher the spatial frequency f 0 is, the larger the diffraction angle θ is. In the case shown in Fig. 6.12, the highest spatial frequency f c which can be intercepted by the lens is D fc , (6.27) 2λd1 where d1 is the distance between the input image and the lens. Therefore, any spatial frequency component higher than f c does not participate in forming the output

Fig. 6.12 The output image through a lens of a finite size. Some of the higher spatial frequency components do not intercept the lens resulting in a loss of the higher frequency component in the output image

6.5 Effects of the Finite Size of the Lens

153

image. The higher frequency components which are needed to fill in the fine details of the output image are missing, and hence the output image is degraded. The degree of degradation with respect to the diameter of the lens is considered in the next section.

6.5.1 Influence of the Finite Size of the Lens on the Quality of the Fourier Transform The relationship between the size of the lens aperture and the quality of the Fourier transformed image is studied [6.5]. As shown in Fig. 6.13, the input transparency is pressed against the lens and the Fourier transformed image is obtained at the back focal plane. The aperture of the lens can be expressed by the pupil function P(x, y) defined in Sect. 4.6. The focal length of the lens is f . The result of Sect. 6.3.1 may be used, the only difference being that g(x0 , y0 ) P(x0 , y0 ) is used instead of g(x0 , y0 ). From (6.11), the field distribution on the screen is xi2 + yi2 1 xi yi xi yi exp jk f + G , , ∗ P¯ . E(xi , yi , f ) = jλ f 2f λf λf λf λf (6.28) where the bar denotes Fourier transform, and F {P (x, y)} = P¯ ( f x , f y ). By comparing (6.28) with (6.11), the difference between the finite and infinite lens sizes is the effect of the convolution operation of the Fourier transform with P¯ (xi /λ f, yi /λ f ). If the image is convolved with P¯ (xi /λ f, yi /λ f ), the image is blurred. A quantitative “degree of blurring” will be calculated for the case of a onedimensional aperture. The pupil function for a lens of lateral dimension D and its Fourier transform are

Fig. 6.13 The Fourier transform by a lens of a finite size

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6 Lenses

Fig. 6.14 The blurring due to the finiteness of the diameter of the lens

x P(x) = , D xi xi P¯ = D sinc D . λf λf

(6.29) (6.30)

Figure 6.14 illustrates the convolution of G(xi /λ f ) with D sinc (D xi /λ f ). The “degree of blurring” is determined by the width f λ/D of the main lobe of sinc (D xi /λ f ). The smallest size ∆xi that is meaningful in the Fourier transform image formed by this lens is approximately the width of the main lobe, i.e., ∆xi = 2

f λ. D

(6.31)

For instance, for D = 10 mm, λ = 0.6 × 10−3 mm and f = 50 mm, we get ∆xi = 6 × 10−3 mm. When the aperture is circular in shape with diameter D, the smallest resolvable dimension ∆l is, from the results in Sect. 4.5, ∆l = 2.44

f λ. D

(6.32)

6.5.2 Influence of the Finite Size of the Lens on the Image Quality When a lens is used for the purposes of image formation, it is useful to know the relationship between the lens size and the resolution of the output image. As shown in Fig. 6.15, a real image is formed on a screen placed at a distance d2 behind a lens, whose focal length is f , and pupil function is P(x, y). The input image g(x0 , y0 ) is placed at a distance d1 in front of the lens. The propagation path is divided into two: from input to the aperture of the lens, and from the aperture of the lens to the screen. From (3.36), the distribution gl (x, y) at the aperture is gl (x, y) = " x 02 + y02 x 2 + y2 1 x2 + y 2 exp −j k exp j k d1 + F g (x0 , y0 ) exp jk P(x, y). jλd1 2d1 2d1 2f just in front of the lens just behind the lens (6.33)

6.5 Effects of the Finite Size of the Lens

155

Fig. 6.15 Influence of the finiteness of the lens size on the resolution of output image

Note that the equation given in Fig. 6.15 is one-dimensional because of limitations on printing space available. Rewriting (6.33) yields ' ( 1 1 1 1 2 2 exp jk d1 + − (x + y ) gl (x, y) = jλd1 2 d1 f y x × Gd 1 , P(x, y) where (6.34) λd1 λd1 x 2 + y2 gd 1 (x, y) = g(x, y) exp j k , (6.35) 2d1 F {gd 1 (x, y)} = G d 1 ( f x , f y ). By using (3.36) the field distribution resulting from the propagation from the aperture to the screen is given by xi2 + yi2 1 exp jk d2 + E(xi , yi ) = jλd2 2d2

x 2 + y2 × F gl (x, y) exp jk 2d2 Equations (6.34, 35) are inserted into (6.36) to yield

" f x = xi /λd2 , f v = yi /λd2

(6.36)

156

6 Lenses

xi2 + yi2 2d2 ' k 1 1 1 × F exp j − + (x 2 + y 2 ) 2 d1 f d2 ( y x ×G d 1 , P(x, y) f x = xi /λ d2 , λd1 λd1

E(xi , yi ) = −

1

λ2 d1 d2

exp jk

d1 + d2 +

(6.37)

f v = yi /λ d2

If the imaging condition is satisfied, (6.37) becomes xi2 + yi2 1 E(xi , yi ) = − exp jk d1 + d2 + M 2d2 " x (xi2 + yi2 yi yi xi i ¯ × g − , − , exp jk ∗P M M 2Md2 λd2 λd2 (6.38) where M = d2 /d1 is the magnification. The operation of the convolution is performed, which yields " xi2 + yi2 1 1 exp jk d1 + d2 + 1+ E(xi , yi ) = − M 2d2 M ∞ ' xi − ξ yi − η × , g M M −∞ 2π π (ξ 2 + η2 ) × exp −j (xi ξ + yi η) + j λd2 M λ d2 M ( ξ η × P¯ , dξ dη. (6.39) λd2 λd2 ¯ i /λd2 , The computation of (6.39) is messy but can be simplified if the value of P(x yi /λd2 ) is sharply peaked at certain values of xi and yi . This method of simplification is demonstrated for a one-dimensional aperture using typical numbers. Let the aperture width be D, so that x and (6.40) P(x, y) = D xi ¯ . (6.41) P(x, y) = D sinc D λd2 A typical set of numbers such as D = 60 mm, d1 = d2 = 100 mm, −3

λ = 0.6 × 10

mm

and

(6.42)

6.5 Effects of the Finite Size of the Lens

157

is inserted into (6.39) (xi2 + yi2 ) 1 E(xi , yi ) = − exp jk d1 + d2 + M 2M f ∞ × (6.43) exp [−j 100 (xi ξ + yi η) + j 50 (ξ 2 + η2 )] −∞ ( d1 ξ d1 × g − (xi − ξ ), − (yi − η) sinc dξ. d2 d2 10−3 The value of sinc (ξ/10−3 ) becomes negligible as soon as |ξ | exceeds 10−3 , so that the error involved in changing the limits of the integration to −10−3 , 10−3 from −∞, ∞, would be very small. In the region |ξ | < 10−3 , the value of exp [−j 100 (xi ξ + yi η) + j 50 (ξ 2 + η2 )] can be considered to be unity as long as |xi | + |yi | < 10. Under these conditions, (6.43) can be approximated by (xi2 + yi2 ) 1 E(xi , yi ) = − exp jk d1 + d2 + M 2M f xi yi xi × g − , − ∗ sinc D . (6.44) M M λd2 Making a general inference from this typical example, (6.38) can be approximated as (xi2 + yi2 ) 1 exp jk d1 + d2 + E(xi , yi ) = − M 2M f xi yi yi ¯ xi × g − , − , ∗P . (6.45) M M λd2 λd2 ¯ i /λd2 , yi /λd2 ) becomes It can be seen that if the diameter of the lens is infinite, P(x a δ function, and a perfect image can be obtained provided the lens has no aberration. However, if the size of the lens is finite, a perfect image can never be obtained even if the lens is aberration free. The resolution is limited by the convolution with ¯ i /λd2 , yi /λd2 ) which is associated with the diffraction from the aperture. A P(x system is called diffraction limited if the system is perfect and the diffraction effect is the sole cause of limiting the resolution. In astronomy, the number of the stars one can observe is often limited by the incident light energy. In this case, the system is power limited. The university laboratory is often money limited. Next, by again using the one-dimensional aperture as an example, the phrase “degree of blur” will be further qualified. When the system has an aperture width D, the output image is given by (6.44). The convolution of the image and sinc[D(xi /λd2 )] can be obtained in the manner illustrated in Fig. 6.14. If one assumes that the smallest dimension ∆xi in the output image that can be resolved is more or less the width of the main lobe, ∆xi is

158

6 Lenses

d2 λ. (6.46) D The value given by (6.46) is the resolution in the output image. This resolution can be converted into that of the input image by multiplying by 1/M = d1 /d2 ∆xi = 2

∆x0 = 2

d1 λ. D

(6.47)

The relationship between the highest spatial frequency f c and the limit of the resolution is 1 . fc ∼ = ∆x0 It is seen that this result agrees with that of (6.27).

Problems 6.1. The front surface υ of a meniscus lens is spherical, as shown in Fig. 6.16. The center of the sphere is at the focus of the lens. Prove that the expression for the back surface υ of the lens is r=

f (n − 1) n − cos θ

(6.48)

where f is the distance between the focus and the front vertex, n the index of refraction of the glass, and θ the angle between the lens axis and a line connecting the focus and a point S2 on the back surface υ of the lens. 6.2. Referring to Fig. 6.9b, find the angle of the tilt of the Fourier transform image due to the translational shift x1 of the input object. 6.3. One wants to take a highly reduced picture of a diagram as shown in Fig. 6.17 by using a lens whose aperture stop is F = 1.2 and the focal length f = 50 mm. (F = f /D, D being the diameter and f the focal length of the lens.) The highest spatial frequency of the diagram is 1 line/mm. Derive the maximum obtainable reduction ratio (size of the picture/size of the object). The wavelength of the illuminating light is λ = 0.555 µm, and the camera is assumed to be diffraction limited.

Fig. 6.16 The design of the surface of a meniscus lens

Problems

159

Fig. 6.17 Maximum obtainable reduction ratio of a camera with a lens of a finite size

Fig. 6.18 Elimination of the phase factor associated with the Fourier transform using a convex lens

Fig. 6.19 Optical computing using two convex lenses

Fig. 6.20 Combination of cylindrical and spherical lenses

6.4. As shown in Fig. 6.18, by using a convex lens with focal length f , the Fourier transform of the input g(x0 , y0 ) is to be obtained on the screen placed at the back focal plane of the lens. In order to eliminate the phase factor associated with the Fourier transform, as seen in (6.19), convex lens is placed on the surface of the screen. What should the focal length of the convex lens be?

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6 Lenses

Fig. 6.21 Electrical equivalent circuit of an optical lens system

6.5. As shown in Fig. 6.19, the input images t1 (x0 , y0 ) and t2 (x1 , y1 ) were placed on the surface of the lens L1 and L2 . The focal length of the lens L1 is f 1 = 2a and that of L2 is f 2 = a. The spacings between L1 and L2 and the screen are all 2a. Derive an expression for the light distribution on the screen. What are the applications of such an arrangement? 6.6. The cylindrical lens has the property of Fourier transforming only in one dimension. Describe the nature of the output image when a convex lens and a cylindrical lens are arranged in the manner shown in Fig. 6.20. What are the possible applications? 6.7. Optics and communication theory have a lot in common. Figure 6.21 is meant to be an equivalent circuit for representing the image-forming system shown in Fig. 6.15. Insert the relevant mathematical expressions in the blanks in Fig. 6.211 .

1

In optics, it is quite easy to reduce the size of the image because a lens is available for this purpose. The pulse width of a radar echo can be reduced to improve the range resolution of the radar in a similar manner by electronically realizing a circuit, such as is shown in Fig. 6.21.

Chapter 7

The Fast Fourier Transform (FFT)

The Fast Fourier Transform (FFT) is one of the most frequently used mathematical tools for digital signal processing. Techniques that use a combination of digital and analogue approaches have been increasing in numbers. This chapter is for establishing the basis of this combined approach in dealing with computer tomography, computer holography and hologram matrix radar.

7.1 What is the Fast Fourier Transform? One can immediately see from a table of Fourier transforms that only a limited number of functions can be transformed into closed analytic forms. When a transform is not available in an analytic form, it must be estimated by numerical computation. The numerical approximation to the Fourier transform integral, which can be obtained by summing over small sections of the integrand, takes time even when a high speed computer is used. In 1965, Cooley and Tukey published a new algorithm which substantially reduced the computation time [7.1]. For instance, an 8,192 point Discrete Fourier Transform (DFT), which takes about 30 minutes of computer time when conventional integration programming is used, can be computed in less than 5 seconds with the algorithm. Although first known as the Cooley-Tukey algorithm, it has since become practically synonymous with FFT [7.2, 4]. Figure 7.1 shows the ratio of the computation time between normal programming and FFT algorithm programming. Besides the economic advantages of FFT, there are certain applications where high-speed processing is essential. Real-time radar-echo processing is one such example. The Fourier transform pair is defined as ∞ g(x) e−j2π f x d x (7.1) G( f ) = −∞ ∞ g(x) = G( f ) e j2π f x d f. (7.2) −∞

161

162

7 The Fast Fourier Transform (FFT)

Fig. 7.1 Substantial reduction in number of computations is demonstrated; One curve with FTT the other with ordinary computation

Similarly, the Discrete Fourier Transform (DFT) is defined as Gl =

N −1

gk e−j2πkl/N

(7.3)

k=0 N −1 1 G l e j2π kl/N . gk = N

(7.4)

l=0

Notice that both (7.3) representing the transform and (7.4) representing the inverse transform are quite similar mathematically. The only difference is whether the exponent contains a positive or negative sign, and whether or not the normalizing factor N is present. Because the two expressions are basically equivalent, the discussion will proceed solely with (7.3). This equation is rewritten in a simpler form as Gl =

N −1

gk W kl

k=0 −j2π/N

W =e

.

where l = 0, 1, 2, 3, . . . , N − 1

(7.5) (7.6)

In the analysis to follow, it will be convenient to use matrix notation. The expression (7.5) for the DFT, is written in matrix form below. ⎞ ⎛ ⎞ ⎛ 0 ⎛ ⎞ W0 W0 W0 . . . W0 W g0 G0 1 2 3 ⎟ ⎜ 0 ⎜ W N −1 ⎟ ⎜G 1 ⎟ ⎜ W 0 W 2 W 4 W 6 . . . ⎟ ⎜ g1 ⎟ 2(N −1) ⎜ ⎟ ⎜W ⎟ ⎟ ⎜ W W W ... W ⎜G 2 ⎟ ⎜ ⎟ ⎜ g2 ⎟ ⎜ ⎟ = ⎜W 0 W 3 W 6 W 9 ⎟ ⎜ g3 ⎟ . (7.7) ⎜G 3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ · ⎟ ⎜ · ⎟ · · · ⎟ ⎜ ⎟ ⎝ · ⎠ ⎜ ⎠ ⎝ · ⎠ ⎝ · · · · · g N −1 W (N −1)(N −1) W 0 W N −1

7.1 What is the Fast Fourier Transform?

163

Thus the DFT can be thought of as a kind of linear transform. A normal Fourier transform can be considered as the special case of N being infinity. The number of G’s which can be uniquely determined is identical to the number of g’s, namely, if there are only N sampled points then there are only N spectrum points. All the elements of the N × N matrix in (7.7) are derived from (7.6) and have peculiar characteristics which can be used for simplifying the matrix. The FFT is, in fact, a systematic method of matrix simplification. The principle of FFT will be explained by taking an example of an 8 point DFT. Inserting N = 8 into (7.5, 7) gives G 0 = W 0 g0 + W 0 g1 + W 0 g2 + W 0 g3 + W 0 g4 + W 0 g5 + W 0 g6 + W 0 g7 G 1 = W 0 g0 + W 1 g1 + W 2 g2 + W 3 g3 + W 4 g4 + W 5 g5 + W 6 g6 + W 7 g7 G 2 = W 0 g0 + W 2 g1 + W 4 g2 + W 6 g3 + W 8 g4 + W 10 g5 + W 12 g6 + W 14 g7 G 3 = W 0 g0 + W 3 g1 + W 6 g2 + W 9 g3 + W 12 g4 + W 15 g5 + W 18 g6 + W 21 g7 G 4 = W 0 g0 + W 4 g1 + W 8 g2 + W 12 g3 + W 16 g4 + W 20 g5 + W 24 g6 + W 28 g7 G 5 = W 0 g0 + W 5 g1 + W 10 g2 + W 15 g3 + W 20 g4 + W 25 g5 + W 30 g6 + W 35 g7 G 6 = W 0 g0 + W 6 g1 + W 12 g2 + W 18 g3 + W 24 g4 + W 30 g5 + W 36 g6 + W 42 g7 G 7 = W 0 g0 + W 7 g1 + W 14 g2 + W 21 g3 + W 28 g4 + W 35 g5 + W 42 g6 + W 49 g7 (7.8) In its present form, (7.8) requires 64 multiplications and 56 additions. Since computer multiplication is essentially the repetition of additions, the key factor to reducing the computer time is to reduce the number of multiplications that must be performed. The value of W kl ranges from W 0 to W 49 but they all can be reduced to one of 8 values between W 0 and W 7 . This is shown in Fig. 7.2 by using a phasor diagram in the complex plane. The magnitude of W kl = exp(−j2π kl/N ) is unity, and for N = 8, the phase angle associated with W kl is an integral multiple of π/4 radians. In the phasor diagram, W kl is a vector whose end point lies on a unit circle. Starting

Fig. 7.2 Representation of W k on the complex plane

164

7 The Fast Fourier Transform (FFT)

at kl = 0 which corresponds to W 0 = 1, the vector is seen to rotate clockwise by π/4 radians each time kl is increased by one. In going from W 7 to W 8 the vector arrives once again at the starting point and the cycle is repeated so that there are only 8 uniquely determined points on the circle. Notice that the W kl ’s diametrically opposite each other on the circle differ only in their sign. Referring to Fig. 7.2, equivalent values of W kl are factored out of the terms in the DFT. This saves on the number of multiplications as can be seen by comparing G 2 and G 3 below with those from (7.8), i.e., G 2 = (g0 + g4 ) + W 2 (g1 + g5 ) + W 4 (g2 + g6 ) + W 6 (g3 + g7 ) G 3 = (g0 − g4 ) + W 3 (g1 + g5 ) + W 6 (g2 + g6 ) + W 6 (g3 + g7 ). The results can be summarized as follows ⎞ ⎛ 1 1 1 1 G0 ⎜G 2 ⎟ ⎜1 W 2 W 4 W 6 ⎜ ⎟ ⎜ ⎜G 4 ⎟ ⎜1 W 4 W 8 W 12 ⎜ ⎟ ⎜ ⎜G 6 ⎟ ⎜1 W 6 W 12 W 18 ⎜ ⎟=⎜ ⎜G 1 ⎟ ⎜0 0 0 0 ⎜ ⎟ ⎜ ⎜G 3 ⎟ ⎜0 0 0 0 ⎜ ⎟ ⎜ ⎝G 5 ⎠ ⎝0 0 0 0 G7 0 0 0 0 ⎛

0 0 0 0 1 1 1 1

⎞ 0 0 0 0 0 0 ⎟ ⎟ 0 0 0 ⎟ ⎟ 0 0 0 ⎟ ⎟ W1 W2 W3 ⎟ ⎟ W3 W6 W9 ⎟ ⎟ W 5 W 10 W 15 ⎠ W 7 W 14 W 21

⎞ g0 + g4 ⎜g1 + g5 ⎟ ⎟ ⎜ ⎜g2 + g6 ⎟ ⎟ ⎜ ⎜g3 + g7 ⎟ ⎟ ⎜ ⎜g0 − g4 ⎟ . ⎟ ⎜ ⎜g1 − g5 ⎟ ⎟ ⎜ ⎝g2 − g6 ⎠ g3 − g7 ⎛

(7.9)

In the above matrix, only two of the 4 × 4 submatrices are non-zero, and the number of multiplication operations in (7.9) is one half of (7.8). Taking the reduction process one step further, a similar method may be applied to reduce each of these non-zero 4 × 4 submatrices into two 2 × 2 non-zero matrices. Systematic reduction in the number of multiplication operations is thus made, which is the goal of FFT. The FFT can be divided into two major methods, and each of them will be explained in the following sections.

7.2 FFT by the Method of Decimation in Frequency Suppose that the sampled values are g0 , g1 , . . . gk , . . . , g N −1 and that the DFT of these values is sought. First, these sampled values are split into two groups; one group containing values g0 through g N /2−1 , and the other group containing values g N /2 through g N −1 . These two groups are designated as follows: gk = gk h k = gk+N /2

where

k = 0, 1, 2, 3, . . . N /2 − 1.

(7.10)

7.2 FFT by the Method of Decimation in Frequency

165

Equation (7.10) is inserted into (7.5), G1 = =

N /2−1

gk W

+

kl

N /2−1

k=0

k=0

N /2−1

N /2−1

gk W kl +

k=0

gk+N /2 W (N /2+k)l h k W kl W Nl/2

and

k=0

2π N l = e−jπl = (−1)l W Nl/2 = exp −j N 2 and therefore Gl =

N /2−1

[gk + (−1)l h k ] W kl .

(7.11)

k=0

The second term inside the bracket of (7.11) keeps changing its sign depending on whether l is even or odd. Thus, first of all, the G l ’s with even number l are pulled out and calculated, and then those with odd number l are calculated. This explains why this method is called method of decimation in frequency. The G l ’s with even l are given by G 2l =

N /2−1

(gk − h k )(W 2 )kl

(7.12)

k=0

where l ranges from 0 to N /2 − 1. Carefully comparing (7.12) with (7.5, 7.6), it can be seen that (7.12) is again in a form of DFT but the number of sampling points has been reduced to N /2. Next, the G l ’s with odd l are considered. From (7.11) one obtains N /2−1 (gk − h k )W k (W 2 )kl (7.13) G 2l+1 = k=0

where l ranges from 0 to N /2−1. Equation (7.13) is again in the form of a DFT with N /2 sampling points. So far, the N -point DFT has been split into two N /2-point DFT’s. If this process of reducing the sampling points is repeated, each N /2-point DFT is further reduced to two N /4-point DFT’s. If one repeats the procedure n times, the N point DFT is reduced to several N /2n point DFT’s. The number of repetitions of the procedure necessary to reduce to the 1 point DFT is N = 1. 2n

(7.14)

Therefore the number of necessary repetitions is n = log2 N .

(7.15)

The signal flow graph, which is a convenient guideline for programming the FFT algorithm, will be explained next. Again, take as an example the case N = 8.

166

7 The Fast Fourier Transform (FFT)

Fig. 7.3 Signal flow graph representing the calculation of (7.12). Black dot represents the summing of the two numbers transferred by the arrows. A number written on the side of a line means the written number is multiplied by the number brought by the line

Figure 7.3 is a signal flow graph representing the calculation of (7.12). An arrow denotes transfer of the number to a new location. A black dot denotes the summing of the two numbers transferred by the arrow. A number written on the side of a line means that the written number should be multiplied by the number brought by the line. A dotted line denotes transfer of the number after changing its sign. This symbolism will be better understood by looking at a specific example. The black dot to the right of g0 in the upper left part of Fig. 7.3 has two arrows pointing at it. This black dot signifies the operation of g0 + g4 . In Fig. 7.4, the black dot and W 3 to the right of g7 signify the operation of the product (g3 − g7 )W 3 . Figure 7.3 shows that G 0 , G 2 , G 4 and G 6 can be obtained by performing the 4 point DFT of (g0 + g4 ), (g1 + g5 ), (g2 + g6 ), and (g3 + g7 ). The method of performing this 4-point DFT is not specified, but is instead represented by a plain square box in Fig. 7.3. Figure 7.4 is a similar diagram except for the odd l of G l ’s. Suppose that the square boxes of Figs. 7.3, 4 are calculated without using special methods to reduce the number of multiplications. Each of the square boxes has four inputs. Therefore, the number of multiplications needed for the G l with even l is (4)2 = 16 and those with odd l is again (4)2 = 16. The total number of multiplications therefore, is 32. This is half the amount of the direct transform using (7.8) which is 82 = 64.

7.2 FFT by the Method of Decimation in Frequency

167

Fig. 7.4 Same as Fig. 7.3 but for (7.13). A dotted line means transfer the number after changing its sign

In order to further reduce the number of operations, each 4 point DFT is split into two 2-point DFT’s. Figure 7.5 illustrates this method. In order to simplify the expression, the g’s and G’s are labeled as follows q0 q1 q2 q3

= g0 + g4

Q0 = G0

= g1 + g5 = g2 + g6 = g3 + g7

Q1 = G2 Q2 = G4 Q3 = G6,

where, the 4 point DFT of the qk ’s is expressed by the Q l ’s. As before, the Q l ’s are separated into even l’s and odd l’s. Q 0 and Q 2 are the 2-point DFT of q0 +q2 and q1 +q3 , Q 1 and Q 3 are the 2 point DFT of (q0 − q2 ) W 0 and (q1 − q3 ) W 2 . As seen from (7.5, 6) with N = 2, the 2 point DFT is quite simple and is given by the sum and difference of the inputs. The sequence of operations for obtaining the 4-point DFT is pictured in Fig. 7.5. Thus, the operation in the square box in Fig. 7.4 can be completed in a similar manner and results in the diagram shown in Fig. 7.6. The combination of Figs. 7.5, 7.6 results in Fig. 7.7. If the bent lines in Fig. 7.7 are straightened and the square frames removed, the signal flow graph shown in Fig. 7.8 is obtained. After close observation, one would find that not only does the

168

Fig. 7.5 The 4-point DFT of the sample points q0 , q1 , q2 , q3 are further split into 2-point DFT

Fig. 7.6 Same as Fig. 7.5 but for r ’s

7 The Fast Fourier Transform (FFT)

7.2 FFT by the Method of Decimation in Frequency

Fig. 7.7 Signal flow graph of 8-point DFT by a method of decimation in frequency

Fig. 7.8 Signal flow graph obtained by removing the frames and straightening the lines of Fig. 7.7 (method of decimation in frequency)

169

170

7 The Fast Fourier Transform (FFT)

signal flow graph in Fig. 7.8 reduce the amount of computation, but it also has a special property that plays a key role in saving memory space. For instance, the A and that of g4 stored at the location value of g0 which is stored at the location B are added, the sum is stored at the location A and the difference between g0 and g4 is stored at the location B . The rule is that the computed results are always stored at the memories located on the same horizontal level as the location of the inputs. Every computation follows this rule. Figure 7.9 is a signal flow graph drawn to emphasize this point. The signal flow graph in Fig. 7.9 can be conveniently used at the time the computer circuits are A A wired. The points B B in Fig. 7.9 correspond to those in Fig. 7.8. A fine line (——) and a broken line (–·–·) both represent the inputs. The heavy line (——) represents the sum of the two inputs and a dashed line (- - -) represents the difference of the two inputs. O indicates an arithmetic unit which calculates the sum and the difference. The W k is always associated with the difference which is represented by the dashed line, and the product of W k and the difference is made. The reason that the inputs are designated by two different kinds of lines is that it is necessary to

Fig. 7.9 Signal flow graph redrawn from that of Fig. 7.8 with the emphasis on using it as an aid to wire the FFT computer. (Method of decimation in frequency)

7.2 FFT by the Method of Decimation in Frequency

171

indicate which of the two inputs is subtracted from the other. The input represented by the broken line is subtracted from that represented by the fine line. For example, the connection by the broken line in the upper left of Fig. 7.9 means g0 − g4 and not g4 − g0 . The input to the computer is first put into the input register and the two numbers, A and g4 stored at the memory location e.g., g0 stored at the memory location B, A and are used to make the sum and the difference. The result g0 + g4 is stored at the g0 − g4 is stored at B . After a similar operation is repeated three times, the values of all G l ’s are obtained. At first, it may seem as if four columns of memory storage locations are needed, but in fact only one column of memory can manage all the computations. This is a major advantage in building an FFT computer. This advantage will be explained by A and A is used taking the example of the operation at B . The value g0 stored at to calculate g0 + g4 and g0 − g4 ; after that, there is no more need to use g0 by itself. The same holds true for g4 stored at B . Hence g0 can be erased from the memory A and be replaced by g0 + g4 . Similarly g4 can be erased and replaced by location A and g0 − g4 . This means that there is no need for the memory locations at B. In fact, all the memory locations expect the first column can be eliminated, and the construction of the FFT computer is simplified. One may notice that the order of the G l ’s is mixed up in Fig. 7.9. The G l ’s as they appear in Fig. 7.9 are G 0 , G 4 , G 2 , G 6 , G 1 , G 5 , G 3 , G 7 . Expressing the subscripts as binary numbers while maintaining the same order as above gives 000, 100, 010, 110, 001, 101, 011, 111. Taking each of these binary numbers individually and writing them backwards, as for example, rewriting 110 as 011, gives 000, 001, 010, 011, 100, 101, 110, 111. The decimal numbers corresponding to these binary numbers are 0, 1, 2, 3, 4, 5, 6, 7 and are in proper order. In the computer, by issuing the control signal of “bit reversal”, the output can be read out in proper order. Next, the total number of operations is counted. In Fig. 7.9 there are 3 columns of operations. In one column there are 4 additions, 4 subtractions and 4 multiplications. This is a substantial reduction in the total number of operations. Recall the direct computation of (7.8) required 56 additions and 64 multiplications. In general, the number of repetitions of operations needed to reach the 1 point DFT was calculated to be (7.15), and in each repetition there are N additions and subtractions and N /2 multiplications. The total number of operations for the FFT is (3/2) N log2 N .

(7.16)

Either from (7.16) or its graph in Fig. 7.1, it is seen that a remarkable reduction in computation is achieved for a large number N of sampling points.

172

7 The Fast Fourier Transform (FFT)

7.3 FFT by the Method of Decimation in Time In this section, the other method of FFT is used. Let the N sampled values be g0 , g1 , g2 , g3 , . . . , g N −1 and their DFT spectrum be G 0 , G 1 , G 2 , G 3 , . . . , G N −1 . The first step is to separate the input gk ’s into those with even k and those with odd k, and designate them as f k = g2k

h k = g2k+1

where

k = 0, 1, 2, 3 . . .

N −1 . 2

(7.17) (7.18)

The DFT is separated into two parts as follows Gl =

N /2−1

N /2−1

g2k W 2kl +

k=0

g2k+1 W (2k+1)l .

(7.19)

k=0

Inserting (7.17, 18) into (7.19) yields Gl =

N /2−1

f k (W 2 )kl + W l

N /2−1

k=0

h k (W 2 )kl .

(7.20)

k=0

The l’s are also split into two groups: one is from 0 to N /2 − 1 and the other from N /2 to N − 1, i.e., Gl =

N /2−1

f k (W 2 )kl + W l

N /2−1

k=0

G N /2+l =

< for 0 < = l = N /2 − 1

h k (W 2 )kl

(7.21)

k=0 N /2−1

2 k(N /2+l)

f k (W )

+W

(N /2+l)

k=0

N /2−1

h k (W 2 )k(N /2+l)

k=0

< for 0 < = l = N /2 − 1.

(7.22)

From the definition of W in (7.6), one obtains W N /k = 1,

W N /2 = −1.

Hence G N /2+l can be further rewritten as G N /2+l =

N /2−1 k=0

f k (W ) − W 2 kl

l

N /2−1 k=0

h k (W 2 )kl .

(7.23)

7.3 FFT by the Method of Decimation in Time

173

Therefore the final results of (7.21, 22) are G l = Fl + W l Hl

(7.24)

G N /2+l = Fl − W Hl

(7.25)

l

where Fl and Hl are the N /2 point DFT of f k and h k : Fl =

N /2−1

f k (W 2 )kl

< N 0< = l = 2 − 1.

h k (W 2 )kl

< N 0< = l = 2 − 1.

k=0

Hl =

N /2−1 k=0

(7.26)

Equation (7.24) shows that the first half of G l is the sum of the N /2-point DFT of the even terms of gk and the N /2-point DFT of the odd terms of gk multiplied by W l . Equation (7.25) shows that the second half of G l is the difference of these DFT’s. This method is called method of decimation in time. The signal flow graph in Fig. 7.10 illustrates the method of calculating DFT using decimation in time for

Fig. 7.10 Signal flow graph of the first stage of the method of decimation in time; First, DFT Fl and DFT Hl are calculated from the sampled values of g2k and g2k+1 and next, G l is found by combining Fl and Hl in the specified manner

174

7 The Fast Fourier Transform (FFT)

Fig. 7.11 The signal flow chart of the method of decimation in time

the case N = 8. In the same manner as outlined in the previous section, the 8-point DFT is first converted into 4-point DFT’s and finally converted into 1-point DFT’s. Figure 7.11 shows the completed signal flow graph. When the bent lines in Fig. 7.11 are straightened, Fig. 7.12 is obtained. Figure 7.13 is the signal flow graph redrawn from Fig. 7.12 to be used as a wiring aid for an FFT special purpose computer. The keys for the lines used in Fig. 7.13 are the same as those used in Fig. 7.9. In the case of the method of decimiation in time, the inputs are fed into the computer in bit reversed order. All other aspects of the FFT such as the capability of writing over the same memory locations in order to save memory space, and the reduction in the total number of computations, are the same as those in the method of decimation in frequency.

7.4 Values of W k Since only a finite number of W k are needed in an FFT special purpose computer, the values of the W k ’s are not recalculated every time they are needed but they are usually stored in the Read Only Memory (ROM). In general, when an N point DFT

7.4 Values of W k

175

Fig. 7.12 Signal flow graph obtained by removing the frames and straightening the lines of Fig. 7.11. (Method of decimation in time)

Fig. 7.13 Signal flow graph redrawn from that of Fig. 7.12 with the emphasis on using it as an aid to wire the FFT computer. (Method of decimation in time)

176

7 The Fast Fourier Transform (FFT)

is performed, N values of W k are needed. For convenience, the definition of W is once more written here. 2π k W = exp −j k N 2π 2π W k = cos k − j sin k . (7.27) N N The computer needs 2 N memory locations to store both the real and imaginary parts of (7.27). If the properties of W k are exploited, the number of memory locations can be substantially reduced. This will be explained by taking N = 16 as an example. W is expressed by a unit circle in the complex plane as in Fig. 7.14. Since the values of W k ’s located diametrically opposite each other on the circle are the same in magnitude but different in signs, the values of the W k ’s in the range of 180 ◦ − 360 ◦ are obtained from those in 0 ◦ − 180 ◦ by simply changing the sign. If the W k in 0 ◦ − 90 ◦ are compared with those in 90 ◦ − 180 ◦ , the only difference is in the sign of cos (2π/N )k, thus the W k ’s in 0 ◦ − 90 ◦ can be used to represent both. If the W k in 0 ◦ − 45 ◦ are further compared with those in 45 ◦ − 90 ◦ , the values of cos (2π/N )k and sin (2π/N )k are merely interchanged, thus the W k ’s in 0 ◦ –45 ◦ contain sufficient information to define all the W k . Finally, as shown in Fig. 7.14, only two complex numbers are to be stored in ROM for N = 16, excluding the value of W 0 which is 1 and need not be stored.

Fig. 7.14 Reduction of the number of values of W k to be stored in a ROM (Read Only Memory)

Problems

177

Problems 7.1. Given input values g0 , g1 , g2 , g3 , g4 , g5 , g6 , g7 , find the values of G 1 , G 5 , G 2 and G 6 of the DFT using the signal flow graph in Fig. 7.7 and compare the results with those obtained by the signal flow graph in Fig. 7.9. 7.2. The DFT G −l in the negative region of l can be obtained by transferring one half of the G l in the positive region of l, as shown in Fig. 7.15, namely, G −l = G N −l . Prove this relationship. Figure 7.15 was obtained by plotting only the real part of G l with real inputs. 7.3. Find the 4-point DFT when all the input gk ’s are unity (Fig. 7.16). Perform the Fourier transform of the function g(x) shown in Fig. 7.17. Compare the result of the DFT with that of the Fourier transform. 7.4. Find the DFT of the input g0 = 0, g1 = 1, g2 = 2, g3 = 0, as shown in Fig. 7.18. Using this result find the DFT of the input gk ’s which were obtained by shifting gk to the right, as shown in Fig. 7.19. Compare the two results. 7.5. As shown in Fig. 7.20, the inputs consist of two periods of the same shape. Using the signal flow graph in Fig. 7.9 prove that one half of the 8-point DFT spectrum of such inputs is zero.

Fig. 7.15 Finding the values of G −l by translating the spectrum near the end

Fig. 7.16 Sampled values of gk = 1

178

7 The Fast Fourier Transform (FFT)

Fig. 7.17 Curve of g (x) =

< 1 for − 0.5 < = x = 3.5 0 for allothers

Fig. 7.18 Value of gk

Fig. 7.19 Value of gk

Fig. 7.20 Value of gk

Problems

179

7.6. Making use of the chart of decimation in frequency (Fig. 7.9), calculate the DFT’s of the following a) gk = [0, 1, 1, 1, 1, 1, 1, 1] b) gk = [0, 0, 1, 1, 0, 0, 1, 1] c) gk = [1, A, A2 , A3 , A4 , A5 , A6 , A7 ] where A = exp(jπ/4).

Chapter 8

Holography

A photograph records the real image of an object formed by a lens. A hologram, however, records the field distribution that results from the light scattered by an object. Since there is a one-to-one correspondence between the object and its scattered field, it is possible to record information about the object by mapping the scattered field. Actually the recording of the scattered field provides much more information about the object than that of the real image recorded in a photograph. For instance, one hologram can readily generate different real images that correspond to different viewing angles. It would seem possible to record the scattered field by just placing a sheet of film near the scattering object, but unfortunately, phase information about the scattered field cannot be recorded in this way. Nevertheless this approach is used in certain cases, such as the determination of crystal structure from the scattered field of x-rays. The missing information about the phase has to be supplemented by chemical analysis. In holography, in order to record the phase information, a reference wave coming directly from the source to the film is superimposed on the scattered field coming from the object to the film.

8.1 Pictorial Illustration of the Principle of Holography Since any object can be considered as an ensemble of points, the principle of holography will be explained by using a point object. As shown in Fig. 8.1, the incoming plane wave is scattered by a point object O, from which a new diverging spherical wave front emerges. Figure 8.1 is a drawing of the instantaneous distribution of the incident and scattered fields. The contours of the maxima are drawn in solid lines and those of the minima are shown by dashed lines. The points where lines of the same kind intersect become extrema and the points where dissimilar lines intersect correspond to intensity nulls (zero). By connecting the extrema, a spatial fringe pattern between the incident and the scattered field is drawn. These fringes indicate a spatial standing wave as shown by the heavy lines in Fig. 8.1 a. 181

182

8 Holography

Fig. 8.1 a, b. Illustration of the principle of holography. (a) Fabrication of hologram. (b) Reconstruction of the image from hologram

The spatial standing wave pattern can be recorded easily by using a photographic plate. The contours of the extrema will show up as dark lines on the plate. Figure 8.1 a is a cross sectional view of the plate placed perpendicular to the z axis. The dark points a, b, c, . . . , f, are the cross sections of the extrema lines. The transmittance pattern of such a photographic film is more like that of a Fresnel zone plate. This photographic plate is a hologram of the point object. If the object consists of more than just one point, the hologram of such an object is the superposition of zone plates each of which corresponds to a point on the object. When the hologram is illuminated by a parallel beam, the points a, b, c, . . . , f become scattering centers, as shown in Fig. 8.1 b. The difference in path lengths between dO and eO is exactly one wavelength, hence the phase of the scattered wave from d is in phase with that from e. Similarly, the waves scattered from a, b, c, . . . , f are all in phase. Thus, the exact phase distribution that a point object located

8.2 Analytical Description of the Principle of Holography

183

at O would have generated in the hologram plane is re-established by the scattered field from a, b, c, . . . , f. An eye looking into the hologram towards the illuminating source, as shown in Fig. 8.1 b, cannot distinguish whether the light originates just from the surface of the hologram, or indeed from the original object, because the distributions of the light to the eye are the same for both cases, and it would appear as if the original object were present behind the hologram. Moreover, all the field originally scattered in the direction within the angle subtended by the original object at the photographic plate is simultaneously generated, and slightly different scenes are seen by each eye of the observer, which is necessary for the observer’s brain to interpret the scene as three dimensional. Whenever the observer moves his eyes, he sees that portion of the scattered field intercepted by the viewing angle of his eyes, and the scene therefore changes. As a result of these effects, the viewer has the illusion that he is actually seeing a three-dimensional object in front of him. Since the light does not actually converge to the location of the image, the image at O is a virtual image. In addition to this virtual image, a real image to which the light converges is formed. Referring to Fig. 8.1 b, at the point O , which is symmetric to point O with respect to the photographic plate, all the scattered waves are in phase and the real image of the point object is created. One interpretation of holography is that the field scattered from an object is intercepted by a photographic plate, and the scattered field pattern is frozen onto the plate. The scattered pattern is stored in the plate as a hologram. Whenever the photographic plate is illuminated by a laser beam, the frozen fringe pattern recreates the original scattered field pattern moving in the original direction. Because of the zone-plate-like properties of a hologram, the image can still be reconstructed even when only a portion of the hologram is used. The resolution of the image, however, is reduced because the number of available fringe lines on the hologram is reduced, but the location and shape of the image will not be affected.

8.2 Analytical Description of the Principle of Holography An outline of the analysis will be given and then the principle of holography will be discussed in a more quantitative manner [8.1, 2]. Figure 8.2 shows an arrangement for producing a hologram. A laser beam is split into the object beam which illuminates the object and the reference beam which illuminates the photographic plate directly. The recording of the superposition of the object beam O and the reference beam R is a hologram. In order to achieve uniform illumination of the object while avoiding specular reflection from the object, a diffuser such as a sheet of ground glass is inserted between the illuminating source and the object. If one assumes that the amplitude transmittance of the photographic film is proportional to the intensity of the incident light, the distribution of the amplitude transmittance of the hologram is |O + R|2 = |O|2 + |R|2 + OR∗ + O∗ R.

(8.1)

184

8 Holography

Fig. 8.2 Arrangement of components for fabricating a hologram

When the hologram is illuminated by a laser beam, the image is reconstructed. Assuming that the reconstructing beam is the same as the reference beam used to make the hologram, the distribution of light transmitted through the hologram is given by (8.2) |O + R|2 R = |O|2 R + |R|2 R + O |R|2 + O∗ R2 . The 3rd term of (8.2) is of particular interest. If R represents a plane wave, |R|2 is constant across the photographic plate, and the 3rd term of (8.2) is proportional to the distribution O of the object beam. Therefore, the distribution of the transmitted light through the hologram is identical to that of the field scattered from the original object, and it will have the same appearance as the original object to an observer. This procedure will now be analyzed in a more quantitative manner. The x − y coordinates are taken in the plane of the photographic plate. The field distributions of the reference and object beams on the photographic plate are represented by R(x, y) and O(x, y), respectively. When the photographic film characteristics and contrast reversal are taken into consideration, the expression for the amplitude transmittance t(x,y) will be given via (8.1) by (8.3) t (x, y) = t1 (x, y) + t2 (x, y) + t3 (x, y) + t4 (x, y) where t1 (x, y) = −β|O(x, y)|2 , t2 (x, y) = β[c − |R(x, y)|2 ], t3 (x, y) = −β R∗ (x, y) O(x, y), t4 (x, y) = −β R(x, y) O∗ (x, y), β being the slope of the linear portion of the film’s transmittance versus exposure curve, and c a constant.

8.2 Analytical Description of the Principle of Holography

185

The fields O(x,y) and R(x, y) will first be calculated by using the Fresnel approximation. Assume that the reference beam is a plane wave incident upon the photographic plate with its direction of propagation in the x − z plane as shown in Fig. 8.2. Then, R(x, y) can be expressed as, R(x, y) = R0 ejkx sin θr ,

(8.4)

where θr is the angle between the direction of propagation and the z axis. The object is considered to be made up of slices, and the field diffracted from only one of the object slices, O(x, y), located in the plane z = z 0 , is considered first. When such a field is observed at z = z, the field distribution is expressed by O(x, y, z) = O(x, y) ∗ f z−z 0 (x, y), where " 1 x 2 + y2 exp jk (z − z 0 ) + . f z−z 0 (x, y) = jλ(z − z 0 ) 2(z − z 0 )

(8.5)

For a photographic plate placed at z = 0, we have O(x, y, 0) = O(x, y) ∗ f −z 0 (x, y), where f −z 0 (x, y) =

j exp{−jk[z 0 + (x 2 + y 2 )/2z 0 ]}. λz 0

(8.6)

(8.7)

By inserting (8.4, 6, 7) into (8.3), an expression for the amplitude transmittance of the hologram is obtained. Next, the procedure for obtaining the reconstructed image is described mathematically. The image is reconstructed by illuminating the hologram with a reconstructing beam P(x, y). This beam is assumed to be a plane wave incident upon the hologram tilted again only in the x direction at an angle θp , as shown in Fig. 8.3, namely (8.8) P(x, y) = P0 ejkx sin θp . The Fresnel diffraction pattern of the illuminated hologram is E(xi , yi ) = [P(xi , yi )t (xi , yi )] ∗ f zi (xi , yi ).

(8.9)

The expression for t (x, y) consists of four terms. The term t2 (x, y) is related to the uniform beam of light which propagates straight through the hologram. The term t1 (x, y) is associated with noise. Terms t3 (x, y) and t4 (x, y) contain the factor O(x, y) and pertain to the image. The expression for E3 (x, y) associated with t3 (x, y) can be rewritten by using (8.3, 4, 6–8) as E3 (xi , yi ) = −β[P(xi , yi ) R∗ (xi , yi ) O(xi , yi )] ∗ f zi (xi , yi ) = −βP0 R0 {exp[jk xi (sin θp − sin θr )][O(xi , yi ) ∗ f −z 0 (xi , yi )]} ∗ f zi (xi , yi ). (8.10)

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Fig. 8.3 Reconstructing of the image from a hologram

In order to simplify (8.10), the double operation of Fourier transform followed by an inverse Fourier transform will be performed: E3 (xi , yi ) = − βP0 R0 ( ' sin θp − sin θr −1 ¯ ×F ∗ O( f x , f y ) F−z 0 ( f x , f y ) Fzi ( f x , f y ) δ fx − λ ' sin θp − sin θr −1 ¯ = − β P0 R0 F , fy O fx − λ ( sin θp − sin θr ×F−z 0 f x − (8.11) , f y Fzi ( f x , f y ) , λ where F {O(x, y)} = O( f x , f y ). Using (3.39), one obtains

E3 (xi , yi ) = − β R0 R0 exp −j k (z 0 − z i ) + jπ λz 0 ×F

−1

' sin θp − sin θr ¯ O fx − , fy λ

sin θp − sin θr λ

2

× exp[jπ λ (z 0 − z i ) ( f x2 + f y2 )]

× exp [−j2π z 0 (sin θp − sin θr ) f x ] .

When z i = z 0 , (8.12) becomes

(8.12)

sin θp − sin θr 2 E3 (xi ,yi ) = −βP0 R0 exp jπ λz 0 λ sin θp − sin θr × O(xi , yi ) exp j2π xi ∗ δ(xi − z 0 (sin θp − sin θr )), λ

8.2 Analytical Description of the Principle of Holography

187

and finally, sin θp − sin θr z 0 (sin θp − sin θr ) E3 (xi , yi ) = − βP0 R0 exp j2π xi − λ 2 (8.13) × O(xi − z 0 (sin θp − sin θr ), yi ) which means that the image is reconstructed at z = z 0 . Furthermore, it is located exactly where the object was placed during the fabrication if the condition θp = θr is satisfied. However, if θp = θr , there is a shift in the xi direction by z 0 (sin θp −sin θr ). Since the image looks exactly the same as the object, the image is called orthoscopic or true image. So far, only the term t3 (x, y) has been considered. The term t4 (x, y) associated with the conjugate image will be considered next. The formula which corresponds to (8.12) is sin θp − sin θr 2 E4 (xi , yi ) = − βP0 R0 exp jk (z 0 + z i ) − jπ λz 0 λ ' sin θp + sin θr × F −1 O¯ ∗ − f x + , − fy λ × exp [−j π λ (z 0 + z i ) ( f x2 + f y2 )] × exp [j2 π z 0 (sin θp + sin θr ) f x ] .

(8.14)

In the plane z i = −z 0 , the expression simplifies to sin θp + sin θr z 0 (sin θp + sin θr ) E4 (xi , yi ) = − β P0 R0 exp j2π xi + λ 2 ∗ (8.15) × O (xi + z 0 (sin θp + sin θr ), yi ). The interpretation of (8.15) is that a conjugate image is formed at z i = −z 0 . This means that the conjugate image is on the other side of the hologram from the original object. There is a shift in the xi direction by z 0 (sin θp + sin θr ) even when θp = θr , but this shift disappears when θp = −θr or θp = θr + π . The conjugate image is a real image; light is actually focused on this location and the projected image can be observed if a sheet of paper is placed at this location.1 The conjugate image has a peculiar property and is called a pseudoscopic image. The pseudoscopic image looks to an observer as if the image is inside out. Figure 8.4 illustrates why it looks inside out. Since the position of the focused beam is z i = −z 0 , a point on the object closer to the hologram is focused to a point further away from the observer. For example, referring to Fig. 8.4, point F which is further away from the observer at E1 than N , is focused to the point F which is closer to the observer than N . To the observer, the nose of the doll is seen further than the 1

If O is a point source, then O = [A exp (jkr )]/r , and O∗ = [A exp (−jkr )]/r . The expression for O∗ is that of spherical wave converging into one point. It is then clear that an image formed with O∗ is a real image and not a virtual one.

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Fig. 8.4 Images reconstructed from a hologram

forehead. Besides, the hatched section (which was not illuminated when the hologram was fabricated) is not present in the reconstructed image. Thus, the observer has a strange sensation of seeing the inside out face of the doll. The phase distributions of the reconstructed image and the original object are exactly the same except for a phase shift of 180◦ indicated by the negative signs in (8.13 and 15). However, since the human eye is only sensitive to the intensity, one cannot recognize this phase shift. The spatial frequency component of the noise term t1 (x, y) is approximately twice that of the object, and the direction of the diffraction from this term is at twice the diffraction angle of the true image. By properly selecting the values of θr and θp , the directions of the true and virtual images can be adjusted so as not to overlap with that of the noise term.

8.3 Relationship Between the Incident Angle of the Reconstructing Beam and the Brightness of the Reconstructed Image Because of the finite thickness of the photographic plate, the brightness of the reconstructed image is influenced by the angle of incidence of the reconstructing beam. Figure 8.5 shows the cross section taken by an electron microscope of the photographic emulsion of a hologram [8.3]. Platelets are arranged like the fins of a venetian window blind. Each platelet acts like a small mirror. These platelets are oriented in the plane of the bisector of the angle between the object and reference

8.3 Relationship Between the Incident Angle of the Reconstructing Beam

189

Fig. 8.5 Photograph of the crosssection of a Kodak 469 F holographic plate taken by an electron microscope [8.3]

Fig. 8.6 The reconstruction of an image from a thick hologram. The same source which was used for the reference beam is used for the reconstructing beam

beams (Fig. 8.6). For instance, the surface of the platelet at the point M0 is in a plane which bisects the angle

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the reference beam used for fabricating the hologram as the reconstructing beam. In this case, the orientations of all the platelets satisfy the above condition and the brightest possible reconstructed image is obtained.

8.4 Wave Front Classification of Holograms According to the shape of the wave front of the object beam, holograms can be classified as: (1) Fresnel hologram, (2) Fourier transform hologram, (3) image hologram, and (4) lensless Fourier transform hologram.

8.4.1 Fresnel Hologram When the distribution of the object beam on the hologram is the diffraction pattern in a Fresnel region, the hologram is called a Fresnel hologram. This is the most common type of hologram, and all the holograms described previously are of this type.

8.4.2 Fourier Transform Hologram This hologram is made such that the field distribution of the object beam on the hologram is the Fourier transform of the object. One way to achieve this is to place the object in the far-field region, but in practice the distance required is too great, so that the hologram is fabricated by placing a photographic plate on the back focal plane of a converging lens. Usually a converging lens is used to form the reconstructed image on the back focal plane of the lens. As seen from Fig. 8.7 a, information about the object is concentrated in the vicinity of the focal point and the area needed to record the hologram is small compared to other types of holograms. The Fourier transform hologram is often used for the purpose of high-density recording [8.4, 19].

8.4.3 Image Hologram A hologram made by placing a photographic plate near the location of the real image formed by a lens (Fig. 8.7 a) is called an image hologram [8.5]. Since the position of the reconstructed image is on the hologram itself, the distance from the hologram to the image is very short, and consequently, the coherency of the source is not very critical. A simple explanation is as follows. If a hologram is made by light of one of two different frequencies, and the image is reconstructed using both of these frequencies, two images slightly displaced from each other are formed. The amount

8.4 Wave Front Classification of Holograms

191

Fig. 8.7 a, b. The classification of holograms according to the shapes of the wave front. (a) Geometries for Fourier transform hologram, image hologram, Fresnel hologram. (b) Lensless Fourier transform hologram

of displacement decreases as the distance between the image and the hologram decreases. Therefore, the image hologram, with its short hologram to image distance, is insensitive to the coherency of the reconstructing beam. This principle is used for white-light holograms, to be mentioned later in this chapter.

8.4.4 Lensless Fourier Transform Hologram As shown in Fig. 8.7 b, a point source is used as the reference beam and the object is placed in the same plane as that of the point source. In this configuration, the distribution OR∗ becomes the Fourier transform of the object, thereby allowing a Fourier transform hologram to be fabricated without the use of a lens. Such a hologram is called a lensless Fourier transform hologram.

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The expression for OR∗ will be examined in more detail in order to demonstrate that the hologram formed is indeed a Fourier transform hologram. Let the location of the point source used as the reference beam be (0, 0, −z r ) and the location of the film be z = 0. Then OR∗ is given by OR∗ =

R0∗ [O(x, y) ∗ f −z 0 (x, y)] · exp {j k [z r + (x 2 + y 2 )/2z r ]} j λ zr

(8.16)

or R∗ OR = 0 2 (λz) ∗

∞

ξ 2 + η2 O(ξ, η) exp −jk 2z −∞ j2π × exp (xξ + yη) dξ dη λz

(8.17)

where z r = z 0 = z.

(8.18)

∗

OR is further rewritten as " ∗ 2 + y2 R x 0 . F O(x, y) exp −jk OR∗ = 2z (λz)2 f x = −x/λz, f y = −y/λz

(8.19)

Thus one can see that OR∗ is the Fourier transform of O(x, y) exp [−jk(x 2 + y 2 )/2z] and belongs to the category of Fourier transform holograms.

8.5 Holograms Fabricated by a Computer Once the shape of the object is determined, the distribution of its scattered field can be calculated. In fact, it is possible to obtain the distribution of the scattered field using a computer instead of light. An intensity pattern is drawn according to the intensity calculated by the computer, and this pattern is then photographically reduced to form what is known as a computed hologram [8.5–9]. The reconstructed image can be viewed by laser light illumination. In principle, it is possible to view a completed three-dimensional building from an architectural drawing by means of this fabricated hologram. The methods of fabricating a computed hologram can be categorized as follows: I) Recording the amplitude and phase of the scattered field. II) Recording the square of the sum of the object and reference beams.

8.5 Holograms Fabricated by a Computer

193

III) Simulating the phase distribution by modulating the positions of small holes made on a mask. IV) Simulating phase and amplitude distributions by modulating both the position and the size of the holes made on the mask. V) Using holes with equal spacing. VI) Using the computer to make sketches from different viewing angles. Method I makes use of the calculated result of t3 (x, y) of (8.3), which can be expressed by the complex number t3 (x, y) = A(x, y) e jφ(x,y) . The phase and amplitude information are recorded on separate photographic plates. The reconstructed image is obtained by transmitting through the two photographic plates. The amplitude A(x,y) can be easily recorded by modulating the transmittance of the photographic plate, but the recording of φ(x, y) is somewhat more difficult. One method of recording φ(x, y) is through bleaching of a photographic plate. When a photographic plate is bleached, the index of refraction of the plate is changed according to the darkness of the plate, and one can empirically relate the transmittance before the bleaching with the phase after the bleaching. As a matter of fact, by illuminating just one plate modulated by φ(x, y), the viewer can see the reconstructed image. This phase only hologram has a special name and is called a kinoform. Method II is based on the same principle as that of a normal hologram. The pattern of |R(x, y) + O(x, y)|2 is calculated by a computer, and its pattern is painted on a sheet of paper. The darkness of the paint is modulated according to the computed field intensity. Fabrication of the hologram is completed by photographically reducing the size of the drawn pattern. The image is reconstructed by illumination from a laser beam. The reconstructed image obtained by Method II will appear fainter compared with that by Method I, but Method II is simpler because it uses only one photographic plate. Both Methods I and II use a laser beam to reconstruct the image from the computed holograms. For either microwave or acoustic holography, it is possible to use the computer itself to generate the reconstructed image by coupling the measured field intensity directly to the computer. Method III will now be explained. In this method, the desired phase distribution is approximated by properly arranging the positions of holes on a mask. It eliminates the need to draw patterns as in Methods I and II. All that is needed is punching holes of equal size at the calculated positions. Figure 8.8 illustrates the principle of this method. As shown in Fig. 8.8 a, an opaque mask is placed at z = 0. A light beam illuminates the mask at an angle θ. The distribution of the light field on the surface of the mask is (8.20) A ejkx sin θ Hence the phase distribution along the x axis is similar to Fig. 8.8 b. With an increase in x, the phase periodically changes between −π and π .

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Fig. 8.8 a–d. Principle of a computer-generated hologram made by modulating the position of the array holes. (a) Incident wave. (b) Phase distribution of incident light. (c) Array of holes. (d) Phase of the point sources

By selecting the locations of the holes, the field with the desired phase can be sampled. If a hole is placed at 1/4 of a period away from the origin, the resulting field has a phase of π/2 radians with respect to the field at the origin. If one hole is made in each period, labelled as a, b, c, . . . , f in Fig. 8.8 c, an array of point sources is generated whose phases are indicated by a , b , c , . . . , f in Fig. 8.8 d. A hologram made in this manner is a kind of kinoform. Method IV is a revised version of Method III, and records not only the phase but also the amplitude distribution of the object beam. The hologram is made in a manner similar to that shown in Fig. 8.8 c except that the size of the holes is made proportional to the field amplitude. With Methods III and IV the positions of the holes were varied according to the phase, but with Method V the holes are made up of groups of four equally spaced holes. Since the spacing between the holes is fixed, one parameter is reduced. Figure 8.9 illustrates this method. A plane wave is incident upon a mask at an angle θ as in the previous case. The phase distribution on the mask is periodic, as shown in Fig. 8.9 b. In each period, four holes are made with equal spacing. The sizes of the four holes are chosen so that the amplitude of the transmitted light from these four holes is proportional to certain calculated values, indicated by g1 , g2 , g3 and g4 in Fig. 8.9. The phase of the light from each hole is 0, π/2, π and 3π/2, respectively, where the phase at the origin is assumed to be zero radian. The resultant light amplitude from the four holes is

8.5 Holograms Fabricated by a Computer

195

Fig. 8.9 a–c. Principle of a computer-generated hologram which was made by modulating the size of the holes located at equal intervals. (a) Incident wave. (b) Phase distribution of the incident wave. (c) Array of holes

g = (g1 − g3 ) + j (g2 − g4 ).

(8.21)

Thus, by controlling the sizes of the four holes, an arbitrary complex number can be represented. A hologram can be made by recording the object beam O(x, y) in this manner. Figure 8.10 a is the original picture used as the object, while Fig. 8.10 b is the reconstructed image of the hologram made by this method. The picture was first divided into 128×128 picture elements and then the intensity of each element was fed into a computer in order to calculate the object beam O(x, y). From the calculated value of O(x, y), the holes were made in the manner described in Method V. The brightness of the reconstructed image of a hologram made by such Methods as III, IV and V depends on the number of holes made. In general, the number of holes needed to make a clear reconstructed image is very large and the amount of computation is also quite large. For instance, to obtain the quality of the reconstructed image shown in Fig. 8.10 b, a memory of 131,072 words is necessary, which puts a sizable demand on one computer. One way around this is Method VI. In method VI, the diffraction pattern is obtained by using light, and only a sketch of the object viewed from different angles is made by computer. The computer time needed for this method is less than 1/1000 of that of the previous method. The principle is based on the fact that when one sees the reconstructed image only a limited portion of the hologram is being used. In other words, referring to Fig. 8.11, when

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Fig. 8.10 (a) Original and (b) reconstructed images from a computer generated hologram of equally spaced holes, modulating the size of the holes [8.7]

Fig. 8.11 The portions which contribute to reconstruction of the image

the viewer’s eyes are at a point A, the shape of the reconstructed image is more or less determined by the portion of the hologram indicated by a −a . When the viewer moves his eyes to a new location B, the shape of the image is formed only by the portion of the hologram indicated by b − b . This method makes use of the fact that only a small part of the hologram is viewed, once the position of the viewer’s eyes is set. The computer makes up a sketch of the object, as viewed from the point A. It is then drawn in one frame of a film. Once a sketch viewed from the point A is made, it is easy to draw a sketch viewed from B, because the computation involved is only the rotation of the coordinate system. Similarly, the sketches viewed from slightly different angles C, D, E, . . . are made and are sequentially recorded on a roll of film. As shown in Fig. 8.12, the laser beam illuminating from behind each frame of the film is the object beam to a small section of the hologram. Each section of

8.6 White-Light Hologram

197

Fig. 8.12 Hologram consisting of a series of small holograms with varying viewing angles

the hologram made corresponds to a particular sequence of the viewing angle. The same reference beam is used for each section of the hologram. When the completed composite hologram is illuminated by a reconstructing beam, and the viewer scans through the reconstructed images with sequentially different viewing angles, the result is an illusion of a three-dimensional object. Figure 8.13 shows a photograph of the reconstructed image from a hologram made in such a manner. The object is a bundle of randomly oriented wires. In this method, a computer was used for making sketches but this is not always necessary. A series of normal photographs taken from sequentially different viewing angles using white light can substitute for the computer. The rest of the procedure is the same. One short-coming of method VI is that the quality of the reconstructed image is degraded as the location of the viewer’s eye deviates from the predetermined position.

8.6 White-Light Hologram A hologram whose image can be reconstructed by white light is called a white-light hologram [8.10, 11]. The types of white-light holograms are the reflection hologram, the image hologram and the rainbow hologram. There are two basic mechanisms in forming an image out of a hologram. In one mechanism, the spacings between the platelets shown in Fig. 8.6 are modulated such that the interference of their scattered waves forms an image. Such a mechanism

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8 Holography

Fig. 8.13 Reconstructed image from a hologram fabricated at the Bell Telephone Laboratories using the computer only for generating the drawings with varied viewing angles of the same object

depends very critically upon the wavelength of the light. The other mechanism is to make the surface shape of the platelet in such a way that the reflected beam forms an image. The platelets serve as mirror surfaces, producing characteristics which are not a sensitive function of the wavelength of the light. The reflection-type white-light hologram uses the latter mechanism as a means of image formation. The required condition is that the area of each platelet be made large while the number of platelets remains small. Such a condition can be satisfied by making the angle between the object and reference beams large when the hologram is fabricated. In producing the reflection hologram, the object beam illuminates the front of the hologram while the reference beam illuminates the back (Fig. 8.14). In reconstructing the image by white light, the mirror-like platelets form the same image regardless of the wavelength of the light. Next, the image-type white-light hologram is described. When the principle of holography was first explained using Fig. 8.1, the wavelengths for producing the hologram and for reconstructing the image were the same and the image was formed exactly at the same place as the original object was. If, however, the image were reconstructed using light with a spread in wavelengths, the angle of diffraction from the fringe patterns in the hologram would also have a spread. The angle of diffraction of the shorter wavelengths would be smaller than that of the longer wavelengths. The

8.6 White-Light Hologram

199

Fig. 8.14 Fabrication of the reflection type white light hologram. The angle θ between object and reference beams is made large

Fig. 8.15 Fabrication of a white light hologram of the image hologram type

blur of the reconstructed image resulting from the spread in diffraction angles can be minimized by choosing an image location as close to the hologram as possible. Figure 8.15 shows the arrangement for making a white-light hologram of this type. The real image of the object is formed right onto the film so as to minimize the distance between the image and the hologram. The last white-light hologram to be mentioned is the rainbow hologram. By projecting a real image of a thin section of a hologram, the overlap of the images due to the spread of wavelengths is removed. The master hologram is first made in the usual manner as arranged in Fig. 8.16 a. The reference beam R1 and object beam O are recorded in the master hologram H1 . Next, only a small horizontal strip of the hologram H1 is illuminated by a reconstructing beam so as to form the real image (Fig. 8.16 b). The reconstructing beam is actually R∗1 which is the conjugate of R1 in Fig. 8.16 a. The conjugate beam can be made by illuminating the hologram strip from the opposite side and converging the beam with lens L1 to the point S1 from which the reference beam initially originated. The second hologram H2 is made using this reconstructed real image from the horizontal strip of the hologram H1 as if it were an object. A beam converging to point S4 is used as the reference beam in forming the hologram H2 . Strickly speaking, the object of the hologram H2 was not the real image of the hologram H1 , but the strip of the hologram H1 itself. The hologram H2 is the desired white-light hologram.

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8 Holography

Fig. 8.16 Procedure for fabricating a rainbow hologram. (a) Fabrication of master hologram. (b) Second hologram H2 is made from real image of H1 . (c) Reconstruction of image using a white light source

8.6 White-Light Hologram

201

In the process of reconstruction of the image, a white-light source is placed at the point S4 so as to illuminate the hologram by the conjugate beam of R2 . The reconstructed image from H2 is the real image of the strip of H1 . If an observer’s eyes are placed in front of this real image of the strip of H1 , it appears as if the strip of H1 is physically in front of the eyes, and one can observe the image of the object. Since the illuminating light is white light, a series of strips of H1 of different colors are formed. The color of the image changes from red to violet as the observer lowers his eye position. This is the reason why this hologram is called a rainbow hologram. The width of the strip has to be made narrow enough to avoid interferences between the images of different colors. Making the strip horizontal sacrifices vertical parallax, while the horizontal parallax is least affected. To human eyes, the loss of vertical parallax is not as noticeable as the loss of horizontal parallax. Figure 8.17 shows the reconstructed image from the rainbow hologram. Since the reconstructing beam illuminates the back of the hologram, and converges to a small area of the real image of hologram H1 , a very bright image is reconstructed.

Fig. 8.17 Reconstructed image from the rainbow hologram. The hologram is displayed in an ordinary laboratory environment, showing the high visibility of the image. The illumination source is a small spotlight above and behind the hologram [8.11]

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8 Holography

8.7 Speckle Pattern The reconstructed image of a hologram has granular texture while that of a white light hologram does not. As a matter of fact, any image formed using coherent light displays this granular texture or speckle pattern [8.12]. Illumination by two coherent plane waves generates a fringe pattern consisting of an array of bright and dark lines, as shown in Fig. 2.8. If there were two pairs of such coherent wave arranged with each set perpendicular to the other, dark lines in a mesh pattern would be generated. When many sources are present at arbitrary locations, the fringe pattern will contain several dark spots of irregular shapes. Wherever the vectorial sum of the total field is zero, a dark spot is generated. In the case of illumination by incoherent white light the situation is different, and the larger the number of sources, the more uniform in general the illumination becomes. The intensities of an incoherent source such as white light always add and there is no vectorial cancellation. Gabor [8.13] classified the speckle patterns into two categories according to their origins. One is “objective speckle” which originates from inhomogeneity already present in the light illuminating the object. When the object is illuminated through a diffuser, as shown in Figs. 8.2 and 18 a, a speckle pattern is projected onto the object. The hologram faithfully reconstructs the shape of the object as well as the projected speckle patterns. One way to cope with objective speckle is to avoid the use of a diffuser. There is another type of objective speckle which is generated by the roughness of the surface of the object. This objective speckle is generated even when a parallel beam is used to illuminate the object. Unless the surface of the object is smooth to the order of the wavelength of light, the reflectivity of a spot on the object critically depends on the relative phases among the wavelets reflected from that point (Fig. 8.18 b). The other kind of speckle is “subjective speckle” which arises when the detector is diffraction limited. For instance, when a human eye is used as the detector for the reconstructed image, the eye is diffraction limited by the size of the pupil. The images of P1 and P2 on the object projected to the retina have some spread (Fig. 8.18 c). The tails of the point-spread functions interfere with each other. This interference is seen as a speckle pattern to the observer. A completely incoherent light, like white light, does not generate speckle patterns, but at the same time does not reconstruct an image of a hologram either, unless the hologram is a white-light hologram (Sect. 8.6). A partially coherent light is a compromise; it effectively reduces speckle patterns and yet reconstructs the image. The coherence of any given spectral line from a sodium arc lamp is lower than that of a laser. Such a source is good for observing the hologram with much less subjective speckle. Another method is the use of a laser light whose spatial coherence is reduced by an external means. A closely spaced pair of diffusers which keep moving with respect to each other is used as a means of reducing spatial coherence. A nematic liquid crystal gate whose minute scattering centers are constantly agitated by the applied voltage is also used for this purpose.

8.7 Speckle Pattern

203

Fig. 8.18 a–c. Causes of speckle patterns of a hologram. (a) Objective speckle due to a diffuser. (b) Objective speckle due to rough surface of the object. (c) Subjective speckle due to the finiteness of the pupil

Although speckle is an annoying problem when reconstructing hologram images, speckle patterns are not all bad, and they can be put to good use. One such area of study, based on the special properties of speckle, is that of speckle interferometry [8.12, 14]. The basic principle of speckle interferometry is illustrated by referring to Fig. 8.19. The movement of a rough surface M is to be analyzed. As explained in Fig. 8.18 b, an optically rough surface generates objective speckle. The objective speckle pattern is photographed by a imaging lens (Fig. 8.19 a). If the illuminating parallel beam is uniform, then the speckle pattern generated on the surface moves with that surface. A double exposure is made before and after the movement. In the double exposed photograph, each speckle spot is paired with its identical speckle spot (Fig. 8.19 b). Such a photographic plate P1 is Fourier transformed by a Fourier transforming lens L2 (Fig. 8.19 c) and the distance of the movement is obtained from the Fourier transformed image.

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Fig. 8.19 a–c. Principle of speckle interferometry. (a) Recording the speckle patterns. (b) Double exposure of the speckle pattern before and after movement. (c) Fourier transform analysis of the movement

Let the distance of the movement be 2ξ and the entire speckle pattern is represented by g(x, y). The Fourier transform of the double exposed plate P2 is F{g (x − ξ, y) + g (x + ξ, y)} = G( f x , f y ) {e−j 2 π f x ξ + e−j 2 π = 2 G( f x , f y ) cos 2 π f x ξ.

fx ξ

} (8.22)

The Fourier transformed image is modulated by cos 2π f x ξ and from the pattern, ξ is determined.

8.8 Applications of Holography The principle of holography has been widely applied to various fields. The applications will be explained with an emphasis on the principle of operation. The application of holography to microwave devices and optical signal processing, however, will be treated in separate chapters.

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8.8.1 Photographs with Enhanced Depth of Field It is very difficult to take a microscopic photograph of a small moving object such as a swimming microbe or a small particle in Brownian motion, because the depth of field of a microscope is so shallow that it is almost impossible to follow the movement by manipulating its focus. Such a photograph can be taken rather easily by using a hologram [8.17]. Holography does not use a lens but records the scattered field from near as well as far objects; once one hologram is made, one can focus the microscope to the reconstructed image of any particular object in a steady state of motion by adjusting the focus of the microscope. Figure 8.20 illustrates a series of photographs showing the swimming motion of a microbe in a water tank. The procedure for taking such photographs is as follows. As shown in Fig. 8.20 a, fine-grain holographic film is installed in a movie camera reel, and a series of holograms are made at short intervals using a pulsed laser. In this particular case, the camera was set at a speed of 70 frames per second. Next (Fig. 8.20 b), the reconstructed image is made from one frame of the hologram at a time. In each frame of the reconstructed image, the focus of the microscope is always adjusted to one particular object to follow its movement. Figure 8.20 c is a series of microscope pictures of the swimming motion of a microbe taken by this method.

8.8.2 High-Density Recording For conventional high-density recording of real images such as microfiche [8.18], even a small dust particle on the film can create a missing portion on the record, and the missing information can never be recovered. Therefore, one has to be extremely careful about dust or scratches on the film. However, when using holograms for high density recording, a scratch on the film will not destroy information but merely causes a slight increase in the noise on the reconstructed image, so that no particular portion of the recording is lost. The highest achievable density of the recording is theoretically the same whether recorded as a real image or as a hologram. Figure 8.21 is the reconstructed image from a small hologram 0.7 mm in diameter containing 800 letters. A density of 2000 letters per square millimeter of hologram is achievable [8.19–21].

8.8.3 Optical Memory for a Computer A black and white pattern like the one shown in Fig. 8.22 can be used as a means of recording the binary numbers of 0’s and 1’s. Figure 8.23 shows an example of

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Fig. 8.20 a–c. A series of reconstructed images to investigate the movement of the microbe. The enhanced field of view of the hologram was applied to make such pictures. (a) Fabrication of series of holograms. (b) Microscope is focused to a particular object in the series of holograms taken. (c) The movement of the microbe was photographed from the reconstructed image of the holograms taken at an interval of 1/70 seconds. The blurred spot on the lower left corner is another microbe and had the microscope been focused to this microbe, the microbe in the center would have been out of focus [8.17]

an arrangement for using such a pattern as a read-only-memory of a computer. Figure 8.23 a illustrates the mechanism for picking a particular hologram from an array of small holograms. Figure 8.23 b is a photograph of the array of small holograms and Fig. 8.23 c shows an image reconstructed from one of the small holograms. The

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Fig. 8.21 The reconstructed image from a hologram 0.7 mm in diameter. The density of the recording is 2000 characters per mm2 [8.19]

Fig. 8.22 The optical memory; the black and white dots represent “0” and “1”, respectively

reconstructed image has 8 × 8 small sections and each small section is again divided into 8 × 8 picture elements. In this photograph, each section is the same and has one “0” bit represented by a black picture element in the center and 63 white picture elements each representing a “1” bit. Therefore, in this one sheet of hologram, 42 × 106 bits of information have been stored [8.22]. The selection of a particular hologram from the array of holograms is made by an X −Y beam deflector, as shown in Fig. 8.23 a. The beam deflector used here has been described in Problem 4.2. The real image reconstructed from the selected hologram is projected onto a matrix of photo-transistors. The size of each photo-transistor is 100 × 100 µm2 and 64 of them are arranged on one wafer (Fig. 8.23 d). In this example, 64 of these wafers are arranged in a square shape. The outputs from the photo-transistors are connected to the electronic circuits of the computer. A highdensity recording is achieved through the use of the Fourier transform hologram described in Sect. 8.4.2. It should be noted that the diffraction pattern of a periodic input like this binary coding mask is characterized by periodic peaks. The darkness of the film at the peak is the limiting factor of the information storage density. One method of raising the limit is to place the hologram slightly away from the focal point when the Fourier transform hologram is used for recording. A better method is to place a sheet of random phase shifter over the bit pattern when the hologram is fabricated. Because

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Fig. 8.23 a–d. Holographic memory applied to computers. (a) Arrangement for using holographic memory for a computer. The reconstructing beams are directed to a particular hologram and its reconstructed image is projected onto an array of photo-transistors to be converted into electrical signals. (b) Array of holograms on the memory plate. (c) Reconstructed image from one particular element of the hologram array. (d) Photo-transistor array forming an 8 × 8 matrix [8.22]

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of the randomization of the phase, the peaks disappear, and the density limitation of the recording is raised [8.19, 20, 21].

8.8.4 Holographic Disk The memory system mentioned in the previous subsection used a two-dimensional recording pattern, but the holographic disk described here uses a one-dimensional recording pattern with rotational movement of the hologram [8.23]. Figure 8.24 shows the arrangement of the holographic disk. An analogue electrical signal is first converted into a digital signal. The digital electrical signal is further converted into a one-dimensional dark-and-light pattern on a liquid-crystal display [8.24]. A Fourier transform hologram of the liquid-crystal display is recorded by lens L1 onto the holographic disk (Fig. 8.24 a). The mask is for limiting the size of the hologram. A series of holograms are recorded as the holographic disk is rotated. When the signal is to be retrieved, the read-out beam is scanned over the rotating holographic disk. The cylindrical lens L3 reconstructs the real image of the signal recorded in the holographic disk onto the photodiode array (Fig. 8.24 b). The photodiode converts the light pattern back into an electrical signal output. The attractive feature of such a system is that the photodiode array is capable of handling a large number of bits simultaneously (parallel processing) and the recording or replaying rate can be made very high.

8.8.5 Laser Machining A high-intensity laser beam with approximately 10 to 100 MW/cm−2 can be easily obtained by focusing a laser beam with a lens. Such a high-intensity laser beam

Fig. 8.24 a, b. Holographic disk. (a) Recording arrangement. (b) Playback arrangement

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can be used for machining or welding. In many cases micro-machining which is difficult to achieve by mechanical tools can be achieved quite easily by laser beam machining [8.25]. In laser-beam machining, two methods are commonly employed. One method involves the use of a hologram and the other uses a Fresnel-zone plate. The hologram method performs the machining, or cutting, by illuminating the object to be machined with the high-intensity reconstructed real image from a hologram. The hologram is made such that the reconstructed image forms a desired pattern. However there is a limit on the laser intensity because the photographic plate is not heat resistive. The other method of machining uses a metallic Fresnel-zone plate to eliminate this limitation on the intensity of the illuminating laser beam, as already shown in Fig. 4.28.

8.8.6 Observation of Deformation by Means of an Interferometric Hologram Holographic interferometry makes it possible to measure extremely small deformations in an object of the order of a wavelength of light [8.15, 16]. Basically, the method involves observing the interference patterns resulting from superimposing the reconstructed images of an object in two different states. This method has a wide range of applications, e.g., it is used to measure such minute changes as, heat expansion, chemical corrosion, plant growth, flow rate of gas or fluid, and detection of damage and fatigue for purposes of quality control. Figure 8.25 shows the principle of this method. A double-exposure hologram is made of a head, before and after being hit. Not only the amplitude but also the phase information of the object is reproduced in the reconstructed image. If the difference between the optical paths SPE and SP E is one half the wavelength of the laser light, the two images add destructively so that the point becomes dark; if the path difference is one wavelength, they add constructively so that the point becomes bright. Thus the contour of the deformation of the object is obtained.

Fig. 8.25 Interferometric holography

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In the case of normal interferometric measurement, two beams of light have to be present simultaneously at the same spot, but in holographic interferometry the measurement can be made even when there is a lapse of time between the first and the second exposures. Because of this property, it is possible to observe the temporal changes of an object such as a growing plant. Another advantage of holographic interferometry is that the three-dimensional pattern of the deformation can be observed. A disadvantage of holographic interferometry, however, is that the analysis becomes complicated if the deformation is a combination of rotation and translation because both of these create a similar interference pattern. Real-time holographic interferometry can be produced by using the object itself to interfere with the reconstructed image of the hologram. Figure 8.26 shows an example of using the interferometric hologram for the repair of old art work. If the art work is warmed by blowing warm air over the surface, the place where the paint is lifted expands more and the spots in need of repair can therefore be detected using this holographic method [8.26]. Interferometric holography is also used in the field of medicine. Deformation by force that takes place in such locations as foot and arm bones, teeth, or skulls has been studied [8.27]. The displacement contours measured by holographic interferometry are used to study pressure distributions.

Fig. 8.26 a, b. A method of finding a repair area of a valuable art work by means of interferometric holography [8.26]. (a) Picture of Santa Caterina by Pier Francesco Fiorentino. (b) Interference of the reconstructed image of the holograms taken at room temperature and that taken at 40 ◦ C is used for detecting the area of the lifted paint

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Fig. 8.27 A method of checking the degree of polish from the interference pattern between the reconstructed image from a computer hologram and that of the surface to be polished [8.28]

Interferometric holograms are also widely used for product quality control. An interferometric pattern is made between the product and the reconstructed image from the hologram. Figure 8.27 shows the configuration for detecting a high spot on a mirror during surface polishing, by means of the interference between the reconstructed image of a computer generated hologram and the surface of the mirror [8.28]. Holographic interferometry is very sensitive and can detect a displacement of a fraction of a light wavelength, but there is an upper limit to the amount of displacement that can be measured by this means [8.15, 16]. If the displacement becomes more than a few micrometers, the interference pattern becomes too fine for detection. The upper limit of speckle interferometry (Sect. 8.7), however, is much larger and is of the order of submillimeters.

8.8.7 Detection of the Difference Between Two Pictures As mentioned in Sect. 8.2, the phase distribution of the reconstructed image is 180◦ out of phase with that of the object. This fact can be used to detect the difference between an object and the reconstructed image of a similar object. If the reconstructed image of the hologram of one picture is laid over another picture, the difference

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213

Fig. 8.28 a–d. Example showing the difference between two similar pictures [8.29]. (a) Picture without a jeep. (b) Picture with a jeep. (c) Only the portion which is different between (a) and (b) is detected; a picture of a jeep. (d) Expanded picture of the jeep

between the two pictures is obtained. Figure 8.28 shows an example of the difference obtained by this method. The difference between a picture with a jeep present and a picture without it was obtained. The presence of the jeep was detected [8.29].

8.8.8 Observation of a Vibrating Object It is mandatory to minimize the movement of the object during the fabrication of a hologram. Since the spatial frequency of the fringe pattern to be recorded on a holographic plate is of the order of a wavelength, a movement of the hologram of the order of a wavelength of light will prevent the reconstructed image from being produced. This fact can be utilized for detecting minute vibrations of the surface of the object. The vibrating part becomes dark in the reconstructed image because no image is formed. In this manner, the standing-wave pattern of the vibration can be observed. Figure 8.29 is a pattern of the vibration of a guitar obtained by this method. In addition, a variation of this method enables one to observe the object during the vibration. If the laser light is strobbed at the same frequency as that of the vibration,

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Fig. 8.29 Picture of a vibration pattern of a guitar taken by holographic means [8.30]

a hologram of the vibrating object with a specified phase angle between the vibration and the strobe light can be obtained. This method is used for measuring the dynamic characteristics of a structure [8.30].

8.8.9 Generation of Contour Lines of an Object By using coherent-source interference patterns, contour lines of equal heights can be made in several ways. The first method uses a spatial standing-wave pattern generated by two point sources. One way to attain this is to use a point source and its image made by a mirror. An alternative approach to creating a spatial standing wave pattern is through a double exposure hologram. The point source is then used as an object. One exposure made with the point source at one location is followed by another exposure with the point source slightly displaced. This makes use of a special property of a hologram, namely the images of two exposures taken at different times can form an interference pattern. The second method utilizes the fact that the location of the reconstructed image shifts from the original location if the hologram is made by one laser wavelength and the image is reconstructed by another wavelength [8.31, 32]. The amount of the shift increases with the distance from the source. One exposure made by one laser wavelength is followed by another exposure made by a different laser wavelength. The image reconstructed by one of the laser wavelengths is a slightly displaced doubled image. The place where the displacement between the two images is an odd multiple of a half wavelength becomes dark, and the place where the displacement is an even multiple of a half wavelength becomes bright. Figure 8.30 shows the contour lines made by this method.

Problems

215

Fig. 8.30 The contour lines generated by making a hologram with two wavelengths and reconstructing it with one of the two [8.32]

The third method which uses a single laser is a modification of the second method which requires the use of two lasers. In order to change the wavelength, the object is submerged in two gasses or fluids of different refractive indices. A double exposure is made first by placing the object in one medium, and then in the other. The reconstructed image has fringe lines indicating equidistance from the source.

Problems 8.1. In this chapter, the size of the hologram was considered infinite; find the expression for the reconstructed image when the hologram dimension in the x direction is reduced to a. Assume that the object is one dimensional and is a function of only x, and assume that θr = θp . 8.2. The reconstructing beam is incident along the z axis. What is the change in the position of the reconstructed image when the plane of the hologram is rotated by an angle α from a position perpendicular to the z axis? 8.3. A hologram was made with a reference beam whose incident angle to the surface of the hologram was θ. This hologram is illuminated by a reconstructing beam

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Fig. 8.31 Variety of orientations of a hologram

Fig. 8.32 a, b. Fourier transform hologram. (a) Fabrication of hologram. (b) Construction of the image from the Fourier transform hologram

whose incident angle to the surface of the hologram is −θ . Find the location of the conjugate image of the hologram. 8.4. Find how the reconstructed image changes when the orientation of the hologram is changed as shown in Fig. 8.31. The reconstructing beam is always incident along the z axis. 8.5. As shown in Fig. 8.32, a Fourier transform hologram is made by placing an object with dimensions l1 × l2 at the front focal plane of the lens L 1 , and a photographic plate at the back focal plane of the lens. The focal length of the lens L1 is f 1 . In order to reconstruct the image, the hologram is placed at the front focal plane of a lens L2 with a focal length of f 2 . Find the location of the reconstructed image. The incident angle of the reference beam is θ.

Chapter 9

Laboratory Procedures for Fabricating Holograms

What may seem simple and easy to understand in theory can often be difficult to put into practice. While space does not permit a description of all the interesting experiments that could be performed to illustrate the theory of this book, the fabrication of a hologram has been chosen as a representative example. This section is intended as a step-by-step guide to the experiment, complete with several illustrative photographs.

9.1 Isolating the Work Area from Environmental Noise A specially prepared bench has to be used to fabricate a hologram. Since details in the fringe pattern which is being recorded are of the order of an optical wavelength, extreme caution must be used to avoid vibration of the optical bench during the time of exposure. The electrical equivalent circuit of the optical bench is a series resonance circuit. The noise reaches the surface of the bench through this resonance filter. The environmental noise spectrum is spread over a very wide range but the predominant component of the noise which reaches the bench top is at this resonance frequency. It is therefore desirable to make this resonance frequency as low as possible, preferably below the inverse of the photographic exposure time. Figure 9.1 shows a drawing of the bench and its electrical equivalent circuit. Let the weight of the bench top be M, and the compliance of an air cushion between the stand and the bench top be K . The resonant frequency of the combination of the bench top and the air cushion is 1 . (9.1) √ 2π M K In order to make f 0 small, the product of M and K has to be made large. Speaking in qualitative terms, it is as if trying to shake a heavy mass through a soft air cushion. Likewise, environmental noise from the floor is not easily transmitted to the bench top. Figure 9.2 shows a photograph of a steel optical bench with f0 =

217

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9 Laboratory Procedures for Fabricating Holograms

Fig. 9.1 Optical bench and its equivalent circuit

Fig. 9.2 Air is being pumped into air cushions which support the stainless steel optical bench

four cushions at the corners. The air cushion is pumped up with a bicycle pump. Figure 9.3 illustrates various improvised optical benches. Figure 9.3 a shows an ordinary bench consisting of braces across the legs of a table with a metal sheet on the top. The entire bench is placed over the inner tube of a tire. When the output power of the laser is large and hence the exposure time is short, even this simple arrangement is practical. When ease of transportation is an important consideration, the bench in Fig. 9.3 b is often used. A metal sheet is placed over an air-cushioned sand box. The weight of the bench would, of course, depend on the amount of sand. The bench in Fig. 9.3 c consists of a metal sheet, bricks and sheets of felt or rubber. This again is designed for easy transportation into the laboratory. Figure 9.3 d shows a bench with special legs. The leg has a servo-loop which automatically absorbs any external vibration. A metal sheet or granite slab is placed over the legs. It is necessary for all the bench tops to be smooth. Roughness in the surface could cause optical elements to rattle, or could result in a change of height as the elements are slid across the bench top, making fine adjustment very difficult.

9.2 Necessary Optical Elements for Fabricating Holograms Requirements vary depending upon the purpose and the desired quality of the hologram, but essential optical elements are:

9.2 Necessary Optical Elements for Fabricating Holograms

219

Fig. 9.3 a–d Construction of various kinds of optical benches for holography experiments

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

optical bench laser beam director spatial filter beam splitter plate holder film front surface mirror variable attenuator light meter lens lens holder.

In the subsections to follow the individual items are examined closely.

9.2.1 Optical Bench As mentioned in Sect. 9.1, select a bench with a resonant frequency as low as possible. The basement of the building is the best place to install optical benches but basements which are too close to subways, street car tracks or roads with heavy traffic should be avoided. One also must be aware of vibration which may come from water pumps or air ventilation ducts. The bench should be carefully adjusted to a comfortable height in order to avoid undue strain on the experimenter. It is desirable to place the bench in the center of the room, rather than a corner, so that the bench is accessible from all sides.

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9.2.2 Laser If the output power of the laser is less than a few milliwatts, the exposure time becomes excessively long and very inconvenient. Some types of lasers have ventilation holes in the external casing where light tends to leak out. A suitable cover or screen should be used to prevent this extraneous light from reaching the film. Those lasers in which ventilation takes place by heat conduction have no holes and do not need covers.

9.2.3 Beam Director In some experimental arrangements, the output beam of the laser is not at the same height as the other optical elements. In these cases, a beam director is used to control the height of the beam. As shown in Fig. 9.4, the beam director consists of two small mirrors and is of similar structure to a periscope. Both the upper mirror M1 and the lower mirror M2 can be rotated with respect to the vertical support and consequently the direction and height of the light beam can be adjusted.

9.2.4 Spatial Filter The spatial filter is comprised of a microscope objective and a pinhole. This is used for the purpose of filtering light whose wave front is disturbed due to scattering, reflection or some other cause. Figure 9.5 shows the structure of the spatial filter. The light beam passing through the microscope objective is focused onto the pinhole. If the size of the pinhole is selected such that only the main lobe of the Fourier transform of the beam can go through, then only the light ray with an angle of incidence parallel to the optical axis can pass through, and all other rays will be eliminated.

Fig. 9.4 Beam director

9.2 Necessary Optical Elements for Fabricating Holograms

221

Fig. 9.5 Spatial filter

When a × N microscope objective is used, a pinhole with a diameter given by λ [µm] (9.2) N should be selected, where the unit of light wavelength λ is micrometers. For instance, when a helium neon laser (λ = 0.6328 µm) is combined with ×10 lens, a hole 20 µm in diameter should be chosen. Dp = 300

9.2.5 Beam Splitter A beam splitter is a sheet of glass which is partially transparent and partially reflecting, and is used for splitting the beam into two, the reference and object beams. Most convenient for hologram experiments is the beam splitter whose transmittance varies linearly along the glass length. This beam splitter allows easy adjustment of the ratio of the intensities of object and reference beams.

9.2.6 Photographic-Plate Holder Since the photographic-plate holder must be manipulated in the dark, it should be simple and be able to hold the film firmly in the same position every time. The reason being that when a real time interferometric hologram is made, it is necessary to put the developed film back exactly in the original position.

9.2.7 Film Since the details in the fringe pattern are of the order of the wavelength of light, high-resolution film is needed to record the hologram [9.1]. Specially made holographic film is available commercially with typical resolutions of 2000 to 2500 lines/mm, which should be sufficient for most experiments. Care should be taken to select film with a good spectral response at the wavelength of the laser light. To minimize deformation of the film during the developing process, the following method may be used. The unexposed film is immersed in a water bath in a transparent developing tank. An exposure is taken, after which developer is added.

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9.3 Photographic Illustration of the Experimental Procedures for Hologram Fabrication To start with, the allocation of the optical elements is drawn on paper with the following points in mind [9.2, 3]: 1) The optical paths of the object and reference beams should be equal in length. 2) The angle between the object and reference beams should be neither too large nor too small. Angles between 30◦ and 60◦ are adequate. 3) It is easier to manipulate the photographic plate in the dark when it is placed near the edge of the optical bench. 4) The viewing angle of the reconstructed image would be the same as that through a window of the same size and location as the particular hologram. Make sure that the scene viewed by looking through the film holder frame is the same as the scene desired to be recorded. 5) The laser does not necessarily have to be placed on the optical bench. It can be placed on a reasonably steady normal bench. A possible arrangement of the optical elements is shown in Fig. 9.6. After making sure the optical bench is level, start putting the optical components, biggest first, onto the optical bench, as shown in Fig. 9.7. Figure 9.8 is a photograph showing the initial stage of the arrangement of the optical components. A photographic emulsion cross-section in thickness of the fringe pattern of a hologram looks like a venetian blind. When the distortion of the fringe pattern is

Fig. 9.6 Arrangement of optical components for hologram fabrication

9.3 Photographic Illustration of the Experimental Procedures for Hologram Fabrication

223

Fig. 9.7 The largest item, usually the laser, is placed first on the optical bench

Fig. 9.8 Installation in progress; items from left to right are: photographic plate holder, beam director, laser, and mirror

critical, as in the case of an interferometric hologram, the incident angle θr of the reference beam to the film is made equal to the incident angle θo of the object beam to the film. Then the surface of each element of the “venetian blind” fringe pattern becomes perpendicular to the surface of the photographic emulsion. A procedure for making θr equal to θo is shown in Fig. 9.9. First, any used photographic plate is attached temporarily onto the plate holder to act as a mirror. The plate is then rotated so that the reference beam reflected from the plate meets the object beam on the surface of a sheet of paper located where the object will be placed. When the two beams meet, the shaft of the plate holder is tightened, completing the adjustment. Figure 9.10 illustrates this procedure, showing the two beam spots on the sheet of paper. In Fig. 9.11, the sheet of paper is replaced by the object. The distances travelled by the object and reference beams from the laser to the film should be made equal. As shown in Fig. 9.12, the distances are checked by a measuring tape. This is done so that the interference patterns on the photographic plate are formed with two light beams, each of which left the laser at the same time. By so doing, the requirement for temporal stability of the laser becomes less critical. If, however, the distance traversed by either reference or object beam is

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Fig. 9.9 Orientation which minimizes the distortion of the reconstructed image due to shrinkage during the development.

Fig. 9.10 The film holder containing a blank plate is rotated until the reflection of the reference beam hits the paper at the location where the object beam hits

Fig. 9.11 Place the object at the location determined by the method illustrated in Figs. 9.9, 10

longer than the other, the beam travelling the longer distance interferes with the other beam which has left the laser later and travelled the shorter distance. The allowable difference in the distances depends upon the stability of the laser used, but it is desirable to be within a few centimeters and at most not more than a couple of tens of centimeters. Next, it is necessary to keep the beam at the same height from the surface of the optical bench everywhere (Fig. 9.13). Almost all optical elements are designed under the assumption that the optical axis is parallel to the surface of the bench.

9.3 Photographic Illustration of the Experimental Procedures for Hologram Fabrication

225

Fig. 9.12 The total distance from the source to the photographic plate of the reference beam should be the same as that of the object beam

Fig. 9.13 The height of the beam from the bench should be made as constant as possible everywhere

Fig. 9.14 The spatial filter is used not only for producing a uniform wavefront but also for expanding the diameter of the beam

Thus, if the beam is not perfectly horizontal, adjusting the elements becomes troublesome. For optimum illumination of object and film plate, the reference and object beams are expanded by placing spatial filters in the beam paths (Fig. 9.14). If the beam is expanded more than necessary, stray light scattered either from the surface of the bench or supports for the optical elements becomes intolerable and the quality of

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9 Laboratory Procedures for Fabricating Holograms

the hologram suffers. The size of the expanded beam can be adjusted by properly locating the spatial filter. Since the wave front of the reference wave must be undisturbed, the spatial filter for the reference beam is inserted in the path just before the photographic plate. Optical elements such as mirrors should not be inserted between the spatial filter and the photographic plate because they will most certainly disturb the wave front. This, however, is not applicable to the object beam since the object beam is to be scattered by the object before it reaches the photographic plate. If the illumination is too strong and sharp shadows form behind the object, illuminate the object through a piece of ground glass. Moreover, the ground glass can be an alternative to a spatial filter. The next step is to measure the ratio of the intensities of the reference beam to the object beam on the surface of the photographic plate. When the path of the incident beam is not perpendicular to the surface of the photographic plate, the power at normal incidence is first measured and its result is multiplied by the cosine of the incident angle. This must be done because most photometers give correct values only when the beam is at normal incidence. In Fig. 9.15, the light power is being measured at its normal incidence. The standing wave fringe pattern peak-to-valley ratio of the light intensity becomes maximum, and infinity, when the amplitudes of the object and reference waves are made equal. In this case, the light intensity at the minimum becomes zero, but the characteristic curve expressing the relationship between exposure time and transmittance of the film is very nonlinear near zero exposure. Therefore the ratio of the intensities of the reference and object beam should be something other than unity. The intensity of the reference beam is normally taken 4 to 8 times that of the object beam. Stray light must be avoided as much as possible. A means of doing so is shown in Fig. 9.16. A black sheet of paper is pasted over the unused portion of the mirrors and a black optical fence is set up to intercept the stray light. If there is any leakage of stray light from the laser, such as on those types with ventilation holes, a cover should be made.

Fig. 9.15 Measurement of the ratio of the intensity of the reference beam to the object beam

9.4 Exposure Time

227

Fig. 9.16 Unused parts of mirrors are covered with black paper to minimize undesirable reflections. If the laser has air ventilation holes, they should be covered to eliminate stray light

Fig. 9.17 The shutter is installed so that there is no direct physical contact with the optical bench during the plate exposure

The shutter should not be installed directly onto the optical bench. As shown in Fig. 9.17, its support should be outside the bench so that no vibration propagates into the bench when it clicks. It is important to ensure that the laser beam reflected from the shutter diaphragm in its closed state does not illuminate unexpected areas.

9.4 Exposure Time The types of safety lamps that can be used in a dark room depend upon the type of film which is used. Safety lamps are not recommended when Kodak 649F or Agfa 8E75 are used, because these films are sensitive over a wide range of the visible light spectrum. When a photographic plate is installed onto a plate holder, the emulsion side should be directed toward the incoming laser light. The emulsion side of a glass plate can be determined by rubbing the surface with a moistened finger. The side which is more viscous is the emulsion side. In films, the convention is to indicate

228

9 Laboratory Procedures for Fabricating Holograms

Fig. 9.18 Method for determining the emulsion side

Fig. 9.19 Chart for optimum exposure time for various films used in holography

the emulsion side by notches. When the notches are positioned at the bottom righthand edge, the emulsion side faces the experimenter, as illustrated in Fig. 9.18. In the process of installing either a photographic plate of a film onto the plate holder, vibrations may be induced in the optical bench and optical components. It is best to wait a short while before making an exposure so that these vibrations can die down. Figure 9.19 can be used to estimate the exposure time for a number of commonly used films. Once a rough estimate of the exposure time is made, the method shown in Fig. 9.20 can be used to refine this estimate. In Fig. 9.20, the exposure time is

9.5 Dark-Room Procedures

229

Fig. 9.20 By sliding the black paper after each exposure, several test exposures can be made using one photographic sheet

determined by using a single photographic plate and sequentially intercepting portions of the photographic plate by a black sheet of paper. The amount of interception is decreased stepwise after each trial exposure. Three to five test exposures can be made with one photographic plate. At no time during these exposures should the sheet of paper, or the hand holding the sheet of paper, touch any part of the optical bench. A rule of thumb governing the darkness of the developed photographic plate is that the fine print of a newspaper should be legible through it. Care should be taken that the hologram is not too dark.

9.5 Dark-Room Procedures The procedures for developing a holographic plate are similar to those for fine-grain film [9.4]. The photographic plate is developed by immersing it into chemicals in the following order: 1) Kodak D-165 developer for 3 ∼ 5 min 2) stop bath 30 s ∼ 1 min 3) fixer bath 4 ∼ 7 min 4) hypo eliminator 2 min 5) rinsing by water 30 min ∼ 1 h 6) Kodak Photo-Flo 30 s ∼ 1 min 7) drying 30 min.

9.5.1 Developing The photo-sensitive material used for film is a silver halide compound. When the exposed film is put into the developer, only the silver halide in the exposed portion is hydrated and changed into silver grains, forming the image. The developer does not react with the silver halide in the unexposed portion.

230

9 Laboratory Procedures for Fabricating Holograms

Fig. 9.21 The γ value of the film decreases with an increase in the developing time

A fine-grain developer is used for developing holographic films. There are various kinds of developers but it is safer to use the kind recommended by the film manufacturer. When preparing a developer, first warm half of the required quantity of water to around 50◦ C so that it will be able to dissolve the chemicals more effectively. The rest of the water is added cold to bring the final temperature to as close to 20◦ C as possible. One should be aware that some chemicals do not dissolve if the order of mixing is not adhered to. The developing time is influenced by temperature. The time is shortened by 30 seconds with a 1◦ C increase in the developer temperature. With an increase in the developing time, the value of the slope (called γ ) of the photographic density versus exposure time curve is increased, as shown in Fig. 9.21. In order to prevent uneven development it is necessary to agitate the developer. For the first few second it is necessary to agitate it as thoroughly as possible. The developer is soon oxidated by the oxygen in the air and becomes ineffective. It is, therefore, desirable to use new developer every time a new plate is developed.

9.5.2 Stop Bath The developer is alkali and chemical reaction stops when it becomes even slightly acid. The stop bath, as its name implies, stops the development of the film, and consists primarily of acetic acid. Rinsing with water can be substituted for the stop bath, but the stop bath halts the developing action more evenly. In addition to putting an end to the developing action, the stop bath can prevent the gelatin of the film from swelling up, as well as prolonging the life of the fixer, which is also acid.

9.5 Dark-Room Procedures

231

9.5.3 Fixer The silver halide in the unexposed portion is not affected by the developer. The fixer converts the unaffected silver halides into soluble acid. It does not react with the developed silver grain. The major ingredient of fixer is sodium thiosulfate Na2 S2 O3 · 5(H2 O), or ammonium thiosulfate ((NH4 )2 S2 O3 ). It is sometimes called hypo. When the film is transferred into the fixer, the unexposed silver portion first changes into a transparent substance, and then into a soluble substance. The fixing process should be continued until the silver halide has been transformed completely into soluble substance. The total fixing time is approximately twice the time required for the film to become transparent. The time required to become transparent can be used as a criterion to determine the degree of exhaustion of the fixer. When the time required to reach the transparent stage becomes twice as long as the fresh fixer, the fixer is considered to have reached the exhaustion point and should be replaced by fresh fixer.

9.5.4 Water Rinsing The fixer should be washed off by immersing it in running water at 18–20◦ C for 30 minutes. The rate of water flow should be such that the water is completely replaced every 5 minutes. If the film is immersed in hypo eliminator for two minutes, the water rinsing time can be cut in half.

9.5.5 Drying After removing streaks of water from the photographic plate with a soft squeegee, it is suspended by a corner and dried. It is not good to attempt to accelerate the drying by directly blowing hot air because the gelatin may distort its shape. If the film is immersed in Kodak Photo-Flo for 30 seconds to 1 minute it is easier to squeeze out the water.

9.5.6 Bleaching A phase hologram can be fabricated by bleaching after rinsing the fixed hologram [9.5]. The bleach converts the transmittance variation of the fringe pattern into phase variation. The brightness of the reconstructed image from the phase hologram is much greater than that of the unbleached hologram. When using Kodak chromium intensifier bleach, immerse the hologram in A solution and B solution, wash for 5 to 10 minutes, and then immerse in a copper

232

9 Laboratory Procedures for Fabricating Holograms

chloride (CuCl2 ) solution for about 5 minutes. The copper chloride solution is prepared by dissolving 5 grams of copper chloride in 1 litre of water. The fabrication of the hologram is complete after drip drying.

9.6 Viewing the Hologram Searching for the location of the reconstructed image can be facilitated by placing the developed hologram back into the original film holder, and illuminating it with the same reference beam used during the fabrication. Look through the hologram, and the reconstructed image will appear right where the object was originally situated. It is such a thrilling experience to take a glimpse of your first hologram. Optics is indeed light work!

Chapter 10

Analysis of the Optical System in the Spatial Frequency Domain

In the field of electrical engineering, analysis can be divided into two broad categories that are related by Fourier transforms, namely, analysis in the time domain and analysis in the frequency domain. If phenomena are known in one domain, those in the other domain can be found by means of an appropriate transformation. Similarly, in the field of optics, analysis can be carried out either in the space domain or in the spatial frequency domain. By knowing one, the other can be found by using a Fourier transform relationship. Previous chapters have dealt with problem solving in optics from the space domain viewpoint. In this chapter, treatment in the spatial frequency domain is introduced.

10.1 Transfer Function for Coherent Light Frequency response is widely used to characterize the quality of electrical products such as video recorders, microphones and audio amplifiers. Similarly, spatial frequency response is used to characterize imaging systems such as cameras, microscopes, telescopes, copying machines, and facsimiles.

10.1.1 Impulse Response Function The expression for the output from a system to which an impulse (delta function) signal is applied is called the impulse response function of that system. The impulse response function not only specifies the quality of the system, but also provides an expression for the output of the system for an arbitrary input signal. Only an ideal imaging system can reproduce an infinitesimally small point as an infinitesimally small point. Hence, the impulse response function is a delta function only if the system is ideal. The impulse response functions of all non-ideal systems will always have finite widths. Generally speaking, the system quality degrades with 233

234

10 Analysis of the Optical System in the Spatial Frequency Domain

Fig. 10.1 a–c. Illustration of the impulse response function h(x, y). (a) Point source at origin. (b) Point source at (x 0 , y0 ). (c) Spatially spread source g (x0 , y0 )

increasing response width. As a matter of fact, the width of the impulse response function is used as a system quality criterion. The Fourier-transform spectrum of the delta function spreads across the entire Fourier transform domain with a constant magnitude. If the system is incapable of responding to the entire range of spatial frequencies, the delta function can never be reproduced by the system. It is therefore pertinent to consider the response of an imaging system to a point source input. First, a point source delta function described by δ(x0 )δ(y0 ) is placed at the origin of the input coordinates, as shown in Fig. 10.1 a. Let the impulse response function of this imaging system be h(xi , yi ). Then, the delta function source is moved from the origin to a new point (x0 , y0 ). Assuming a system magnification of unity, the position of the output image should move by the same amount. For simplicity, the sense of the output coordinates is taken such that the magnification is always plus one. The moved image is then expressed by u (xi , yi ) = h (xi − x0 , yi − y0 ).

(10.1)

As a further step, the delta function source is now replaced by a distributed source g, as shown in Fig. 10.1 c. If the distributed source is considered as an ensemble

10.1 Transfer Function for Coherent Light

235

of point sources, the amplitude of the constituent point source at a particular point (x0 , y0 ) is naturally g(x0 , y0 ). The output due to this constituent point source is then g (x0 , y0 )h(xi − x0 , yi − y0 )

(10.2)

and the output image u(xi , yi ) due to all point sources comprising g is ∞ u (xi , yi ) =

g (x0 , y0 )h(xi − x0 , yi − y0 ) d x0 dy0 .

(10.3)

−∞

Since (10.3) is exactly in the form of a convolution, it can be rewritten as u (xi , yi ) = g (xi , yi ) ∗ h(xi , yi ).

(10.4)

Once the impulse response function h(xi , yi ) is known, the output image for any arbitrary input function can be calculated by (10.4). In this respect, the impulse response function is one of the most important functions used to characterize the system. However, it should be noted that (10.3) is true only when the ratio of the translation of the input source to that of the output image is always the same regardless of the location of the source, or in other words, only when the system is shift invariant. In actual optical systems, this condition is satisfied only in very limited regions of the input coordinates.

10.1.2 Coherent Transfer Function (CTF) In the case of an audio amplifier, its frequency characteristics can be easily determined by finding the frequency spectrum distribution of the output by means of a spectrum analyzer (or Fourier transform) when a delta function signal is applied to the amplifier input. This is possible because the input delta function signal has a constant magnitude throughout the entire frequency spectrum. The same analogy is applicable to optical systems and the Fourier transform H ( f x , f y ) of the impulse response function h(x0 , y0 ) specifies the system. In optics, H ( f x , f y ) is called the coherent transfer function or CTF [10.1]. Because (10.4) expresses the output image in the form of a convolution, its Fourier transform is the product of the Fourier transforms of each of the functions being convolved. This fact is one of the advantages of analysis in the spatial frequency domain as opposed to that in the space domain. The Fourier transform of (10.4) is U ( fx , f y ) = G ( fx , f y ) H ( fx , f y ) U ( fx , f y ) . H ( fx , f y ) = G ( fx , f y )

or

(10.5) (10.6)

236

10 Analysis of the Optical System in the Spatial Frequency Domain

Fig. 10.2 Calculation of the coherent transfer function (CTF) of a camera

Exercise 10.1. Using coherent light, a picture is taken by a camera according to the geometry shown in Fig. 10.2. Find the CTF of the camera and determine the highest spatial frequency resolved by this camera when the diameter of its lens is D. For simplicity, treat the question as a one-dimensional problem. It is assumed that the camera is diffraction limited (Sect. 6.5.2). Solution The results obtained in Sect. 6.5.2 can be applied directly. Let the onedimensional pupil function be P(x) = (x/D). From (6.44) the light distribution u(xi ) on the film is x D i u (xi ) = K g − xi ∗ sinc (10.7) M λ d2 where the first factor of (6.44) was assumed constant and represented by K . Comparing the form of this formula with (10.4), the impulse response function h(xi ) is D xi . h (xi ) = K sinc λ d2 Therefore, by Fourier transforming, the coherent transfer function H ( f x ) is obtained as λ d2 λ d2 f x . (10.8) H ( fx ) = K D D The value of this equation is ⎧ D λ d2 ⎪ ⎨K for | f x | < = 2λd D 2 H ( fx ) = (10.9) D ⎪ ⎩0 for | f x | > . 2 λ d2 From (10.9), the CTF has its cut-off at f imax =

D 2λd2

(10.10)

and f imax is the maximum resolvable spatial frequency of the output image. The corresponding maximum resolvable spatial frequency f 0max of the input image is obtained by multiplying f imax by the magnification ratio d1 /d2 , i.e.,

10.2 Spatial Coherence and Temporal Coherence

237

f 0max = D/2λd1 . For the more general case of the two-dimensional pupil function p(x, y), the coherent transfer function is obtained by Fourier transforming both sides of (6.45), so that E¯ ( f x , f y ) = K G (−M f x , −M f y ) · P(λd2 f x , λ d2 f y )

(10.11)

where the first factor of (6.45) was assumed constant. Aside from a constant factor, comparing (10.11) and (10.5) gives H ( f x , f y ) = P (λd2 f x , λ d2 f y ).

(10.12)

An interesting result has been obtained. The coherent transfer function of an imaging system by a lens of finite aperture is obtained simply by replacing the variables of the pupil function by λd2 f x and λd2 f y , aside from a constant factor. This fact makes the calculation of the coherent transfer function of a diffraction-limited imaging system very simple.

10.2 Spatial Coherence and Temporal Coherence Consider a fringe pattern made by Young’s double pinhole, as shown in Fig. 10.3 a. If the phase of the field at one of the pinholes is perfectly coherent with that of the other, a fringe pattern of 100% modulation is observed on the screen. As the coherency between the two pinholes deteriorates, the contrast of the fringe pattern decreases. When the coherency is totally lost, the fringe pattern disappears. Thus Young’s fringe pattern is very often used as a means to measure the coherency between two points in space [10.2]. To make a quantitative measurement of spatial coherency, the spacing between the pinholes is varied. When the spacing is increased, two things happen. One is an increase in the spatial frequency of the fringe pattern and the other is a decrease in the fringe contrast. The limit of the pinhole spacing beyond which the fringe pattern can no longer be observed is called the spatial coherence width and is used as a measure of the spatial coherence between the locations of the pinholes. Next, Fig. 10.3 b shows an arrangement for measuring the temporal coherence of the source. The light from the source is split into two rays. One ray goes straight to the photo-detector while the other ray goes through a delay path made of a movable prism before it reaches the photo-detector. As the prism is moved, the amount of delay in the prism path with respect to the straight path is varied. The output from the photo-detector goes through a series of maxima and minima with the movement of the prism, as shown in the graph in Fig. 10.3 b. At any particular moment, these two rays incident upon the photo-detector will have left the light source at two different times. Consequently, the stability of the source is essential for the two rays to interfere with each other at the location of

238

10 Analysis of the Optical System in the Spatial Frequency Domain

Fig. 10.3 a, b. Methods of measuring spatial and temporal coherence. (a) Young’s fringe pattern for measuring spatial coherency. (b) Arrangements to measure temporal coherency

the photo-detector, thereby producing a modulation of the detector output with the movement of the prism. If the source is not stable i.e., not perfectly temporally coherent, the degree of the modulation of the output decreases as the prism is moved away until finally no modulation with the movement of the prism can be observed. The furthest distance l where the modulation with respect to the movement just disappears is used as a measure of the coherency of the source. This length l is known as the coherence length. The difference t in the departure times of the two rays at this limit of coherency is t = l/c, where c is the speed of light, and t is called the coherence time. The arrangement just described for measuring temporal coherence is often used to measure the stability of a monochromatic source such as a laser. The coherence length of the laser source is an important consideration when fabricating a hologram. The difference between the total path of the reference beam from the laser to the film and the total path of the object beam from the laser to the film must be much less than the source coherence length in order to avoid a washing out of the fringe pattern on the hologram. In fact, the size of the field of view of the hologram is determined by l obtained above.

10.3 Differences Between the Uses of Coherent and Incoherent Light

239

10.3 Differences Between the Uses of Coherent and Incoherent Light One often wonders what are the differences in the resolution of an imaging system when a coherent light source and incoherent light source is used. We will make a comparison in this section. Let the output intensity distribution be I (xi , yi ) when a spatially distributed source g(x0 , y0 , t) is applied as an input to the imaging system. The case of the spatially coherent source is precisely what we have so far dealt with and there is no problem. The output intensity distribution is I (xi , yi ) = |g (xi , yi ) ∗ h (xi , yi )|2 .

(10.13)

When the distributed source is spatially incoherent it is quite a different matter. The rigorous analysis is quite complicated. A simplified version of the analysis will be presented here. The output intensity is given by I (xi , yi ) = u u ∗

(10.14)

where u is the light amplitude on the output plane, and u ∗ is the complex conjugate of u [10.1, 3]. The brackets represent the operation of averaging in time. The source is time dependent and is expressed by g(x0 , y0 , t). Inserting (10.4) into (10.13), the intensity of the output image from the system due to the input field g(x0 , y0 , t) is obtained as1 1

To be more exact, in the process of obtaining u(xi , yi , t), one more convolution with respect to t is necessary ⎡ ⎤ t ⎣ (10.15) g (x 0 , y0 , τ )h(xi − x 0 , yi − y0 , t − τ ) d x 0 dy0 ⎦ dτ. u (xi , yi , t) = −∞

A

In the case of a monochromatic (coherent) source, the time delay due to the system is simply expressed by one phase constant. In the incoherent case, the source is a arbitrary function of t, and the input is normally treated as a chain of impulses whose amplitudes follow the shape of the source function. The response of the system to each one of the impulses has its own trail (impulse response). In order to determine the output, the contributions of all the trails have to be summed up from −∞ to the present time t. Now, (10.15) is expressed in terms of the frequency spectrum: ∞ u (xi , yi , t) =

u(x ˜ i , yi , f ) e j2π f t d f

(10.16)

−∞

where the symbol “∼” means the Fourier transform with respect to time. If (10.16) is used in (10.15), one obtains ∞ u (xi , yi , t) = ej2π f 0 t −∞

g˜ (x 0 , y0 , f ) h˜ (xi − x 0 , yi − y0 , f ) e j2π ( f − f 0 ) t d x 0 dy0 d f

A

(10.17) where f 0 is the center frequency of the spectrum. ˜ i − x0 , yi − y0 , f ) is not necessarily frequency independent. It should be recognized that h(x

240

10 Analysis of the Optical System in the Spatial Frequency Domain

5 I (xi , yi ) =

g (x0 , y0 , t)h(xi − x0 , yi − y0 ) d x0 , dy0 A

×

g

∗

(x0 , y0 , t)h ∗ (xi

−

x0 , yi

−

y0 ) d x0 dy0

6 .

A

Since the choice of a small element area d x0 dy0 for integration is not related to the choice of d x0 dy0 , the area segments d x0 dy0 and d x0 dy0 are independent, and the above integration can be rewritten as 5

h(xi − x0 , yi − y0 ) h ∗ (xi − x0 , yi − y0 )

I (xi , yi ) = A

A

× g (x0 , y0 , t) g

∗

(x0 , y0 , t) d x0 dy0 d x0 dy0

6 .

(10.19)

If it is assumed that the propagation medium is not subject to disturbances such as air turbulance or haze, and does not change with time, (10.19) can be rewritten as h (xi − x0 , yi − y0 ) h ∗ (xi − x0 , yi − y0 ) I (xi , yi ) = A

A

× g (x0 , y0 , t) g ∗ (x0 , y0 , t) d x0 dy0 d x0 dy0 .

(10.20)

Note that t designates the departure time from the source rather than the arrival time at the output plane. In the incoherent case, the phase at one point of the input field has no relationship with that of another point. The time average of the integrand in (10.20) becomes zero, unless the two points are identically the same point, and the radiating source power per unit area is thus expressed by g(x0 , y0 , t) g ∗ (x0 , y0 , t) = |g (x0 , y0 )|2 δ(x0 − x0 )δ(y0 − y0 ).

(10.21)

A spectrometer separates the output spectrum by the action of a grating or a prism whose ˜ i − x0 , yi − y0 , f ) strongly depends upon the frequency. Even with a simple aperture h(x diffraction, the transfer function depends upon the wavelength. If a severe restriction is imposed of a narrow frequency spectrum and a narrow spread of the phase angle (2π f /c)d 1, where d is the maximum distance involved in the geometry, then (10.17) can be rewritten as g (x 0 , y0 , t)h˜ (xi − x 0 , yi − y0 ) d x 0 dy0 (10.18) u (xi , yi , t) = A

˜ i − x0 , yi − y0 , f 0 ) was suppressed. where f 0 in h(x

10.4 Transfer Function for Incoherent Light

241

The value of I (xi , yi ) for the spatially incoherent case is calculated by inserting (10.21) into (10.20) h(xi − x0 , yi − y0 ) h ∗ (xi − x0 , yi − y0 ) I (xi , yi ) = A

A

× |g(x0 , y0 )|2 δ(x0 − x0 ) δ (y0 − y0 ) d x0 dy0 d x0 dy0 . Integrating with respect to x0 and y0 gives h(xi − x0 , yi − y0 ) h ∗ (xi − x0 , yi − y0 ) |g (x0 , y0 )|2 d x0 dy0 I (xi , yi ) = A

=

|h (xi − x0 , yi − y0 )|2 |g (x0 , y0 )|2 d x0 dy0 A

or using the convolution notation, I (xi , yi ) = I0 (xi , yi ) ∗ |h (xi , yi )|2

where

(10.22)

I0 (x0 , y0 ) = |g (x0 , y0 )|2 . The difference between (10.13) for coherent light and (10.22) for incoherent light is the order in which the square modulus and the convolution occur. For coherent light, the intensity is the square modulus of the convolution between the input function and the impulse response function, whereas for incoherent light, the intensity is the convolution between the square modulus of the input function and the square modulus of the impulse response function.

10.4 Transfer Function for Incoherent Light Similar to the case of spatially coherent light, a transfer function is defined for spatially incoherent light and is used for characterizing the optical system. However, for incoherent light, the transfer function is defined in terms of the Fourier transform of the intensity rather than the Fourier transform of the amplitude. Fouriertransforming both sides of (10.22) gives F {I (xi , yi )} = F {|g (xi , yi )|2 } F {|h (xi , yi )|2 }.

(10.23)

Equation (10.23) corresponds to (10.5) of the coherent case. Equation (10.23) is usually normalized with respect to the field intensity of the background light which is the component of light with f x = f y = 0. Incorporating this normalization, (10.23) is rewritten as

242

10 Analysis of the Optical System in the Spatial Frequency Domain

I ( f x , f y ) = I0 ( f x , f y ) H ( f x , f y ) F {I (xi , yi )} I ( fx , f y ) = F {I (xi , yi )} f x = f y =0

where

(10.24) (10.25)

I0 ( f x , f y ) =

F {|g (xi , yi )|2 } F {|g (xi , yi )|2 } f x = f y =0

(10.26)

H ( fx , f y ) =

F {|h (xi , yi )|2 } . F {|h (xi , yi )|2 } f x = f y =0

(10.27)

The expression H ( f x , f y ) is called the Optical Transfer Function (OTF). When the intensity |g(xi , yi )|2 of the input light is a delta function, then I0 ( f x , f y ) becomes unity and I ( f x , f y ) becomes H ( f x , f y ). The optical transfer function is therefore the Fourier transform of the output intensity when an impulse intensity input is applied to the system. Next, it will be demonstrated that the auto-correlation of the CTF is the OTF. The correlation operation, denoted by the symbol , is defined as ∞ X ( fx , f y ) Y ( fx , f y ) =

X (x, y) Y ∗ (x − f x , y − f y ) d x d y.

(10.28)

−∞

For the special case when X and Y are equal, this operation is called auto-correlation. The numerator of the OTF defined in (10.27) can be rewritten as F {|h (xi , yi )|2 } = F {h (xi , yi ) h ∗ (xi , yi )} = H ( f x , f y ) ∗ H ∗ (− f x , − f y ).

(10.29)

Equation (10.29) can be further rewritten using the notation of the correlation operator as (10.30) F {|h (xi , yi )|2 } = H ( f x , f y ) H ( f x , f y ). The denominator of (10.27) is ∞ F {|h (xi , yi )| } f x = f y =0 = 2

H (x, y) H ∗ (x − f x , y − f y ) d x d y| f x = f y =0

−∞ ∞

=

|H (x, y)|2 d x d y.

(10.31)

−∞

Combining (10.27, 30, 31) gives H ( fx , f y ) =

H ( fx , f y ) H ( fx , f y ) . ∞ 77 |H (x, y)|2 d x d y −∞

An example of how to use (10.32) follows.

(10.32)

10.4 Transfer Function for Incoherent Light

243

Exercise 10.2. Find the OTF of the system described in Exercise 10.1 when spatially incoherent light is used in place of coherent light. Also, find the maximum resolvable spatial frequency that this camera can resolve, and compare this resolution with that of coherent light. Solution Inserting (10.8) into (10.32) gives H ( fx ) =

λd2 D 7∞

K

−∞

2

Π λdD2 f x 2 8 2 λd2 x dx K λdD2 D

and therefore,

Π

H ( fx ) =

λd2 f x D

λ d2 fx . D

A comparison between the result of this exercise and that of the previous one is made in Fig. 10.4. The cut-off spatial frequency of the incoherent light shown in Fig. 10.4 b is twice as large as that of coherent light shown in Fig. 10.4 a. Glancing at these figures for the CTF and OTF, it would appear that the resolution of the camera becomes twice as good when incoherent light is used, but this is not necessarily true. It must be kept in mind that, in order to reproduce an output image which resembles the input, the transfer function should have a bandwidth wide enough to cover the entire bandwidth of the spectrum of the input. The spectral bandwidth of the intensity of the input, in most cases, is twice as wide as that of the amplitude. Hence for the same resolution of the output, the OTF has to be twice as wide as the CTF. Unless the bandwidth of H ( f x , f y ) is more than twice that of H ( f x , f y ), the resolution of the output image is more or less the same for either light. Next, the optical transfer function is obtained by a graphical method. Equation (10.12) which defines the coherent transfer function of a diffraction limited imaging system with pupil function P(x,y) is combined with (10.28, 32) to express the optical transfer function. The result is

Fig. 10.4 a, b. Comparison between (a) the coherent transfer function and (b) optical transfer function for Exercises 10.1, 2 respectively

244

10 Analysis of the Optical System in the Spatial Frequency Domain

Fig. 10.5 a, b Graphical method of finding the optical transfer function (OTF). (a) Geometry to assist in the computation of the numerator of H ( f x , f y ). (b) Illustration for the case of a square pupil

7∞ H ( fx , f y ) =

−∞

P (λ d2 x, λ d2 y) P (λ d2 (x − f x ), λ d2 (y − f y )) d x d y 7∞ −∞

[P (λ d2 x, λ d2 y)]2 d x d y

(10.33) where P(λ d2 x, λ d2 y) is real valued. Putting λ d2 x = ξ, λ d2 y = η, (10.33) becomes ∞ 77

H ( fx , f y ) =

−∞

P (ξ, η) P (ξ − λ d2 f x , η − λ d2 f y ) dξ dη ∞ 77 −∞

. [P

(ξ, η)]2 dξ

(10.34)

dη

Figure 10.5 shows an example of graphically obtaining the OTF of an imaging system with a square pupil. Since the pupil function is unity throughout the aperture, the denominator represents the area of the aperture, and the numerator represents the area of overlap of two pupil functions, one of which is displaced by λd2 f x in the ξ direction and by λd2 f y is the η direction. Equation (10.34) can be generalized to the statement of OTF =

area of overlap of displaced pupil functions . complete area of pupil function

(10.35)

Exercise 10.3. The image of an input mask whose distribution function is g(x) = | cos 2π f 0 x| was formed by using an imaging system whose cut-off spatial frequency is 1.2 f 0 . Would it be better to use coherent light illumination or incoherent light illumination?

10.4 Transfer Function for Incoherent Light

245

Fig. 10.6 a,b Comparison of the images of g (x) = | cos 2π f 0 x| when formed by (a) coherent light and (b) incoherent light

Solution g(x), and I0 (x) = |g(x)|2 can be rewritten as | cos 2π f 0 x| =

∞ 2 (−1)n e j2π n·2 f0 x 2 π 1 − (2 n) n=−∞

| cos 2π f 0 x|2 = cos2 2π f 0 x =

1 (2 + e−j4π f0 x + e j4π f 0 x ). 4

The spatial frequency spectra obtained by Fourier transforming these functions are indicated in the top diagrams of Fig. 10.6 a and b. The CTF and OTF, respectively, are indicated by the middle diagram of the same figure. The diagrams at the bottom of the figure are the spatial frequency spectra of the output amplitude or intensity, as the case may be, which were formed by taking the product of the upper and middle diagrams. In the coherent case, U ( f x ) = G( f x )H ( f x ) winds up with only one spectral component located at f x = 0. This means that the output becomes nothing but a plain background. In the incoherent case, on the other hand, there are two spectral components at f x = ±2 f 0 besides the one at f x = 0, and the image can be recovered even though its contrast is not as good as the original. This example is somewhat unusual in that the fundamental spatial frequency of the amplitude is identical to that of the intensity. Usually the latter is twice the former.

246

10 Analysis of the Optical System in the Spatial Frequency Domain

10.5 Modulation Transfer Function (MTF) It is difficult to establish an exact measure of image quality since quality is an entity which usually finds itself subject to human judgement. Nevertheless, certain criteria have been set forth in an attempt to quantify image quality, one of these being the cut-off spatial frequency. This, however, gives no information about the shape of the spatial frequency curve. The MTF (or OTF), which is about to be described, is a better criterion since it applies to the shape of the entire spatial frequency range. The contrast m is defined as m=

Imax − Imin Imax |Imin

(10.36)

where Imax and Imin are the maximum and the minimum of the light intensity. Suppose the input light intensity I (x) is modulated as I (x) = 1 + A cos 2π f x.

(10.37)

Then when A is positive and real, the maximum and minimum intensities are Imax = 1 + A

Imin = 1 − A.

Inserting these into (10.36) the relationship m=A

(10.38)

is obtained. This means that, when the amplitude of the background is taken as unity, m immediately indicates the amplitude of modulation. This fact makes such a definition of m more useful. In general, the contrast of any image which has gone through a normal imaging system is worse than that of the input image. The ratio of the contrast of the output image to that of the input image is defined as the modulation transfer function (MTF) [10.4] contrast of output image . (10.39) MTF = contrast of input image The change in contrast by passing through an optical system is expected to have a lot to do with the optical transfer function that specifies the quality of the system. In the next section, the exact relationship between the MTF and OTF is explored.

10.6 Relationship Between MTF and OTF When such an image as is expressed by (10.37) is put through an imaging system, the output image can be calculated by (10.22),

10.6 Relationship Between MTF and OTF

247

I (xi ) =(1 + m cos 2π f xi ) ∗ |h(xi )|2 ∞ ∞ m |h(x)|2 d x + ej2π f xi |h (x)|2 e−j2π f x d x I (xi ) = 2 −∞

+

−∞

m −j2π f xi e 2

∞ |h(x)|2 ej2π f x d x −µ

∞ |h(x)|2 d x +

=

(10.40)

−∞

m j2π f xi H H + (e j2π f xi H H )∗ ]. [e 2

(10.41)

Since H H is in general a complex number, it is written as H H = |H H | e jφ .

(10.42)

By inserting (10.42) into (10.41) ∞ I (xi ) =

|h(x)|2 d x + m|H H | cos(2π f xi + φ)

(10.43)

−∞

is obtained. The maximum and minimum of (10.43) are ∞ Imax =

|h (x)|2 d x + m |H H |

(10.44)

|h (x)|2 d x − m |H H |.

(10.45)

−∞ ∞

Imin = −∞

Inserting (10.44, 10.45) into (10.36), the contrast of the output image is m out =

m |H H | . 7∞ 2 |h (x)| d x

(10.46)

−∞

Using the definition of (10.39), the modulation transfer function is written as MTF =

m out |H H | . = ∞ 7 m 2 |h (x)| d x −∞

By applying Parseval’s theorem ∞

∞ |h (x)|2 d x =

−∞

|H ( f x )|2 d f x −∞

(10.47)

248

10 Analysis of the Optical System in the Spatial Frequency Domain

to (10.47) and comparing with (10.32), it is easy to see that MTF = |OTF|.

(10.48)

In conclusion, the modulation transfer function has been found to be identical to the absolute value of the optical transfer function. From (10.42) it is also noticed that there is a phase shift of φ, which is called the phase transfer function (PTF). The PTF is sometimes an important quantity to consider. For instance, when φ = π, the contrast completely reverses. If h(x) is symmetric with respect to the origin, then H ( f x ) is a real quantity. (Problem 10.2) and φ = 0 so that no phase shift takes place.

Problems 10.1. Prove that the auto-correlation A A is symmetric with respect to the origin when the function A( f x , f y ) is a real function. 10.2. Prove that the optical transfer function (OTF) is real if the impulse response function h(x, y) is symmetric with respect to the origin. 10.3. Figure 10.7 shows an arrangement to determine the impulse response function. The image of a knife edge placed at ξ = 0 is formed by the imaging system under test. Show that intensity I (x) along x is related to the square of impulse response |h(x)|2 by 1 d I (x) , (10.49) |h (x)|2 = I0 d x where I0 is the intensity of the incident light. 10.4. Find the optical transfer function of a diffraction-limited camera whose pupil function consists of two square apertures side by side, as shown in Fig. 10.8. The

Fig. 10.7 A method of determining |h(x)|2 from the image of a knife edge

Problems

249

Fig. 10.8 Pupil made up of two squares side by side

Fig. 10.9 Octagonal shutter with edge dimension a

Fig. 10.10 Calculation of the overlapped area of two circles for the purpose of determining the optical transfer function of the system with a circular pupil

squares have 1 cm sides and the spacing between the centers of the squares is 3 cm. The wavelength used is centered at 0.5 µm(5 × 10−4 mm). The distance from the lens to the film is 10 cm. 10.5. Figure 10.9 shows a camera shutter in the shape of an octagon with edge dimension a. The distance from the lens to the film is d2 and the wavelength used is λ. Sketch the curve of the optical transfer function along the f x axis by a graphical method.

250

10 Analysis of the Optical System in the Spatial Frequency Domain

10.6. The object of this problem is to determine the optical transfer function of an imaging system with a circular aperture. (a) Show that the shaded portion of the overlapped area of the two circles in the Fig. 10.10 is expressed by d 2 a d d a2 1− . (10.50) cos−1 − S(d) = 2 2a 4 2a (b) Making use of (10.50), find the expression for the optical transfer function of the system and draw its curve. The distance from the lens to the screen is d2 and the wavelength is λ.

Chapter 11

Optical Signal Processing

Optical signal processing has become a useful tool in a wide variety of fields including pattern recognition, computer aided vision, computed tomography, aerial photography, meteorology, image storage, and image improvement, to name a few. As various techniques for optical processing are available that range from the simple to the sophisticated, examples representative of this wide range will be described.

11.1 Characteristics of a Photographic Film Photographic film is one of the most common means for optical recording, so that a study of film characteristics is beneficial for the understanding of optical signal processing. Film is composed of a base material such as glass, acetate, or paper, which has been coated with a photographic emulsion consisting of a suspension of silver halides in gelatin. The silver halides used are silver chloride AgCl, silver bromide AgBr, and silver iodide AgI, where silver iodide is the most sensitive to light and silver chloride is the least sensitive. The silver halides found in print paper are usually silver chloride and silver bromide, whereas those in the negative film are silver bromide and silver iodide [11.1, 2]. When film is subjected to light and developed, the exposed grains of silver halides are changed into metallic silver which creates a change in the light transmittance of the film. The intensity transmittance τ which is defined as the ratio of the transmitted light intensity to the incident light intensity decays exponentially with respect to the increase in d which is a quantity proportional to the amount of metallic silver. τ = Ae−d or d = ln(1/τ ) + lnA. In order to define the degree of opaqueness of the film, a quantity somewhat proportional to the metallic silver content is used D = log10 (1/τ )

(11.1)

where D is called optical density. 251

252

11 Optical Signal Processing

Fig. 11.1 Typical H-D curve used to characterize a photographic film

F. Hurter and V. C. Driffield represented the characteristics of a film by plotting the optical density as a function of the logarithm of the exposure. This characteristic curve is known as the H-D curve and has been widely used. Figure 11.1 shows a typical H-D curve. The important features of this curve will be examined more closely. Even when unexposed, the film is not completely transparent. Consequently, the H-D curve never crosses the origin, but the curve becomes parallel to the horizontal axis near the origin. This region of the curve is called fog, and the turning point is called the toe. As the exposure is increased beyond the toe, the curve becomes linear and it is this linear region that is preferred for most applications. The slope of the linear region is denoted by the symbol γ . The point at the top of the linear region is called the shoulder. Beyond the shoulder there is a region where the optical density decreases (rather than increases) with the exposure time. This phenomenon is called solarization. The formula relating the exposure to the transmittance can be obtained from Fig. 11.1, which yields D = γ [log10 (E) − log10 (E 0 )] so that from (11.1)

(11.2)

E −γ τ= . (11.3) E0 The √ amplitude transmittance t is defined in terms of the intensity transmittance τ as t = τ , and (11.3) can also be expressed as −(γ /2) E . (11.4) t= E0 To be more exact, a phase factor which accounts for the relief and the change in the refractive index in the film should be included, but was omitted for simplicity. The t − E curve demonstrates the relation between transmittance and exposure and is an alternate way of representing film characteristics. Figure 11.2 shows an example of a typical t − E curve.

11.2 Basic Operations of Computation by Light

253

Fig. 11.2 Typical t − E curve used to characterize film

11.2 Basic Operations of Computation by Light Optical signal processing can be described as a combination of certain basic operations which are summarized in this section [11.3, 4].

11.2.1 Operation of Addition and Subtraction Let g1 (x, y) and g2 (x, y) be the input functions for which the operations of addition and subtraction are to be performed. For convenience, these functions will be abbreviated as g1 and g2 provided there is no cause for ambiguity. The input from two perpendicular collimated light beams can be added by means of a beam splitter oriented at 45◦ with respect to the beams, as shown in Fig. 11.3. The lens forms the images of both g1 and g2 on the screen, and the sum g1 + g2 is obtained. It should be noted that if the lens is eliminated, the distribution on the screen becomes the addition of the diffraction patterns of g1 and g2 rather than the addition of the images of g1 and g2 . If the optical path difference of g1 and g2 is made an odd multiple of half wavelengths, the operation becomes a subtraction whereas if the path difference is made an even multiple, the operation becomes an addition. If the glass plate in the left part of Fig. 11.3 is rotated, the path length of the light in the glass changes and the phase of the transmitted light is altered. In applications where performing real-time operations is not essential, the hologram provides another method for adding input functions. The photographic plate is first exposed to the sum of the reference R and the input g1 , and is then reexposed to the sum of the reference R and the input g2 . The transmittance of the hologram, or strictly speaking, the opacity of the hologram, is |R + g1 |2 + |R + g2 |2 =2|R|2 + |g1 |2 + |g2 |2 + R ∗ [g1 + g2 ] + R[g1∗ + g2∗ ].

(11.5)

The reconstructed image from the fourth term in (11.5) represents the sum of g1 and g2 .

254

11 Optical Signal Processing

Fig. 11.3 Optical system for addition of g1 and g2

Fig. 11.4 A method of multiplying g1 by g2

11.2.2 Operation of Multiplication Real-time multiplication of the inputs g1 and g2 can be performed easily by laying one input on top of the other. However, if one input plate is to be moved with respect to the other, damage could result from close contact and a set of lenses is used to avoid scratching the plates, as shown in Fig. 11.4. Lenses L1 and L2 form an inverted magnified, real image of g1 on the plane P2 where the magnification is determined by f 2 / f 1 . (Recall from Sect. 6.4 that an image formed by only one lens introduces a phase error.) The product of g1 and g2 can be obtained by placing g2 on the plane P2 . The lens L3 forms the real image of the product on the screen. A holographic approach may also be used to perform the product operation. In this method, g1 (x, y) and g2 (x, y) are incident on a photographic plate at two different angles as shown in Fig. 11.5. The transmittance of the plate is |g1 (x cos α, y)e−jkx sin α + g2 (x, y)|2 = |g1 (x cos α, y)|2 + |g2 (x, y)|2 + g1 (x cos α, y)e−jkx sin α + g2∗ (x, y) + g1∗ (x cos α, y)ejkx sin α g2 (x, y)

(11.6)

where α is the angle of incidence of the input g1 . When the hologram is illuminated by a reconstructing beam incident perpendicular to the surface of the hologram, the distribution of g1 (x cos α, y)g2∗ (x, y) exp(−jk x sin α) is obtained in the direction defined by the angle α. It should be noted that this product is made by the complex conjugate g2∗ rather than by g2 . The value of the factor exp (jkx sin α) can be made unity by using a prism.

11.2 Basic Operations of Computation by Light

255

Fig. 11.5 a, b A method of multiplication by means of holography. (a) Fabrication of g1 g2∗ . (b)-Reconstruction of g1 g2∗

11.2.3 Operation of Division The operation of division g1 /g2 is equivalent to the product operation of g1 and 1/g2 . The problem of division is thus reduced to obtaining the transmittance of the inverse. For the operation of the inverse, two plates are needed corresponding to the two factors in the right hand side of (11.7), namely 1 ∗ 1 = g . g2 |g2 |2 2

(11.7)

The first factor, 1/|g2 |2 can be realized by referring to (11.4) and using a photographic film with γ = 2. The second factor, g2∗ can be realized by making a hologram with g1 = 1 so that the third term in (11.6) yields the desired g2∗ .

11.2.4 Operation of Averaging Obviously, averaging can be performed by using the operation of addition mentioned in Sect. 11.2.1 [11.5]. However, in that method there is a limit on the number of inputs. The method described below is a real-time operation and has the merit that any unwanted input can be attenuated by observing the output as the inputs are being added. Figure 11.6 illustrates the principle of this operation. As shown in Fig. 11.6 a, the input “A” is placed in the front focal plane of a lens L1 and a mask of III (x/a) made by an array of pinholes is placed in the back focal plane of the lens. The output from the mask is x x ¯ . (11.8) III EM = K A λ f1 a If this output is further Fourier transformed by the lens L2 , the result is f1 ax1 x = K A − ∗ III , E¯ λ f2 f2 λ f2

(11.9)

256

11 Optical Signal Processing

Fig. 11.6 a – c Arrangement for taking an average of several inputs. (a) Formation of many “ A” images. (b) Formation of many “AB” images. (c) Adjustment of d0 such that “A” and “ A” of “A A” image overlap

where the bar represents Fourier transform. In this way, a large number of images of the input “ A” can be created on the screen. Now suppose that the input “A” is replaced by “ AB”. Naturally the output on the screen becomes a series of “ AB” images. As the spacing between “A” and “B” is increased, the “A” starts to overlap the adjacent “B”, as shown in Fig. 11.6 b. After the spacing between “A” and “B” has been adjusted so that the image of “A” exactly overlaps that of “B”, the input letter “B” is replaced by “ A”, then the output becomes a series of “ A” overlapped with “A” images, as shown in Fig. 11.6 c. Extending this concept to an input made up of N number of “A” s, the output on the screen will consist of an array of images, and, except for the outermost elements of the array, each image is the addition of N number of “A” s. The photographs in Fig. 11.7 illustrate the two-dimensional analog of the operation in Fig. 11.6 a. In the case of Fig. 11.7, one lens L1 plays the dual role of Fourier transforming and image forming. Figure 11.8 shows the average of 100 faces, each of which is incomplete in the input stage. As a result of the averaging, all faces become complete.

11.2 Basic Operations of Computation by Light

257

Fig. 11.7 A method for generating several output images from a single input image

Fig. 11.8 Averaging of the 100 imperfect faces

11.2.5 Operation of Differentiation At first glance, forming the derivative of an input g(x) according to (11.10) seems quite a formidable task, i.e. ∂ g (x) = g(x). (11.10) ∂x The method described here makes use of the fact that g (x) = F −1 {F [g (x)]}.

(11.11)

258

11 Optical Signal Processing

Fig. 11.9 a, b A composite mask realizing “2π f x ” consisting of a mask for |2π f x | overlapped with that of the step function for the sign. (a) 2π f filter. (b) Means of achieving 2π f filter

Fig. 11.10 Arrangement for the derivative operation by light

It turns out that taking the Fourier transform of the derivative is much easier than performing the differentiation directly. Once F [g (x)] is obtained, the derivative g (x) can be obtained through the inverse transform according to (11.11). The Fourier Transform of (11.10) is F {g (x)} = j2π f x G( f x )

(11.12)

so that 2π f x G( f x ) is obtained by laying the mask of 2π f x over that of G( f x ). Since the value 2π f x ranges from negative to positive, it has to be realized by two masks as shown in Fig. 11.9. The transmittance of one of the masks is proportional to |2π f | and that of the other is a phase plate step function distribution for attaining the correct sign. Figure 11.10 shows the lens arrangement used to perform the differentiation. The lens L1 performs the Fourier transform of the input on the 2π f x mask, and the lens L2 further Fourier transforms 2π f x G( f x ) onto the screen. The image on the screen is a f 2 / f 1 magnified, inverted image of the derivative, where the inversion results from applying F F instead of F −1 F in (11.11). Figure 11.11 shows an example of the results of the derivative operation [11.6]. Differentiation concludes the analysis of the basic operations. In the following sections, signal processing by combinations of these basic operations will be considered.

11.3 Optical Signal Processing Using Coherent Light

259

Fig. 11.11 a, b By differentiating the input pattern (a), the output pattern (b) is obtained. The derivative was taken only in the horizontal direction [11.6]

11.3 Optical Signal Processing Using Coherent Light A filter is a device used for optical signal processing. Filters used in the spatial frequency domain are called spatial frequency filters or simply spatial filters. This section will be devoted to spatial filtering techniques using coherent light. In the next section, similar techniques using incoherent light will be presented.

11.3.1 Decoding by Fourier Transform Decoding by Fourier transform is a technique based on periodic features in a given input [11.7]. As an example, consider the meteorologist who wishes to find out oceanic wind conditions. The direction and the magnitude of the wind on the surface of the ocean can be determined from the shape of the ocean waves. The period and the direction of the waves are decoded from an aerial picture as follows. The Fourier transform of a portion of an aerial photograph is made, as shown in Fig. 11.12. The transform pattern has a spread in a direction perpendicular to the ocean wave pattern and the amount of the spread is used to determine the spatial frequency distribution of the ocean wave. Figure 11.13 shows the aerial picture amalgamated with the Fourier transform of its local regions obtained in the manner described above. From the figure, the direction of the surface wind and period of the ocean waves is interpreted.

260

11 Optical Signal Processing

Fig. 11.12 A method to decode the wind speed direction from the Fourier transform of the ocean wave pattern

Fig. 11.13 From the Fourier transform of a local region of an aerial picture, the wind direction and magnitude can be decoded [11.7]

11.3.2 Inverse Filters The Fourier transform of an image using a coherent source is given by (10.5), namely (11.13) U ( f x , f y ) = G( f x , f y )H ( f x , f y ) where U ( f x , f y ) is the Fourier transform of the output, G( f x , f y ) is the Fourier transform of the input, and H ( f x , f y ) is the coherent transfer function. For an ideal optical system, the value of H ( f x , f y ) is constant and unity for all spatial frequencies, but in real systems there is a cut-off at some finite spatial frequency and the output is not a faithful reconstruction of the input. The inverse filter is one means of improving the quality of the output image. When the spectrum of the output is multiplied by 1/H ( f x , f y ) we get U ( fx , f y ) = U ( fx , f y )

1 = G( f x , f y ) H ( fx , f y )

(11.14)

11.3 Optical Signal Processing Using Coherent Light

261

so that the output can be made identical to the input. Such a filter with a transmittance T ( f x , f y ) = 1/H ( f x , f y ) is called an inverse filter. The fabrication of the inverse filter is similar to that of the two-mask filter made for the division operation in Sect. 11.2.3. Thus T ( fx , f y ) =

1 1 H ∗ ( f x , f y ). = H ( fx , f y ) |H ( f x , f y )|2

Exercise 11.1 investigates the problem of designing an inverse filter to suit a particular purpose. Exercise 11.1. Suppose that a certain doubly exposed picture consists of identical images separated from each other by 2a. Design an inverse filter to reconstruct a single image from the doubly exposed images. Solution The impulse response function of the double exposure h(x, y) is h(x, y) = δ(x + a) + δ(x − a).

(11.15)

The Fourier transform of (11.15) is H ( f x ) = e−j2π f x a + e−j2 π f x a = 2 cos 2π f x a.

(11.16)

The graph of H ( f x ) appears in Fig. 11.14 a. The transmittance T ( f x , f y ) of the inverse filter is 1 . (11.17) T ( fx , f y ) = 2 cos 2π f x a Equation (11.17) is shown in Fig. 11.14 b. Since there is a section that becomes negative, it has to be realized by a combination of an amplitude mask and a phase mask as shown in Fig. 11.14 c. Even though the principle of the inverse filter is simple, it has the following drawbacks. 1) Sometimes the transmittance of the mask should be infinity, as indicated by Fig. 11.14 b, but as long as passive elements are used, the transmittance can never be larger than unity. It has to be approximated by a distribution such as that of Fig. 11.14 c. 2) The inverse filter has to be designed for individual configurations. 3) It is necessary to match the amplitude and the phase plates properly. 4) The image quality becomes poor near the region where T ( f x , f y ) should be infinite. 5) This method is impractical when the transfer function is time varying in an unpredictable manner, such as the case of propagation through turbulent media. The last two drawbacks in the above list can be compensated for by the Wiener filter which is explained in the next section.

262

11 Optical Signal Processing

Fig. 11.14 a–c The design of an inverse filter to reconstruct a single image from a picture with double images. (a) Coherent transfer function. (b) Transmittance of inverse filter. (c) Amplitude and phase masks making up inverse filter

11.3.3 Wiener Filter Undesirable characteristics that cause a masking of the true image are denoted as noise. When taking aerial photographs, for example, the atmosphere in the form of clouds, mists, and turbulences represents a source of noise which tends to mask the true picture that would be obtained in the absence of atmospheric disturbances. The Wiener filter uses information about the statistical distribution of the noise spectrum in order to improve the noisy image [11.8, 9]. In almost every case, there are some peculiar characteristics in the distribution of the noise spectrum that can be used for processing. The transmittance of the Wiener filter is modulated according to the noise-spectrum distribution by attenuating where the probability of noise is high and by accentuating where the noise probability is low. The transmittance is designed in such a way that the mean-square deviation of the corrected image from the true image is minimized. Let the amplitude distribution of the image with noise be represented by u(x, y) = g(x, y) ∗ h(x, y) + n(x, y)

(11.18)

where g(x,y) is the true image, h(x,y) is the impulse response function in the absence of noise, and n(x,y) is the noise. Let the amplitude of the corrected image be represented by

11.3 Optical Signal Processing Using Coherent Light

g (x, y) = u(x, y) ∗ t (x, y)

263

(11.19)

where t(x,y) is a function whose Fourier transform gives the desired transfer function. The filter is designed such that the mean square deviation of g (x, y) from g(x,y) is minimized, i.e., the quantity E= |g (x, y) − g(x, y)|2 d xd y (11.20) is minimized. The transmittance T ( f x , f y ) that makes the value of E minimum is 1 T ( fx , f y ) = H ( fx , f y )

Φgg ( f x , f y ) Φnn ( f x , f y ) Φgg ( f x , f y ) −2 Φnn ( f x , f y ) + |H ( f x , f y )|

(11.21)

where H ( f x , f y ) is the coherent transfer function of the noiseless system, and Φgg ( f x , f y ) and Φnn ( f x , f y ) are the power spectral densities of the signal and the noise, respectively. Their values are the Fourier transforms of the autocorrelations Rgg (ξ, η) and Rnn (ξ, η), i.e. ⎫ Φgg ( f x , f y ) = F {Rgg (ξ, η)} ⎪ ⎪ ⎬ (ξ, η)}, where Φnn ( f x , f y ) = F {Rnn 7 ∗ (11.22) Rgg (ξ, η) = g g = 7 g (x, y)g(x + ξ, y + η) d xd y ⎪ ⎪ ⎭ Rnn (ξ, η) = n n = n ∗ (x, y)n (x + ξ, y + η) d xd y. Equation (11.21) is the transmittance of the Wiener filter and its derivation and an excellent description are found in [11.8]. The first factor of (11.21) is identical to the inverse filter and the second factor is a consequence of the statistical treatment of the design. The experimentally measured value of H ( f x , f y ) can be combined with the computed value of the second factor of (11.21) to determine the transmittance distribution of the filter. Since the filter has rotational symmetry, a light source shaped according to (11.21) is imaged onto an unexposed filter film, and the film is rotated in order to achieve the desired two-dimensional transmittance. Figure 11.15 illustrates image processing for a picture taken through a perturbing medium. The result using the Wiener filter is compared with that using the inverse filter. It can be seen that the Wiener filter gives a much better picture [11.9].

11.3.4 A Filter for Recovering the Image from a Periodically Sampled Picture In addition to explaining image recovery from sampled pictures, this section provides a good opportunity to illustrate the principles of sampling theory [11.10–12]. An example of a periodically sampled picture is the screened photograph used for half-tone printing in newspapers. The sampled picture is expressed by

264

11 Optical Signal Processing

Fig. 11.15 a–c. Comparison of the results with an inverse filter and Wiener filter [11.9]. (a) A normal picture taken through perturbing medium. (b) Picture processed by an inverse filter. (c) Picture processed by the Wiener filter

gs (x, y) = III

x

III

y

a b If this is Fourier transformed by a lens, it becomes

g(x, y).

G s ( f x , f y ) = ab[III(a f x )III (b f y )] ∗ G( f x , f y )| f x =xi /λ f,

(11.23)

f y =yi /λ f

(11.24)

where a constant phase factor was ignored. The general shape of (11.24) is shown in Fig. 11.16. In the f x , f y plane, the function G( f x , f y ) appears in the form of a two-dimensional array. Each one of the array elements is the Fourier transform of the original picture. A filter made up of an opaque mask with a pinhole in its center is placed in the Fourier transform plane so that only one of the array elements is transmitted. The original image is recovered by inverse Fourier transforming the beam transmitted through the mask. Figure 11.17 shows the optical system for recovering g(x, y). The required condition for the feasibility of such a processing is that the spectral width of G( f x , f y ) be finite. The highest spatial frequency f x B in the x-directon and the highest spatial frequency f y B in the y-direction have to satisfy the conditions 1 1 (11.25) 2 f x B and 2 f yB . a b If (11.25) is not satisfied, the array spectra overlap each other, and G( f x , f y ) can not be retrieved. In other words, the spacing of the sampling in the x-direction has

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265

Fig. 11.16 Fourier transform of gs consisting of a grid of identical functions G( f x , f y )

Fig. 11.17 An optical system used to recover the original image from a periodically sampled image

to be shorter than 1/2 f x B and that in the y-direction has to be shorter than 1/2 f y B . These are precisely the Nyquist’s criteria for the sampling spacing in the sampling theorem. As long as these conditions are satisfied, a perfect image can be recovered. This is a special characteristic of this method.

11.3.5 Matched Filter This filter is used for locating a particular picture g(x, y) in the midst of several other similar pictures. Figure 11.18 demonstrates how to use this filter [11.7]. The particular picture g(x, y) is put at the front focal plane of a lens L1 to produce its Fourier transform G( f x , f y ) in the back focal plane P2 . In that plane P2 , various trial masks of Fourier transforms G ∗ ( f x , f y ) are inserted one by one for examination. When the mask G ∗ ( f x , f y ) is matched with G( f x , f y ), the reading from a meter located in the back focal plane of the lens L2 will be maximum. When a mask of

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Fig. 11.18 Illustration of how a matched filter works. The reading from the light meter becomes maximum when the mask transmittance distribution matches the light distribution immediately in front of the mask

any other distribution is used, the reading from the light meter will be less than this value. Thus the searched for shape can be identified. In better quantitative terms, when the transmittance of the mask is matched and is G ∗ ( f x , f y ), the light transmitted through the mask in Fig. 11.18 is |G( f x , f y )| ejφ ( f x , f y ) | G( f x , f y )| e−jφ ( f x , f y ) . Fourier transform of the input

(11.26)

Transmittance of the mask

The amplitude distribution of the input matches up with that of the transmittance and the output becomes maximum when both distributions are identical. The phase of (11.26) becomes zero throughout the mask and the distribution beyond the mask becomes a parallel beam so that all the output from the lens L2 is converged into the light meter. This system is phase matched as well as amplitude matched, thus providing a great degree of sensitivity. The matched filter can be also used with incoherent light as described in Sect. 11.6. Phase-matching cannot be used in the case of incoherent light. The above method is also applicable to searching for a particular figure hidden inside one large picture. As a specific example, a search for the letter “a” in a picture containing the letter “a” and “b” will be undertaken. As shown in Fig. 11.19, both letters are located on the y axis with “a” being located at (0, l) and “b” at (0, −l). Let the transmittance distribution of the letters “a” and “b” be represented by a(x, y) and b(x, y), respectively. Then the transmittance of the input mask is f (x, y) = a(x, y − l) + b(x, y + l). This input mask is placed in the front focal plane of a lens L1 to obtain its Fourier transform in the back focal plane. The Fourier transform is F( f x , f y ) = e−j2π f y l A( f x , f y ) + ej2π f y l B( f x , f y ) where F {a(x, y)} = A( f x , f y ), F {b (x, y)} = B( f x , f y ). As shown in Fig. 11.19 a, a mask with the transmittance distribution A∗ ( f x , f y ) is placed at the back focal plane of the lens L1 . The lens L2 Fourier transforms the product F( f x , f y )A∗ ( f x , f y ) and projects it onto the output plane. The light distribution on the output plane is

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267

Fig. 11.19 a, b A search for the location of a particular letter by means of a matched filter: (a) search for “a”, (b) search for “b”

a ∗ (x, y) ∗ a(−x, −y − l) + a ∗ (x, y) ∗ b(−x, −y + l) = [a(x, y) a(x, y − l) + a(x, y) b(x, y + l)]∗

(11.27)

where the definition of the correlation (11.29) was used. The first term in (11.27) is the auto-correlation of a(x, y) as defined by ∞

a(ξ, η)a ∗ (ξ − x, η − y − l)dξ dη.

a(x, y) a(x, y − l) =

(11.28)

−∞

The value of a a in (11.28) reaches its maximum ∞ |a(ξ, η)|2 dξ dη −∞

at x = 0, a = −l, and a sharp peak is generated at (0, −l). The second term in (11.27) is the cross-correlation of a(x, y) and b(x, y) as defined by ∞ a(x, y) b(x, y + l) =

a(ξ, η)b∗ (ξ − x, η − y + l)dξ dη.

−∞

The value of the cross-correlation has a small peak at x = 0, y = +l. In conclusion, a large peak is formed at a point (0, −l) and a small peak at (0, l), as shown in Fig. 11.19 a. If the mask is replaced by another mask whose transmittance distribution is B ∗ ( f x , f y ), a similar result is obtained, namely

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Fig. 11.20 a–e The matched filter is used to trace cloud movement [11.7]. (a) Original picture of the cloud. (b) Another cloud picture a short time after (a) was taken. (c) A small section of the cloud picture (a). (d) The correlation between (a) and (c). (e) The correlation between (b) and (c). By using the difference in the positions of the peaks in (d) and in (e) the wind velocity can be calculated

[b(x, y) a(x, y − l) + b(x, y) b(x, y + l)]∗ .

(11.29)

The first term of (11.29) generates a small peak at (−l, 0) and the second term a large peak at (l, 0), as shown in Fig. 11.19 b. The position of the two peaks are the reverse of the previous case. By means of the location of the larger peak, one can find the location of the sought after letter. Figure 11.20 illustrates a method for obtaining the wind velocity from the correlation peaks of cloud pictures taken from the satellite ATS III. The vector diagrams of the wind velocities obtained by this method [11.7] are shown in Fig. 11.21.

11.4 Convolution Filter Since the convolution filter is placed in the Fourier transform plane, it belongs to the category of spatial frequency filters [11.4, 11]. However, filter design is sometimes simpler if it is considered from the point of view of the space domain rather than the spatial frequency domain. The convolution filter is one such filter [11.13]. The design procedure for the filter will be explained by using a specific example. One wants to recover a good photograph from a picture which was mistakenly exposed four times. The four identical images are displaced from one another by the length ∆. The impulse response of such a system consists of 4 delta functions

11.4 Convolution Filter

269

Fig. 11.21 a, b The wind vector diagram obtained from the movement of the correlation peaks [11.7]. (a) In order to determine the cloud movements, the circled sections were first correlated with the picture they belong to and then with a picture taken at a short time later. (b) The vector diagram of the wind obtained from the movement of the correlation peaks

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Fig. 11.22 Impulse response of the quadruple exposure and the operation of h w (x, y) ∗ t (x, y)

spaced by ∆, as shown in the top portion of Fig. 11.22. The mathematical expression for the impulse response h w (x, y) is h w (x, y) =

3

an δ(x − n∆).

(11.30)

n=0

The image with quadruple exposure is given by u w (x, y) = g(x, y) ∗ h w (x, y)

(11.31)

where g(x, y) is the image that would have been obtained with a single exposure. The way to extract g(x, y) from (11.31) is to find a function t (x, y) whose convolution with u w (x, y) becomes g(x, y), namely uw ∗ t = g ∗ hw ∗ t = g

(11.32)

where all functions are understood to be functions of x and y. The condition for (11.32) to be satisfied is (11.33) h w ∗ t = δ(x). Suppose that t (x, y) also consists of 4 delta functions with the same spacing as h w (x, y) and is expressed by t (x, y) =

3

bm δ(x − m∆)

(11.34)

m=0

where the bm are unknown and are to be determined such that (11.33) is satisfied. To perform the operation of the convolution h ∗ t, as shown in Fig. 11.22, t is first flipped with respect to the vertical axis, and then t and h are multiplied together as t is slid in the horizontal direction. If it is assumed that the four exposures are identical and an = 1, (11.33) becomes

11.4 Convolution Filter

271

b0 b0 + b1 b0 + b1 + b2

=1 =0

=0 or b0 + b1 + b2 + b3 = 0 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 1000 b0 ⎢1 1 0 0⎥ ⎢b1 ⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣1 1 1 0⎦ ⎣b2 ⎦ = ⎣0⎦ . b3 0 1111

(11.35)

The solution of this set of simultaneous equations is b0 = 1 b1 = − 1 b2 = 0 b3 = 0.

(11.36)

By inserting (11.36) into (11.34) the transmittance distribution of the filter is found to be like that shown in Fig. 11.23 a. The expression h w (x, y) ∗ t (x, y) now has one extra delta function at x = 4 besides the one at x = 0, as shown in Fig. 11.23 b, so that the output of the filter has two images. The two output images will be separated from each other if the image g(x, y) is not too wide, as shown in Fig. 11.23 c. It will be left as a problem to show that, by increasing the number of terms in t (x, y), the separation between the two delta functions in Fig. 11.23 b is increased.

Fig. 11.23 a–c. Design of a filter to recover the original image from a photograph with four exposures. (a) The transmittance t (x, y) of the convolution filter. (b) Graph of h w (x, y) ∗ t (x, y). (c) Graph of u w (x, y) ∗ t (x, y)

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Figure 11.24 shows an arrangement for picture processing using this filter. The input image u w (x, y) is placed in the front focal plane of the lens L1 . The Fourier transform Uw ( f x , f y ) of the input image is made at the focal plane and illuminates the convolution filter T ( f x , f y ) placed in the same plane, where T ( f x , f y ) = F {t (x, y)}. The output from the filter which is the product of Uw ( f x , f y )·T ( f x , f y ) is then Fourier transformed by the lens L2 to give the final image. An arrangement for fabricating the convolution filter T ( f x , f y ) is shown in Fig. 11.25. The Fourier transform of t (x, y) can be obtained by means of the Fourier transform hologram described in Sect. 8.4.2 where the object in this case corresponds to two pinholes, one of which is reversed in phase by means of a halfwavelength plate. The reference beam is formed by the lens combination of L0 and L1 . Figure 11.26 shows the results of processing a picture of quadruple exposure.

Fig. 11.24 Optical system to recover the original image from a picture with quadruple exposure

Fig. 11.25 Arrangement for fabricating the convolution filter with transmittance T ( f x , f y ). The filter is a Fourier transform hologram of the two pinholes, one of which undergoes a phase reversal

Fig. 11.26 a, b. The recovered image from a picture of quadruple exposure. (a) The picture of the quadruple exposure. (b) The processed image of (a) by means of a convolution filter [11.13]

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273

Fig. 11.27 a–c. A method for designing a convolution filter to process a blurred image. (a) Impulse response function of a moving camera. (b) Transmittance distribution of the filter for processing a blurred image. (c) Graph of h b (x, y) ∗ t (x, y)

A picture blurred by camera motion during the exposure can be treated in the same manner. The blurred picture can be considered to be made up of several closely spaced multiple exposures. The design of the filter is exactly the same as for the quadruple exposure, except that the spacing ∆ has to be made as small as practically possible. The impulse response function h b (x, y) and the transmittance distribution t (x, y) of the convolution filter for a blurred image are presented in Fig. 11.27 a and b. Figure 11.27 c shows the result of the convolution t (x, y) ∗ h b (x, y). The expression for the blurred image is (11.37) ub = g ∗ hb and when this is convolved with t, the image g is recovered, namely u b ∗ t = g ∗ h b ∗ t = g ∗ δ (x) = g.

(11.38)

The rest of the procedure is exactly the same as the convolution filter for the quadruple exposure. Figure 11.28 shows an example of the result of processing a blurred image by means of the convolution filter. The two recovered images are located in positions corresponding to the beginning and the end of the blurred image.

11.5 Optical Signal Processing Using Incoherent Light Optical signal processing with incoherent light cannot utilize the phase information of the image and the sensitivity is then generally smaller than that achieved using coherent light, but the input requirements are less stringent. With incoherent light,

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Fig. 11.28 a–c. The processed image obtained from the blurred image by means of a convolution filter [11.13]. (a) The original image. (b) The processed image obtained by using a convolution filter

it is not necessary to ensure optical flatness of the input and even images printed on such optically irregular surfaces as paper can be used directly as an input.

11.5.1 The Multiple Pinhole Camera A pinhole camera is advantageous because of its almost infinite depth of focus and its usefulness over a broad spectrum of sources, which range from visible light to xrays and γ -rays [11.14]. However, a major disadvantage of the single-pinhole camera is its low sensitivity, which is a consequence of the fact that only a very small fraction of the incident rays are transmitted through the pinhole. The multiple pinhole camera alleviates this disadvantage by letting in more rays. This is particularly important for x-ray and γ -ray sources since it reduces the amount of time necessary for irradiation. The picture taken by this camera is similar to the image of a multiple exposure. The peculiarity of this camera is that the positions of the pinholes are randomly distributed, and this randomness is taken advantage of in processing the image. Figure 11.29 shows the impulse response function of a multiple pinhole camera. Figure 11.30 a shows an arrangement used for taking a picture of the object “F” illuminated by incoherent light using a multiple pinhole camera. If the light intensity distribution of the object is designated by F0 (x, y), the intensity distribution Fm on the camera film is then Fm (x, y) = F0 (x, y) ∗ h m (x, y)

(11.39)

where |h(x, y)|2 = h m (x, y) is called the impulse response function1 . Notice that each pinhole produces an image of the input. Because of a curious property in the 1

In coherent systems, the term “impulse response function” refers to the output amplitude from a system to which an amplitude delta function has been applied. In incoherent systems, impulse response function usually refers to the intensity output from a system to which an intensity delta function has been applied.

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275

Fig. 11.29 Impulse response h m (x, y) of a multiple pinhole camera. The impulse response function h m (x − ξ, y) which is slightly displaced from h m (x, y) is included to show the relationship h m (x, y) h m (x, y) = δ(x)

Fig. 11.30 a, b The multiple image taken by a multiple pinhole camera. (a) Image taken by the multiple pinhole camera. (b) Processing of the image by using the same multiple pinhole camera

geometry, the image can be processed by using the same pinholes that were used for taking the picture. Figure 11.30 b shows this arrangement. The intensity Fc (x, y) of the reconstructed image is Fc (x, y) = F0 (x, y) ∗ h m (x, y) ∗ h m (−x, −y).

(11.40)

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When h m (x, y) is real, we get Fc (x, y) = F0 (x, y) ∗ [h m (x, y) h m (x, y)].

(11.41)

If it can be shown that h m (x, y) h m (x, y) in (11.41) is a delta function, then Fc (x, y) will be identical to F0 (x, y). Recall that the meaning of the h m (x, y) is just the light-intensity distribution on the film when a point source is used as the object, as shown in Fig. 11.29. The distribution h(x − ξ, y) which is shifted from h(x, y) by ξ in the x direction, is included in Fig. 11.29 as an aid to understanding the convolution operation of (11.41). When ξ = 0, the two distribution functions precisely overlap each other and the product of the function becomes maximum. As ξ is gradually increased from ξ = 0, the product of the two functions sharply decreases and becomes zero as soon as the value of ξ exceeds the diameter of the pinhole. The fact that the product becomes zero is a consequence of the random distribution of the pinholes, which ensures negligible probability of two holes overlapping for ξ greater than the pinhole diameter. In short, the correlation h m (x, y) h m (x, y) behaves like a delta function δ(ξ ) δ(η). Another explanation of the operation of the multiple pinhole camera is given in Fig. 11.30. The 3-pinhole camera in Fig. 11.30 a first produces a potograph that has three identical images of the input letter “F”. The same camera is then put to use once again, only this time the photograph taken in Fig. 11.30 a becomes the new input. A second photograph is produced which has a total of nine images, as shown in Fig. 11.30 b. Moreover, when the distance in Fig. 11.30 a is made equal to that in Fig. 11.30 b, three of the nine images will overlap and produce one bright image, while the six remaining images will be randomly distributed. The ratio of the intensity of the brightest image to that of any of the other images increases with the number of pinholes. Fig. 11.31 a shows the image taken by a 7-pinhole camera, and Fig. 11.31 b shows its processed image.

11.5.2 Time Modulated Multiple Pinhole Camera Next, a similar multiple pinhole camera but with a time modulated aperture will be explained. The transmission of each pinhole is varied with respect to time. A specific example of the time modulated pinhole camera is the gamma ray camera illustrated in Fig. 11.32. The principle of the modulation is described as follows. Instead of plugging or unplugging each pinhole, a lead plate with a particular pattern of pinholes is slid over the aperture. As the plate is slid, the transmission of a particular point in the aperture changes between 0 and 1. The change of the transmission at points A, B, and C as a function of time is at the left-hand side of the figure. The mode of the temporal modulation is either random, pseudorandom or sinusoidal. The output image at any specific moment is a superposition of the images that are formed by all pinholes located within the aperture at that moment. The image thus formed is projected onto an array of detectors whose output is connected to

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277

Fig. 11.31 a, b Image taken by a multiple pinhole camera [11.14]. (a) A photograph of the letter “F” taken by a 7-pinhole camera. (b) The reconstruction image obtained by using the photograph in (a) as the camera input

Fig. 11.32 Time modulated multiple pinhole gamma ray camera

the memory of a processing computer. The images are stored preserving the sequence of events. The correlation technique is used to generate a clear image from the superimposed images. The image that would have been formed had only one pinhole been opened, say A alone, is generated by performing the correlation operation between the stored images and the temporal transmission curve for pinhole A. Only the component of the stored images that has the same temporal variation as the transmission curve for A predominates. The contributions of the other components of the stored images to the correlation operation become very small and would eventually reach zero if the temporal modulation curve were absolutely random and infinitely long. Repeating the same with transmission curve B, the image

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that would have been formed had the pinhole B alone been opened and all others closed is obtained. Similarly with all other transmission curves, clear images from the respective pinholes are generated. Now that the images that would have been formed with various locations of the pinholes are available, the depth information of the image can be extracted by computer processing. The depth information cannot be obtained by a singlepinhole camera, because it is a similar situation as that when trying to acquire depth information with only one eye open. This is one of the attractive features of the time-modulated multiple-pinhole camera. Another feature of this method is that the correlation is performed with respect to time rather than with respect to space, and there are no limitations imposed by the finite number of sampling points. The temporal correlation can be continued as long as the dose of γ -ray exposure to the patient or object permits, and as long as the capacity of the computer permits. Figure 11.33 shows the image of a monkey head obtained by this method. A 15 millicurie dose of isotope 43 Tc99 was administered as the γ -ray emitting source. Figure 11.33 a shows the approximate depthwise location of the slices of the image, while Fig. 11.33 b shows the corresponding image slices. This figure demonstrates the image slicing capability of the multiple pinhole γ -ray camera [11.15].

11.5.3 Low-Pass Filter Made of Randomly Distributed Small Pupils An aperture made up of randomly distributed tiny pinholes behaves as a low-pass filter. The cut off frequency of such a filter is calculated from the optical transfer function of each pinhole. The OTF is expressed by the autocorrelation of the pupil function as given by (11.35). Here, the graphical method demonstrated in Fig. 11.5 is used to obtain the OTF. Fig. 11.34 a examines the total overlap area of the two identical pinhole distributions as a function of the relative displacement of the two pupil functions. As the displacement increases, the overlap area decreases and finally becomes zero when the displacement becomes larger than the diameter 2a. That is to say, the integrand of (11.35) becomes zero when λd2 f x > = 2a and the cut-off spatial frequency f i of the output image is fi =

2a . λd2

(11.42)

The OTF of the filter is shown in Fig. 11.34 b. Since the ratio of the sizes of the input to the output images is d1 /d2 , the cut-off spatial frequency f 0 of the input image is f0 = fi

d2 2a = d1 λd1

11.5 Optical Signal Processing Using Incoherent Light

279

Fig. 11.33 a, b Images of monkey head taken by time modulated multiple pinhole gamma ray camera [11.15]. (a) X -ray image of a monkey head to show the approximate location of the reconstructed image. (b) Slices of the reconstructed image

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Fig. 11.34 a, b The pupil function and the OTF of a mask with randomly located tiny pinholes. (a) Two identical pupil functions are slid past each other to calculate the autocorrelation. (b) The OTF of the pupil function shown in (a)

where d1 is the distance between the input image and the lens, and d2 is the distance between the output image and the lens. The procedure for using this filter is simple. The lens is first focussed so as to form the image and then the filter is inserted immediately in front of the lens. Fig. 11.35 shows an example of filtering out the vertical lines from a lined photograph using the low pass filter.

11.6 Incoherent Light Matched Filter It has been already mentioned in Sect. 11.3.5 that the matched filter designed for coherent light can be readily used with incoherent light at the expense of sensitivity. The incoherent matched filter makes use of the intensity matching condition in (11.26), but cannot utilize its phase matching condition.

As pointed out earlier, the advantage of using incoherent light is that optical flatness in the input is no longer necessary. When using coherent light, an input image on a rough sheet of paper such as the page of a book would have to be copied onto a flat photographic plate, but when using incoherent light, this step is eliminated.

11.6 Incoherent Light Matched Filter

281

Fig. 11.35 a, b Filtering of the vertical lines using an incoherent light low pass filter. The filter is made out of randomly located tiny pinholes as shown in Fig. 11.34 a. (a) The original lined photograph. (b) Image whose vertical lines are filtered out by the low pass filter (Courtesy of Le Chateau Champlain, Montreal)

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Another advantage of incoherent light signal processing is that it is not as sensitive to the orientation of the input image as coherent light processing. The disadvantages are relatively poor sensitivity and, in some cases, the difficulty of coupling the output image with other optical systems. Figure 11.36 shows an optical system for processing with an incoherent-light matched filter. This system is quite similar to the coherent-light signal processing system shown earlier in Fig. 11.18. In fact, the matched filter for incoherent light is fabricated by the same method as the coherent matched filter except that the phase information is destroyed by adding a random phase filter. If the phase distribution of the input image is not quite random, a random phase shifter is also placed over the input image. The result obtained using an optical system similar to Fig. 11.36 is shown in Fig. 11.37. The incoherent light matched filter was used to find the location of a cartoon character. A peak in the output of a light meter is obtained as the light meter is scanned across the location of the desired character [11.16].

Fig. 11.36 Optical system for incoherent light matched filtering. The system is quite similar to the case of coherent light filtering shown in Fig. 11.18. The additional plates appearing in this figure are random phase filters

Fig. 11.37 a, b The location of a cartoon character was found by using an incoherent light matched filter [11.16]. (a) The bright spot below the character indicates the location of the character. (b) The output from a light meter scanned over the bright spot

11.7 Logarithmic Filtering

283

11.7 Logarithmic Filtering In this section, a filter which uses photographic film with a logarithmic sensitivity curve is described [11.17]. The logarithmic filter is useful when the signal is contaminated with noise in the form of the product S(x, y) N(x, y) where S(x, y) represents the signal and N (x, y) represents the noise. A picture taken with uneven illumination is one example of noise belonging to this category. In this case, the light distribution g(x, y) of the noisy input is g(x, y) = S(x, y)N (x, y).

(11.43)

The signal is to be extracted using a logarithmic filter. The noisy input image is photographed by a film with logarithmic characteristics. The transmittance t (x, y) of the imaged film is t (x, y) = log S(x, y) + log N (x, y).

(11.44)

The noise term now becomes additive rather than multiplicative. The Fourier transform of (11.44) is formed with a lens, i.e., F {t (x, y)} = F {log S(x, y)} + F {log N (x, y)}.

(11.45)

The term F {log N (x, y)} is removed by means of an opaque absorber painted in the shape of F {log N (x, y)} so that only F {log S(x, y)} is transmitted. A second lens is used to perform another Fourier transform on the transmitted signal so that S(x, y) may be obtained by using film with exponential characteristics. On the other hand, if (11.43) is Fourier transformed, it becomes F {S(x, y)} ∗ F {N (x, y)}.

(11.46)

It is hard to generalize whether it is easier to remove the noise by using the additive form of (11.45) or the convolution form of (11.46). It all depends on the nature of the signal and noise. Quite often, however, the additive form proves more advantageous. Figure 11.38 shows an arrangement for removing the fringe pattern on a picture. The system input consists of a fringe pattern mask superimposed on a transparency made from a photograph of a face. Let the transmittance function of the face be S(x, y) and that of the fringe pattern be N (x, y). The input function is then S(x, y)N (x, y). As preparation for processing, transparencies of log N (x, y) and of log{S(x, y)N (x, y)} are made by photographing the fringe pattern and the input image with logarithmic film. First, the transparency of log N (x, y) is placed in the front focal plane of the lens L1 to obtain the Fourier spectrum F {log N (x, y)} in the back focal plane. In order to fabricate the mask M, a transparent sheet of glass is placed in the back focal plane, and the areas where the Fourier spectrum exists are painted with an opaque absorber. As the next step, the input transparency of log{S(x, y)N (x, y)} is placed in the front focal plane of the lens L1 while the mask M is left in the back focal plane. In this arrangement, the noise spectrum F {log N (x, y)} is blocked by the mask M and

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Fig. 11.38 Logarithmic filtering. The noise component is filtered out by means of a logarithmic filter F {log N (x, y)}

Fig. 11.39 a, b The result of logarithmic filtering [11.17]. (a) Input with a grid pattern. (b) The output image with the grid pattern removed by the logarithmic filter

only the signal spectrum is transmitted into the lens L2 . In the back focal plane of the lens L2 , a photographic plate with exponential characteristics is placed to recover the output image. Figure 11.39 a shows an input image containing vertical lines and Fig. 11.39 b the output image with the lines removed through logarithmic filtering. This method requires the use of films with logarithmic and exponential characteristics. One way of achieving these characteristics is to record the image in two steps; first, it is photographed into a screened photo consisting of tiny dots similar to those used for half-tone photos in a newspaper and next, it is recopied by a regular photographic film. The t − E characteristics of the screened film is quite nonlinear and it happens to approximate the required distribution. By empirically adjusting the two exposure times, the desired characteristics of the film can be achieved.

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11.8 Tomography The Greek word “tomos” means slice or section, and tomography is the technique of producing an x-ray photograph of a selected cross section within the body [11.18]. The picture normally taken by an X-ray camera is a projected shadow of several internal organs, bones, and tissues. For diagnostic purposes, it is often desirable to image a single plane or a single structure without obstructions from structures in other planes. This is accomplished by tomography. This technique was probably first used when the French doctor G. P´elissier (1931) told a patient to keep moving his jaw during the exposure of an x-ray camera so as to eliminate the shadow of the jaw and thus take a clear picture of the patient’s cervical spine located behind the jaw. Tomography can be classified into two broad categories. One is planigraphic tomography or focused tomography, and the other, computed tomography (CT). The former uses blurring of the shadow from the obstructing organ by moving the source and the film in opposite directions. The latter uses a computer to mathematically combine the x-ray images taken from numerous angles and electronically process them to produce a picture of the cross-section or a tomogram.

11.8.1 Planigraphic Tomography Consider the relative geometry between an x-ray source, an object and a film, as shown in Fig. 11.40. With the movement of the x-ray source from left to right, the shadows of the objects move from right to left. The important point is that the

Fig. 11.40 Diagram illustrating the principle of planigraphic tomography

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Fig. 11.41 Geometry for calculating the resolution of the planigraphic tomography

amount of movement of the shadow of the object O is different from that of O . The shadow of O moves from Q 1 to Q 2 , whereas that of O moves from Q 1 to Q 2 . The distance Q 1 Q 2 is longer than Q 1 Q 2 . The speed of the movement of the shadow of O is faster than that of O. If the film is moved to the right at the same speed as that of the shadow of O, a clear picture of O can be formed and that of O is blurred. With movement, images of any points on the surface of the plane f are all sharp and those on all other planes are blurred to varying degrees. Thus, only the objects of the plane f are selectively focused. This is the principle of planigraphic tomography [11.19]. The plane f is called the focal plane. The difference Q 1 Q 2 − Q 1 Q 2 is known as the distance of blurring. The principle can be explained more quantitatively from Fig. 11.41. Because of symmetry, only one half of the movement is considered. The source located at X 0 projects the shadow of the object O to X i . The same source casts the shadow of the object O , which is ∆t away from O, to X i . The thickness 2∆t (which gives the distance of blurring Bm ) is calculated. Using the similarity between the triangles ∆C0 X 0 O and ∆Ci X i O and between ∆C0 X 0 O and ∆Ci X i O one obtains b C0 X 0 a b + ∆t Ci X i = C0 X 0 a − ∆t Ci X i =

(11.47)

where a is the vertical distance from the source to the object O and b is the same from O to the film. The difference between the two equations is Bm b 1 1 X i X i C0 X 0 ∆t + 2 a a b Bm . (11.48) 2∆t = b 1 1 a a + b C0 X 0 The larger the movement 2C0 X 0 , the finer the thickness resolution is.

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Fig. 11.42 a–e. Various types of tomographic movement. (a) Linear, (b) circular, (c) spiral, (d) elliptical, (e) hypercycloidal

If a structure located above or below the focal plane is a linear object, such as a straight bone, and if it is oriented in the same direction as that of the movement, the shadow of such a bone overlaps with the focused image despite the movement and deteriorates the quality of the image. This can be avoided by moving the source in either circular, spiral, elliptical or hypocycloidal paths, as shown in Fig. 11.42. Planigraphic tomography has the advantage that no computation is necessary to see the image, but it has the disadvantage of not being able to effectively remove the shadow effect.

11.8.2 Computed Tomography (CT) Computed tomography is able to remove shadow effects more effectively. As shown in Fig. 11.43, amazingly clear details in the cross-section of a human body can be reconstructed by computer tomography, but they are obtained at the expense of rather involved computation. Consider the case when an x-ray penetrates through a composite material whose layers are characterized by the attenuation coefficients α0 , α1 , . . . , α N −1 (Fig. 11.44). The power I transmitted through the layers is I = I0 exp [−(α0 + α1 + α2 + . . . + α N −1 )∆s]

(11.49)

where I0 is the incident x-ray power, and ∆s is the thickness of each layer. Equation (11.49) can be rewritten as (α0 + α1 + α2 + . . . + α N −1 )∆s] = − ln

I . I0

(11.50)

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Fig. 11.43 Reconstructed image by computer tomography. The spine and ribs are bright white, the liver is the large organ on the left side of the image (patient’s right) and the liquid and gas contents of the stomach are clearly resolved in the upper right (Courtesy of M. J. Bronskill)

The aim is to find the values of α0 , α1 , α2 , . . . , α N −1 by measuring I /I0 . In order to solve for the N unknown values of α0 , α1 , . . . , α N −1 uniquely, it is necessary to measure I /I0 with N different configurations. Expressed in integral form, (11.50) is equivalent to I f (x, y)ds = − ln (11.51) I0 where f (x, y) represents the two-dimensional (or cross-sectional) distribution of the attenuation associated with the layers under test. Each small subdivision shown in Fig. 11.44 is called a picture element, or for short, pixel. Figure 11.45 shows one of the most common modes of scanning used in generating a computed tomogram. Each single scan is called a ray sum. A set of ray sums associated with one direction is called a projection. For example, in the diagram in Fig. 11.45, there are 4 projections, and 8 ray sums in each projection. Computed tomography can be broadly classified into four categories according to the method of computation [11.20]: a) b) c) d)

Back projection method, Algebraic reconstruction technique, Fourier transform method, and Filtered back projection method.

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Fig. 11.44 Ray sum configuration

Fig. 11.45 Scanning geometry of computed tomography

Fig. 11.46 Density distribution f (x, y) of phantom

Throughout the explanation of these four methods, the density distribution f (x, y) shown in Fig. 11.46 will be used as an example. In the terminology of imaging, an object described by a density distribution, such as that shown in Fig. 11.46, is called a phantom. a) Back Projection Method. For simplicity of illustration, only the shadows projected in the two directions corresponding to the x and y axes are considered. The distributions of the projections are shown on the x and y axes in Fig. 11.47. Keep in mind that, in this example, a projected ray value (ray sum) of zero means the ray

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Fig. 11.47 Back projection method

intensity suffered no attenuation in passing through the pixels in its path, while a projected ray value greater than zero indicates the degree of attenuation of the ray as it passed through the pixels. The purpose of this exercise is to extract the shape of the object from the projections. Using Fig. 11.47 as an example, the intensity distributions along the x and y axes are back projected along the x and y directions as shown. The sections where the back projected shadows overlap denote the higher attenuation. The pattern thus obtained is shown by the shading in the squares in Fig. 11.47. Computationally, one can achieve equivalent results to those displayed visually by the shaded areas in Fig. 11.47 as follows. Each pixel is assigned a value which is equal to the total of all the ray sums of the rays passing through it. For example, the upper right pixel in Fig. 11.47 has two rays passing through it, namely, the top ‘row’ ray and the right hand ‘column’ ray. Each of these rays has a ray sum of 1 so that the upper right pixel is assigned a value of 2. Comparing the results of the computation with the density distribution of Fig. 11.46, one finds that the computation correctly predicts that the darkest region is the upper right pixel. However, there is a fictitious attenuation present in two of the computed pixels. The back projection method is simple to implement but it has this disadvantage of creating unwanted streaks. Fig. 11.48 a shows the geometry for deriving a more general expression for the projections. An expression for a straight line passing through a point P at (x, y) is sought. It is to be expressed in terms of r and φ which are the perpendicular distance from the origin to the ray, and its angle to the x axis, respectively. The expression is derived by an elementary vector analysis. Let the position vector of a point P at (x, y) be P, and let the unit vector of the perpendicular distance be rˆ . Then, P = i x + j y, and rˆ = i cos φ + j sin φ. Hence the projection of P to rˆ is given by the scalar product r = P · rˆ = x cos φ + y sin φ. (11.52) This means that for a given (x, y) and φ, the value of r is determined by (11.52).

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Fig. 11.48 Illustration of the back projection method of computed tomography

Let the intensity of the back projected image at a specific point (x, y) be fˆ(x, y). Here, the caret (∧ ) is used to signify the back projected image and, in general, this image will differ somewhat from the original object distribution f (x, y). Notice that the rays in Fig. 11.48 b can be specified uniquely in terms of their r and φ values. As the first step towards computing fˆ(x, y), the computer stores all the ray sums in its memory banks, and at the same time, labels them according to the r and φ values of the appropriate ray. For example, the ray sums passing through the point P in Fig. 11.48 b would be stored as p(r1 , φ1 ) and p(r2 , φ2 ). For each point (x, y) in fˆ(x, y), the computer searches through the data in its memory to find the ray sums that have passed through this point. These ray sums are added together to form fˆ(x, y) thus giving the intensity of the back projected image at the point (x, y). Thus, the back projected image at P in Fig. 11.48 b is p(r1 , φ1 ) + p(r2 , φ2 ). A general expression for the back projection method is fˆ(x, y) =

N −1

px,y (ri , φi )

i=0

where N is the number of projections passing through (x, y). Noting that r can be calculated from x, y, and φ using (11.52), the back projected image at (x, y) can also be written as2

2

All Hi in Fig. 11.48 b are on the same circle and φ ranges from φ = φ0 to φ N covering π radians.

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fˆ(x, y) =

φ0 +π

px,y {r (φ), φ}dφ.

(11.53)

φ0

These expressions will be used later. The merit of this method is that the principle is straight forward but, unfortunately, it does not get rid of streaks in the reconstructed image. b) Algebraic Reconstruction Technique. If there are four unknown intensities of pixels, as shown in Fig. 11.8.2, four ray-sum equations are sufficient to find a unique solution.3 While this is simple when only a small number of pixels are involved, the situation becomes more complicated as the number of pixels increases. For example, if there are 256 × 256 = 65, 536 unknown pixels, this means that 65,536 simultaneous equations must be solved, clearly a formidable task! Instead of this brute force method, a more practical iterative method is considered. This iterative method is actually a combination of the back projection method and the simultaneous equation method. The procedure for the zeroth iteration is exactly identical to the back projection method. The image obtained by the zeroth iteration is then used for the first iteration, as described in the following. New ray sums are calculated from the zeroth-order distribution. These calculated ray sums are then compared with the original exper-

!! ! ! ! ! !1 1 0 0! ! A ! !1! !! ! ! ! ! !0 1 0 1! ! B ! !1! !! ! ! ! ! !0 0 1 1! ! C ! = !0! !! ! ! ! ! !1 0 1 0! ! D ! !1! Solution are A = C = D = 0, B = 1 Fig. 11.49 Simultaneous equations to solve for pixel intensities 3

In formulating simultaneous equations, one should make sure that the equations are independent. For instance, simultaneous equations made by choosing such projections as shown in Fig. 11.50 do not have a unique solution because the equations are not all independent. This can be checked by the determinant, which must be non-zero for a unique solution to exist. Simultaneous equations made by the choice of projections shown in Fig. 11.50 are !⎡ ⎤ ⎡ ⎤ ! !1 1 0 0! A S1 ! ! !0 1 0 1! ⎢ B ⎥ ⎢ S2 ⎥ !⎢ ⎥ = ⎢ ⎥ ! !0 0 1 1! ⎣ C ⎦ ⎣ S3 ⎦ ! ! !0 1 1 0! D S4 The determinant is zero. The determinant of the simultaneous equations in Fig. 11.8.2 is non zero.

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Fig. 11.50 Projections which do not give solutions

Fig. 11.51 a–f. Example for ART tomography. (a) Projection, (b) back projection. (c) First iteration to row pixels. (d) Result of the first iteration to row pixels. (e) First iteration to column pixels. (f) Result of the first iteration to column pixels

imentally obtained ray sums. The difference between two values is divided among all the pixels intercepted by the ray so that the modified ray sum exactly matches the experimentally obtained ray sum. Since a pixel is shared by more than one ray, the iteration in one ray undoes the iteration in the other ray that intercepts the same pixel. Iteration is repeated until a reasonably converged value is reached. Such an iterative method is sometimes called Algebraic Reconstruction Technique (ART). An example of ART processing using the density distribution in Fig. 11.46 is shown in Fig. 11.51. The zeroth iteration is the same as that of the back projection indicated in chart (b). The first iteration is performed with the rows of the matrix. The sum of the first row of chart (b) is 3, and it is 2 in excess of the measured projected ray value shown in chart (a). This excess is evenly taken out from each pixel of the first row as shown in chart (c). The same procedure is repeated with the

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second row. The result with both rows iterated is shown in chart (d). Note that, as far as the sum of the pixels in each row of chart (d) is concerned, they match the measured projection shown in chart (a), but the sums in the columns do not match with the measured projection. The same procedure is repeated with the columns of chart (d) as is illustrated by chart (e) with the result shown in chart (f). Usually, the iteration in the columns undoes the result of the iteration in the rows, hence, only the result of either row or column, in general, matches with the measured projections. With this particular example, however, the values both in row and column match the measured projections and the differences to be used for the next iteration are all zero. The values have converged already after a small number of iterations. As can be seen by comparing the final result in chart (f) of Fig. 11.51 with Fig. 11.46, the values obtained by iteration do not necessarily converge to the correct values even after the values both in row and column have matched the measured projections. This is one of the drawbacks of this method. c) Fourier Transform Method. This analysis is made in the spatial frequency domain. First, f(x, y) is transformed by using an x − y rectangular coordinate system in which the orientation of the axis is taken in an arbitrary direction. The twodimensional Fourier transform with respect to these coordinates is ∞ F( f x , f y ) =

f (x, y) exp (−j2π f x x − j2π f y y) d xd y

(11.54)

−∞

where f x , f y are the Fourier spectra in the x and y directions and where the f x and f y axes are parallel to the x, y axes, respectively. Special attention is focused on the value of F( f x , f y ) when f y = 0. In that case (11.54) becomes ∞ F( f x , 0) = −∞ ∞

= −∞

f (x, y)e−j2π f x x d xd y ⎡ ⎣

∞

⎤ f (x, y) dy ⎦ e−j2π f x x d x.

(11.55)

−∞

The meaning of (11.55) is that the integral in the square bracket is the projection of f(x, y) along x = x in the y direction onto the x axis. This means that the value of the two-dimensional Fourier transform along the f x axis is obtained by Fourier transforming the projection p with respect to x, namely F( f x , 0) = F { f (x)}.

(11.56)

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In the above, the x axis was taken in an arbitrary orientation with respect to the object, and the value of the Fourier transform along any other straight line can be found in the same manner. If the direction of the x axis is rotated in steps through a total angle of 2π radians, the two-dimensional Fourier transform F( f x , f y ) is obtained for the entire f x − f y plane. Once the two-dimensional Fourier transform in the entire plane is known, it is easy to reconstruct f (x, y), which can be obtained by simply inverse Fourier transforming F( f x , f y ). A slightly different explanation concerning the relationship between the Fourier transform and the projection can be given. A double integral is the limit of a sum of area elements, each element of which is multiplied by the value of the integrand at the location of the element. The integrand of (11.54) can be separated into two factors; the first factor is f(x, y) and the second factor exp(−j2π f x x − j 2π f y y). These factors are indicated in Figs. 11.52 a, b. Note that the second factor is precisely the expression of a plane wave propagating in the − f direction, or in the direction connecting the origin to the point at ( f x , f y ) (Sect. 2.3). A new rotated coordinate system (r, s) is introduced, and the r axis is taken along the f direction, ⎫ f = −( f x i + f y j ) = ˆf f ⎪ ⎪ ⎪ f x = fcos φ, f y = f sin φ ⎪ ⎬ (11.57) 2 2 | f | = fx + f y ⎪ ⎪ ⎪ ⎪ f ⎭ φ = tan−1 y . fx

The expression for the second factor now can be written as exp(−j2π f r ). Since there is no restriction on how to slice the bounded area, one way of slicing is in the direction parallel to the equiphase line of the plane wave, as shown in

Fig. 11.52 a, b. Graphical illustration of the performance of the double integral F ( f x , f y ) = 7∞ f (x, y) exp(−j2π f x x − j2π f y y) d xd y. (a) First factor f (x, y). (b) Second factor

−∞

exp[−j2π ( f x x + f y y)]

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Fig. 11.52 a. The double integral can then be calculated by the following two steps. First, the calculation of the projection along the s axis is performed, p(r, φ) = f (x, y)ds. (11.58) s

Next, all the projections are summed along the r axis, ∞ F( f x , f y ) = P( f, φ) =

p(r, φ)e−j2π f r dr.

(11.59)

−∞

The separation into two steps was possible because the value of exp (−j2π f r ) was constant along the s axis. Thus, the single Fourier transform of the projection is the double Fourier transform along the projected line. By repeating the same procedures at each step of the rotation of r , the Fourier transform is obtained in the entire f x , f y space. The Fourier transform method is very elegant, but one difficulty with this method is that the points at which the two-dimensional Fourier transform is determined are in a polar pattern (Fig. 11.53 a). In order to reconstruct f (x, y), F( f x , f y ) must be inverse Fourier transformed. This is often done by using the Discrete Fourier Transform (DFT), outlined in Sect. 7.1; however, the DFT applies to samples of F( f x , f y ) in a Cartesian pattern, as shown in Fig. 11.53 b. A possible solution is to interpolate or extrapolate in order to estimate the values in the rectangular pattern from those in the polar pattern.

Fig. 11.53 a, b. The available DFT in polar coordinates is represented by and those in cartesian coordinates by x. (a) Dots represent the location of the sampled points. (b) A rectangular grid in the spatial frequency domain (only the first quadrant for clarity) superimposed on the available DFT in Polar coordinates

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Fig. 11.54 Two-point interpolation for the value c at P

The method of two point interpolation is illustrated in Fig. 11.54. The value c at point P is to be interpolated from the values a and b of the points located, respectively, d1 and d2 away from the point P. The idea is to weigh the values of a and b according to their proximity to c as follows, a/d1 + b/d2 d2 d1 d1 +a = . (11.60) c=a +b = (b − a) d1 + d2 d1 + d2 d1 + d2 1/d1 + 1/d2 In a similar manner, the interpolated value F(P) at a point surrounded by the values a, b, c and d located d1 , d2 , d3 and d4 away, respectively, as shown in Fig. 11.53 b, is F (P) =

a/d1 + b/d2 + c/d3 + d/d4 . 1/d1 + 1/d2 + 1/d3 + 1/d4

(11.61)

The interpolation needs sizable computation time and is therefore a drawback in this method. An example of the Fourier transform method will be demonstrated in Fig. 11.55 by again using the same object distribution as in Fig. 11.46. From the projections to the x and y axis, the DFT’s along the f x and f y axes are calculated4 using (7.3) with N = 2. Simply using two projections 90 degrees apart is not enough to determine the value at f x = f y = 1 which is needed to find the inverse DFT later. The projection at 45◦ is used to determine the value near f x = f y = 1. In performing the Fourier transform in the diagonal direction, the spacing between the pixels is now √ 2 times as long as that of either in the x or y direction (Fig. 11.55 a). A property of Fourier transforms is that an elongation in one domain results in a contraction in the corresponding Fourier transform domain. Hence increasing the spacing by √ √ 2 in the spatial √ domain results in a contraction of 1/ 2 and expansion in the magnitude √ spatial frequency domain [Recall the similarity theorem: √ by 2√in the F { f (s/ 2)} = 2F( 2 f )]. The results of the Fourier transform are summarized in Fig. 11.55 b. The value of the DFT at f x = f y = 1 is approximated as −0.8 by using the interpolation formula (11.61). It is interesting to compare this interpolated result with the exact value shown in Fig. 11.55 c obtained by Fourier-transforming f (x, y) directly. It is evident that 4

The easiest way to perform the two-dimensional Fourier transform is to first arrange the inputs in a matrix form and perform the one-dimensional transform along the row elements and put back the Fourier transformed results in the respective row of the matrix. Using these new matrix elements, the same process is then carried out for the column elements. The result is the twodimensional Fourier transform.

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Fig. 11.55 a–d. Charts of Fourier transform method of computer tomography. (a) Projections in x, y and diagonal directions. (b) Fourier transform from the projections. (c) Exact Fourier transform of object distribution. (d) Inverse Fourier transform of (b)

there is a sizable error in the interpolated value at (1, 1). Needless to say, it is not usually possible to make this comparison, simply because the object distribution is not known. The expression f (x, y) is finally obtained by inverse Fourier transforming the function F( f x , f y ) and the result is shown in Fig. 11.55 d. The sizable difference between the charts in Figs. 11.46 and 11.55 d is produced primarily by interpolation errors. d) Fourier Transform in Polar Coordinates and Filtered Back Projection. These methods are based on Fourier analysis and back-projection. Writing the distribution function as a Fourier transform gives ∞ F( f x , f y ) e j2π( f x x+ f y y) d f x d f y .

f (x, y) =

(11.62)

−∞

The Fourier transform operation will be manipulated in the polar coordinate system. Equation (11.62) is rewritten using (11.57) as

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2π ∞ f (x, y) =

F( f cos φ, f sin φ) 0

× exp [j2π f (x cos φ + y sin φ)]| f |d f dφ.

(11.63)

Recall that F( f cos φ, f sin φ) was obtained from the single Fourier transform of the projection on the r axis. The axis r is oriented at an angle φ to the x axis, as shown in Fig. 11.52. Now, (11.63) is further rewritten by using (11.52, 59) as ⎫ ⎧ ⎤ ⎡ π ⎨ ∞ ∞ ⎬ ⎣ p (r, φ) e−j2π f r dr ⎦ | f | ej2π f r d f dφ. f (x, y) = ⎭ ⎩ −∞

−∞

measured projection

“filter” in Fourier domain

One dimensional Fourier transform P ( f, φ) Inverse Fourier transform Addition from all directions passing through (x, y)

(11.64)

In order to express (11.64) exactly in the form of an inverse Fourier transform with respect to f , the lower limit of the integral with respect to f was changed to −∞ and the upper limit of the integral with respect to φ was changed to π , and f in the integrand was changed to | f |. It is necessary to use | f | rather than f because the element of area f d f dφ was defined for positive values of f . There are two ways one can approach the calculation of (11.64). The more direct approach (Fourier Transform method in polar coordinates) is described first. This procedure is to obtain the one-dimensional Fourier transform P( f, φ) along the r axis from the measured projections and then multiply the Fourier transformed value by | f |. The next step is to inverse Fourier transform the above quantity with respect to f . The last step is to sum up all the contribution from all angles φ to obtain the final result. In summary, the computation of (11.64) for one particular point (x, y) can be separated into the following steps: 1) Obtain P( f, φ) by Fourier transforming the projection, 2) multiply P( f, φ) by | f |, 3) evaluate the inverse Fourier transform of P( f, φ)| f | at r (φ), which is the projection from the point (x, y) to a particular direction φ, 4) repeat step (3) for different φ’s, 5) sum up all the values obtained in step (4) to find f (x, y) at (x, y) (in a way similar to the back projection), and 6) in order to fill up the values for the entire x, y plane, repeat the same procedures for other points in the entire plane.

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Fig. 11.56 a–d Example for filtered back projection method. (a) Projections. (b) Fourier transform P( f, ϕ) of the projections. (c) Product of p( f, ϕ)| f |. (d) Inverse Fourier transform of P( f, ϕ)| f |. (e) Back projection of (d)

It should be recognized that, but for | f |, (11.64) is exactly the same as (11.53) of the back projection method. The factor | f | is interpreted as a filter with a frequency characteristic of | f |. A filter with such a characteristic attenuates lower spatial frequency components and accentuates higher spatial frequency components. The intensity of a streak is constant across the image plane and it contains more of the lower spatial frequency components, whereas the edges of the contour of the images have abrupt changes and contain more of the higher frequency components. The present filter emphasizes the higher frequency components and is suitable for improving the quality of such an image. The procedure will be illustrated using once again the distribution in Fig. 11.46. In step (1), P( f, φ) is obtained by Fourier transforming the projection in Fig. 11.56 a. The result is shown in Fig. 11.56 b. In the step (2), multiplication by | f | is performed. The origin is taken at (0,0), and the result is shown in Fig. 11.56 c. In step (3), P( f, φ)| f | is inverse Fourier transformed in the φ = 0◦ and φ = 90◦ directions and the result is given in Fig. 11.56 d. In step (4), the final result (Fig. 11.56 e) is obtained by back projecting the distribution (Fig. 11.56 d). It is seen in this example that the contrast of the upper right pixel against the surrounding pixels is stronger than obtained by the method of back projection. Caution should be used when applying the DFT. In digital form, | f | can be represented as in Fig. 11.57 a. It extends from l = −(N /2 − 1) to l = N /2. Now, suppose that pk are the points sampled from the projection and that the Pl are the DFT of these points. If the procedure described in Chap. 7 is used to perform the DFT, the value of the l index for Pl will be either positive or zero. In other words, the Pl values are normally plotted to the right

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Fig. 11.57 a,b. Method of calculating product of Pl and a filter | f | (a) when P−l was generated from PN −l and (b) when | f | was reformed

of the origin (Fig. 11.57 b). Since | f | was plotted symmetrically with respect to the origin, this results in a possible source of confusion when one attempts to multiply Pl by | f |. One approach for removing the confusion is to use the relationship given in Problem 7.2 in order to find the Pl values for negative l. This relationship P−l = PN −l allows the Pl to be transferred to the left of the origin as shown in Fig. 11.57 a. Alternatively, the left half of | f | can be transferred to the right of the origin (Fig. 11.57 b). In the case of the 2 point DFT, the transfer becomes ambiguous and | f | was arbitrarily represented by the sequence of {0, 1} in this example. The second approach to calculating (11.64), called the filtered back projection, will now be described. The filtered back projection has the advantage of shortening the number of steps required through manipulation of (11.64). As far as the integral in f is concerned, (11.64) is in the form of an inverse Fourier transform of a product, so that (11.64) can be rewritten as π f (x, y) =

p (r, φ) dφ

where

(11.65)

p (r, φ) = p(r, φ) ∗ h(r ) ∞ | f |ej2 πr f d f. h(r ) = −∞

(11.66) (11.67)

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Fig. 11.58 Expression of | f | when the range is restricted to −B < f < B

Because this expression is in the form of a convolution, the filtered back projection method is sometimes called the “convolution” method. In a digital computation, the limit of integration −∞ to ∞ in (11.67) is not practical. The assumption is then made that the image has a spatial frequency spectrum extending only up to f = B, so that the image is assumed to be band-limited. In that case, the limits of the integration can be safely replaced by −B to B. With this assumption, (11.66) can be further calculated. The Fourier transform of (11.66) is P ( f, φ) = P( f, φ) · | f |.

(11.68)

Referring to Fig. 11.58, the function | f | with the range limit of B is represented by f f Λ . |f| = B Π 2B B

(11.69)

Inserting (11.69) into (11.68) gives

P ( f, φ) = B P( f, φ) Π

f 2B

f −Λ . B

(11.70)

Note that P( f, φ) is band-limited and is zero outside the band, hence the first term of (11.70) is simply B P( f, φ). Using (2.38), the inverse Fourier transform p (r, φ) becomes p (r, φ) = Bp(r, φ) − B 2 p(r, φ) ∗ sinc2 (Br ). (11.71) The computation of (11.71) is not as bad as it looks. If the spacing ζ of the sampled points is taken as ζ = 1/(2B), which satisfies the sampling condition, then rn , which is the nth sampled point, becomes n/(2B) and ⎧ ⎫ ⎨ 1 n = 0⎬ sinc2 (Brn ) = (nπ1 )2 , oddn . (11.72) ⎩ ⎭ 0, evenn Using (11.72), the calculation of (11.71) is straightforward.

Problems

303

The Filtered Back Projection method uses the same expression (11.64) as used for the Fourier transform method in polar coordinates. The difference is the means of computation. Equation (11.66) is the expression of the projection p(r, φ) filtered by h(r ) whose frequency spectrum is | f |. Equation (11.65) is the expression of the back projection. The Filtered Back Projection method immediately filters the measured projected data and then back projects. More realistically, a band limited filter, such as expressed by (11.69), is used in (11.67). The computation procedure for the filtered back projection is 1) Calculate h(r ) using such an expression as (11.69). 2) Perform the convolution (11.66) between p(r, φ) and h(r ) which is a calculation in one dimension. 3) Use the result of the convolution for the operation (11.65) of the back projection. As a matter of fact, the filtered back projection method is probably the most frequently used method by the manufacturers of computed tomography equipment.

Problems 11.1. The operation of averaging is to be made by using the arrangement shown in Fig. 11.6. The spacing between the A’s is 0.5 mm. The wavelength of the light used is 0.63 µm and the focal length of the lens L1 is 50 cm. What should the spacing a between the pinholes in the mask be? 11.2. In Sect. 11.4, the number of delta functions used to represent the point response function t (x, y) was 4. What would happen if 8 delta functions were used? 11.3. In processing the picture with quadruple exposure in Sect. 11.4, all an ’s were assumed to be unity. What happens if the an ’s are assigned the values: a0 = 1, a1 = 2, a2 = 3, and a3 = 4? 11.4. Explain why the tiny pinholes in the low pass filter of Sect. 11.5.3 should be randomly distributed. What happens if they are arranged periodically? 11.5. Two approaches were described as a means of removing the grid lines from a picture, such as shown in Fig. 11.35 a. One method was to use a filter for periodic noise as described in Sect. 11.3.4, and the other employed a logarithmic filter. Compare the restriction on the value of the sampling period a. 11.6. Figure 11.59 is a picture of a tiger in a cage. One wants to make a picture of the tiger without the cage from this picture. Design two coherent light systems and one incoherent light system for this purpose. The parameters are: a = 0.4 mm, b = 0.5 mm, c = 20 mm, and d = 10 mm. The highest spatial frequency in the picture is f B = 0.8 lines/mm. The focal lengthof the lens to be used

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11 Optical Signal Processing

Fig. 11.59 A picture of a tiger in a cage

Fig. 11.60 Density distribution

is 100 mm. The wavelength of the coherent light to be used is λ = 0.6 µm, and the incoherent light wavelength is centered at λ = 0.6 µm. 11.7. Try all the computer tomographic methods using the density distribution shown in Fig. 11.60. 11.8. Derive (11.72) from (11.71).

Chapter 12

Applications of Microwave Holography

The principle of holography is applicable not only to light waves but also to microwaves and acoustic waves. Light, microwave and acoustic waves are similar in that they are all wave phenomena but they are quite different in the way they interact with material media. For example, a medium which is opaque to light waves may be transparent to microwaves and acoustic waves. There are certain advantages in using microwaves in place of light waves in holography, one of which is that the wavelength of the microwave is about 105 times larger than that of light, thus making it easier to manipulate the fringe pattern. Another advantage is that microwave sources radiating with long-term stability are readily available.

12.1 Recording Microwave Field Intensity Distributions Microwave holography can be classified into three categories according to the methods of reconstructing the image. One method is to reconstruct the image by illuminating the photographically reduced microwave hologram with laser light. The second method is to illuminate the microwave hologram directly with microwaves without photographic reduction. The real image is then projected onto microwave sensitive sheets. The last method uses a computer to form the reconstructed image. In the case of optical holography, recording the light intensity distribution is no problem; it is simply done by means of a photographic plate. However, in microwave holography, the recording of the microwave intensity distribution is a much more complex matter. Numerous procedures for recording the microwave intensity distribution have been proposed. In the following subsections, these procedures will be introduced.

305

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12 Applications of Microwave Holography

12.1.1 Scanning Probe Method Figure 12.1 illustrates a setup for scanning a probe antenna across a frame. The output from the probe antenna is detected and amplified. The amplified signal is used to modulate the intensity of a small lamp attached to the probe carriage. The probe carriage is scanned in the dark and a camera aperture is set open to take the entire intensity distribution of the microwave field. Figure 12.2 shows a microwave field mapped by this method. The sensitivity of the system can be varied by adjusting the gain of the amplifier. It is often 60–70 dB higher than for most of the methods which will be described in this chapter. The limitation of the system is primarily due to mechanical vibration and a slow mapping speed [12.1–4].

Fig. 12.1 Field mapping by a scanning probe

Fig. 12.2 Microwave field mapped by a scanning probe [12.2]

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307

In addition to planar scanning of the probe, there are also non-planar scanning methods where the probe movement is either spiral or cylindrical. Other variations include scanning the object while keeping the probe fixed, or scanning both. The following methods are non-scanned, area mapping types. They all utilize the heat generated by the microwave dielectric loss.

12.1.2 Method Based on Changes in Color Induced by Microwave Heating When a thin film is placed in an E field, a current density i = σ E is induced in the thin film. The film is heated in proportion to the square of this induced current. The heat sensitive film will then change in color [12.5–8]. The first method utilizes filter paper immersed in cobalt chloride (CoCl2 − 6H2 O). The color of this filter paper is light pink but the color changes to blue when the temperature is raised by microwave heating. Figure 12.3 shows an example of microwave intensity distributions recorded by this technique. The pattern is a mode pattern inside a horn antenna connected to a waveguide. The second method utilizes a temperature dependence of the developing time of a Polaroid film. The Polaroid film starts the developing action as soon as the film is pulled through a narrow gap of rollers and the developing reagent fills a layer between the positive and negative. As soon as developing starts, the film is put into the microwave field in question. The developing reagent is a water based material with high dielectric loss, and a good microwave absorber. Those parts of the film irradiated by microwaves raise in temperature. The heated portions develop faster than the unheated portions. If positive and negative are separated at a premature stage of development, the fringes of the color density distribution representing the microwave field is obtained. Figure 12.4 shows a microwave field pattern at 35 GHz measured in this manner using type 58 Polaroid film. The sensitivity of the method is about 10 ∼ 20 mW/cm2 . The third method utilizes the dependence of color on temperature in liquidcrystal film. The simplicity of the preparation is an advantage of this method. Preparation consists of merely painting the liquid crystal over a film of microwave absorber. The sensitivity of the method is about 10 mW cm−2 .

Fig. 12.3 a, b Microwave field mapped by cobalt chloride sheet [12.5]. (a) Inside a microwave horn. (b) Inside a waveguide

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12 Applications of Microwave Holography

Fig. 12.4 Microwave field mapped by Polaroid technique

The fourth method utilizes the color fading rate of photochromic film. The photochromic film is transparent to visible light, but when illuminated with ultraviolet light it changes its color to dark blue. When the ultraviolet light is removed, the color starts fading and becomes transparent again. The fading rate is accelerated by an increase in temperature. By backing the photochromic film with a layer of microwave absorber, the variation of fading rate with temperature is used to map the microwave distribution. Since the photochromic is of non-granular texture, the advantage of this method is that the reconstruction of the microwave holographic image can be achieved by directly illuminating the photochromic sheet with laser light.

12.1.3 Method by Thermal Vision A thermal vision camera can be used to map the heat distribution generated by the microwave illumination on a microwave absorber film [12.9]. The mechanism of the thermal vision is similar to a television camera except that the center of the sensitivity spectrum of the thermal vision camera is not in the visible region but in the infrared region. Figure 12.5 shows the appearance of the thermal vision camera system. It has the capability of detecting a temperature difference of 0.05 ◦ C from a few meters away.

12.1 Recording Microwave Field Intensity Distributions

309

Fig. 12.5 Photo of the Thermal Vision of AGA Corporation

12.1.4 Method by Measuring Surface Expansion When microwaves are directed onto a microwave absorber, a small heat expansion takes place according to the amount of microwave energy absorbed. This surface heating effect can be used to record the microwave intensity distribution, provided a highly sensitive measuring technique is available to detect the minute expansion [12.10, 11]. Interferometric holography discussed in Sect. 8.8.6 is well suited for detecting such a minute change. Best results are obtained when the microwave absorber is fabricated out of a material with a large heat expansion coefficient such as wax or silicone rubber mixed with carbon. Figure 12.6 is a block diagram describing the method. The shaded portions indicate the microwave holography and the remaining unshaded portions refer to the optical interferometric holography. The surface expansion can be recorded by making an interferometric hologram with and without the microwave source on. Figure 12.7 a is the object “R” made out of aluminum sheet. Figure 12.7 b is the pattern recorded by this interferometric method and Fig. 12.7 c is the reconstructed image of the microwave hologram of the object “R”. Another highly sensitive technique for mapping the surface expansion is the Moir´e technique. The Moir´e pattern is best described by giving an example. Suppose one has two nylon stocking fabrics, and one nylon stocking fabric is placed over the other. If each one of the threads in one fabric is precisely laid over those of the other fabric, no pattern is seen, but as soon as one fabric moves even slightly, a big grid pattern which represents the weaving pattern becomes visible. The pattern moves very rapidly with a small movement of the fabric and the amplified movement of the fabric is observed. The same amplification action as described in this simple fabric example is used to detect the minute surface expansion of a microwave absorber due to microwave heating. The microwave intensity distribution thus mapped

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12 Applications of Microwave Holography

Fig. 12.6 Method of mapping microwave field by measuring the expansion distribution of the microwave illuminated absorber by means of an interferometric hologram

Fig. 12.7 (a) A target made of aluminium sheet. (b) A microwave hologram made from a surface expansion pattern measured by an optical interferometric hologram. (c) Reconstructed image obtained by illuminating a reduced microwave hologram with laser light

12.2 Microwave Holography Applied to Diagnostics and Antenna Investigations

311

Fig. 12.8 a, b. Reconstructed image of Moire microwave hologram [12.11]. (a) Object. (b) Reconstructed image of Moire microwave hologram of the object

is used for fabricating a microwave hologram. The optical image of a grid pattern is projected onto the surface of a microwave absorber. A double-exposed photograph of the grid pattern is made with the microwave source on and off. A Moir´e pattern is observed in the portion where the expansion has taken place. The reconstructed image of a microwave hologram made by the Moir´e technique is shown in Fig. 12.8. The object is a small pair of scissors.

12.2 Microwave Holography Applied to Diagnostics and Antenna Investigations Microwave holography can be used to extract information about materials which is not obtainable by optical means alone. Microwave holography has also proven valuable in resolving dilemmas in antenna theory, as well as providing new antenna designs.

12.2.1 “Seeing Through” by Means of Microwave Holography Since microwaves can penetrate into several materials which are optically opaque, microwave holography has been used as a “seeing through” device [12.12–14]. An example of a large scale application is the investigation of the internal structure of the crust of the moon by holographically processing the signal transmitted and received by an orbiting space craft. The largest difference between microwave holography and x-rays as a “see through” device is the film position. The x-ray film must always be behind the object because the x-ray image is the shadow of the object, whereas the film position in a

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12 Applications of Microwave Holography

Fig. 12.9 a, b. Seeing through by microwave holography. (a) Inserting a Kennedy coin and a triangular metal sheet into a purse. (b) Result of seeing through the purse by microwave holography

microwave hologram is almost arbitrary. This flexibility in positioning the film is clearly an advantage in such applications as nondestructive testing and prospecting for the location of mining ore. Figure 12.9 shows the detection of a coin and a triangular metal piece in a purse by means of microwave holography. In this example, the image is reconstructed by microwaves instead of by a laser beam, elaborated as follows. The microwave field intensity distribution is first recorded as a liquid-crystal film color pattern (Sect. 12.1.2). By using properly set color filters, this color pattern is converted into a black and white pattern with an optical density proportional to the microwave field intensity. This black and white pattern is further changed into a screened photo consisting of tiny dots similar to those used for half-tone photos in a newspaper. This screened photo is printed onto a printed circuit board. The printed circuit board is then used as a hologram when the image is reconstructed by an illuminating microwave. The transmittance of the microwave is low in the area of larger

12.2 Microwave Holography Applied to Diagnostics and Antenna Investigations

313

dots, and high in the area of smaller dots. When the printed circuit board is illuminated by microwave, the real microwave image is reconstructed. This reconstructed microwave image can be observed as a color pattern by placing liquid crystal film at the location of the image formation.

12.2.2 Visualization of the Microwave Phenomena Most of the time, the microwave hologram is photographically reduced by 10−4 ∼ 10−5 and the image is reconstructed by laser light. This means that phenomena occurring in the microwave wavelength region are translated to that of a visible wavelength region, as if our eyes had become sensitive to microwaves. Visualization of a radiating monopole is one example of this kind [12.15]. This extended vision into the microwave region has helped resolve the two theories of interpretation of the mechanism of wave radiation from a linear antenna. One theory states that every portion of the wire is responsible for radiation. The other theory states that only the portions at the ends and the center driving-point, i.e. the areas of discontinuity, are responsible. In other words, the question is: does the reconstructed image of a monopole look like a corrugated sausage proportional to a sinusoidal curve, or does it consist of just two bright spots at both ends of the antenna? Figure 12.10 shows the result. The result confirms the latter theory. Once one

Fig. 12.10 a–e. Visualization of microwave phenomena. (a) A radiating monopole antenna used as an object. (b) Reconstructed image in the plane of the antenna. (c) Reconstructed image in the near field zone of the antenna. (d) Reconstructed image in the far field zone. (e) Block diagram for fabricating a microwave hologram

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12 Applications of Microwave Holography

Fig. 12.11 By interlacing the positive and negative films of the microwave hologram, a subtractive hologram is fabricated

hologram is fabricated, the radiation pattern at other arbitrary planes can be formed by the proper combination of the locations of a lens and a screen. This is another attractive feature of this method. The radiation pattern of a half wave dipole is a figure eight and that of a longer antenna produces a pattern similar to a chrysanthemum flower. These radiation patterns can be nicely explained in terms of an interference pattern of three point sources, one at each of the two ends, and a point source in the center.

12.2.3 Subtractive Microwave Holography Because the size of the microwave fringe pattern is about 105 times larger than that of light, it is possible to manipulate the fringe patterns of the microwave hologram to play tricks. An example of one of these tricks is subtractive microwave holography. By making a contact print of the hologram, a hologram with reversed contrast is easily made. The shape of the reconstructed image from the hologram, after the reversal of contrast, is the same as before but the phase of the reconstructed image is reversed [12.16]. A subtractive microwave hologram is made by interlacing a reversed and a regular hologram to form a mosaic type hologram as shown in Fig. 12.11. This hologram is then photographically reduced in size and illuminated by laser light. The resulting reconstructed image represents the difference between the two original constituent images. In order to demonstrate this principle, two holograms were made. In the first, the object consisted of three coins, and in the second, only two coins were used. The phase of the latter hologram was reversed. The subtracted image is shown in Fig. 12.12. By making a microwave hologram based upon this principle, the image of a plasma discharge, free from interference by the reflected wave from the container, is obtained, and the accuracy of the diagnosis is superior to other microwave diagnoses

12.2 Microwave Holography Applied to Diagnostics and Antenna Investigations

315

Fig. 12.12 a–c. Subtractive microwave holography. (a) Reconstructed image of three coins in a purse. (b) Reconstructed image of two coins in a purse. (c) The image of the difference of the above

Fig. 12.13 Visualization of the plasma discharge enclosed in a glass tube. By means of subtractive microwave holography, the electron density of the plasma without the interference of the glass tube can be obtained

Fig. 12.14 a–c. Reconstructed image of the microwave hologram of a plasma tube. (a) Reconstructed image of only the glass tube. (b) Reconstruction of the subtracted image when the plasma discharge is not on. (c) Reconstruction of the subtracted image when the plasma discharge is on

of the plasma discharge. The experimental arrangement is shown in Fig. 12.13 and the results are shown in Fig. 12.14. Figure 12.14 a shows the reconstructed image of the glass tube only, Fig. 12.14 b shows the complete cancellation of the image of the glass, and Fig. 12.14 c shows the image of the plasma discharge only.

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12 Applications of Microwave Holography

Fig. 12.15 a, b. Design principle of a holographic antenna. (a) Fabrication of hologram. (b) Formation of a parallel beam by the combination of the horn antenna and the hologram

12.2.4 Holographic Antenna A holographic antenna is an antenna designed to make use of the duality principle of holography. When a hologram is illuminated by the reference beam, the object beam is reconstructed and, when it is illuminated by the object beam, the reference beam is reconstructed [12.17, 18]. This principle of holography is expressed by the formulae reference beam + hologram → object beam object beam + hologram → reference beam. Figure 12.15 shows a procedure for designing a holographic antenna. As shown in Fig. 12.15 a, a parallel beam from a paraboloid disc is used as a reference beam and a beam from a horn antenna is used as an object beam. Both beams illuminate the holographic plate. The field-intensity distribution across the holographic plate is then measured. Based upon this information, a microwave hologram is made. The microwave hologram in this case is simply a sheet of aluminum with many small windows. The size of the windows is modulated in such a way that its transmittance is proportional to the measured field intensity at that spot. By illuminating this hologram with the diverging beam from the horn antenna, a parallel beam is obtained. Figure 12.16 shows the measured pattern of the holographic antenna fabricated in this manner.

12.2.5 A Method of Obtaining the Far-Field Pattern from the Near Field Pattern The radiation pattern of an antenna is normally measured in the far-field region of the antenna. The distance R, beyond which it is considered to be far field (Sect. 3.3) is D2 , R= λ

12.2 Microwave Holography Applied to Diagnostics and Antenna Investigations

317

Fig. 12.16 Radiation pattern of a holographic antenna [12.18]

Fig. 12.17 Determination of the far-field pattern of a large aperture antenna

where D is the diameter of the antenna under test, and λ is the wavelength of the radiating field. In the case of a very large aperture antenna, however, the value of R which satisfies the far-field condition is so large that it becomes too cumbersome to move the probe antenna around at such a large radius [12.19, 20]. By fabricating a hologram out of the microwave signal measured in the near field (Fig. 12.17), the far-field pattern can be optically observed. The measured amplitude of the near field and the theoretical amplitude of a reference beam are first added. After that, a fringe pattern is drawn with an intensity modulation given by the value of the sum, and this pattern is photographically reduced in size to make a hologram. The microwave image is optically reconstructed by illuminating the hologram by laser light. The field intensity distribution across the antenna aperture is visualized when one looks into the hologram, and the far-field pattern is visualized when the transmitted light is projected onto a screen located some distance away. The distance to the screen can be reduced by observing it at the back focal plane of a converging lens. The optical system is not necessary for obtaining the final results. The results can be computed by using a simplified scattering theory. The far-field pattern of the antenna can be obtained by Fourier transforming the near-field distribution, and the

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12 Applications of Microwave Holography

antenna aperture field can be obtained by inverse Fresnel transforming the near-field distribution. The inverse Fresnel transform is explained as follows. From (3.34), the relationship between the field distribution g(x, y) across the antenna aperture, and the field distribution gn (x, y) across a plane located a distance d in front of the aperture, is given by (12.1) gn (x, y) = g(x, y) ∗ f d (x, y). If the value of g(x, y) is desired, it can be obtained by convolving (12.1) with f −d (x, y), namely gn (x, y) ∗ f −d (x, y) = [g(x, y) ∗ f d (x, y)] ∗ f −d (x, y) = g(x, y) ∗ [ f d (x, y) ∗ f −d (x, y)] = g(x, y),

(12.2)

g(x, y) = gn (x, y) ∗ f −d (x, y).

(12.3)

and therefore Figure 12.18 a shows the antenna used for this experiment, while Fig. 12.18 b shows the field distribution across the aperture obtained by the inverse Fresnel transform of the near field. Figure 12.18 c shows the comparison between the measured far-field pattern and the calculated far-field pattern obtained by Fourier transforming the near-field pattern by a computer. The agreement between the two values is quite good.

12.3 Side Looking Synthetic Aperture Radar The side looking synthetic aperture radar is a radar with a power of resolution which is enhanced by processing the radar echo in a special manner [12.21–23]. The power of resolution of a radar is influenced by the beam width of the antenna. In order to obtain a narrow radiation beam and hence a high resolution, one generally needs an antenna with a large aperture. A real aperture antenna funnels all the information into the system at once, but, for instance, if the target is a ground object that does not change with respect to time, it is not necessary to process the incoming information immediately. In the side looking synthetic aperture radar, all the necessary information about the incoming signal is picked up by scanning a small probe antenna, where the distance scanned is called the synthetic aperture (Fig. 12.19). The resolution of this type of antenna is equivalent to that of an antenna with an aperture equal to the scanned distance. For instance, if the total distance of the scanning is 1 km, and the wavelength is 10 cm, the beam width is found from (4.2) to be 1.14×10−2 degrees, which is an extremely high power of resolution. Unlike other radar antennas, the antenna for the side looking radar has a wide fanshaped beam. When an aircraft uses this antenna to map the terrain below, the targets at larger distances are illuminated for a longer time than those at shorter distances.

12.3 Side Looking Synthetic Aperture Radar

319

Fig. 12.18 a–c. A method of obtaining the far-field pattern out of the near-field pattern of a large aperture antenna [12.20]. (a) The 3 m paraboloid antenna under test. (b) From the measured distribution across a plane located some distance away from the paraboloid antenna under test, the distribution of the field on the paraboloid was calculated and mapped onto facsimile paper. (c) The far-field calculated from the near-field pattern is compared with the measured far-field [12.19]

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12 Applications of Microwave Holography

Fig. 12.19 The beam width of a synthetic aperture antenna is equivalent to that of a large real aperture antenna

As a result, unlike the case of a pulse radar, resolution does not deteriorate with the distance to the target. This is a very desirable feature of the side looking synthetic aperture radar. The recording of the radar signal is performed on a running film as a black and white fringe pattern. The image of the target is reconstructed from this film by means of an optical system. A disadvantage of the radar is that it takes some time before the image is displayed after the signal is received. Moreover, various mechanical devices have to be improvised, at high cost, to fly the probe antenna straight.

12.3.1 Mathematical Analysis of Side Looking Synthetic Aperture Radar Suppose that an aircraft carrying a probe antenna flies over land to make a map. In order to make an accurate map, resolution is required in the direction of the flight path (azimuth) as well as in the direction perpendicular to it (ground range). The power of resolution of the side looking synthetic aperture radar is obtained by the pulse method in the ground range direction and by the synthetic aperture antenna in the azimuthal direction. For simplicity, only one target, located at (x0 , y0 ) in the coordinate system in Fig. 12.20, is considered. Let the distance from the aircraft to the target be r , and let the wave transmitted from the aircraft be represented as S(t) = ae−jωt . The received signal R(t) scattered from the target is 2r R(t) = a exp −jω t − c

(12.4)

(12.5)

12.3 Side Looking Synthetic Aperture Radar

321

Fig. 12.20 Geometry of the synthetic aperture radar

where a is a complex number which absorbs all physical constants including the electrical properties of the scatterer and the transmission loss. The received signal R(t) is first amplified and then mixed with the transmitter signal S(t). The mixed signal is then square-law detected. The output I (x) from the square-law detector is 1 [S (t) + R (t)] [S (t) + R (t)]∗ 2 1 1 = |a|2 (1 + ej2kr + e−2jkr ) 2 2

I (x) =

(12.6)

where

ω c and where, in order to simplify the expression, the amplifier is set such that a is equal to a. Referring to Fig. 12.20, an approximate expression for r is 1 (x − x0 )2 . (12.7) r = r02 + (x − x0 )2 r0 + 2 r0 k=

Inserting (12.3.1) into (12.6) gives I (x) = |a|2

(x − x0 )2 1 + cos k 2r0 + r0

.

(12.8)

Equation (12.8) is equivalent to the expression for a one-dimensional zone plate (Sect. 4.5 and Problem 4.5) with a focal length of r0 /2 centered at x = x0 . Consequently, a straight horizontal line drawn on a film with its opacity modulated in accordance with (12.8) has the properties of a one-dimensional zone plate. When the film is illuminated by a thin parallel beam, the transmitted light will be focused to one vertical line. In reality, there will be other targets at distances other than r0 ; some closer and some further from the plane, and it is necessary to select only the signals associated with the distance r0 , otherwise the resultant signal will not have the form of (12.8).

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12 Applications of Microwave Holography

Fig. 12.21 Recording mechanism of a side looking radar

Details of the selection process are illustrated in Fig. 12.21. The microwave pulse from the oscillator is radiated through the antenna. The returned echo from the targets is fed through a circulator to the detector amplifier. The output from the amplifier is used to modulate the raster of a scope. The raster is scanned only along one line. By using a converging lens, the image of the raster is projected onto film running perpendicular to the scanning direction of the raster. As soon as a microwave pulse is emitted, the scanning of the raster on the scope starts from S to Q along a straight line. The intensity of the raster is stepped up as soon as the signal returns after being bounced from the target. The position of the brightened raster indicates the distance to the target. For instance, let us say that the raster was brightened at R, so that the length SR represents the time taken for the wave to make a round trip to the target. The further away the location of the target, the longer the length SR; and the stronger the reflection from the target, the brighter the point R becomes. Thus, the echoes from the targets at equidistances from the aircraft are recorded along a horizontal line on the film. The horizontal line from one target at the distance r0 , modulated according to (12.8), is represented by the line R0 in Fig. 12.22. A cylindrical convex lens is placed in front of R0 and illuminated from behind by a narrow parallel beam. The light passing through the cylindrical lens undergoes a phase shift of x2 . exp −jk 2f

12.3 Side Looking Synthetic Aperture Radar

323

Fig. 12.22 Principle of reconstructing the image of the synthetic aperture radar

Light passing through both the lens and the film can be expressed as ! ! x2 x2 ! 2! I (x) exp −jk = !a ! exp −jk (12.9) 2f 2f " x02 1 1 1 2x0 2 + exp jk 2r0 + + − x x − 2 r0 r0 2f r0 " x02 1 1 1 2x 0 + exp −jk 2r0 + + + x x2 − 2 r0 r0 2f r0 where in order to simplify the analysis, the speed of the running film is taken to be the same as that of the airplane, and I (x) itself is taken as the transmittance of the film disregarding the film characteristics. If the focal length f of the lens is selected to be r0 /2, the second exponential term in (12.9) becomes x02 1 exp jk 2r0 + exp (−j k x sin θ0 ) where (12.10) 2 r0 sin θ0 =

2x0 , r0

This is an expression for a plane wave propagating at an angle θ0 with respect to r0 , which later becomes the optical axis z in Fig. 12.23. All other terms in (12.9) represent non-planar waves. As far as the y direction is concerned, there is no focusing action and the wave front is cylindrically divergent. Next, the final stage of reconstructing the image from the film will be described using Fig. 12.23. The equi-phase surface of the light transmitted through the lens L1 is at an angle θ0 to the x direction and divergent in the y direction. The shape of the equi-phase surface is similar to year markings on a tree-trunk cross-section in a wedge shape, with the center of the year markings as its apex. If the center of the year markings is placed at an angle θ0 to the x axis, the year markings would then correspond to equiphase surfaces of the transmitted light.

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Fig. 12.23 Method of obtaining the image from the recording of a synthetic aperture radar by means of a lens system

Fig. 12.24 Reconstructed image of the glacier taken by a synthetic aperture radar (Courtesy of the Environmental Research Institute of Michigan)

If a cylindrical lens L2 is placed with its front focal plane at R0 , the wave front of the wave from L2 becomes parallel to both the y direction and the direction angled at θ0 with respect to the x axis, and can be focused to one point by the convex lens L3 . The same can be said about the line R1 in Fig. 12.22. If the focal length f is selected to be r1 /2, the beam propagating in the θ1 direction is obtained. As shown in Fig. 12.23, if a convex lens L1 of a trapezoidal shape is placed such that the focal length decreases as the bottom end is approached, the transmitted wave across the film becomes a combination of plane waves emanating from different locations in the film and propagating in various directions. It can be used to focus multitargets to their respective points and the over-all image is formed. Figure 12.24 shows an image made by a side looking synthetic aperture radar.

12.4 HISS Radar

325

12.4 HISS Radar HISS radar (Holographic Ice Surveying Systems) is a short-range radar which measures distance by detecting the radius of curvature of the reflected wave [12.24]. Using Fig. 12.25 as an analogy, when a stone is thrown into a pond not too far from the bank, the radius of curvature of the wave ripple observed on the bank is small; on the other hand, when the stone is thrown far from the bank, the ripple wave front looks like a plane wave and the radius of the curvature approaches infinity. It is, therefore, possible to determine the distance by knowing the radius of curvature. HISS radar is based upon this principle. Figure 12.26 shows the block diagram of the radar. The principle of the HISS radar is quite different from that of a pulsed or an FM radar which measures distance by the lapse of time. The HISS radar measures distance by the shape of the wave front of the returned signal. Referring to Fig. 12.27, a sharp beam of radiation is formed by the transmitting antenna array. This beam is reflected from all scattering centers S0 (y) located underneath the top surface of the ice. The field distribution along the receiving antenna array from all these scatterers is given by ∞ H (x) = K

∞ 2 S (y) S0 (y) x 0 exp j k y 2 + x 2 dy K exp jk dy 2y y2 y2

y0

y0

(12.11) where S0 (y) is the reflection coefficient of the target at (0,y), and where the approximation, (12.3.1), was used. S0 (y) absorbed exp (jky). The presence of the term y −2 is due to the round-trip divergence of the beam. By changing variables X = x2 Y = (2y λ) (12.11) becomes H (X ) = K

(12.12) −1

(12.13)

∞ S1 (Y ) ej2πX Y dY.

(12.14)

Y0

Equation (12.14) is mathematically recognizable as a form of Fourier transform. H (X ) is the inverse Fourier transform of S1 (Y ). The distribution of the reflection

Fig. 12.25 Simplified illustration of the principle of HISS radar

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12 Applications of Microwave Holography

Fig. 12.26 Block diagram of HISS radar

Fig. 12.27 Coordinate system of a HISS radar

coefficients S1 (Y ) of the targets can therefore be obtained by measuring H (X ) along the X axis and calculating its Fourier transform. The reason that the receiving antenna elements in Fig. 12.27 are not equally spaced is that, by arranging them quadratically, the transformation of (12.12) can be done automatically.

12.4 HISS Radar

327

The other special feature of this radar is that the array elements need not be excited all at once and in fact, the performance can be improved if both transmitting and receiving elements are excited sequentially forming a so-called hologram matrix. Elements of the hologram matrix consist of the signals with every combination of the transmitter and receiver elements. The information about the distance is derived by double FFT of the elements of the hologram matrix. The next subsection explains how to manipulate the hologram matrix in a general manner to obtain the distance information.

12.4.1 Hologram Matrix An ordinary hologram is made with a fixed position of illumination. The hologram matrix is generated by scanning the illuminating source as well as the receiver, thus acquiring information about the object with every geometric configuration of source, receiver and object. The concept of the hologram matrix is best realized in the case of microwave holography because the distribution of the phase of a microwave is much more easily measured than that of light. Consider an antenna array equally spaced along the X axis as shown in Fig. 12.28. Suppose only the mth antenna emits a spherical wave to illuminate the object and only the nth antenna records the signal of the complex scattered field which is denoted by Hnm . If the field Hnm is recorded for each receiving antenna n = 0, 1, 2, 3, . . . , N −1 with every transmitting antenna m = 0, 1, 2, 3, . . . , N −1, one obtains an N × N matrix. This is called the hologram matrix. Equation (12.15) is a 4 × 4 hologram matrix, and represents the case of 4 element transmitting and 4 element receiving antennas,

Fig. 12.28 Antenna array to form hologram matrix

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12 Applications of Microwave Holography

R0 R1 R2 R3

T0 H00 H10 H20 H30

T1 H01 H11 H21 H31

T2 H02 H12 H22 H32

T3 H03 H13 . H23 H33

(12.15)

The element H13 , for example, is the complex field when the field emitted by transmitter 3 is recorded by receiver 1. The manner in which the HISS radar derives the depth information from the hologram matrix will be explained using Fig. 12.28. The signal received by R1 is delayed from that by R0 by 2π ∆φ = ∆l, (12.16) λ where ∆l is the difference between the path from the target to R0 and the path from the target to R1 . Those of 2, 3, . . . , N − 1 are delayed by 2∆φ, 3∆φ, . . . , (N − 1)∆φ. However, the value of the phase shift ∆φ decreases with the distance h of the target and is given from Fig. 12.28 by 2π √ 2 h + L2 − h ∆φ = N 1−1 λ 1 π L2 . (12.17) N −1 λ h The signals received by each antenna are X 0 = Eh X 1 = E h ej∆φ X 2 = E h ej2∆φ .. . Xn X N −1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ jn∆φ ⎪ ⎪ = Eh e ⎪ ⎭ j(N −1)∆φ = Eh e

.

(12.18)

If the phase shifters W0 , W1 , . . . , W N −1 are installed, one to each port as indicated in Fig. 12.28, the output y from the array is y=

N −1

X n Wn .

(12.19)

n−0

The output y will become a maximum if the values of the phase shifters are chosen in such way to exactly compensate the delays at each port, i.e. ⎫ W0 = 1 ⎪ ⎪ ⎪ ⎪ W1 = e−j∆φ ⎪ ⎪ ⎪ −j2∆φ ⎬ W2 = e . (12.20) .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ Wn = e−jn∆φ ⎪ ⎭ −j(N −1)∆φ W N −1 = e

12.4 HISS Radar

329

In other words, the antenna array is focused by the phase shifters to the distance h specified by (12.17) and the focal distance can be controlled by the values of the phase shifters. In fact, if a digital system is employed, the phase shifters, in a physical sense, are not needed. One can simply multiply numerically the input X n by the complex number Wn specified by (12.20). Notice that (12.18 and 12.20) are remarkably similar to certain expressions developed in connection with the DFT in Chap. 7. Equation (12.18) forms a geometric series which looks exactly the same as the inverse Discrete Fourier Transform and (12.20) forms a series which looks exactly the same as the weighting factors of the DFT. Let us take the DFT of (12.18), ym = E h

N −1 n=0

2π exp j ∆φ − m n . N

(12.21)

At the particular distance h [or ∆φ which is related to h by (12.17)] such that the innermost bracket in the exponent becomes zero, the value of ym peaks, and the antenna is focused to this distance. The term (2 π/N ) m n in (12.21) can be considered as the n th phase shifter. The mth spectrum of (12.21) represents the array output when the focused distance h (or ∆φ) is 2π m and from(12.17) N L2 N 1 hm = . 2 (N − 1) λ m

∆φ =

(12.22)

The spectrum (y0 , y1 , y2 , ym , y N −1 ) represents the outputs when the array is focused to the distances (h 0 , h 1 , h 2 , h m , h N −1 ). Now let us return our attention to the manipulation of the hologram matrix of (12.15). The DFT of the first column, denoted by H¯ m0 , F0 H¯ 00 H¯ 10 H¯ 20 H¯ 30

(12.23)

H¯ (N −1)0 represents the output from the array when the receiver array is focused to the distances (h 0 , h 1 , h 2 , h m , . . . , h N −1 ) and only one transmitter T0 is used to irradiate the targets. Similarly, the DTF of the second column denoted by

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12 Applications of Microwave Holography

F1 H¯ 01 H¯ 11 H¯ 21 H¯ 31 .. .

(12.24)

H¯ (N −1)1 represents the output from the array when the receiver array is focused to similar distances but when the transmitter T1 alone is used to irradiate the targets. Exactly the same argument holds for the entire transmitter array except that the direction of propagation is reversed. The transmitting beam can be focused to various distances by installing the phase shifters. The focal distance h and the amount of the phase shift is likewise determined by (12.17). Thus, the Fourier transform of the row elements of the matrix R0 H¯ 00 , H¯ 01 , H¯ 02 , H¯ 0m , H¯ 0N −1

(12.25)

corresponds to the array output when the transmitter array is focused to the distances (h 0 , h 1 , h 2 , h m , h N −1 ) and only one receiver R0 is used to receive the reflected signal from the target. Therefore, the double DFT of both column and row elements of the matrix represents the signal when both transmitting and receiving arrays are focused. The diagonal elements of F0 F1 F2 F3 R0 H¯ 00 H¯ 01 H¯ 02 H¯ 03 R1 H¯ 10 H¯ 11 H¯ 12 H¯ 13 (12.26) R2 H¯ 20 H¯ 21 H¯ 22 H¯ 23 R3 H¯ 30 H¯ 31 H¯ 32 H¯ 33 represent the output of the radar as the focii of both transmitter and receiver are moved together along the y axis. Exercise 12.1. Show the principle of operation of a hologram matrix radar taking the case of N = 2 as an example. Solution The geometry shown in Fig. 12.29 is considered. If the signal from T0 to R0 is taken as unity, the signal from T0 to R1 is ejφ = A, and that from T1 to R1 is A2 , where π L2 2π 2 h + L2 − h . (12.27) ∆φ = λ λ h The hologram matrix is then T0 T1 R0 1 A R1 A A 2

(12.28)

12.4 HISS Radar

331

Fig. 12.29 Two element hologram matrix radar

The two point Fourier transform of the column is F0 F1 R0 1 + A A + A 2 , R1 1 − A A − A 2 and the further transform in the row is F0 F1 R0 (1 + A)2 1 + A2 . R1 1 − A2 (1 − A)2

(12.29)

If the object is located at h 0 = ∞, then A=1 and the matrix elements become F0 F1 R0 4 0 R1 0 0

(12.30)

The first element of the diagonal elements becomes large indicating the location of an object at h = ∞. If, however, the object is located at h = L 2 /λ or A = −1 the matrix elements become F0 F1 R0 0 0 (12.31) R1 0 4 indicating that the location of the object corresponds to m = 1, and this location is given by (12.22) as h 1 = L 2 /λ.

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12 Applications of Microwave Holography

Fig. 12.30 Photograph of HISS radar

Fig. 12.31 a, b. Profile of the thickness of ice measured by the HISS radar. (a) Echo from top and bottom surfaces. (b) Intensity modulation of (a)

Figure 12.30 shows the appearance of the HISS radar and Fig. 12.31 shows the results obtained. Images of high quality can be obtained by this radar. An additional feature of this radar is that it needs neither pulse shaping nor frequency modulation as does a pulse or FM radar.

Chapter 13

Fiber Optical Communication

Fiber optical communication is based on the principle of light transmission through a fine glass fiber by total internal reflection. A first demonstration of the guiding of light by total internal reflection was given by John Tyndall in 1870. In front of an audience of the Royal Academy of London, he demonstrated that light illuminating the top surface of water in a pail can be guided along a semi-arc of water streaming out through a hole in the side of the pail. Tremendous progress has been made since then, and thin glass fiber is now a viable means of transmission of light for communications.

John Tindall’s experiment in 1870

Figure 13.1 shows the basic components of a fiber optical communication system. At the transmitting end, an electrical signal is converted into a modulated light signal which is then transmitted through a fiber optical cable. At the receiving end, the signal is converted back into an electrical signal.

333

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13 Fiber Optical Communication

Fig. 13.1 Block diagram of fiber optical communication

13.1 Advantages of Optical Fiber Systems The optical fiber is a very versatile transmission medium. The advantages itemized below explain why optical fibers have become such an attractive alternative to conventional transmission media in applications such as telecommunications.

13.1.1 Large Information Transmission Capability Since the frequency of light is roughly 104 times than that of microwave frequencies, the information transmitted by an optical fiber could theoretically be increased by four orders of magnitude. In other words, ten thousand microwave communication systems in parallel may be replaced by just one thin optical fiber. This large information capacity can best be utilized in the trunk lines of telecommunications.

13.1.2 Low Transmission Loss In the case of ordinary window glass, it takes only a few centimeters of transmission for the light intensity to drop to half of the incident intensity. This attenuation is largely due to impurities in the glass. The manufacture of ultra-pure glass fibers has been refined to such an extent that a good optical fiber takes more than 10 kilometers of transmission before the intensity drops to a half. This low transmission loss means a longer span between repeater stations, thus reducing the maintenance cost in a long haul telecommunication network.

13.2 Optical Fiber

335

13.1.3 Non-Metallic Cable Since the optical fiber transmission medium is non-metallic, it does not suffer from electromagnetic noise problems. With optical fibers, there is no stray electromagnetic pick-up, electromagnetic tapping, or ground potential problems. There is no risk of sparks or overheating which may occur with short-circuited metallic lines. These special features of a fiber optical link are utilized in many an adverse environment. The fiber optical link is used at electrical power stations where the high noise level associated with high-voltage corona discharge is a problem. The electrical power station also benefits from fiber-optic links in that these links help prevent faulty functioning of the control systems caused by a sudden ground potential rise associated with a surge current. The fiber optical link also provides an advantage in petrochemical factories where electrical sparks or over-heating of the communication cable could cause an explosion. Yet another industry where optical fibers would prove useful is the aircraft industry. Inside the fuselage of an aircraft a number of signal cables are fully packed. In order to avoid interference between the cables they are armored with heavy shielding metals. This weight can be substantially reduced by replacing them with fiber-optical links. A disadvantage of fiber optical communication, however, is that even though the fiber is reasonably flexible, it is not as flexible as copper wire, so that splicing of the fiber cable is less easily achieved and needs more precision than connecting copper wires.

13.2 Optical Fiber One of the parameters that specifies the optical fiber is its numerical aperture (NA) which is a measure of the maximum angle of acceptance for which light will be guided along the fiber when the fiber is illuminated at its end. The step index fiber consists of core glass with a higher refractive index surrounded by cladding glass of a lower refractive index. Referring to Fig. 13.2, the incident light ray will be guided through the core if it undergoes total internal reflection at point Q on the core-cladding boundary. In order to determine the numerical aperture, the incident angle φc to the fiber end surface is calculated such that the angle θ at the point Q on the core-cladding boundary is the critical angle for reflection. When θ is the critical angle, (13.1) n 1 sin θ = n 2 where n 1 and n 2 are refractive indices of core and cladding. Snell’s law is applied at P to obtain (13.2) n sin φc = n 1 sin(90◦ − θ ) = n 1 cos θ = n 1 1 − sin2 θ where n is the refractive index of the external medium in which the end of the optical fiber is immersed. Equations (13.1, 2) are combined to give

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13 Fiber Optical Communication

Fig. 13.2 Definition of numerical aperture: NA = n sin ϕc =

n sin φc =

n 21 − n 22 = NA.

n 21 − n 22

(13.3)

Equation (13.3) defines the numerical aperture (NA) and is an indicator of the fiber’s light acceptance angle. The NA value is extremely important in optical fiber systems not only as an indication of the acceptance angle but also as a measure of the dispersion of the fiber, as described in the next section.

13.3 Dispersion of the Optical Fiber When an input light pulse is launched into an optical fiber, the light energy is distributed among beams of various quantized angles (Chap. 5). Each beam has a different arrival time at the output end depending on these angles. This difference in arrival time is called dispersion and results in a spread in the received pulse width. Dispersion determines the highest rate of pulses (or the narrowest spacing between pulses) that can be transmitted per second through the fiber. In other words, if the rate is too high, two successive pulses can no longer be recognized as two separate pulses because of the spread. While there are no steadfast criteria, for the pulse spread of ∆T , the minimum pulse width T should satisfy ∆T < 0.7T.

(13.4)

The bit rate of transmission B is defined as B = 1/T . The amount of dispersion or pulse spreading is a function of the refractive index profile of the fiber core. It will be shown that the information transmission capability of the Selfoc fiber (Sect. 5.7) is much higher than that of a step-index multimode fiber. First, the dispersion of a step-index multimode fiber is considered. Referring to Fig. 13.3 it is easy to see that the total length of the zigzag path within a unit length of fiber is 1/ cos γ , γ being the angle of the ray to the fiber axis [13.1, 2]. The time that the light takes to travel a unit length of fiber is called the group delay τ . The difference ∆τ in the group delay of the zigzag path and that of the straight path is 1 n1 −1 . (13.5) ∆τ = c cos γ

13.3 Dispersion of the Optical Fiber

337

Fig. 13.3 Calculation of the total zigzag path of a step index

The term ∆τ is the modal pulse spread for a unit length of step-index multimode fiber. Since γ is normally very small, cos γ ∼ = 1 − 12 γ 2 and n1 ∆τ ∼ (13.6) = γ 2. 2c When a ray following a zigzag path strikes the core-cladding boundary at the critical angle, (13.3) applies and NA = n sin φc = n 1 sin γ ∼ NA . γ = n1

hence (13.7)

Inserting (13.7) into (13.6), ∆τ finally becomes ∆τ ∼ =

1 (NA)2 . 2cn 1

(13.8)

Thus, the modal pulse spread ∆τ of a step-index multimode fiber is proportional to (NA)2 . This means that the capacity for transmission of information through the fiber is inversely proportional to the square of the numerical aperture. The value of NA, however, cannot be made indefinitely small because, as seen from (13.3), the acceptance angle of the incident beam decreases with the value of NA and the difficulty associated with launching the incident light into the fiber increases. Thus, the number of bits that can be sent through a step index fiber can be obtained from (13.4, 8). Next, the pulse spreading of a graded-index fiber, specifically of a Selfoc fiber, will be calculated in order to compare this result with that of the step-index multimode fiber. The pulse spreading of the Selfoc fiber can easily be found using the earlier result for the group delay derived in Chap. 5. From (5.155), the group delay difference is n c 1 + cos2 γ −1 ∆τ = c 2 cos γ =

n c (1 − cos γ )2 . c 2 cos γ

(13.9)

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13 Fiber Optical Communication

Using (13.7), and the small-angle approximation for cos γ gives ∆τ ≈

1 (NA)4 . 8c n 3c

(13.10)

Thus, the pulse spreading of the Selfoc fiber is proportional to the fourth power of NA and the dispersion of the Selfoc fiber is extremely small since its NA is typically 0.1 ∼ 0.3. From (13.4), the information carrying capacity of the fiber is inversely proportional to the pulse spreading. By taking the ratio of (13.8) and (13.10), it is seen that the information carrying capacity of the Selfoc fiber is 4(n 3c /n 1 ) NA−2 times (10 ∼ 100 times) larger than that of a step-index multi-mode fiber for the same values of NA. The Selfoc fiber is thus well suited to long-distance, high density information applications. The pulse spreading mentioned above is called mode dispersion because it is caused by differential delays between the straight path of the lowest-order mode and the zigzag paths of the higher-order modes. It should be noted that mode dispersion is not the sole source of pulse width spreading. There are two other mechanisms for dispersion: material dispersion and waveguide dispersion. Such dispersion takes place only when the light is composed of more than one wavelength. In contrast, mode dispersion will occur even if the light is purely monochromatic. Material dispersion is caused by the change in the refractive index of glass as a function of the wavelength of the light. Waveguide dispersion is caused by the fact that the quantized angles of the zigzag path change with respect to the wavelength of the light. In fact, the two dispersions cannot be clearly separated because a change in the refractive index also changes the quantized zigzag angles. The pulse spreading due to the spread in frequency, k = ω/c, is ∆τ =

dτ ∆k. dk

(13.11)

From Fig. 13.3 the group delay of a step index multimode fiber is τ=

n1 1 . c cos γ

(13.12)

Inserting (13.12) into (13.11) gives ∆τ =

n1 d 1 dn 1 1 ∆k + c dk cos γ c dk

material dispersion

1 cos γ

∆k .

(13.13)

waveguide dispersion

One approach to reducing dispersion is to use a single-mode fiber. As mentioned in Chap. 5, the core size of a single-mode fiber is reduced so that only the lowestorder mode (straight path) can be excited, thus eliminating mode dispersion completely. The other alternative is to use a light source with a narrow spectral width,

13.4 Fiber Transmission Loss Characteristics

339

such as a laser diode, so as to minimize the material and waveguide dispersions. Another approach is to operate within a wavelength region where the first and second terms of (13.13) cancel each other. For instance, with a single-mode fiber made of fused silica, the first term changes sign near 1.27 µm and the dispersion effect is significantly reduced.

13.4 Fiber Transmission Loss Characteristics The causes of attenuation in an optical fiber can be broadly grouped into materialrelated and structure-related areas [12.3, 4]. First, the material-related causes are described. Typical curves of the dependence of fiber loss on wavelength are shown in Fig. 13.4. The curves are approximately V-shaped, and can be divided more or less into four regions according to the attenuation mechanisms which are i) ii) iii) iv)

Rayleigh scattering loss, OH− ion absorption loss, inherent absorption loss, impurity absorption loss.

i) The Rayleigh scattering loss is caused by irregularity in the refractive index of the fiber with dimensions of the order of 0.001 to 0.01 µm, which are produced

Fig. 13.4 Attenuation characteristics of a fiber as a function of wavelength

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13 Fiber Optical Communication

in the fiber drawing process. An approximate expression for the scattered intensity from objects whose size is much smaller than the wavelength was derived by Rayleigh who showed that this intensity is proportional to λ−4 . This type of scattering, also known as Rayleigh scattering, is predominant in the shorter wavelength region and follows very closely the λ−4 line. ii) The O H − ion absorption loss arises from the presence of water molecules in the glass. The OH− ion has its fundamental absorption line at λ = 2.8 µm with second and third harmonics at 1.4 and 0.95 µm, respectively. The second and the third harmonics of this absorption line are within the range shown in Fig. 13.4. These peaks disappear as the content of OH− ions becomes less than 1 ppb (parts per billion). The glass with lower peaks is sometimes called “dry glass” and the glass with higher peaks, is then refered to as “wet glass”. iii) The third region, i.e. the inherent absorption region, is on the right shoulder of the V-shape. This loss is caused by the absorption bands of molecules that are an essential part of the fiber’s composition (as opposed to unwanted impurities) and hence the term inherent loss. The refractive index profiles of the core and cladding are controlled by adding dopants to the glass. Additives such as germanium and phosphorus raise the refractive index, while additives such as boron lower the refractive index. These dopants have molecular absorption bands in the inherent loss region. iv) The impurity absorption loss is due to impurities introduced unintentionally during the process of fabrication. They are primarily transition metal ions. The absorption band of Fe++ ions ranges from 0.7 to 1.3 µm and Cr +++ ions ranges from 0.6 to 0.7 µm. We consider next the structure-related losses, which are summarized in Fig. 13.5. When the cable is bent excessively, some of the light rays will strike the corecladding boundary at angles larger than the critical angle and these rays will be lost in the cladding, as indicated in Fig. 13.5 a. Microbending is a special case of bending. When a protective plastic coating is applied to the bare fiber, the loss increases. This is due to small bends located at intervals of about one millimeter due to irregular stresses built in during the curing process. This loss is called microbending loss. Even though the angles involved are small, the number of bends are large enough so that the total loss cannot be ignored. Irregularities in the boundary surface between the core and the cladding glass also acts as a loss mechanism. Any ray scattered at an angle larger than the critical angle will be lost into the cladding glass, as shown in Fig. 13.5 b. This loss is smaller in graded-index fibers than in step-index fibers. This happens because not all the rays in the graded-index fiber reflect at the outer region of the fiber, so that irregularities in the outer region do not contribute as much to the loss.

13.4 Fiber Transmission Loss Characteristics

341

Fig. 13.5 a–e. Structure related loss of the optical fiber due to (a) bending, (b) boundary irregularities, (c) irregularities in the core, (d) misalignment, (e) imperfect splicing

Any irregularities in the core glass made during the fabrication of the fiber obviously scatter the light, as shown in Fig. 13.5 c, and contribute to the loss. In many practical situations it is necessary to join two fibers together. If the cores of the two fibers are misaligned, this will give rise to a loss at the connection (Fig. 13.5 d). Misalignment either axially or angularly causes the loss. A splice refers to a special type of connection in which two fiber ends are fused or bonded together. This joining process may result in deformations at the location of the splice. Imperfect splicing, as shown in Fig. 13.5 e, creates scattering and leads to loss. Likewise, core diameter variations along the fiber will contribute to fiber loss in a manner similar to that illustrated in Fig. 13.5 e.

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13 Fiber Optical Communication

13.5 Types of Fiber Used for Fiber Optical Communication Step-index fiber, graded-index fiber and single-mode fiber are most widely used for fiber-optical communication systems. Figure 13.6 shows the light path and the radial distribution of the refractive index of these three optical fibers. The step-index multimode fiber has a large core diameter (50 ∼ 75 µm) which has the advantage that coupling light into the fiber and splicing fibers together is much easier. However, the information transmission capacity is low, because hundreds of modes are excited in the fiber and the pulse spreading is as large as 50 ns/km, which means that the signal bit rate is of the order of Mb/km (13.4). The graded-index fiber has much less dispersion than a step-index multimode fiber. According to the conclusion of Sect. 13.3, the information carrying capacity of a graded index fiber is 4(n 3c /n 1 ) NA−2 times larger than that of a step-index multimode fiber. With a fiber of NA = 0.2, the capacity of the graded-index fiber is approximately 200 times higher. The group delay spread ∆τ of the graded-index fiber is of the order of 1 ns/km and the bandwidth is approaching 1 GHz/km.

Fig. 13.6 a–c. Three basic types of optical fiber. (a) Multimode step index. (b) Multimode graded index. (c) Single mode step index

13.6 Receivers for Fiber Optical Communications

343

The single-mode fiber permits only one mode to propagate, and the mode dispersion is completely eliminated, thus achieving a bandwidth of several GHz/km. The disadvantage of single-mode fibers is the difficulty of coupling and splicing as a consequence of its small core diameter.

13.6 Receivers for Fiber Optical Communications The detectors used for fiber optical communication are, in a sense, photon-electron converters [13.5–12]. Solid-state devices are by far the most common detectors because they have such good features as small size, long life, durability, and low electrical power consumption [13.6]. The PIN photodiode and the avalanche photodiode (APD) are the major solid state detectors used.

13.6.1 PIN Photodiode A heuristic explanation of the principle of operation of the PIN photodiode will be made using the microscopic views in Fig. 13.7. A PIN photodiode consists of three regions: a p-type region, an intrinsic region and an n-type region [13.7]. Figures 13.7 a and b show the microscopic views of these regions before and after they are put together to complete the fabrication. The crystal structure of each region will be explained separately. First, the intrinsic region in the middle section is addressed. It is an uncontaminated region of silicon (Si) atoms. Each silicon atom has four outer shell electrons of its own and the atoms are held together to form a crystal by sharing the outer shell electrons of four neighboring silicon atoms. The silicon atoms strive to fill their outer shells and form a complete set of eight electrons by this valence electron sharing. In the intrinsic region, the valence electrons in the outer shell of the atoms are trapped in the crystal lattice and cannot freely move around, so that the conductivity of the intrinsic region is very low. These valence electrons, however, can be broken loose to produce free electrons when the crystal is exposed to either high thermal agitation or irradiation of light. These freed electrons play an important role in photo detection. Next, in the n-type region, group V atoms like arsenic atoms are doped into the intrinsic crystal. The arsenic (As) atoms have five valence electrons. When this atom forms a lattice with neighboring silicon atoms having four valence electrons, one electron is left unpaired since only eight electrons are needed to complete the outer shell. This excess electron is only loosely attached to the As atom. Even at room temperature, thermal energy will loosen some of these weakly bound electrons so that they become free electrons. Although the electron becomes mobile, the As atom remains immobile and a plus charge ⊕ is created at the site of the As atom after its electron has left. In order to differentiate the mobile (free) charges from

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13 Fiber Optical Communication

Fig. 13.7 a, b. Illustration of PIN photodiode. (a) Microscopic view. (b) Field distribution

the immobile charges, the latter charges are circled in the figures. The fact that one type of charge is mobile and the other is not is essential to the functioning of semiconductor devices. In the p-type region, group III atoms such as gallium (Ga) are doped into the intrinsic crystal. The gallium atoms have only three valence electrons. As a result, when such an atom forms a lattice with neighboring silicon atoms having four valence electrons, the outer shell is incomplete since one electron is lacking. Even at room temperature, thermal energy may loosen a valence electron from a neighboring silicon atom in order to fill in the vacancy in the silicon-gallium pair leaving again a missing valence electron behind. As far as the gallium atom is concerned, when the missing valence electron is filled an extra immobile negative charge is created because Ga was electrically neutral originally. As far as the neighboring Si atom that has lost the valence electron is concerned, a + charge was effectively created.

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The missing electron can be readily filled by electrons from further neighboring Si atoms, and this mobile effective + charge is called a hole. Next, these three regions are put together to form a PIN photodiode. As soon as they are joined together, the highly populated free electrons in the n-type region start to diffuse and spread into the intrinsic region, as do the holes in the p-type region. This diffusion process, however, does not go on indefinitely. For every electron that departs from the n-type region, an immobile ⊕ charge is left behind in the n-type region. The amount of ⊕ charge increases as the number of departing free electrons increases. In a similar manner, as the holes depart from the p-type region, immobile charges build up in the p-type region. As a result, a potential difference exists between the two regions. This potential is observed externally as a contact potential. An electric field E, which is the slope of the potential versus distance curve, is established as shown in Fig. 13.7 b. The force exerted on the electrons by this electric field E is directed toward the n-type region (to the right) while the force exerted on the holes by E is directed toward the p-type region (to the left), exactly in directions opposite to the directions that the electrons and holes want to diffuse. The diffusion stops when the diffusion tendency and the force due to this electric field E are balanced. The density of mobile charges drops rapidly as one moves away from the p − i and n − i boundaries toward the middle of the intrinsic region. That portion of the intrinsic region which has a scarcity of mobile charges is called the depletion region. The PIN photodiode is usually operated with a bias voltage. A positive potential is applied to the n-type region and a negative one to the p-type region (reverse bias). With this bias, an additional electric field is created which reinforces the internal electric field and forces the mobile electrons and holes back towards their native n and p regions. The depletion region expands as a result of the reverse bias. When a photon h v is injected into the intrinsic region, there is a high probability that the photon will be absorbed by a silicon atom. As a result of the photon absorption, the silicon atom gains energy. If the photon energy exceeds a certain critical value, then the energy gained by the silicon atom will liberate one of the valence electrons creating a mobile electron. A mobile hole is created simultaneously as a result of the vacancy in the silicon valence shell. This process is called electron-hole pair production. While the conductivity of the intrinsic region is low, that of either the p- or n-type regions is high because of an abundance of mobile charges. When an external potential is applied, this conductivity distribution causes the electric field to build up primarily in the intrinsic region and not in the p- or n-type regions. This applied E field swiftly sweeps the newly produced electrons to the n-type region and the holes to the p-type region creating a signal current in the external circuit. The intrinsic region is made long because the light absorption coefficient of silicon is rather small so that long distances are needed for the majority of incident photons to participate in electron-hole production. In conventional PIN photodiodes, the photons enter the intrinsic region through the p-type region. The p-type region is made thin because any electron-hole pairs generated in this region diffuse in random directions as a consequence of the absence

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of an electric field due to high conductivity. The electrons generated in this region take a long time to reach the positive terminal. This not only deteriorates the response time but also becomes a source of noise.

13.6.2 Avalanche Photodiode The avalanche photodiode (APD) utilizes the avalanche multiplication process to achieve higher electric output. The structure of the APD is shown in Fig. 13.8 a. The APD resembles a PIN photodiode except for the insertion of an additional p-type multiplying region. Most of the incident photons are absorbed in the intrinsic region and electron-hole pairs are first generated in this region as in the PIN photodiode. As before, the holes drift toward the p-type region and the electrons toward the n-type region. As shown in Fig. 13.8, the electrons, before reaching the n-type region, pass through a p-type multiplying region. If the electrons have undergone sufficient acceleration on reaching the multiplying region, new electron-hole pairs are generated by the collision-ionization process. The newly generated electronhole pairs themselves generate more electron-hole pairs, thus initiating the process

Fig. 13.8 a, b. Principle of the avalanche photodiode (APD). (a) Mechanism of the avalanche effect. (b) Field distribution

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known as avalanche multiplication. It may produce 10 to 100 or even 1000 times more electron-hole pairs than is produced by photon absorption alone. The intrinsic region is slightly doped to reduce its resistivity to around 300 #cm, so that the region of the highest resistivity is concentrated near the junction between the n-type region and the multiplying p-type region when reverse bias is applied. The highest field concentrates in the region of the highest resistivity when an external potential is applied and fields higher than 105 V/cm can be established in the vicinity of the junction where most of the avalanche multiplication takes place. The n-type region is made thinner but is more highly doped than the multiplying region. The field distribution in the APD is shown in Fig. 13.8b. Next, an expression for the output signal current from the photo-detector is calculated. The number of photons in the light having total energy E is E/ hv where hv is the energy of a single photon, h = 6.63 × 10−34 J s is Planck’s constant, and v is the frequency of the light. The total charge generated in the detector is η (E/ hv) e M, where η is the quantum efficiency, e is the elementary charge of 1.6 × 10−19 C, and M is the multiplication factor which is unity for a PIN photodiode, and 10–1000 for an APD. The quantum efficiency is the probability that an incident photon will generate an electron-hole pair in the intrinsic region and is a number less than or equal to unity. The output current i is thus obtained by dividing the charge by time, namely e (13.14) i = ηp M [A] hv where p is the optical power. The value of hv is 2 × 10−19 J for light with λ = 1 µm. The value of M depends critically on the applied voltage and the temperature. As a rule of thumb, the output current from a PIN photodiode is i = 0.5 A/W and an average APD has an output current of i = 50 A/W. Two major sources of noise in the PIN photodiode and the APD are thermally generated noise and shot noise. Thermal noise is independent of signal current whereas the shot noise is dependent on the current. Thus, both the current and the signal bandwidth must be specified in order to specify the photodiode’s signal to noise ratio (S/N ratio). The S/N ratio is normally used for analog signals. In the case of digital signals, the bit error rate (BER) is used to specify the quality of the system instead of the S/N ratio. The bit error rate is the ratio of the number of error pulses in the receiver to the number of transmitted pulses. In general, the BER is a function of the rate at which pulses are transmitted, as well as being a function of optical power. Figure 13.9 shows the optical power needed as a function of pulse rate in order to maintain a BER of 10−8 .

13.6.3 Comparison Between PIN Photodiode and APD The PIN photodiode is more stable than the APD in the presence of fluctuations of temperature and applied voltage. The fact that the APD is very sensitive to changes

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Fig. 13.9 Average optical power required to maintain BER 10−8 [13.17]

in applied voltages can be used advantageously to control the output of the APD by feeding an automatic gain control (AGC) signal directly to the applied voltage. The gain of the APD is 10–1000 times higher than that of the PIN photodiode because of the internal avalanche effect. Another advantage of the PIN photodiode is that it requires less applied voltage, typically 15–40 V. The applied voltage of an APD is 100–150 V.

13.7 Transmitters for Fiber Optical Communications By virtue of their small size, high reliability and stability, modest power supply requirements, long lifetime, ease of modulation and low cost, solid-state light sources are the preferred light sources for fiber optical communication systems [13.13]. These same considerations are likewise responsible for the common use of solidstate receivers. The solid-state transmitters which are discussed in this section are the light emitting diode (LED) and the laser diode (LD).

13.7.1 Light Emitting Diode (LED) The LED and PIN photodiode are photon-electron converters having exactly opposite functions. With the PIN photodiode, the input is photons and the output is electrical current. The input photons liberate valence electrons from the outer shell of the atoms of the crystal to create free electrons. The free electrons are the output signal. By contrast, in an LED, electrons and holes are injected into the junction region of the p − n junction diode by forward biasing the LED (recall that the PIN

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Fig. 13.10 Light emitting diode (LED)

Fig. 13.11 Light intensity of LED as a function of wavelength

photodiode and the APD are reverse biased), as shown in Fig. 13.10. When the injected electrons recombine with the holes, i.e., when the electrons settle down into the positions of missing valence electrons in the crystal, the electron’s energy is released. The released energy E g is used for either emitting photons (radiative recombination) or for heating the crystal lattice (non-radiative recombination). The wavelength of the emitted light is λ = h c/E g . The conversion rate of the LED is about 10–20 mW/A and the output is reasonably linear with respect to the injected current. The maximum light output is a few milliwatts. The highest frequency of modulation is determined by the recombination time, which is the time the injected electrons take to recombine. The highest practical frequency of modulation is several hundred Megahertz. Not all the recombination takes place with the release of the same amount of energy because of the non-uniformity of the energy of the electrons. This nonuniformity results in a spread in the frequency spectrum. The half power width of ˚ = 10−4 µm), as shown in the LED is as wide as a few hundred Angstroms (1 A Fig. 13.11, and as a result, the LED is useful for only short to moderate transmission distances because the wider the frequency spread, the greater the dispersion in the optical fiber. Another disadvantage of the LED is that the light radiation is emitted over a large range of angles. This makes efficient coupling of the light into

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Fig. 13.12 a–e. Various methods of coupling the emission of an LED to the optical fiber. (a) Formation of image of source by a lens. (b) Formation of image of source by a Selfoc micro lens. (c) Auxiliary optical guide along active layer. (d) Spherical tip. (e) Coupled from close to active region (Burrus type)

an optical fiber difficult. Lenses of various types, such as shown in Fig. 13.12, are used to couple the output of the LED to the fiber. In Sect. 13.6, silicon was chosen as the intrinsic semiconductor, and gallium and arsenic were chosen as dopants but a number of other elements are equally well suited to constructing semiconductor devices. Germanium can be used instead of silicon as an intrinsic semiconductor. Boron B, aluminum Al, and indium In can replace gallium as p-type dopants. Nitrogen N, phosphorus P, and antimony Sb can replace arsenic as n-type dopants. These are shown in Table 13.1. In addition to using elements as semiconductor building blocks, certain compounds can also be used. For example, the compound gallium arsenide GaAs which is made up of the group III element Ga and the group V element As can be used as a semiconductor. By changing the proportions of the elements in the compound, the wavelength of the emission can be varied. The combinations shown by the lines in Table 13.1 are used for fabricating semiconductors.

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Table 13.1 Summary of Compounds and elements used for semiconductors [13.3]

13.7.2 Laser Diode (LD) Emission of light with frequency v 21 = (E 2 − E 1 )/ h, occurs when a transition from an upper energy state E 2 to a lower one E 1 takes place. On the other hand, when the transition of the reversed direction, namely from E 1 to E 2 takes place, energy is absorbed. There are two types of emission of light; one is stimulated emission and the other, spontaneous emission. Stimulated emission takes place when the light whose wavelength exactly corresponds to v 21 (seed photons) impinges onto the atoms. The spontaneous emission is the emission which takes place regardless of the external illumination. The degree of the stimulation is proportional to the intensity Pi of the illuminating light. The expression for the total emission is P0 = AN2 + Pi (B21 N2 − B12 N1 ) where A is Einstein’s constant, N1 , N2 are the electron populations in the lower and upper energy levels, B21 and B12 are the transition probabilities from E 2 to E 1 and that from E 1 to E 2 , respectively. If equal degeneracies in these two levels is assumed, the transition probabilities from E 2 to E 1 and that from E 1 to E 2 are equal1 and B = B21 = B12 . The expression of the total emission becomes 1

In a degenerate system, the transition probabilities are related as [13.14, 15] g1 B12 = g2 B21

where g1 and g2 are the degeneracies of the levels 1 and 2. A degeneracy gi means that there are gi independent states (ways) in which the level i of the system can have the same energy. A good example is the spin of an electron. There are two ways for an electron to spin, i.e., clockwise or counterclockwise about its axis (2 states). Both states can have the same energy,

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P0 = AN2 + Pi (N2 − N1 ) Thus, the populations of the electrons determine whether emission or absorption takes place. In the case when the population of the upper state is larger and N2 > N1 , emission occurs, and when the population of the lower state is larger and N2 < N1 , we have absorption. The populations are related by Maxwell-Boltzmann statistics at thermal equilibrium as N2 = N1 exp [−(E 2 − E 1 )/k T ]. Thus at thermal equilibrium, power is always absorbed. In the laser diode, N2 is larger than N1 due to the injection of mobile electrons (upper state electrons), thus maintaining emission. In photo-detection, N2 is kept smaller than N1 by removing the mobile electrons to the external circuit by reverse bias so that the incident photons are readily absorbed. Radiation from the LED is predominantly due to spontaneous emission, whereas that from the LD is due to stimulated emission, amplified by positive feedback in a laser cavity. Both ends of the LD are highly polished so that light is reflected back and forth through the active region of the LD. Each time light passes through the diode, it stimulates emission. As the emission increases, the number of photons available to stimulate more emission also increases (positive feedback), so that the light radiation builds up very quickly. Another important difference between spontaneous and stimulated emission is that the photons emitted spontaneously have no preferred phase whereas photons emitted by stimulated emission have not only the same frequency but also follow the same phase as the seed photon. The wavelength spectrum of the LD is very narrow and suitable for high information density, long-transmission distances because a narrower wavelength spread means less dispersion in the optical fiber. In the following subsections, the LD is described in more detail.

13.7.3 Laser Cavity and Laser Action Figures 13.13 a, b show the geometry of a laser diode. The length of the laser cavity L is a few hundred micrometers and is formed by cleaving the crystal along two parallel crystalline planes. The region in which lasing takes place is called the active region. For instance, when the refractive index of the active layer of the LD is 3.5, and using Fresnel’s formula for the reflection coefficient at normal incidence, r = (n − 1)/(n + 1), the cleaved ends each have a reflection coefficient r = 0.56. and in this case, the degeneracy of the level associated with these spinning electrons is two. In fact, this degeneracy can be removed by applying an external magnetic field. A spinning electron can be considered as a small magnet whose polarity is determined by the direction of the electron spin. The magnetic energy of a small magnet in an external magnetic field depends upon the polarity of the magnet. Thus, in the presence of a magnetic field, the two electron spin states no longer have the same energy, and the degeneracy is removed.

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Fig. 13.13 a, b. Laser diode (LD). (a) Emission in Fabry-P´erot laser cavity. (b) Structure of the laser diode

The refractive index n 1 of the active layer is chosen higher than that of the cladding layer n 2 so as to form a light guide. The thickness 2d of the active layer is made thin (0.3–0.4 µm) so that only one mode is excited in the transverse (thickness) direction. The transverse direction is taken as the x-direction in Fig. 13.13. The electrode is stripe shaped so that the laser action is confined to a narrow region W of the order of 10 µm (in the y-direction). The resonance condition of the cavity in the longitudinal z-direction is β N L = pπ

(13.15)

where p is an integer, and β N is the propagation constant of the N th mode in the longitudinal direction. If the thickness 2d is made small enough that only the lowest mode is excited, then 2π (13.16) n1. βN ∼ = λ

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From (13.15, 16), the lasing wavelength must satisfy the condition λ p = L. 2n 1

(13.17)

As indicated in Fig. 13.14 a, there are a number of wavelengths that satisfy (13.17) and these wavelengths are known as resonant wavelengths. The spacing ∆λ between adjacent resonant wavelengths is ∆λ =

λ2 . 2n 1 L

(13.18)

The shorter the cavity length is, the wider the spacing ∆λ becomes. Next, the condition for lasing will be found. Lasing occurs when the gain in light intensity per round trip in the cavity exceeds the loss per round trip. For each round trip in the cavity, the amplitude decays by r 2 exp (−2αL), α being the amplitude attenuation constant. The gain in each round trip is exp [2g(i, λ)L], g(i, λ) being the amplitude gain per unit distance; it is a function of the current i and wavelength λ. The threshold condition of oscillation is r 2 e−2αL+2g(i,λ)L = 1.

(13.19)

The threshold gain is 1 1 ln . (13.20) L r The gain g(i, λ) increases with current. Taking i as a parameter, the spectral distribution of the spontaneous emission of the LD is shown in Fig. 13.14 b. In Fig. 13.14 c the loss level, gain curve, and resonant wavelengths are all combined in one diagram. At the current i 1 which is below the threshold, no laser action takes place, the light intensity is low and the spectrum is wide, as shown on the left side in Fig. 13.14 d. With a further increase in current to just below the threshold i 2 , the output increases and the spectrum spreads, as shown in the center in Fig. 13.14 d. There is no laser action yet. As soon as the current exceeds the threshold value, laser action starts taking place. Stimulated emission, which is initiated by the presence of spontaneously emitted photons in the cavity, begins to predominate. It is the nature of stimulated emission that the particular resonant wavelength that first starts to predominate becomes the dominant wavelength of emission. More emission means more stimulation. Emission of one wavelength predominates, as shown on the right side in Fig. 13.14 d. Although the mechanisms are completely different, analogies can be drawn between stimulated emission of light and the rupturing of a strained river dam. An initial small starting spot rapidly leads to a rupture of that portion. In reality, emission from the adjacent resonant wavelengths is also present. The adjacent spectra can however be suppressed by making ∆λ large by choosing a shorter cavity length L in (13.18). The price that must be paid is that the threshold gain in (13.20) increases as L is shortened. Typical output power characteristics as a function of current are shown for the LD and LED in Fig. 13.15. The output power of the LD increases very slowly until g(i, λ) = α +

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Fig. 13.14 a–d. Principle of laser action. (a) Laser cavity resonance. (b) Spontaneous emission with current as a parameter. (c) Threshold of lasing. (d) Spectrum of laser output

the threshold current of 100 mA is reached, and then the optical power rises steeply to 5 mW with an additional 20 mA of current. In comparison, the output power of the LED increases steadily with increasing current. The LED curve was added in the figure for comparison. The threshold current of the LD is strongly influenced by temperature. A special control circuit is necessary to ensure that the LD is operated in the linear region of the output power versus current curve. The control signal is usually derived from the laser beam emitted from the unused end of the device.

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Fig. 13.15 Power emission from a laser diode compared with a light emitting diode

Fig. 13.16 Delay in the start of lasing

Another characteristic of laser oscillation is the delay between the applied current and the start of the emission, as shown in Fig. 13.16. It takes time to reach the threshold density of injected electrons needed to start lasing. Besides the delay, the light intensity goes through relaxation oscillations at the starting time of lasing. These set the upper limit on the modulation frequency to the order of a Gigahertz. The mechanism for the relaxation oscillation is explained as follows. Soon after the injection of electrons into the laser cavity, the electron density surpasses the threshold value and lasing begins. As soon as lasing starts, the electron density begins to decrease. The reduced electron density means reduced emission, and this again allows the build up of the electron density. After a few cycles of this competition between electron density and light emission, the intensity of the emission

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reaches a steady state value. One way of increasing the upper limit of the modulation frequency is to apply a dc bias current just below the threshold value. In the case of an on-off modulation, however, the dc bias deteriorates the on-off ratio of the light emission because of the non-zero light at “off” time. The beam radiation pattern of the LD shown in Fig. 13.13 is wide in the vertical x-direction and narrow in the horizontal y-direction because the beam width is roughly inversely proportional to the dimensions of the source (Chap. 4). A rather interesting practical idea for injecting the LD beam into a fiber guide is shown in Fig. 13.13 b. Here, a short piece of optical fiber is used as a cylindrical lens for focusing the LD beam.

13.7.4 Temperature Dependence of the Laser Diode (LD) The resonance condition of the laser cavity is significantly influenced by temperature [13.16]. Both the refractive index and the dimensions of the cavity change as the temperature varies. As far as the changes in the refractive index are concerned, the temperature influences the refractive index in two ways. First of all, the temperature directly changes the refractive index. Secondly, the temperature changes the wavelength of operation and this change in lasing wavelength in turn changes the refractive index. This cascade can be expressed by the formula dn =

∂n dλ ∂n dT + dT. ∂T ∂λ dT

(13.21)

The resultant change in the wavelength of operation due to the changes in both dimension and the refractive index is obtained by differentiating both sides of (13.17) with respect to λ and using (13.21) to obtain 1 dL 1 ∂n n ∂ T + L dT dλ . =λ λ ∂n dT 1− n ∂λ

(13.22)

It should be remembered that the g(i, λ) curves in Fig. 13.14 are also very critically influenced by temperature. The relative positions of the curves in Fig. 13.14 change as the temperature changes. It is difficult to achieve a stability better than ˚ ◦. 1 A/C

13.7.5 Comparison Between LED and LD This subsection summarizes the comparisons between the LED and the LD mentioned above. One of the most striking differences is their wavelength spectra. While the wavelength spectrum of the radiation from the LED is a few hundred Angstroms

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wide, that of the center spectrum of the LD is only a few Angstroms wide. The peak power of the LED is about one milliwatt but that of the LD is several milliwatts. The radiation beam pattern of the LD is much narrower than the radiation beam pattern of the LED. The output of the LED is linear with bias current, and the ratio of output optical power to input current is approximately 0.01 W/A. The output from the LD is nonlinear near the threshold current. Moreover the output exhibits a time lag. The LED is simpler in structure, simpler to modulate, and has a longer life than the LD.

13.8 Connectors, Splices, and Couplers In order to build a complete fiber optical communication system, the transmitter and receiver must be interfaced with the optical fiber. In this section, the various components and techniques for linking transmitter, fiber and receiver are described [13.5, 11, 12].

13.8.1 Optical Fiber Connector Connectors are installed at the end of the optical fiber for the purpose of connecting two fibers together or for the purpose of connecting the fiber to some other component. For connections between two fibers, the smaller the fiber core diameter is, the tighter the alignment tolerance between the two centers has to be. The alignment tolerance for 50 µm core diameter fibers is of the order of a few micrometers. Both the positioning of the center and the parallelism of the axis are important. One approach to the problem of aligning the core is to align the cladding glass under the assumption that the core and the cladding are concentric. The other approach is the alignment of the core itself. In the former approach, the glass fiber is inserted into a watchmaker’s jewel secured in a metal ferrule. Figure 13.17 a shows an assembly of this type. Even though the connector can be assembled in the field, it is difficult to reduce the insertion loss to less than 1 dB since the connection relies on the intrinsic tolerance of the fiber. This type of connector is sometimes called the FA-type connector (field assembly). The other approach is a C-type (C stands for concentric) connector. The fiber is first put into a sheath to increase the diameter for better handling. Then the sheathed fiber is positioned by a microscope so that the fiber core is accurately centered in a plug. While positioned in the center, the sheathed fiber is cemented to the plug to complete the connector. It is important that the connection loss be the same every time the connection is made. It is much preferable to make a connection by splicing if a permanent connection is intended because splicing is more reliable in the long run than a connector.

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Fig. 13.17 a, b. Assembly of the ferrule of the optical fiber connectors. (a) FA type (field assembly type) connector. (b) C type (concentric type)

Fig. 13.18 a–c. Splicing with bonding agent. (a) V groove method. (b) Method of using an edge of the sleeve. (c) Method of securing by three cylindrical rods

13.8.2 Splicing A permanent joint between two fiber ends is often needed. The joints are first aligned and then are bonded either by a cement or by melting the glass. The alignment is usually made by relying upon the quality of the fiber. Figure 13.18 illustrates the common methods of splicing with bonding agents. The ends of the fiber are first cleanly cleaved and then brought together. It is important to prevent the fibers from touching with any force. Even with a very light force, the pressure per unit area is very large and the ends are easily abraded. A spacing as large as 100 µm practically makes no difference in the loss. The alignment is achieved by either a V groove,

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Fig. 13.19 a–c. Splicing by fusion. (a) Preheating ends of fibers. (b) Ends are aligned by surface tension. (c) Splicing is completed

the corner of a square tube or round cylinders, as shown in Fig. 13.18. They are permanently secured by a bonding agent. Figure 13.19 shows the procedure for splicing by the fusion method. The ends of the fiber are heated by a controlled electric arc. The ends are preheated to the extent that the shape of the end becomes a hemisphere, as shown in Fig. 13.19 a. Then the ends are brought together until they touch. At the moment that the ends touch each other, the connection evolves from the center of the core glass, as shown in Fig. 13.9 b, thus eliminating the chance that air bubbles will become trapped inside the fused region. The surface tension of the glass also helps to align the fiber ends. The fiber ends are brought together at exactly the precalculated distance to give the minimum deformation of the fiber’s cylindrical shape at the location of the splice. Since the diffusion constant of the glass is very low, the original distribution of the refractive index is preserved during the fusion, and the splicing loss is typically less than 0.2 dB.

13.8.3 Fiber Optic Couplers A coupler is a device which couples light from the optical fiber to another fiber or to other optical components, such as monitoring photodiodes. There are three major types of couplers: (i) monitoring couplers, (ii) branching couplers, and (iii) directional couplers. The monitoring couplers are shown in Fig. 13.20 a–c. Since the purpose of these couplers is to monitor the optical power in the transmission line, it is desirable that the coupled power be independent of the type of modes excited in the main line. The ability to adjust the amount of coupling is also desirable. The grating coupler has

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Fig. 13.20 a–f. Optical power divider coupler. All couplers have isolation between (2) and (3) and transmission between (1) and (2). (a) Taper coupler. (b) Grating coupler. (c) Prism coupler. (d) Selfoc micro lens coupler. (e) Tapered butt-joint type. (f) Planar multiple coupler

the advantage of a controllable coupling coefficient, but some modes can be coupled out more than the others and the monitored power represents the power of only a portion of the modes in the main line. The prism coupler can pick up the power of all modes but the amount of coupling cannot be varied. Figures 13.20 d–f are diagrams of the branching couplers which are used for distributing the same information to all branches of the subsystems. Equal light power is distributed among the branches.

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Fig. 13.21 a–c. Light signal scrambler. (a) Star type. (b) Glass cylinder type. (c) Reflection type

Figures 13.21 a–c are diagrams of light scramblers. Each input port carries a different signal. The scrambler adds the input port signals together so that each output carries a fraction of the addition of all the input port signals. Figures 13.22 a, b show optical directional couplers. When the input light is transmitted from port (1) to port (2), some of the input signal from port (1) is coupled into port (3). When the direction is reversed and the light is transmitted from port (2) to port (1), port (3) is isolated and no signal comes out of port (3). The prism coupler shown earlier in Fig. 13.20 c also has a directional property and can be used as a directional coupler. Both the beam splitter and Selfoc half-mirror type directional couplers in Fig. 13.22 can couple all modes without any prejudice, but the star directional coupler in Fig. 13.22 b does not couple all the modes evenly.

13.9 Wavelength Division Multiplexing (WDM) The wavelength division multiplexing technique is used to transmit more than one wavelength of light in one optical fiber [13.4]. The signal transmission capacity of the fiber increases m-fold if an m-fold wavelength multiplexing is employed. For instance, both analog and digital signals can be transmitted in one fiber. Bidirectional transmission also becomes possible.

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Fig. 13.22 a–c. Directional coupler. When transmitting from (1) to (2), (3) is coupled but when transmitting from (2) to (1), (3) is isolated. (a) Beam splitter type. (b) Star directional coupler. (c) Selfoc with half mirror coupler

Figure 13.23 shows the most commonly used components for wavelength division multiplexing. The prism type multiplexing shown in Fig. 13.23a is simple in structure, and the number of available channels can be large. The interference filter type multiplexing shown in Fig. 13.23 b is limited in the number of available channels. The grating type shown in Fig. 13.23 c requires high precision but can handle many channels. Needless to say, the same element can be used for both multiplexing and demultiplexing the signal. Figure 13.24 shows examples of bi-directional transmission in (a) and wavelength division multiplexing in (b). The bidirectional system in Fig. 13.24 a uses the filter F1 with pass band at λ1 to separate the transmitting channel from the receiving channel. The system in Fig. 13.24 b uses filter F1 with pass band at λ1 , and filter F3 with pass band at λ3 . Neither F1 nor F3 can pass λ2 . Using these two filters, the lights of three different wavelengths are combined and sent through one optical fiber. At the receiving end, they are separated using an arrangement identical to the transmitting end.

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Fig. 13.23 a–c. Components for wavelength division multiplexing. (a) Prism type. (b) Interference filter type. (c) Grating type

13.10 Optical Attenuators There are discretely variable and continuously variable attenuators. The attenuation of the discretely variable attenuator is changed by changing the combination of the individual attentuation cells, as shown in Fig. 13.25a. The attenuator film in each of the attenuation cells in Fig. 13.25a is tilted so as to minimize the light reflected back to the incident port. Selfoc micro lenses are used to widen the beam so that any spatial non-uniformity in the attenuator film is averaged out. Figure 13.25b shows a continuously variable attenuator which is useful as a measuring device. The required value of attenuation is obtained by the combination of the step attenuator films and the continuously varying attenuator film.

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365

Fig. 13.24 a, b. Wavelength division multiplexing systems. (a) Bi-directional transmission of light in one optical fiber. (b) Three wavelenghts of light are combined and transmitted in one fiber and separated at the receiver end

13.11 Design Procedure for Fiber Optical Communication Systems In this section, a step by step procedure for designing a fiber optical communication system is presented along with an example problem [13.17, 18]. The procedure is divided into six steps. Steps 1 to 4 consider the distribution of the light power levels at various components, and Steps 5 and 6 consider the signal frequency bandwidth of the system. The normal procedure is to start at the receiver end and work back to the transmitter. Step 1: Determine the required light power to the receiver using Fig. 13.9. The required light power depends on the choice of detectors (PIN photodiode or avalanche photodiode (APD)), the bit rate of the signal, and the allowable bit error rate (BER).

366

13 Fiber Optical Communication

Fig. 13.25 a, b. Optical attenuators. (a) Discretely variable attenuator. (b) Continuously variable attenuator

Step 2: Calculate the total transmission losses which include optical fiber loss, connector loss and splicing loss. Step 3: Make allowances for degradation of the optical power of the source or the sensitiviy of the receiver with either temperature or time. Step 4: Select the source. Step 5: Calculate the required rise time of the system from the required bit rate. Step 6: Determine the rise time of the entire system using the empirical formula for finding the rise time of tandem circuits: (13.23) tr = 1.1 t12 + t22 + t32 + . . . where t1 . . . tn are rise times of components such as the source, optical fiber and detector [13.19]. The details of the procedure will be explained using an exercise.

13.11 Design Procedure for Fiber Optical Communication Systems

367

Exercise 13.1. Design a fiber optical communication system to connect two computers located in buildings 1.5 km apart. The bit rate is 10 Mb/s and the type of coding used is return to zero code (RZ code). The BER should be less than 10−10 . Solution Step 1: If one uses a PIN diode as a detector, the required average optical power is −39 dB m. The values in Fig. 13.9 are for a BER of 10−8 , but a general rule of thumb is that for each decimal of BER an additional dB is required. Hence the required average light power is −37 dB m. Step 2: If 10 dB/km loss fiber is used, the total loss for 1.5 km is 15 dB. Let us say that 2 splices are required to connect the ends of the spools of the optical fiber and allow one additional splice for a possible breakage during installation. The loss of each splice is 0.2 dB so that the total splice loss is 0.6 dB. Connectors are used at both terminals. The connector loss for each is 1.0 dB and the total connector loss is 2 dB. Step 3: The degradation allowances for temperature and time are both 3 dB each and the total allowance is 6 dB. Step 4: The total loss is fiber loss 15 dB splice loss 0.6 dB connector loss 2 dB degradation allowances 6 dB 23.6 dB This means that one has to select a source whose average output power is larger than −37 + 23.6 = −13.4 dB m. This is within the power range of an LED. Sometimes, the LED is rated in terms of peak power. An example of an RZ signal is shown in Fig. 13.26. “1” is represented by a pulse that comes down to zero at the center of the clock pulse. Thus, if there are equal numbers of “1” s and “0” s, the peak power is 6 dB above the average power and an LED whose peak power is higher than −7.4 dB m has to be selected. Step 5: The most commonly used criterion for estimating the required rise time of a pulse transmission system is that the system rise time has to be faster than 70%

Fig. 13.26 Shapes of pulse of non-return to zero (NRZ) and return to zero (RZ) coding

368

13 Fiber Optical Communication

of the input pulse width. The pulse width of a 10 Mb/s signal is 50 ns. Thus the rise time of the system has to be less than 35 ns. If, however, the coding were NRZ, the required rise time would have been 70 ns. Step 6: The rise time of each component is normally found in the manufacturer’s specification sheets. A typical rise time of an LED is 10–20 ns and that of an LD is 1–2 ns. The rise time of a PIN photodiode is 1 ns and that of an APD is 2 ns. The rise time of the optical fiber has a wide range of values depending on the type of the fiber. Assume that a graded index fiber with 3 ns/km was selected. Components

Rise time Rise time square

LED 15 ns Fiber 3 ns PIN photodiode 1 ns

The rise time of the system = 1.1 × time required.

√

225 9 1 235

235 = 16.86 ns which is well within the rise

Problems 13.1. a) What is the modal pulse spread for 1 km of a step index multimode fiber with NA = 0.2 and refractive index of the core glass n 1 = 1.48? b) What is the maximum pulse rate that can be sent if the dispersions other than mode dispersion can be ignored? Non-return to zero (NRZ) pulses are used. 13.2. a) What is the theoretical limit (η = 1.0) for the output current from a PIN photodiode when the power of the incident light is 1 mW and the wavelength is λ = 0.84 µm? b) What value was assigned to the quantum efficiency to obtain the rule of thumb i = 0.5 A/W, which expresses the output current from a PIN photodiode? 13.3. From Fig. 13.9, prove that the number of photons that have to be contained during one pulse to maintain a given BER value is practically independent of pulse rate. How many photons have to be contained in one pulse for an APD to maintain a BER of 10−8 ? (The wavelength of the light is λ = 0.84 µm.) 13.4. Figures 13.27 a, b are partially drawn diagrams of a wavelength multiplexing system. Complete the diagram by filling in necessary optical components in the areas A, B, C, D, and E enclosed by the dotted lines. Complete the diagram using only two kinds of filters: filter F1 with pass band at λ1 and filter F3 with pass band at λ3 . The characteristics of these filters are shown in Fig. 13.24.

Problems

369

Fig. 13.27 a, b. Wavelength division multiplexing systems. (a) Wavelength division multiplexing in both directions. (b) Wavelength division multiplexing loop

13.5. A fiber optical communication system which is capable of transmitting pulses at a rate of 10 Mbits per second is to be designed. The mode of the pulse modulation is return to zero mode (RZ mode). The bit error rate (BER) has to be smaller than 10−8 . Find the spacing between the repeaters if only the system margin of power is considered (dispersion is not considered). A PIN photodiode is used as the detector and an LED whose peak power is −14 dB m is used as the transmitter. The loss of the optical fiber is 5 dB/km. The connector loss is 1 dB each. Splice loss is 0.3 dB for one. The length of one spool of fiber cable is 5 km. Allow for one additional splice for an accidental break of the cable. The allowance for temperature and time degradations combined is 6 dB.

Chapter 14

Electro and Accousto Optics

This chapter reviews the propagation of light waves in anisotropic media because most of the media used for controlling light propagation are anisotropic. This review will be extended to include devices that use the bulk effect of these phenomena. The bulk phenomena often become the basis of thin-film integrated-optics devices, which are described in Chap. 15.

14.1 Propagation of Light in a Uniaxial Crystal When an electric field is applied to an electrooptic material, electrons in the crystal are perturbed from their equilibrium distributions, and the polarizability of the crystal changes, so that the refractive index changes. In addition to the change in refractive index in the direction of the applied field, it is possible for the refractive index to change also in other directions. A simple analog is the soft pencil eraser which deforms in every direction when it is pressed by a thumb in one direction. A property common to almost all of these crystals is that their refractive indices are anisotropic, even in the absence of external fields. (Gallium arsenide, however, becomes anisotropic only when an external electric field is applied to the crystal.) They are sometimes called doubly refractive or birefringent, which implies that the refractive index for one direction of light polarization is different from that for another direction of polarization. To understand the basics of wave propagation in a birefringent crystal, it is instructive to present here a brief review on birefringence. As is well known, the shape of a wave front diverging from a point source in an isotropic medium is a sphere. The wave front, however, is no longer a sphere if the source is in an anisotropic medium because the phase velocity depends upon the direction of polarization of the wave. Figure 14.1 a shows wave fronts of two waves propagating in the same direction in a plane containing the crystal axis but polarized in two perpendicular directions. One of the waves is polarized perpendicular to the page (represented by dots), and the other, in the plane of the page (represented by bars). The rate of advance of the wave front of the former is the same in all directions 371

372

14 Electro and Accousto Optics

Fig. 14.1 a, b. Wave front of a point source in a birefringent crystal. (a) Negative birefringence. (b) Positive birefringence

in the page and the leading wave front lies in a circle, whereas that of the latter depends upon its direction of propagation and the wave front lies in an ellipse. The former wave is called the ordinary wave or o wave, whereas the latter, is called the extraordinary wave or e wave. There is one direction for which both o and e waves have the same phase velocity at the circle and ellipse intercept1 . This direction is called the optical axis of the crystal. The refractive index of the crystal for the o wave is called the ordinary index n o , and that for the e wave whose direction of polarization is parallel to the optical axis is called the extraordinary index n e . There are two types of birefringence. The birefringence is called positive birefringence when n e − n o is positive, and it is called negative birefringence when n e − n o is negative. Figure 14.1a illustrates the case of negative birefringence. (Note that what is shown is the developed wave front some time after the emission, and that the distribution of the refractive index is inversely proportional to the radius). Figure 14.1b is the corresponding situation for positive birefringence. Crystals such as quartz (SiO2 ), and lithium tantalate LiTaO3 show positive birefringence while lithium niobate LiNbO3 , calcite or calcium carbonate (CaCO3 ), potassium dihydrogen phosphate (KH2 PO4 ) or KDP, and ammonium dihydrogen phosphate (NH4 H2 PO4 ) or ADP exhibit negative birefringence. In nature there are two kinds of crystals: uniaxial crystals which have only one optical axis and biaxial crystals which have two optical axes. The analysis of the phase front in a biaxial crystal is much more complicated than that of a uniaxial crystal. Most of the birefringent crystals found in nature are biaxial but fortunately, almost all the crystals used for electro-optic devices are uniaxial. Therefore, subsequent descriptions will concentrate solely on the uniaxial crystal. In order to deal with the general case, the concept of the optical indicatrix, or index ellipsoid, has been found useful. For the uniaxial crystal, the optical indicatrix is an ellipsoid of revolution (Fig. 14.2). Such an ellipsoid is expressed by the equation 1

A crystal in which the circle and ellipse do not quite intercept is called an optically active crystal and is a very important class of crystal that will be discussed in Sect. 14.4.1.

14.1 Propagation of Light in a Uniaxial Crystal

373

Fig. 14.2 Indicatrix of a uniaxial crystal

x2 y2 z2 + + = 1. n 2o n 2o n 2e

(14.1)

The manner in which the indicatrix is used is explained in the following. Consider light propagating with its wave normal in the ON direction. The intercept of the plane perpendicular to ON (hatched section) with the ellipsoid generates an ellipse with minor axis D1 and major axis D2 . When the light propagates in the ON direction, the wave must be polarized in either D1 or D2 with the index of refraction represented by the length of D1 or D2 . The former is the o wave, and the latter is the e wave. No matter what the direction of ON is, the vector D1 is always perpendicular to the optical axis, and its magnitude is always n o . The vector D2 however is different. Its direction changes and approaches the optical axis as ON moves away from the optical axis, and its magnitude increases as D2 approaches the optical axis. The magnitude is maximum (positive birefringence is assumed) when D2 coincides with the optical axis. It is important to note that, when the wave front is propagating along any given direction ON, there are only two allowed directions of polarization, these being the directions of the minor and major axes of the ellipse, D1 and D2 . No other direction of polarization is permitted, except when ON is along the optical axis, in which case the intercept is a circle, and any direction is allowed. A heuristic explanation of this reasoning is given in Fig. 14.3. Imagine a voice is transmitted through a fictitious wall made up of rectangular pebbles arranged neatly like a crystal lattice with their axes slanted. These pebbles are constrained so that they can move only along the length axis. Now, an incident sound wave tries to vibrate the wall in the ξ direction but, due to the anisotropy of the pebbles, a

374

14 Electro and Accousto Optics

Fig. 14.3 Transmission of the voice through an anisotropic wall

displacement D is allowed only in the direction of the length axis of the pebble. Regardless of the direction of ξ , the allowed direction of D is always the same. It may be added that the sound energy mainly transmits in the direction straight across the wall and is not greatly influenced by the direction of D. This corresponds to the fact that the direction of the energy transmission is not necessarily the same as that of the wave normal in an anisotropic crystal. Even when the light incident upon the anisotropic crystal is polarized in the direction of neither D1 nor D2 , the light still transmits through the crystal. In this case, the electric field of the light is decomposed into components parallel to D1 and to D2 , and each component is treated separately. The final result is obtained by combining the two results. When the incident wave is linearly polarized in a direction bisecting D1 and D2 , the two decomposed components each parallel to D1 and D2 have equal magnitudes but they propagate at different phase velocities so that the resultant wave becomes elliptically polarized. At a certain distance, the phase difference becomes π/2 radians, and the wave becomes circularly polarized. A plate cut and polished at this particular length is called a quarter wave plate. It is used for generating a circularly polarized wave from a linearly polarized wave. Figure 14.4 shows two examples of wave front propagation in planes perpendicular and parallel to the optical axis. They were obtained using the indicatrix in Fig. 14.2. When the direction ON of propagation is in the plane perpendicular to

14.2 Field in an Electrooptic Medium

375

Fig. 14.4 a, b. Wave front of light in a uniaxial crystal. (a) Wave front of light whose direction of propagation is in a plane perpendicular to the optical axis. (b) Wave front of light whose direction of propagation is in the plane of the optical axis (in y − z plane)

the optical axis, the wave front behaves, as shown in Fig. 14.4a. The polarization of the e wave is parallel and the o wave is perpendicular to the optical axis, but the orientation of both polarizations with respect to the optical axis remains the same, regardless of the direction of ON, as long as it stays in this plane. Figure 14.4b shows the wavefront when the direction of ON is in a plane containing the optical axis (y − z plane). Referring to the indicatrix in Fig. 14.2, it is seen that as ON approaches the y axis, D2 increases while D1 is invariant. Thus with a scan of ON, the locus of the phase front of the e wave is an ellipse, while that of the o wave is a circle.

14.2 Field in an Electrooptic Medium The change in the refractive index n due to an applied electric field E is expressed as 1 (14.2) = r E + g E 2. ∆ n2 The first term is linearly proportional to the applied electric field and is known as the Pockels effect. The second term, which has a quadratic dependence on the applied electric field, is known as the Kerr electrooptic effect. While the Pockels effect depends upon the polarity of the applied electric field, the Kerr electrooptic effect does not. The most popular crystals which display the electrooptic effect and are used for electrooptic devices are: lithium niobate (LiNbO3 ), lithium tantalate (LiTaO3 ), potassium dihydrogen phosphate better known as KDP (KH2 PO4 ), ammonium dihydrogen phosphate better known as ADP (NH4 H2 PO4 ), and gallium arsenide (GaAs).

376

14 Electro and Accousto Optics

The electrooptic effect can be best represented as a deformation of the indicatrix. If the major or minor axes are lined up with the x, y and z axes, the ellipsoid is expressed in the compact form y2 z2 x2 + + = 1. n 2x n 2y n 2z

(14.3)

However, once the ellipsoid is deformed and these axes move away from the x −y−z axes, the cross terms y z, z x or x y become necessary to represent the new ellipsoid. The general expression for such an ellipsoid is a11 x 2 + a22 y 2 + a33 z 2 + 2 a23 y z + 2 a31 z x + 2 a12 x y = 1.

(14.4)

The values of the coefficients a11 , a22 , a33 , a23 , . . . change according to the applied electric fields. They can be represented by ⎤ ⎡ ⎤ ⎡ a11 − 1/n 2x r11 r12 r13 ⎢a22 − 1/n 2y ⎥ ⎢r21 r22 r23 ⎥ ⎡ ⎤ ⎥ ⎢ ⎥ ⎢ ⎢a33 − 1/n 2 ⎥ ⎢r31 r32 r33 ⎥ Ex ⎥⎣ ⎦ ⎢ z⎥ = ⎢ (14.5) ⎥ ⎢r41 r42 r43 ⎥ E y ⎢ a23 ⎥ ⎢ ⎥ Ez ⎢ ⎦ ⎣r51 r52 r53 ⎦ ⎣ a31 r61 r62 r63 a12 where r11 , r12 , r13 , . . . are called electrooptic (or Pockels) constants; Ex , E y , Ez are the components of the applied electric field in the x, y, z directions; and n x , n y , n z are the refractive indices in the x, y, z directions when the applied field is absent. Note that when the applied field is zero, (14.4) reduces to (14.3). Equation (14.5) looks rather cumbersome, but many of the 18 elements of the matrix are zero and some of the non-zero elements have the same value. Their values depend upon the symmetry of the crystal. Lithium niobate, for instance, has only 4 different values for its 8 non-zero elements. Gallium arsenide has only 1 value for its 3 non-zero elements. Electrooptic constants of representative crystals and their matrices are tabulated in Table 14.1 and 14.2.

14.2.1 Examples for Calculating the Field in an Electrooptic Medium Examples which show how to analyze the electrooptic effect are presented. The general steps to solve the problems are: 1) 2) 3) 4)

find the indicatrix under the applied electric field, find the allowed directions of polarization for a given direction of propagation, find the refractive indices in the allowed directions, and decompose the incident light wave into components along the allowable directions of polarization.

The sum of the components of the emergent wave is the desired solution.

14.2 Field in an Electrooptic Medium

377

Table 14.1 Electrooptic constants [14.1–3] Material

Crystal symmetry

LiNbO3

3m

LiTaO3

3m

GaAs

4¯ 3 m

HgS

32

CdS

6 mm

KDP

4¯ 2 m

ADP

4¯ 2 m

SiO2 (Quartz)

32

r [10−12 m/V] ⎫ = 8.6 ⎪ ⎪ ⎬ = 30.8 = 3.4 ⎪ ⎪ = 28 ⎭ ⎫ r13 = 7.9 ⎪ ⎪ ⎬ r33 = 35.8 r22 = 1 ⎪ ⎪ r51 = 20 ⎭ r41 = 1.2 ( r11 = 3.1 r41 = 1.4 r13 r33 r22 r51

r13 = 1.1 r33 = 2.4 r51 = 3.7 ( r41 = 8.6 r63 = 10.6 ( r41 = 28 r63 = 8.5 ( r11 = 0.29 r41 = 0.2

λ[µm]

Refractive index

0.63

n x = n y = 2.286 = 2.200 nz

0.63

n x = n y = 2.176 = 2.180 nz

0.9

n x = n y = n z = 3.42

0.63

n x = n y = 2.885 = 3.232 nz

0.63 0.63 0.59

n x = n y = 2.46 = 2.48 nz

0.55

n x = n y = 1.51 = 1.47 nz

0.55

n x = n y = 1.52 = 1.48 nz

0.63

n x = n y = 1.546 = 1.555 nz

Table 14.2 Electrooptic matrices 3m ⎛

⎞ 0 −r22 r13 ⎜ 0 −r22 r13 ⎟ ⎜ ⎟ 0 r33 ⎟ ⎜ 0 ⎜ ⎟ 0 r 0 ⎜ ⎟ 51 ⎝r 0 0⎠ 51 −r22 0 0 6 mm ⎛ ⎞ 0 0 r13 ⎜ 0 0 r13 ⎟ ⎜ ⎟ ⎜ 0 0 r33 ⎟ ⎜ ⎟ ⎜ 0 r51 0 ⎟ ⎝r ⎠ 0 0 51 0 0 0

4¯ 3 m ⎛ ⎞ 0 0 0 ⎜0 0 0⎟ ⎜ ⎟ ⎜0 0 0⎟ ⎜ ⎟ r 0 0 ⎜ 41 ⎟ ⎝0 r ⎠ 41 0 0 0 r41 32 ⎛ ⎞ r11 0 0 ⎜−r11 0 0⎟ ⎜ ⎟ 0 0⎟ ⎜ 0 ⎜ ⎟ 0 0⎟ ⎜ r41 ⎝ 0 −r 0⎠ 41 0 −r11 0

4¯ 2 m ⎛ 0 ⎜0 ⎜ ⎜0 ⎜ ⎜r41 ⎝0 0

0 0 0 0 r41 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ 0⎠ r63

378

14 Electro and Accousto Optics

Exercise 14.1. An external electric field Ez is applied along the optical axis z of the uniaxial crystal LiNbO3 , and the light is incident along the y axis. a) Find the allowable directions of polarization and their refractive indices. b) Find the emergent light when the incident light is along the y axis with its polarization at 45◦ from the x axis in the x z plane. What is the condition for the emergent light to be circularly polarized? The length of the crystal is L, the applied voltage is V , and the spacing between the electrodes is d. Solution The matrix for the electrooptic constants of lithium niobate is ⎛ ⎞ 0 −r22 r13 ⎜ 0 r22 r13 ⎟ ⎜ ⎟ ⎜ 0 0 r33 ⎟ ⎜ ⎟ ⎜ 0 r51 0 ⎟ . ⎜ ⎟ ⎝ r51 0 0 ⎠ −r22 0 0

(14.6)

With n x = n y = n o , n z = n e and Ex = E y = 0, (14.4–14.6) lead to 2 −2 2 −2 2 (n −2 o + r13 Ez ) x + (n o + r13 Ez ) y + (n e + r33 Ez )z = 1.

(14.7)

From Table 14.1a, the electrooptic coefficients r are expressed in units of 10−12 m/V. For applied electric fields of practical interest, the product r E is a number much less than unity. Since in each bracket the second term is much smaller than n o or n e , (14.7) can be approximated by using the binomial expansion2 as

x2 no −

n 3o 2 r13 Ez

2 +

y2 no −

n 3o 2 r13 Ez

2 +

z2

ne −

2 n 3e 2 r33 Ez

= 1.

(14.8)

Equation (14.8) is the expression for the indicatrix. As shown in Fig. 14.5, the size of the indicatrix shrinks with an increase in the external field. Since the electrooptic constants of lithium niobate are r33 = 30.8 × 10−12 m/V and r13 = 8.6×10−12 m/V, the shrinkage in the z direction is more than 3 times that in the x or y directions. It also should be recognized that the amount of change in the refractive index is, in general, very small compared to n o or n e . Next, the allowable directions of polarization for light propagating in the y direction are considered. The cross-section of the indicatrix cut by a plane normal to ON is an ellipse. The ellipse is in the x z plane since ON is along the y axis. The allowable directions of polarization are the major and the minor axes D1 and D2 of this ellipse. The corresponding refractive indices are ( n x = n o − 12 n 3o r13 Ez . (14.9) n z = n e − 12 n 3e r33 Ez 2

Note and solve for ∆.

2 −2 (n −2 o + r13 Ez ) = 1/(n o + ∆) n o [1 − 2(∆/n o )]

14.2 Field in an Electrooptic Medium

379

Fig. 14.5 Deformation of the indicatrix of lithium niobate when an electric field is applied in the z direction

b) Finally, the emergent field is considered. The incident field has to be decomposed into components parallel to the allowed directions of polarization. The incident wave is decomposed into E x and E z components. The phases of these components at the exit are V 2π π n o L − n 3o r13 L λ λ d V 2π π n e L − n 3e r33 L. φz = λ λ d φx =

(14.10)

Because of the phase difference ∆φ = φz − φx , the emergent wave is in general elliptically polarized. In particular, it is circularly polarized when ∆φ = π/2, and it is linearly polarized when ∆φ = π . Exercise 14.2. Consider a situation identical to Exercise 14.1, but with GaAs in place of LiNbO3 . a) Find the direction of the optical axes when the field is applied. b) Find the allowable directions of polarization and their refractive indices. Solution The matrix for the electrooptic constants of GaAs from Table 14.1 is

380

14 Electro and Accousto Optics

⎛

0 ⎜0 ⎜ ⎜0 ⎜ ⎜r41 ⎜ ⎝0 0

0 0 0 0 r41 0

⎞

0 0⎟ ⎟ 0⎟ ⎟. 0⎟ ⎟ 0⎠ r41

With n x = n y = n z = n o , and using (14.4, 5), the expression for indicatrix is 2 −2 2 −2 2 n −2 o x + n o y + n o z + 2 r41 Ez x y = 1.

(14.11)

Notice that, in the absence of an external field Ez , the indicatrix is a sphere, which simply means that the GaAs crystal is isotropic. In order to see the shape of the indicatrix more closely, the coordinate system is rotated by 45 degrees around the z axis into a new coordinate system (x , y , z) using the rotation matrix, 1 1 −1 x x . =√ 1 1 y y 2 It is interesting to see how the x y term in (14.11) is uncoupled by the magic power of a change of coordinates, i.e., x y = (x − y )(x + y )/2 = (x 2 − y 2 )/2. The result of the coordinate rotation is

x 2 no −

n 3o 2 r41 Ez

2 +

y 2 no +

n 3o 2 r41 Ez

2 +

z2 = 1. n 2o

(14.12)

As indicated by the dotted line in Fig. 14.6, the indicatrix changes from a sphere to an ellipsoid when an external electric field Ez is applied. The refractive indices in these directions are ⎫ n3 n x = n o − 2o r41 Ez ⎪ ⎬ n 3o (14.13) n y = n o + 2 r41 Ez ⎪ . ⎭ n z = no From Fig. 14.6, the cross-section with the y = 0 plane is a circle, and the y axis is the optical axis. [It can also be seen directly by setting y = 0 in (14.11)]. Similarly, the x axis is also the optical axis. The refractive indices in these directions are n o . It may be added that, if the direction of propagation were along the y axis, the allowed directions of polarization would be the x and z axes with refractive indices given by (14.13). Exercise 14.3. Consider the case of KDP with an external electric field Ez in the z direction. The light is incident at an angle θ from the x axis in the x y plane. Find the allowed directions of polarization and their refractive indices. Solution The matrix for KDP is

14.2 Field in an Electrooptic Medium

381

Fig. 14.6 Deformation of the indicatrix of GaAs when an electric field Ez is applied

⎛

0 ⎜0 ⎜ ⎜0 ⎜ ⎜r41 ⎜ ⎝0 0

0 0 0 0 r41 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ r63

with n x = n y = n o and n z = n e . Inserting the matrix into (14.4, 5) leads to 2 −2 2 −2 2 n −2 o x + n o y + n e z + 2 r63 Ez x y = 1.

(14.14)

The cross-section of the indicatrix cut by the plane perpendicular to the direction of light is needed. It can be found by rotating the x axis to the x axis, which coincides with the direction of light, and then setting x = 0. The rotation of the axis by θ degrees around the z axis can be executed by a rotation matrix x cos θ − sin θ x . = y y sin θ cos θ The application of the matrix to (14.14) with x = 0 leads to y 2

no +

n 3o 2 r63

Ez sin 2 θ

2 +

z2 = 1. n 2e

(14.15)

Thus, the allowed directions of polarization are y and z with refractive indices

382

14 Electro and Accousto Optics

n 3o (14.16) r63 Ez sin 2θ, n z = n e . 2 Note, if the direction of propagation is either along x or y axis there is no influence of Ez . n y = no +

Exercise 14.4. Find the optical axes of a crystal whose indicatrix axes are(n x , n y , n z ). Solution The problem reduces to cutting the indicatrix in such way as to create a cross-section of circular shape. First, pick the shortest axis. Let us say n x is the shortest axis of the indicatrix drawn in Fig. 14.7. The cross-section of the indicatrix cut by the plane normal to the direction of light ON along the z axis is an ellipse. The shape of the ellipse becomes more circular as ON starts to tilt in the x z plane away from the z axis. The minor axis r starts to increase from n x , whereas the major axis n y is fixed. There must be a particular angle θ of inclination such that r equals n y . That is the direction of the optical axis. A point P in the y = 0 plane is represented by ⎫ x = r cos θ ⎬ z = r sin θ . (14.17) ⎭ y=0 If P were to stay on the ellipse in y = 0 plane, (14.17) should satisfy r 2 sin2 θ r 2 cos2 θ + = 1, 2 nx n 2z 1 1 1 1 = 2 + − 2 sin2 θ r2 nx n 2z nx

Fig. 14.7 Method of finding the optical axis of an indicatrix (n x , n y , n z )

hence

14.2 Field in an Electrooptic Medium

and the angle θ that r = n y is nz sin θ = ± ny

383

n 2y − n 2x n 2z − n 2x

.

(14.18)

Notice that there are two angles which satisfy (14.18), and there are two optical axes which are symmetric with respect to the z axis. Such a crystal is called a biaxial crystal. An alternate method of solving the problem is rotation of the coordinates x, z into x , z around the y axis. The cross-section cut by z = 0 is the ellipse concerned. The angle of rotation θ is adjusted so that the major axis equals the minor axis.

14.2.2 Applications of the Electrooptic Bulk Effect A prism deflector, such as shown in Fig. 14.8a, uses the bulk electrooptic effect. The direction of the deflected beam is controlled by an external electric voltage. The angle of deflection can be increased by staggering a number of prisms, as illustrated at the bottom of the figure. Figure 14.8b shows a light deflector which scans in both the horizontal and vertical directions. The angle of deflection is proportional to grad n (Sect. 5.4), and n is proportional to the E field which is – grad V . This deflector with its quadrupole shape deflects the light in linear proportion to the applied voltage [14.4]. Figure 14.8c shows a digital electrooptic deflector. A birefringent crystal like calcite is combined with an electrooptic polarizer like lithium niobate. There are two distinct directions of refraction depending upon the direction of the polarization of the incident light. The direction of the polarization of the incident light is switched between two directions 90 degrees apart by switching on and off the voltage to the electronic polarizer. With the arrangement shown in the figure, the light can be shifted to 4 different locations by various combinations of the control voltages υ0 and υ1 . A deflector with N number of stages can direct the light to 2 N number of locations. A binary coded voltage can be directly used to control the light. This kind of deflector is called a digital light deflector because the light is directed to only discrete locations, whereas deflectors such as the previously described prism and quadrupole gradient deflectors are called analog light deflectors because the amount of deflection is continuously changed by the control voltage. The geometry of an electrooptic modulator using lithium niobate is shown in Fig. 14.8d. The crystal axis (most often called C axis) is placed along the z axis of the coordinate system. The direction ON of propagation is taken along the y axis. The external electric field Ez is in the z direction. This is a geometry similar to Exercise 14.1. This time, however, the explanation will make use of a diagram rather than the formulae. The allowed directions of polarization are the x and z axes. The input polarization has to be decomposed into x and z components. Consider an incident wave which is

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14 Electro and Accousto Optics

Fig. 14.8 a–d. Electrooptic devices. (a) Prism type electrooptic deflector. (b) Quadrupole type deflector. (c) Electrooptic digital light deflector. P: Electrooptic polarizer (LiNbO3 ) to rotate the direction of polarization by 90◦ when the electric field is applied. D: Birefringent crystal like calcite to defelct light. (d) Electrooptic effect used for a light switch or a light modulator

linearly polarized at an angle of 45◦ with respect to the z axis. At the entrance, E x and E z reach their maximum values at the same instant in time or in other words, both oscillate in phase. But once inside the crystal, the velocities for E x and E z are different due to anisotropy, and one starts to lag behind the other. There is a location O where E z becomes a negative maximum at the instant when E x becomes a positive maximum. Namely, the phase of E z lags 180 degrees behind that of E x . At this location, the resultant E t of E x and E z points at −45◦ and the resultant field vibrates along a line which is rotated 90 degrees from the direction of the incident polarization. If one arranges a polarized sheet at the location O with its direction of polarization perpendicular to E t , then the light can be completely switched off. The device can be used as an optical switch. The location O can be calculated using (14.10), which stipulates that the required length L has to satisfy

14.2 Field in an Electrooptic Medium

385

Fig. 14.8 Continued

V π 3 (n r33 − n 3or13 ) λ e d

L = π,

(14.19)

where a constant value (2π/λ)(n e − (n o ) was suppressed. The same device can be used as a light modulator if V is modulated. Electrooptic materials are generally quite temperature sensitive, and special precaution has to be taken to keep the temperature constant. A fluctuation of 1 ◦ C results in a change of 0.26 radians in a 1 mm long lithium niobate crystal at 1 µm wavelength of light [14.5]. The upper and lower electrodes of the modulator in Fig. 14.8d form a capacitor and this usually sets the upper limit on modulation. When the wavelength of the modulating signal is short compared to the length of the electrode, the applied field becomes non-uniform due to the presence of a standing wave. The polarities of two points spaced half a wavelength apart oppose one another and efficiency is reduced. One way to avoid this and raise the frequency limit is to treat the parallel electrodes as a parallel plate transmission line to match the velocity of the modulating signal along the transmission line with that of the light in the crystal. It is also important to terminate it with a load impedance which is the same as the characteristic impedance of the line.

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14 Electro and Accousto Optics

14.3 Elastooptic Effect Materials such as water, lithium niobate, rutile (TiO2 ), and Lucite change their refractive index when they are under stress. With many crystals, the change in refractive index in one direction is different from that in other directions. In other words, the relationship between the amount of change in refractive index and the strain is expressed by strain-optic constants in a tensor form. Since an acoustic wave in a medium is a particular type of strain wave, the terms “acoustooptic effect” and “elastooptic effect” are often used interchangeably.

14.3.1 Elastooptic Effect in an Isotropic Medium With such a medium as water whose elastooptic effect is isotropic, the change n in refractive index due to the strain is expressed by the scalar equation ∆n = −

n 3o pS 2

(14.20)

where n o is the refractive index under no stress, p is the elastooptic constant, and S is strain defined by the deformation divided by the total length. A light deflector, such as shown in Fig. 14.9, is one of the applications of the acoustooptic effect of fluid. An acoustic wave is launched from the bottom of the deflector cell containing an elastooptic medium. A periodic strain distribution is established in the cell. Since the velocity of the acoustic wave is much slower than that of the light waves, the cell can be treated as if a stationary diffraction grating were occupying the cell. When the cell is thin, and the intensity of the acoustic wave is low, the change of direction of the ray during transmission through the cell is small. The emergent wave is still considered to be a plane wave, but with its phase spatially modulated. The diffraction pattern under such a condition is called Raman-Nath diffraction, and was discussed in Problem 4.2. The angle θ of the direction of the maxima of the emergent wave was given by sin θ − sin θ0 = m

λ , λa

(14.21)

where θ0 is angle of incidence, λa is wavelength of the acoustic wave, and m is an integer. Angles are taken as positive when they are counter-clockwise from the normal to the cell surface. As shown in Fig. 14.9, on both sides of a line extending from the direction of the incident ray, a number of maxima associated with m = 0, ±1, ±2, ±3, . . . appear. The angles of diffraction can be steered by changing the frequency of the acoustic wave. The advantage of such a deflector over a deflector using a mechanically moving mirror is that the former can steer the beam with greater speed.

14.3 Elastooptic Effect

387

Fig. 14.9 a, b Acoustooptic deflector. (a) Raman-Nath diffraction. (b) Bragg diffraction

When the thickness of the cell and the intensity of the acoustic wave is increased, as shown in Fig. 14.9b, the mechanism of diffraction of the beam becomes slightly different and is called Bragg diffraction. The cell is interpreted as a stack of reflecting layers spaced by the wavelength of the acoustic wave. A similar analysis is used for treating x-ray diffraction from the layers of atoms. When the path of the ray becomes such that the difference in the optical path between the adjacent layers in the figure equals an integral multiple of the light wavelength, the light rays reflected from each layer will be in phase and the reflected light intensity displays a peak. The Bragg diffraction condition is sin θ = m

λ . 2 λa

(14.22)

The Bragg condition is a special case of the Raman-Nath condition. Since with Bragg diffraction, each layer is considered as a mirror, the additional condition of

388

14 Electro and Accousto Optics

−θ0 = θ

(14.23)

is imposed over the Raman-Nath condition. Insertion of (14.23) into (14.21) leads to the Bragg condition (14.22). It should be noticed that, in the case of Raman-Nath diffraction, several discrete peaks can be observed in the emergent wave corresponding to the various diffraction orders m = 0, ±1, ±2, and so forth. However, when light is incident at the Bragg angle, a single peak in the emergent wave dominates all others for a given Bragg angle of incidence, and this peak is the Bragg diffracted wave. In fact, when certain parameters such as the acoustic intensity, frequency and cell length are properly chosen, a very high percentage of the incident wave can be converted into the Bragg diffracted wave. The efficiency of converting the incident to the Bragg diffracted wave decreases dramatically if the direction of incidence strays from the Bragg angle defined by (14.22). It may be added that, in the case of x-ray diffraction, if the setting of the incident beam to the crystal is arbitrary, absolutely no diffraction pattern is observed. The pattern appears only when the incident beam satisfies (14.22,14.23). In x-ray crystallography [14.6–8], the Laue pattern meets this condition by continuously varying the wavelength, whereas the rotating-crystal method and powder method (that uses a powdered sample rather than a lump sample) meet the condition by varying the incident angle. By turning on and off the acoustic wave, the acoustooptic deflector can be used to switch directions between the beam which passes straight through the cell, and the beam or beams which are diffracted by the cell. The deflector shown in Fig. 14.9 can also be used as a modulator. The amount of light diffracted by the cell depends on the amplitude of the acoustic wave. Therefore, an amplitude modulation of the acoustic wave will produce an amplitude modulation on each of the emergent light beams. Another characteristic of the acoustooptic diffraction is that the diffracted beams are inherently shifted in frequency by an integral multiple of the acoustic frequency. One interpretation of this phenomenon is that the light is being reflected by a moving reflecting plane, and a shift of frequency accompanies. Frequency modulation of the acoustic wave results in a frequency modulation of the deflected waves, so that the device can also be used as a frequency modulator. Compared with the bulk electrooptic modulator, which needs several kilovolts to operate, a bulk acoustooptic modulator needs only several volts. However, the response time is much slower than the electrooptic device because it needs time for the acoustic wave to propagate across the cell. Quite often, the frequency band characteristics are limited by the frequency response of the transducer. If one wants to keep the incident Bragg diffraction angle constant, the range of the modulation frequency has to be restricted.

14.3.2 Elastooptic Effect in an Anisotropic Medium Such elastooptic materials as lithium niobate or tellurium dioxide (TeO2 ) are crystals and the elastooptic effect is anisotropic. The relationship between the applied

14.3 Elastooptic Effect

389

strain and the change in the refractive index is found by calculating the deformation of the indicatrix [14.9]. Almost the same expression as the electrooptic effect (Sect. 14.2) is used for the elastooptic effect. The only difference is that the size of the matrix is 6 × 6 rather than 6 × 3, namely ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ B11 − 1/n 21 p11 p12 p13 p14 p15 p16 s1 ⎢ B22 − 1/n 2 ⎥ ⎢ p21 p22 p23 p24 p25 p26 ⎥ ⎢s2 ⎥ 2⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎢ B33 − 1/n 2 ⎥ ⎢ p31 p32 p33 p34 p35 p36 ⎥ ⎢s3 ⎥ ⎥ ⎢ ⎥ ⎢ 3⎥ = ⎢ (14.24) ⎥ ⎢ p41 p42 p43 p44 p45 p46 ⎥ ⎢s4 ⎥ ⎢ B23 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ p51 p52 p53 p54 p55 p56 ⎦ ⎣s5 ⎦ ⎣ B31 p61 p62 p63 p64 p65 p66 s6 B12 where B11 x 2 + B22 y 2 + B33 z 2 + 2 B23 y z + 2 B31 x z + 2 B12 x y = 1. There are two types of strain. One type is the deformation which takes place uniformly across the plane, as shown in Fig. 14.10a. Such a strain is called the principal strain and is defined as Sx x =

du , dx

(14.25)

where du is the deformation in the x direction in a line element of length dx. The other type is the deformation which is not uniform across the plane, such as shown

Fig. 14.10 a–d Strain components. (a) Principal strain. (b) Shearing strain in y. (c) Shearing strain in x and y. (d) Combination of principal strain and shearing strain

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14 Electro and Accousto Optics

in Fig. 14.10b. Here, the deformation du varies with y. Such a deformation is called the shearing strain. The shearing strain Sx y in Fig. 14.10b is defined as ∂u/∂ y. Figure 14.10c shows the shearing strain in the general case. The shearing deformation takes place in both the x and y directions. The shearing strain in this case is defined as 1 ∂u ∂υ + (14.26) Sx y = 2 ∂y ∂x where dv is the deformation in the y direction. Note that ∂u/∂ y is approximately φ1 which is the angle associated with the deformation in the x direction. Similarly ∂v/∂ x is approximately φ2 which is associated with the deformation in the y direction. Thus, Sx y is one half of the angle associated with the total shearing strain taking place in the x y plane. The general case of deformation in the x y plane (Fig. 14.10d) can be represented by a combination of these two types of strain. The components of strain are summarized as ⎫ ∂u 1 ∂υ ∂w ⎪ , S yz = + ,⎪ Sx x = ⎪ ⎪ ∂x 2 ∂z ∂y ⎪ ⎪ ∂υ 1 ∂u ∂w ⎬ S yy = , Sx z = + , (14.27) ∂y 2 ∂z ∂x ⎪ ⎪ ⎪ ∂w 1 ∂u ∂υ ⎪ ⎪ Szz = , Sx y = + .⎪ ⎭ ∂z 2 ∂y ∂x These can be expressed by one formula ∂u j 1 ∂u i + , (14.28) Si j = 2 ∂x j ∂ xi where Si j is sometimes called Cauchy’s infinitesimal strain tensor. A common convention is to express the subscripts by numerals as Sx x S yy Szz S yz Sx z Sx y ↓ ↓ ↓ ↓ ↓ ↓ S1 S2 S3 S4 S5 S6 . Exercise 14.5. An acoustic shear wave whose direction of propagation is in the z direction and whose direction of oscillation is in the y direction is launched into a lithium niobate crystal. Find the expression of the indicatrix under the influence of the acoustic wave. The matrix of the elastooptic constants of lithium niobate is ⎤ ⎡ 0 p11 p12 p13 p14 0 ⎢ p12 p11 p13 − p14 0 0 ⎥ ⎥ ⎢ ⎢ p31 p31 p33 0 0 0 ⎥ ⎥ ⎢ ⎢ p41 − p41 0 p44 0 0 ⎥ ⎥ ⎢ ⎣ 0 0 0 0 p44 − p41 ⎦ 0 0 0 0 p14 p66 with n x = n y = n o = 2.286, n z = n e = 2.2, p11 = −0.02, p12 = 0.08, p13 = 0.13, p14 = −0.08, p31 = 0.17, p33 = 0.07, p41 = −0.15, p44 = 0.12, p66 = −0.05.

14.3 Elastooptic Effect

391

Fig. 14.11 Excitation of shear wave

Solution Figure 14.11 shows an exaggerated view of the deformation of the elastooptic medium due to the acoustic wave. The strain created by the acoustic wave is in the z direction and is a function of y, as shown in Fig. 14.11. Such a strain is the shearing strain taking place in the y z plane. Thus, only S4 is nonzero, and the expression for the indicatrix becomes 2 −2 2 −2 2 (n −2 o + p14 S4 ) x + (n o − p14 S4 ) y + n e z + 2 p44 S4 y z = 1.

(14.29)

One method of eliminating the y z cross term is to rotate the coordinates around the x axis by such an angle that this term disappears, but here another method is employed. First consider a general expression of the form A y 2 + 2 B y z + C z 2 = 1.

(14.30)

It is more complicated to find the major axes when A = C as in (14.30). The axis will be found by an eigenvalue method. The normal to this ellipse is found from the gradient, ∇g(y, z), where g(y, z) is the left-hand side of (14.30). A vector N in the direction of the normal is Ay+Bz A B y N= = (14.31) B y+C z BC z where the factor of 2 contained in ∇g(x, y) is immaterial for expressing the direc y tion and is suppressed. As shown in Fig. 14.12, a vector υ = connecting the z origin to the point (y, z) on the ellipse lines up with the normal N from that point if the point is on either the major or minor axis. When both vectors are lined up, A B y y =λ (14.32) BC z z should be satisfied. In order that a non-trivial solution exist, the determinant of ! ! !A − λ B ! ! ! ! B C − λ!

392

14 Electro and Accousto Optics

Fig. 14.12 A way of finding the major axes of an ellipse

should vanish so that (A − λ)(C − λ) − B 2 = 0 and hence, λ1,2 =

1 [(A + C) ± (A + C)2 − 4(AC − B 2 )]. 2 −1/2

(14.33)

−1/2

and λ2 , and the unit vectors The major and minor axes3 of the ellipse are λ1 expressing the axis are obtained from (14.32, 14.33) as 1 B , u1 = B 2 + (λ1 − A)2 λ1 − A 1 B . (14.34) u2 = B 2 + (λ2 − A)2 λ2 − A 3

Note that (14.30) can be rewritten as an inner product of the normal vector N and the position vector υ, A B y . = 1, where M= (y, z) M B C z If the position vector υ is expressed by new coordinates (Y, Z ) along u1 and u2 such that y then = Y u1 + Z u2 , υ= z υ t M υ = (Y ut1 + Z ut2 ) M (Y u1 + Z u2 ) = (Y ut1 + Z ut2 )(λ1 Y u1 + λ2 Z u2 ) = λ1 ut1 u1 Y 2 + λ2 ut2 u2 Z 2 , where the superscript t means transpose. Therefore λ1 Y 2 + λ2 Z 2 = 1, where the eigenvector relationship (14.32) and the orthogonality condition ut1 u2 = ut2 u1 = 0 were used. The orthogonality can be easily checked by calculating this product using (14.33, 14.34).

14.4 Miscellaneous Effects

393

This procedure is the same as finding eigenvalues and eigenvectors of a matrix. Finally, the expression of the indicatrix in (14.29) becomes λ0 x 2 + λ1 Y 2 + λ2 Z 2 = 1, λ0 =

n −2 o

where

+ p14 S4 ,

and λ1 and λ2 are calculated using (14.33) with A = n −2 o − p14 S4 ,

B = p44 S4 ,

C = n −2 e .

14.4 Miscellaneous Effects Besides the above-mentioned Pockels, Kerr, and elastooptic effects, there are a few other effects that are important.

14.4.1 Optical Activity With materials such as a quartz crystal or a sugar solution, the phase front surfaces for the o and e waves shown in Fig. 14.1 do not touch each other even when propagation takes place along the optical axis. When a linearly polarized wave is incident along the optical axis of such a material, the direction of polarization rotates as it propagates, as shown in Fig. 14.13. This rotation of polarization takes place even without an applied field. Such a phenomenon is called optical activity. Fresnel’s explanation of the effect is that it is due to the difference between the refractive indices nl and nr for left and right circularly polarized waves. A linearly polarized wave can be decomposed into two circularly polarized waves rotating in opposite senses, as shown in Fig. 14.14. In other words, the resultant of the two circularly polarized waves rotating in opposite directions always oscillates along a

Fig. 14.13 Rotation of polarization due to optical activity

394

14 Electro and Accousto Optics

Fig. 14.14 a, b. Rotation of a linearly polarized wave. (a) When nl = nr . (b) When nl = nr

fixed line, provided that the speed of rotations of both circularly polarized waves are the same. When one of the circularly polarized waves rotates slightly faster than the other, the polarization of the resultant wave rotates as it oscillates. The amount of rotation of the resultant wave per unit length of material is known as the rotary power. The fact that the sense of rotation and the rotary power are peculiar to a type of material is used for analyzing biochemical substances. A saccharimeter uses this effect to determine sugar content in solutions.

14.4.2 Faraday Effect As a phenomenon, the Faraday effect is similar to optical activity. With the Faraday effect, rotation of the polarization takes place only when an external magnetic field is applied. Materials that exhibit the Faraday effect include sodium chloride (NaCl), zinc sulphide (ZnS), and yttrium iron garnet (YIG). The relationship between the angle of polarization rotation and the applied magnetic field H is = V H l cos γ (14.35) where V is the Verdet constant, l is the length of the material, and γ is the angle between the direction of the magnetic field and the direction of propagation of the light beam. The Faraday effect is one of the few magnetooptic effects that are large enough to be utilized for devices. The sense of rotation depends solely on the direction of the magnetic field H . Reversing the direction of H reverses the sense of rotation. Reversing the direction of light propagation has no effect on the sense of rotation.

14.4 Miscellaneous Effects

395

Fig. 14.15 a, b. Principle of the optical isolator. (a) Direction of transmission. (b) Direction of isolation

Figure 14.15 shows an example of how this characteristic of the Faraday effect can be used to make an optical isolator. The optical isolator consists of a magnetooptic material, a magnetic coil, a polarizer P1 and an analyzer P2 . The direction of the polarizer P1 is rotated from that of analyzer P2 by 45 degrees. Figure 14.15a shows how the light passes through from left to right with minimum attenuation and Fig. 14.15b shows how the light in the reverse direction is blocked. As shown in Fig. 14.15a, the polarization of the incident light is oriented in the same direction as the polarizer P1 for easy passage. The direction of the polarization is rotated by −45◦ (clockwise) in the Faraday medium. Thus the direction of the polarization becomes the same as the analyzer P2 for easy passage. Next, referring to Fig. 14.15b, a wave propagating in the opposite direction (reflected wave) is considered. It can go through P2 , but the Faraday effect rotates it by −45◦ , the same as before, and the direction of the polarization becomes perpendicular to that of P1 so that it cannot go through P1 . Thus, the optical isolator prevents the reflected wave from reaching the source. It may be added that exactly the same arrangement can be used for modulating the light intensity by modulating the magnetic field. The inductance of the magnetic coil limits the maximum frequency of such a modulator.

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14 Electro and Accousto Optics

14.4.3 Other Magnetooptic Effects The Cotton-Mouton effect is a magnetic effect which is observed in liquids such as nitrobenzene (C6 H5 NO2 ). When a magnetic field is applied perpendicular to the direction of light propagation, the material behaves similarly to a uniaxial crystal whose optical axis lies along the external magnetic field. This effect is more or less a magnetic counterpart of the electrooptic Kerr effect, and the effect is proportional to the square of the magnetic field. A similar effect observed in a vapor state like lithium gas is called the Voigt effect. The Kerr magnetooptic effect is observed when linearly polarized light is incident upon a magnetic pole-piece. The reflected wave becomes elliptically polarized.

14.4.4 Franz-Keldysh Effect The Franz-Keldysh effect is an electro-absorptive effect which is caused by the shift of the absorption edge of a semiconductor due to an external electric field [14.10– 12]. Figure 14.16 shows a plot of the absorption coefficient of a GaAs semiconductor with respect to the quanta of light. There is an abrupt increase in the absorption around 1.4 eV. This is called the absorption edge. Thus the quantity 1.4 eV is the minimum energy required to release a bound electron from its orbit thereby producing a free electron. (In terms of energy-band theory, the photon energy is used to kick an electron from the valence band into the conduction band, thereby creating

Fig. 14.16 Absorption edge of a semiconductor

Problems

397

a hole in the valence band and an electron in the conduction band. The absorption edge is used for determining the band structure of the semiconductor). When an external electric field is applied to the semiconductor, the absorption edge shifts, as indicated by the dotted line in the figure, and thus the absorption of light at a frequency near the absorption edge increases. A modulator based upon the electroabsorptive effect is quite similar to a laser diode in structure, as shown in Fig. 13.13. The differences are that the composition of the layers of the modulator is designed so that the absorption edge appears at the desired wavelength, and that the modulator device is reverse biased instead of forward biased. When it is biased to almost breakdown voltage, an extinction ratio defined as the leakage power (in the off state of the switch) divided by the transmitted power (in the on state) of about 10 dB is obtainable.

Problems 14.1. Consider the case where an external electric field E x is applied along the x axis (perpendicular to the crystal axis) of a quartz crystal, and the direction of the incident light is along the y axis. Find the allowed directions of polarization and their refractive indices. 14.2. An electrooptically controlled optical guide is to be fabricated using a lithium niobate substrate. The direction of the light propagation is in the z direction, and the directions of polarization of the guided modes are as shown in Fig. 14.17. How should the crystal be cut in order to achieve the largest electrooptic effect? What would be the configuration of the electrodes for applying the external field corresponding to the following situations:

Fig. 14.17 a, b. Configuration of an electrooptic guide of LiNbO3 . (a) TE0 mode. (b) TM0 mode

398

14 Electro and Accousto Optics

Fig. 14.18 a, b. Strain gauge. (a) Plan view. (b) Cross-sectional view

a) when the guide is excited with E field as shown in Fig. 14.17a? b) when the guide is excited with E field as shown in Fig. 14.17b? 14.3. A strain gauge is to be made of a lithium niobate substrate. The external electric field needed to compensate the elastooptic effect is used to determine the strain. 14.4. A Mach-Zehnder interferometer such as shown in Fig. 14.18 is proposed. A C-cut crystal is used and a mode polarized in the y direction is excited. The strain is applied in the direction normal to the surface of the substrate. Find a formula relating the external electric field to the applied strain.

Chapter 15

Integrated Optics

In the case of an electronic integrated circuit (IC), e.g., for a shift register IC, one normally does not worry about how each flip-flop unit is functioning. The only concern is whether or not the input and output electrical signals are correct. The IC’s are packaged in small black plastic cases, and they are literally looked upon as black boxes. A similar objective has been the goal of integrated optics. When this goal is reached, no one will have to worry about vibration, coupling between components, temperature, dust, or moisture. Various optomodules could then presumably be integrated on a small substrate the size of a microscope deck glass for easy handling. The optomodules would contain thin-film devices such as modulators, directional couplers, switches, bistable optical devices, lenses, A/D converters and the spectrum analyzer. This chapter begins with a fundamental waveguide analysis and an introduction to coupled mode theory. Following this analysis is a description of the basic thin film devices which form the building blocks of integrated optics. A summary of the thin-film techniques that were specifically developed in connection with integrated optics is also included. Before proceeding, a word of caution is advised concerning categorization. Quite often, a single integrated-optics device may provide a variety of functions. For example, the same basic integrated optics component may be used as a modulator, a switch, or a directional coupler. Therefore, to use the phrase “directional coupler” as the name of a subsection may be misleading. Despite this drawback, every attempt has been made to present the material in an order which is easy to comprehend.

15.1 Analysis of the Slab Optical Guide The slab geometry is one of the most fundamental configurations in integratedoptics technology. Consider the propagation of a wave in a core glass with index of refraction n 1 , when it is sandwiched by a cladding glass with index of refraction n 2 , as shown in Fig. 15.1. When n 2 is smaller than n 1 , the light propagates along the 399

400

15 Integrated Optics

Fig. 15.1 Slab optical guide

core glass in a zigzag path due to the total reflection at the upper and lower boundaries. The optical guide will be analyzed using both wave optics and geometrical optics.

15.1.1 Differential Equations of Wave Optics The wave optics treatment is slightly more complicated but provides accurate information about the guide. First, the wave equations are derived from Maxwell’s equation. Consider, an electromagnetic wave propagating in the z direction such that E = E ej(βz−ωt) ,

H = H ej(βz−ωt) .

(15.1)

By inserting these into Maxwell’s equations one obtains ∇ × H = −j ω ε0 εr E,

∇ × E = j ω µ H.

(15.2)

An assumption is made that no component of the field varies in the y direction, or in other words ∂ = 0. (15.3) ∂y The field is decomposed into TE and TM waves. The waves that contain no electric field in the direction of propagation are TE waves, namely E z = 0.

(15.4)

Similarly, waves that contain no magnetic field in the direction of propagation are TM modes, namely (15.5) Hz = 0. Equations (15.3–5) significantly simplify the curl operation of (15.2).

15.1 Analysis of the Slab Optical Guide

401

With the assumption of E z = 0, (15.2) gives the TE-mode solution as j ∂ Hz ∂ Hx E x = 0, E y = − , E z = 0, ω ε0 εr ∂z ∂x j ∂ Ey −j ∂ E y , Hy = 0, Hz = , (15.6) Hx = ωµ ∂z ωµ ∂ x ∂2 2 2 ∂2 √ + 2 n k E y = 0 where ω2 µ ε0 εr = n 2 k 2 , k = ω µ ε0 . 2 ∂x ∂z Similarly, with the assumption of Hz = 0, the TM-mode solution is obtained, Ex =

−j ∂ Hy , ω ε0 εr ∂z

∂ Hy j E y = 0, E z = , ω ε0 εr ∂ x j ∂ Ex ∂ Ez Hy = − , Hz = 0. ω µ ∂x ∂z

Hx = 0, ∂2 ∂2 + + k 2 n 2 Hy = 0. ∂x2 ∂z 2

(15.7)

The expressions for the TE and TM modes contain many similarities. First, the solutions for the TE modes are sought.

15.1.2 General Solution for the TE Modes As seen from (15.6), once an expression for E y is obtained, all other components of the TE wave are automatically found. A good starting point for finding an expression for E y is the differential equation at the bottom of (15.6). Making use of the fact that z enters (15.1) as an exponential function, the differential equation becomes 1 ∂2 Ey 2 2 2 + n k − β =0 Ey ∂x2

(15.8)

The method of separation of variables will be used to solve (15.8) for E y . Since the second term of (15.8) is constant, the first term also has to be constant. As far as the values of the constant is concerned, regardless of whether a positive number or a negative number is picked, (15.8) still holds true, but the solution differs depending on which is chosen. The solution is either trigonometric or exponential. Let us first pick a negative number −K 2 as the value of the first term. In order that (15.8) be true, the second term has to be K 2 namely ∂2 Ey + K 2 E y = 0 with ∂x2 K 2 = n2k 2 − β 2. The solution of (15.9) is

(15.9) (15.10)

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15 Integrated Optics

E y = A cos K x + B sin K x.

(15.11)

Next, let us pick a positive number γ 2 as the value of the first term in (15.8), then, ∂2 Ey − γ 2 Ey = 0 with ∂x2 γ 2 = β 2 − n2k 2.

(15.12) (15.13)

The solution of (15.12) is E y = Ce−γ x + Deγ x.

(15.14)

Both (15.11, 14) are equally valid as a solution of (15.8). In fact the sum of (15.11, 14) is also a solution. The choice of the solution depends on which one of the solutions suits the physical situation best. (One way is to use the sum of (15.11, 14) as a solution and, depending on the situation, any of A, B, C, or D is (are) made zero. As long as A, B, C or D are constants, including zero, these solutions hold true.) For the case of a slab optical guide, (15.11) is assigned as the solution in the core glass because of its oscillatory nature and (15.14) is assigned as the solution in the cladding glass because its exponential decay nature is suitable for representing the wave in the cladding glass. With this arrangement of solutions, (15.10, 13) become K 2 = n 21 k 2 − β 2 ,

(15.15)

γ =β

(15.16)

2

2

− n 22 k 2 .

Note that n 1 has to be used in (15.15), whereas n 2 appears in (15.16). For the sake of simplicity, the two terms in (15.11) are treated separately and added later. They are E y = A cos Kx, E y = B sin Kx.

(15.17) (15.18)

The solution expressed by (15.17) is called an even mode and that by (15.18) is called an odd mode, because cos Kx is an even function with respect to x and sin Kx is an odd function. In fact, the optical slab guide can be excited by either even modes alone, or odd modes alone, or a combination of the two. The ratio of A and B is determined by the type of incident wave to the optical guide. If the field distribution of the incident wave is symmetric with respect to the center plane of the slab, only the even modes are excited. If the distribution of the incident wave is antisymmetric, the slab will be excited only by the odd modes. If the incident field is arbitrary distribution, both modes will be present so that the incident field distribution and the excited field in the slab guide are smoothly connected.

15.1.3 Boundary Conditions The propagation constant and the field distribution of each mode will be found. The treatment for even and odd modes are similar, and the even modes are considered

15.1 Analysis of the Slab Optical Guide

403

first. The values of the constants A, B, C and D in (15.11, 14) are found from the boundary conditions. As shown in Fig. 15.2, there are four boundaries; two at plus and minus infinity and two others at the upper and lower core-cladding boundaries. First, the boundary at x = +∞ is considered. So as to avoid (15.14) growing infinity for large x, D has to be zero in the region x > d and thus, E y = Ce−γ x ,

x > d.

(15.19)

On the other hand, in the lower cladding region, C has to be set to zero for the same reason, (15.20) E y = De yx , x < d. The boundary condition at x = d is now considered. Tangential components of both E and H have to be continuous across the boundary. Continuity of E y is set using (15.17, 19), (15.21) A cos KD = Ce−γ d . Continuity of Hz is set by incorporating the formula for Hz in (15.6) with (15.17, 19), (15.22) KA sin K d = Cγ e−γ d . The ratio of (15.21) to (15.22) gives γ d = K d tan K d.

(15.23)

Equation (15.23) is called the characteristic equation of the optical guide. Next, the values of γ d and K d will be determined. One more equation is necessary besides (15.23). This equation can be obtained from (15.15, 16), (K d)2 + (γ d)2 = V 2 , V = kd n 21 − n 22 .

Fig. 15.2 Boundary conditions

where

(15.24) (15.25)

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15 Integrated Optics

The V parameter is a function of such physical constants as the thickness of the slab optical guide, the wavelength of the light, and the refractive indices of the core and cladding glasses, and is sometimes called the normalized thickness of the guide. So as to solve the simultaneous equations of (15.23, 15.24), γ d is plotted in the vertical axis, and K d, in the horizontal axis in Fig. 15.3. The curve of (15.24) is a circle with radius V , and that of (15.23) has a shape similar to that of tan K d. These two curves are shown as solid lines in Fig. 15.3. The positions of the intersection of the two curves are determined by the value of V . In this example, there are three intersections. Each intersection corresponds to one mode. Starting from the intersection nearest the origin they are named as even TE0 mode, even TE1 mode, even TE2 mode and so on. In this example, the light energy propagates in three different modes with different propagation constants. Any of the three modes can be excited in this guide, depending on the configuration of excitation. The corresponding propagation constants are βN =

(n 1 k)2 − K N2 ,

(15.26)

where K N denotes the discrete values of K determined by the point of intersection in Fig. 15.3. Next, the distribution of the amplitude E y is considered. The even TE0 mode is considered first. Since the value of K d of the even TE0 mode is slightly smaller than π2 , let it be represented by ( π2 − ∆). From (15.17), the amplitude E y inside the core is π 2 − x. E y = A cos d

Fig. 15.3 TE mode solution in a step index optical guide

15.1 Analysis of the Slab Optical Guide

405

The curve for this equation is shown in Fig. 15.4 a. It is a cosine curve inside the core glass. Inside the cladding glass the magnitude of E y is very small, but as long as ∆ is not zero, E y is not zero either, and the light penetrates into the cladding glass. Because this field decays exponentially with x, as seen from (15.19), this decaying field near the boundary is called an evanescent wave. When the intensity distribution |E y (x)|2 is observed from the end of the guide, the even TE0 mode displays one bright spot in the center, and the even TE1 mode displays 3 bright spots, as seen from Fig. 15.4 b. Generalizing, even TEn modes display 2n + 1 bright spots. As the value of γ is further decreased, the penetration inside the cladding increases. When γ = 0 the light is present throughout the cladding glass. This can be interpreted as if all the light leaks into cladding glass, which corresponds to the cut-off condition of the guide. Speaking of the example in Fig. 15.3, if the value of V is decreased to 2π radians, the even TE2 mode goes to the cut-off condition. Similarly the cut-off condition of the even TEn mode occurs when V = nπ radians. The cut-off of the even TE0 mode is very special. This cut-off does not occur until V = 0, and for the region 0 < V < π2 only the even TE0 mode is excited. The mode which is capable of being the sole excited mode is called the dominant mode.

Fig. 15.4 a, b. General trend of the field E y as cutoff is approached. (a) Even TE0 mode. (b) Even TE1 mode

406

15 Integrated Optics Table 15.1 The E y components of even and odd TE modes Even TE modes

Odd TE modes

E y = A cos K d e−γ (x−d) E y = A cos K x E y = A cos K d eγ (x+d)

E y = A sin K d e−γ (x−d) E y = A sin K x E y = A sin K d eγ (x+d)

in upper cladding in core in lower cladding

The even TE0 mode is thus the dominant mode of the optical guide. The even TE1 cannot be a dominant mode because, whenever the even TE1 mode is excited, the even TE0 mode can be excited at the same time. Hence the even TE1 mode cannot be excited alone, and this is also true with all other higher-order modes. Next, the odd TE modes are considered. The approach is quite similar to that of the even TE modes. To start with, the continuity at x = d is considered. The continuity of E y is applied to (15.18, 19) and one obtains A sin K d = Ce−γ d .

(15.27)

Similarly, the continuity of Hz , which is obtained by using (15.6), gives KA cos K d = −Ce−γ d .

(15.28)

The ratio of (15.27) to (15.28) gives γ d = −K d cot K d,

(15.29)

which is the characteristic equation of the odd TE modes. This characteristic equation is plotted in Fig. 15.3 in dashed lines. These lines are almost parallel to those of the even TE modes, and interlace with them. One has to consider both even and odd modes to account for all TE modes. The spacing between the curves is approximately π2 . The solutions of E y for even and odd TE modes are summarized in Table 15.1. All other components of the TE modes in each region are obtained using (15.6).

15.1.4 TM Modes Procedures that are exactly the same as for the TE modes are applicable to the TM modes. The results are summarized here. The expression for Hy is exactly the same as those in Table 15.1 with E y replaced by Hy . The other components of the TM modes are found using (15.7). The characteristic equations, however, are slightly different from those of the TE modes. The characteristic equation of the even TM modes is

n2 n1

2 kd tan K d = γ d,

(15.30)

15.1 Analysis of the Slab Optical Guide

and that of the odd TM modes is 2 n2 − kd cot K d = γ d. n1

407

(15.31)

If these characteristic equations are plotted in Fig. 15.3, together with those of the TE modes, the characteristic curves of the TM modes follow those of the TE modes but slightly below them. In practice, n 2 differs from n 1 by only a fraction of one per cent and the two sets of curves are close together.

15.1.5 Treatment by Geometrical Optics In geometrical optics, the transmission of the light inside the slab optical guide is interpreted as a channelling of light by the repeated total reflections at the upper and lower boundary layers, as shown in Fig. 15.5 a. If the propagation constant n 1 k of the zigzag wave is decomposed, as shown in Fig. 15.5 b into K and β, in the x and z directions, respectively, then K is (15.32) K = (kn 1 )2 − β 2 . Equation (15.32) is rewritten as (K d)2 + (β d)2 = (kn 1 d)2

(15.33)

which represents a circle (Fig. 15.6). There are upper and lower limits that the value of βd can take. The upper limit is obviously kn 1 d β d from (15.33). The lower

Fig. 15.5 a, b. Component waves in the optical guide. (a) Zigzag wave in the optical guide. (b) Decomposition of the propagation constant

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15 Integrated Optics

Fig. 15.6 K d − βd diagram to determine modes

limit is determined by the condition of total reflection. The critical condition at the boundary between the core and cladding glasses is n 1 cos α > n 2 . From Fig. 15.5b, n 1 cos α = β/k, hence the lower limit of βd is βd > kn 2 d, and the hatched region in Fig. 15.6 should be excluded. The parameter β/k is sometimes called the effective refractive index n e . Next, a quantization is introduced. The angle of incidence to the boundaries between the core and cladding glass is quantized in the manner described earlier in Sect. 5.8.1. The quantization condition is 2K N d = N π,

N = 0, 1, 2, 3, . . . .

where

(15.34)

The intersections between the circle and horizontal solid lines in Fig. 15.6 are the quantized values of K N d. The propagation constant β N for the N th mode is given from (15.33, 34) by β N = k n 1 1 −

Nπ 2k n 1 d

2 .

(15.35)

15.2 Coupled-Mode Theory

409

15.1.6 Comparison Between the Results by Geometrical Optics and by Wave Optics The results obtained by wave optics will be rewritten in a form that can be easily compared with that obtained by geometrical optics. Equation (15.17) of the wave optics treatment is put into (15.1) and rewritten as E y =

1 A exp j (K N x + βz − ωt) + A exp j (−K N x + βz − ωt) , 2 (15.36)

where K N denotes the discrete values of K obtained from Fig. 15.3. According to the vector expression (2.17) of plane wave propagation, the first term of (15.36) is a plane wave propagating in the direction expressed by $

$

k = K n i + β k, and the second term is another plane wave in the direction $

$

k = −K N i + β k. The vector diagrams for these propagation constants are the same as those shown in Fig. 15.5b, except for the slightly different value of K . In the wave optics approach, K takes on the discrete values K N which are determined from the intersections between the characteristic equation and the normalized thickness V in Fig. 15.3, whereas the values of K N in the geometrical optics approach were determined by (15.34). The values of K N d approach π2 N for large V . The solutions by geometrical optics approach those by wave optics as V is increased. The values of K N are indicated by the dashed lines in Fig. 15.6. For low values of V , the wave optics approach is more accurate. It should be recognized that the electric field distribution inside the core glass is nothing but a standing wave pattern of the two component waves of (15.36). The higher the mode number N , the larger the angle α N , and hence a greater number of loops and nulls of the field pattern exist with higher modes.

15.2 Coupled-Mode Theory Coupled mode equations are useful for analyzing such components as directional couplers, light switches, light modulators, and wavelength filters [15.1, 2]. It is so widely used that it is worthwhile to describe it in some detail here. Referring to Fig. 15.7, the coupling taking place between two parallel lines is considered. Each line is located in the region of the evanescent wave of the other line, and both are loosely coupled to each other. The wave propagating along the lone guide 1 in the absence of guide 2 is expressed by

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15 Integrated Optics

) * E 1 = A exp j β10 z − ωt

(15.37)

and satisfies the differential equation d E1 = jβ10 E 1 . dz

(15.38)

When another guide 2 is introduced in the vicinity of guide 1, then two things will happen. First, the propagation constant changes because of the change in geometry. Secondly, power is exchanged between the two lines because each line is located in the region of the evanescent wave of the other. The coupled differential equations thus become d E2 d E1 = jβ1 E 1 + k12 E 2 , = k21 E 1 + j β2 E 2 , where dz dz β2 = β20 − K 22 , β1 = β10 − k11 ,

(15.39) (15.40)

and where β10 and β20 are the propagation constants of the isolated case for guides 1 and 2, respectively; k11 and k22 are the changes in propagation constants of guides 1 and 2, respectively, when an adjacent guide is introduced; and k12 and k21 are the coefficients associated with the transfer of energy from guide 2 to guide 1 and from guide 1 to guide 2, respectively. The basis of coupled-mode theory lies in the solution of these differential equations. Assume a solution of the form E 1 = A eγz ,

E 2 = B eγz .

In order to find the value of γ , (15.41) is inserted into (15.39) to obtain

Fig. 15.7 Coupled optical guide

(15.41)

15.2 Coupled-Mode Theory

411

A (γ − j β1 ) −k12 = 0. −k21 (γ − j β2 ) B

(15.42)

The condition that non-trivial solutions of A and B exist is that the determinant vanishes. This condition leads to β1 − β2 2 β1 + β2 γ1,2 = j . (15.43) ± k12 k21 − 2 2 Thus, the general solutions for E 1 and E 2 are E 1 = A1 eγ1 z + A2 eγ2 z 1 A1 (γ1 − j β1 ) eγ1 z + A2 (γ2 − j β1 ) eγ2 z E2 = k12

(15.44) (15.45)

where (15.45) was obtained by inserting (15.44) into (15.39). Next, the values of k12 and k21 are closely investigated. If one could impose the condition that the energy which has leaked out of one guide is coupled completely into the adjacent line without loss, the sum W of the energy of the two guides does not alter along the line [15.2]; dW = 0 where dz W = (E 1 E 1∗ + E 2 E 2∗ ).

(15.46)

This leads to dW = (E 1 E 1∗ + E 1 E 1∗ + E 2 E 2∗ + E 2 E 2∗ ). dz

(15.47)

Using (15.39) and reversing the order of complex conjugation and derivative, one obtains ∗ E 1∗ E 2 = 0. (15.48) Re k12 + k21 For (15.48) to be true for any values of E 1 and E 2 , ∗ k12 = −k21

(15.49)

has to be satisfied. This is the coupling condition. When (15.43) is written using (15.49), one obtains γ1,2 = j (β ± βb ) where β1 − β2 β1 + β2 , ∆β = , β= 2 2 β1 = β + ∆,

β2 = β − ∆,

(15.50) ∆β ≡ ∆, βb = |k12 |2 + ∆2 .

(15.51)

Figure 15.8 indicates the relationship among β1 , β2 and β. The symbols ∆β and ∆ are used equivalently. Now everything except the constants A1 and A2 has been found. A1 and A2 are determined by the conditions of excitation. In the following, a few examples are given.

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15 Integrated Optics

Fig. 15.8 Definitions of β’s

Exercise 15.1. Find the coupled mode solutions of two identical guides when only one of the guides is excited with amplitude U at the end (unbalanced excitation). Solution The following conditions hold 1) ∆ = 0, β1 = β2 = β, 2) E 1 (0) = U, E 2 (0) = 0.

(15.52)

First, the values of A1 and A2 are found. Inserting (15.52) into (15.44, 45) gives U = A1 + A2 ,

0 = A1 − A2 .

(15.53)

The solutions for A1 and A2 are A1 = A2 =

U . 2

Using this result and (15.52) in (15.44, 45) gives jβz E u1 (z) = U (cos |k∗ 12 |z) e , E u2 (z) = −U k12 /k12 (sin |k12 |z)ejβz .

(15.54) ( (15.55)

Whereas the amplitude of E u1 (z) is modulated by cos |k12 |z, that of E u2 (z) is modulated by sin |k12 |z. This means that, when E u1 (z) is at an extremum, E u2 (z) is zero (and vice versa) as shown in Fig. 15.9. The energy in guide 1 is completely transferred to guide 2 after a transmission of |k12 |z = π/2, and the energy in guide 1 becomes zero. At the distance |k12 |z = π , however, the energy in guide 2 is completely transferred back to guide 1. The energy is tossed back and forth as it propagates along the guides. Such a phenomenon resembles the oscillation of two pendula connected with a weak spring as in Fig. 15.10. Let us suppose that one initiates the swing by pushing only pendulum 1. As time passes on, pendulum 2 starts gaining energy from pendulum 1 and finally reaches full amplitude when pendulum 1 is at rest. If time passes further, the situation reverses and pendulum 1 reaches full amplitude and pendulum 2 will be at rest, as illustrated at the bottom of Fig. 15.10. In this case, it is important to realize that complete energy transfer takes place only when pendulum 1 is identical to pendulum 2, and that the period of energy exchange is longer the weaker the coupling coil is.

15.2 Coupled-Mode Theory

413

Exercise 15.2. Find the solutions of the coupled guide when both guides have identical dimensions and are excited by the same input light amplitude B (balanced excitation). Solution The conditions of excitation are 1) ∆ = 0, β1 = β2 = β 2) E 1 (0) = E 2 (0) = B. Insertion of (15.56) into (15.44, 45) gives

Fig. 15.9 Intensity distribution along a coupled guide

Fig. 15.10 Motion of coupled pendula

( (15.56)

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15 Integrated Optics

B = A1 + A2 , ∗ /k (A − A ), B = j k12 12 1 2

( (15.57)

which leads to A1 =

a+1 B, 2a

A2 =

a−1 B, 2a

∗ /k . a = j k12 12

Inserting (15.56, 58) into (15.44, 45) gives ∗ sin |k | z) ejβz , E b1 (z) = B(cos |k12 |z + k12 /k12 12 jβz ∗ /k E b2 (z) = B(cos |k12 |z − k12 12 sin |k12 | z) e .

(15.58)

(15.59) (15.60)

It is interesting to note that, if an assumption is made that k12 is a pure imaginary number, then E b1 = B ej(β+|k12 |)z , j(β−|k12 |)z

E b2 = B e

.

(15.61) (15.62)

The two waves have the same amplitude but different phase velocities. Comparing the results of Exercises 15.1, 2, the amount of exchange of energy between the two guides does not simply depend upon the geometry of the guides, but it depends to a great extent upon how the guides are excited. Exercise 15.3. Find the solutions of the coupled guides when both of the guides are identical, but are excited differently by E 1 (0) and E 2 (0). Solution The results of above two exercises will be used to solve this problem. The excitation can be decomposed into two modes of excitation; one is single-side feeding or unbalanced feeding, as in Exercise 15.1, and the other is identical feeding or balanced feeding as in Exercise 15.2. Using the law of superposition, the excitation can be decomposed in a convenient manner. An arbitrary excitation can be considered to consist of a combination of an unbalanced excitation U and a balanced excitation B. Each case is solved separately, and the two solutions are summed to obtain the final result. Referring to Fig. 15.11, E 1 (0) and E 2 (0) are expressed in terms of U and B as (15.63) E 1 (0) = U + B, E 2 (0) = B. Thus U and B are found to be U = E 1 (0) − E 2 (0),

B = E 2 (0).

(15.64)

The solution for E 1 (z) and E 2 (z) are written in terms of the results of the last two exercises as E 1 (z) = E u1 (z) + E b1 (z), Using (15.55, 59, 60, 64, 65) gives

E 2 (z) = E u2 (z) + E b2 (z).

(15.65)

15.2 Coupled-Mode Theory

415

Fig. 15.11 Decomposition of an arbitrary excitation into an unbalanced and a balanced excitation

∗ sin |k | z E 1 (0) E 1 (z) k12 /k12 12 ∗cos |k12 | z = E 2 (z) E 2 (0) − k12 /k12 sin |k12 | z cos |k12 | z

(15.66)

where a factor of exp(jβz) was suppressed. The matrix in (15.66) is called the transfer matrix. The next exercise treats the coupling between non identical guides. Exercise 15.4. Find the solutions of the coupled guides when the dimensions of the guides are not identical, and only one of the guides is excited. Solution The boundary conditions are 1) ∆ = 0, γ1,2 = j(β ± βb ), 2) E 1 (0) = U E 2 (0) = 0.

( (15.67)

First, the coefficients A1 and A2 are determined by using these conditions of excitation. The use of (15.44, 45, 50, 51, 67) leads to U = A1 + A2 , 0 = (βb − ∆)A1 − (βb + ∆)A2 .

(15.68)

The solutions for A1 and A2 are A1 =

βb + ∆ U, 2 βb

A2 =

βb − ∆ U. 2 βb

(15.69)

Inserting (15.69) in (15.44, 45), the final results are obtained after some rather straight forward manipulation; ∆ (15.70) E u1 (z) = U cos βb z + j sin βb z ejβz , βb k∗ E u2 (z) = − 12 U (sin βb z) ejβz . (15.71) βb

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15 Integrated Optics

Let us look at the solution of Exercise 15.4 more closely. In order to see the energy transfer from this result, E 1 E 1∗ and E 2 E 2∗ are calculated, 2 ∆ ∗ 2 2 2 E u1 E u1 = |U | cos βz z + sin βb z βb |k12 |2 2 2 = |U | 1 − sin βb z and (15.72) |k12 |2 + ∆2 ∗ E u2 E u2 = |U |2

|k12 |2 sin2 βb z. |k12 |2 + ∆2

(15.73)

As anticipated, the sum of the energy of the two guides is |U |2 and is independent ∗ and E ∗ of the distance z. As shown in Fig. 15.12, the fluctuations of E u1 E u1 u2 E u2 compliment each other. It should be realized that complete transfer of energy from one guide to the other takes place when |k12 |2 = 1. |k12 |2 + ∆2

(15.74)

The condition of (15.74), however, can be satisfied only when ∆ = 0 which means only when the two guides are identical. Another interesting fact is that an increase in ∆ decreases the amount of transfer of energy. In other words, the amount of transfer can be adjusted by increasing the dissimilarity of the guides. It is appropriate to conclude this section on coupled-mode theory by remarking on how widely used the equations are. They are a powerful tool in analyzing the interaction between two states in a conserved system. The applications include the conversion of power between two modes in a waveguide, the lower- and higherfrequency components of second-harmonic generation, and even the interaction between light and atomic systems where total energy is conserved. It is also used in connection with the population of atoms in upper and lower states, where the total number of atoms is conserved. In short, it is an extremely useful theory in dealing with coupling phenomena in a conserved system.

15.3 Basic Devices in Integrated Optics Sections 15.3–6 are the climax of this and the previous chapter combined. Using the basic knowledge so far developed, various devices will be explained.

15.3.1 Directional Coupler Switch Based upon coupled mode theory, an optical directional coupler that can be electrically switched has been fabricated. The structure of the switch is shown in

15.3 Basic Devices in Integrated Optics

417

Fig. 15.12 Power transfer between two guides with different characteristics as a function of electrical coupling length. Input is fed to only guide 1

Fig. 15.13. It consists of two closely spaced parallel guides imbedded in an electrooptic substrate like LiNbO3 or GaAs. Electrodes are placed on top of each guide so that an external field can be applied to the guides. The refractive index of the guides is changed by the electrooptic effect. The applied electric fields in guide 1 and 2 are arranged to be in opposite directions so that a difference ∆ in propagation constants between the two guides is created by the applied voltage. The light entering terminal (1) is channeled out to either (1 ) or (2 ) depending on the applied voltage. The switching operation is explained by referring back to the diagram in Fig. 15.12. The applied electric field primarily changes ∆ but not k12 . As mentioned in the previous section, complete cross over takes place only when ∆ is zero. When the electrical coupling length defined by |k12 |2 + ∆2 L is an odd integral multiple of π2 , all the energy entering (1) crosses over and comes out at (2 ). Let this situation be called the cross state and be designated by ⊗, thus π (15.75) ∆ = 0, |k12 | L = (2 n + 1) , 2 where n is an integer. On the other hand, when all the energy entering (1) comes out at (1 ), this situation is called the straight through state, or bar state, and is designated by = . Even when ∆ is non zero, as long as the electrical coupling length is an integral multiple of π , the bar state is obtained, (15.76) |k12 |2 + ∆2 L = m π,

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15 Integrated Optics

Fig. 15.13 a, b. Directional coupler switch. (a) Plan view. (b) End view

where m is an integer. Thus, the switching action can be achieved by designing the length of the guide to satisfy exactly (15.75) when no voltage is applied. This establishes the ⊗ state. The = state is obtained by increasing the applied voltage so as to satisfy (15.76). The graph of (15.76) in the |k12 |−∆ plane consists of circles while that of (15.75) contains points on the |k12 | axis. From these two graphs, a switching diagram such as the one shown in Fig. 15.14 is made. Equations (15.75, 76) were normalized as (Note ∆L means product of ∆ and L and not a small displacement in L.): ; " 2∆L L = 2 n + 1, state π = 0, l L 2 2∆L 2 (15.77) + π = (2 m)2 , = state l where l is the length needed for the first ⊗ state in Fig. 15.12 and is l=

π . 2|k12 |

(15.78)

The distance l is sometimes called the transfer length. The cross states occur only at isolated points marked by ⊗ on the vertical axis in the Fig. 15.14, whereas the bar states occur at any point along the concentric circles. The length of the coupler is designed to be on one of ⊗ marks; and as the applied voltage is increased, i.e., ∆ is increased, the point of operation moves horizontally and meets the bar condition. A difficulty with this coupler is that since the ⊗ states appear only as points in Fig. 15.14, a lossless cross state is achieved only by accurate fabrication. But a coupler with an alternating polarity of ∆ can be electrically tuned and relieves this stringent requirement. The details of this coupler will be discussed next.

15.3 Basic Devices in Integrated Optics

419

Fig. 15.14 Switching diagram for a single section [15.3]

15.3.2 Reversed ∆β Directional Coupler A directional coupler that has the capability of electrically tuning both cross and bar states is shown in Fig. 15.15. Such a capability is achieved by splitting the section into two halves and applying the voltage to each half with reversed polarity [15.3, 4]. The reversed ∆β is equivalent to increasing the electrical coupling length without affecting the magnitude of ∆β it is as if the length L of the balanced directional guides were stretched with the applied voltage. In order to understand this electrical adjustability, the analysis developed in Sect. 15.2 will be further extended. In the left section of the reversed ∆β directional coupler, only one end of the guides is driven, and the result of Exercise 15.1 is immediately applicable. In the right section, the ends of both guides are driven, and a more generalized expression is needed. The solution with general excitation will be obtained as the sum of unbalanced and balanced excitations in a manner similar to that described in Exercise 15.3. Since the solution with unbalanced excitation has already been obtained, only the solution with balanced excitation has to be found. The boundary condition for the balanced excitation at z = 0 is E 1 (0) = E 2 (0) = B.

(15.79)

Inserting (15.79) into (15.44, 45) gives B = A1 + A2 , j [(βb − ∆) A1 − (βb + ∆) A2 ]. B= k12

(15.80)

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Fig. 15.15 Electrode configuration of a reversed ∆β directional coupler

The solutions for A1 and A2 are A1 =

βb + ∆ − jk12 B, 2 βb

A2 =

βb − ∆ + jk12 B. 2 βb

(15.81)

Insertion of (15.81) into (15.44, 45) gives

(∆ − j k12 ) sin βb z E b1 (z) = B cos βb z + j βb

∗ ) (∆ − j k12 E b2 (z) = B cos βb z − j sin βb z βb

ejβz ,

(15.82)

ejβz .

(15.83)

The rest of the procedure is quite similar to Exercise 15.3 and the final result is

A jB E 1 (0) E 1 (z) = , jB ∗ A∗ E 2 (z) E 2 (0)

(15.84)

where

∆ A = cos βb z + j βb

sin βb z,

k12 B = −j βb

sin βb z,

(15.85)

and a factor of exp(jβz) was suppressed. This completes all that is necessary to find the transfer matrix M of the reversed ∆β directional coupler shown in Fig. 15.15. Let M + and M − be the transfer matrices for the left and right sections of the coupler, respectively. Between the sections, β1 and β2 are exchanged and the sign of ∆ is reversed. The only difference between M + and M − is whether or not A has a complex conjugate sign. The desired transfer matrix is the product of the two matrices, namely

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M = M− M+ =

Ar j Br

j Br∗ A∗r

Ar = | A|2 − |B|2 ,

,

where

Br = 2A∗ B.

(15.86) (15.87)

Using the transfer matrix of (15.86), the switching condition of the reversed ∆β directional coupler is obtained. First, the condition for the bar state when guide 1 is excited is considered. The condition is given from (15.87), by 2A∗ B = 0.

(15.88)

Either A∗ or B has to be zero to satisfy this condition. Since A∗ is a complex number, for A∗ to be zero, both real and imaginary parts have to vanish, ∆ = 0, L π |k12 | = (2 n + 1) , 2 2 where L/2 is the length of the subsection. This condition is represented by a series of isolated points at L/l = 2, 6, 10, . . ., with an interval of 4 along the vertical axis of the switching diagram, as shown in Fig. 15.16. If B is to be zero, then L |k12 |2 + ∆2 = m π, (15.89) 2 which can be rewritten as 2 2∆L 2 L + = (4 m)2 . l π

(15.90)

Equation (15.90) represents concentric circles on the switching diagram with radii 4, 8, 12,. . ., at intervals of 4. Finally, the condition for the cross state is considered. From (15.85, 87), Ar is Ar = 1 − 2

βb L |k12 |2 sin2 . 2 βb2

(15.91)

The condition that Ar = 0 is |k12 |2 L sin2 2 |k12 |2 + ∆2

π |k12 |2 + ∆2 = sin2 (2 m + 1) . 4

(15.92)

Equation (15.92) is a transcendental equation and cannot be solved analytically, except along the vertical axis. The computer calculated graph [15.3] is shown in Fig. 15.16. The curve starts at 1, 3, 5, . . ., on the vertical axis and ends on the same axis.

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Fig. 15.16 Switching diagram for two alternating ∆β sections [15.3]

From the diagram in Fig. 15.16, the operation of the switch is understood. By increasing the applied voltage, the point of operation moves horizontally. Conditions of crosses and bars can be achieved with almost any length (L/l) of the guide, and the stringent requirement on fabrication is relieved. By increasing the number of subsections, the required voltage is reduced. A control signal of 3 V can switch the states of a directional coupler of 6 reversed ∆β sections formed in Ti diffused LiNbO3 [15.1]. Finally, a heuristic interpretation is given for the reason why a perfect cross state does not exist with ∆ ∓ 0 in the case of a non-reversed coupler, while a perfect cross state does exist in the case of a reversed ∆β coupler. In the non-reversed case, the condition for a perfect cross state is E 1 (z) = 0 with E 1 (0) = 0 and E 2 (0) = 0, and hence A = 0 in (15.84). Notice that A expressed by (15.85) has real and imaginary terms (in phase and quadrature components), and both terms have to be zero simultaneously. The output from the reversed ∆β coupler is investigated closely. The condition of the ⊗ state is that the output from (1 ) is zero. The outputs from the left half section (at the mid-point in Fig. 15.15) are E 1+ = AU and E 2+ = j B ∗ U . Since A and B are complex numbers expressed by (15.85), the phase of E 1+ is delayed by L ∆ tan βz φ A = tan−1 βb 2 from that of U . The similar delay of E 2+ is φ B , which depends on the value of k12 . The quantity k12 is in general a complex number. The right half section is excited by E 1+ and E 2+ and the contribution of the former to the output is A∗ E 1+ and that of the latter is j B E 1+ . This time, the phase of E 1+ is advanced by exactly the same amount φ A . The phase of E 2+ is also advanced by

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exactly the same amount φ B . Thus, the output (1 ) becomes a real quantity including an additional parameter k12 to play with, which makes the zero condition less severe, and Ar can be made zero with more varieties of combinations of ∆, L/2, and k12 . The directional coupler can be also used as a modulator in addition to its use as a switch but the requirements are slightly different. A modulator normally does not require a perfect cross-over state, but a switch needs it to minimize cross talk. Another difference is that a switch has to include a dc state. The upper limit of the modulation frequency of such a modulator can be raised by designing the electrodes as a transmission line, as was explained for the bulk modulator (Sect. 14.2.2).

15.3.3 Tunable Directional Coupler Filter The tunable directional coupler filter combines the wavelength dependence of the propagation constant of the guide (dispersion of the guide) and the critical dependence of the cross-over state upon the value of ∆. The pass band of the filter is tuned by a control voltage [15.5]. The following is a qualitative illustration of the operation of such a filter. A directional coupler such as that described earlier in Fig. 15.13 is modified. In order to accentuate the imbalance between the guides, the dimensions of guides 1 and 2 are purposely made different. Assume that the wavelength dependence of the propagation constants β1 and β2 of the guides is represented by the curves at the top of Fig. 15.17. The solid lines refer to no applied voltage, and the dashed lines refer to an applied voltage. The middle graph shows the difference |β1 − β2 | of the propagation constants with respect to wavelength, with and without applied voltage. If it is further assumed that the guides are cut at the transfer length, and that the coupling constant k12 is essentially constant with respect to the applied voltage, then the center frequency of the maximum cross over efficiency (∆ = 0) moves according to the applied voltage. The condition for good operation is rather critical, but by using the reversed ∆β directional coupler with independent voltage control to each electrode, the operation can be made less critical.

15.3.4 Y Junction Figure 15.18 shows a Y junction of an optical guide. An optical guide is imbedded in an electrooptic substrate like lithium niobate. Guide (1) is symmetrically divided into output guides (2) and (3). External electrodes are deposited along the guides. In the absence of an external electric field, the light power is divided into equal portions, but when an electric field is applied such that the index of refraction of one guide is increased whereas the other is decreased due to electrooptic effect, the light tends to be steered toward the direction of the higher index of refraction (Sect. 5.4). The application of the electric field acts to switch one guide on and the other guide off.

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Fig. 15.17 Qualitative explanation of the operation of a tunable directional coupler filter

Fig. 15.18 Y junction.

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Even though the structure is simple, an on-off ratio of 17 dB is obtainable [15.6]. The output ports can be increased to more than two by increasing the number of output arms [15.7], or by staggering many Y junctions in the form of a tree branch.

15.3.5 Mach-Zehnder Interferometric Modulator This modulator uses the interference between two light beams which have travelled through separate paths en route to a common output port [15.8]. Figure 15.19 shows the layout of the device. An optical guide is laid out on an electrooptic material. Single-mode guides (TE0 mode) are used. The input is first divided into two optical guides of equal length. One of the guides is sandwiched by gold strip electrodes. A potential applied to the electrodes changes the propagation constant of the guide between the electrodes. If the input light is divided equally between guides (1) and (2) in Fig. 15.19, and if a voltage is applied such that the phase difference between the outputs of the two guides is an odd multiple of π radians, the outputs cancel each other. Correctly speaking, the field distribution of the output light is like the one shown in the circular insert in Fig. 15.19. This kind of field distribution belongs to a higher-order mode rather than the lowest mode excited at the input guide whose field distribution is uniform. Since the dimension of the output guide is designed such that the higher modes are cut-off, there is no output. If the output guide were widened, total cancellation would not occur. By switching on and off the applied voltage, the device switches on and off the optical power. The optical output at an intermediate voltage is expressed by Iout =

Iin (1 + cos φ) 2

(15.93)

where the angle φ is the phase difference between the controlled and uncontrolled arms. The expression for φ is easily found if use is made of Exercise 14.1. Let us

Fig. 15.19 Mach-Zehnder interferometric modulator

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say that the LiNbO3 crystal is X cut (meaning that the cut plane is perpendicular to the X axis of the crystal. A C cut means that the cut plane is perpendicular to the crystal axis. X is taken perpendicular to C). The external electric field is in the z direction and the polarization of the TE0 mode is parallel to the surface and also in the z direction. The geometry is exactly the same as Exercise 14.1 and from (14.10) φ = φ0 − K L V,

with

K =

α π 3 n r33 , λ e d

where V is the applied voltage, L is the length of the guide, α is a number between 0 and 1 depending upon the geometry which influences the effectiveness of the external voltage, and φ0 absorbs all the phase shift which is independent of V . Thus, if the external voltage V is changed linearly with time in the form of a triangular wave, the optical output power is modulated sinusoidally according to (15.93) [15.8]. The device is simple in structure but demands high accuracy in fabrication if it is to be used as an optical on-off switch. The reason for this is that the extinction ratio depends critically upon how well the amplitudes of the output from the two guides are matched. The Mach-Zehnder interferometric modulator modified to be an optical exclusive “OR” gate [15.9] is shown in Fig. 15.20. The upper guide has two φ = π control electrodes and the lower guide has one φ = π control electrode. Optical power output is present only when either V1 or V2 is on, and the optical power output is absent for all other combinations of V1 and V2 . Figure 15.21 shows the Mach-Zehnder interferometric modulator when it is modified to be a digital electrical to analog optical converter. The least significant bit of the electrical signal is connected to the shortest electrical control, and the most significant bit of the electrical signal is connected to the longest. The lengths of the control electrodes are progressively increased by doubling the length of the preceding lower-order bit. The dynamic range of such a converter is limited to the range in which output power can be approximated as proportional to the length of the control electrodes.

Fig. 15.20 Optical exclusive “OR” logic

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Fig. 15.21 Digital electrical-analog optical converter

15.3.6 Waveguide Modulator The modulator that is perhaps the simplest in structure is the waveguide cut-off modulator. This consists of an optical guide which operates near the cut-off region and is switched to the stop band by electrooptically changing the refractive index of the guide. A demerit of such a modulator is the rather high on state transmission loss, which cannot be avoided when operating so close to the cut-off of the optical guide.

15.3.7 Acoustooptic Modulator Acoustooptic devices based upon the bulk effect have been already covered in Sect. 14.3. In this section, a similar device using the surface acoustic wave (SAW) is discussed. The SAW device is lighter and more compact, and its diffraction efficiency for a given acoustic energy is better than the corresponding bulk-effect devices [15.10]. Figure 15.22 shows various ways in which the SAW device is used. The surface acoustic wave is launched from an interdigital transducer deposited on the surface of a piezoelectric material such as lithium niobate. The interdigital transducer has a comb-like structure. The teeth of the comb establish an electric field whose polarity is spatially alternating. This creates an alternating electrically induced strain due to the piezoelectric effect which excites a surface acoustic wave. It should be noted that both a periodic variation of the refractive index and a surface deformation contribute to the diffraction effect. The variation of refractive index penetrates about one acoustic wavelength from the surface. The relative orientation of the light beam with respect to the acoustic beam may vary depending on the intended use of device.

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Fig. 15.22 a–c. Various uses of a SAW device. (a) Light scanning by Bragg diffraction. (b) Filtering by selective reflection. (c) Probing the SAW by diffraction of the ray

In Figure 15.22 a, the device is used for such functions as steering, modulating or switching the light beam. The light is directed so that it satisfies Bragg’s condition. The direction of the Bragg diffracted wave is swept by sweeping the acoustic frequency. The intensity of the diffracted ray is controlled by the intensity of the acoustic wave. Figure 15.22 b shows an arrangement in which a SAW device is used as a mode filter or a frequency band pass filter. It is a special case of Bragg diffraction with θ = 90◦ in (14.22). The modes (or frequencies) of light whose one half wavelength is exactly one acoustic wavelength are selectively reflected back, and cannot pass through the guide. The center frequency of the stop band is tunable by changing the acoustic frequency. This band-stop filter can be used as the reflector of a laser cavity to make an electrically tunable laser or to remove the side lobes of the laser diode.

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Fig. 15.22 d. Processing of a radar return

Figure 15.22 c shows a light beam being focused onto the surface of the SAW device in order to probe the excitation. The wavelength and degree of excitation can be determined from the diffraction beam. Both diffracted and transmitted beams can be used to gather information. Figure 15.22 d shows how the signal from a chirp radar is processed by the SAW device. Because of the slow propagation speed of the surface acoustic wave in the medium (3,485 m/s in LiNbO3 ), a time segment of an electrical signal can be converted to a spatial distribution of the variation of the refractive index of the SAW device. A chirp radar uses pulses whose carrier frequency is progressively increasing with respect to time during each pulse. If the radar echo of such pulses is used to launch a surface acoustic wave on the SAW device, a surface acoustic wave whose wavelength is progressively decreasing propagates across the device toward the end and is lost at the end. The variation of the refractive index on the device resembles that of a zone plate. A parallel beam of light incident upon the device will converge to a single point. If the device is made long enough to store one entire pulse, the device displays one single point for each pulse duration. This is the basis for compressing a finite length pulse into a single point. The chirp modulation circumvents the mutually conflicting requirements of a longer pulse for a larger energy of the returned signal (longer range of the radar) and a shorter pulse for higher target resolution. The SAW device can also be used as a convolver. Two SAW beams V1 and V2 propagating in opposite directions can be generated by depositing two interdigital transducers at both ends of the substrate. The amplitude of the light diffracted from the SAW beams is proportional to V1 + V2 . The output from a square law detector

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Fig. 15.22 e. Disk delay line

includes the product of 2 V1 V2 . The spatial integral of this product as V1 and V2 are sliding in opposite directions performs the convolution of V1 and V2 . There are some functions such as the delay line, the band-pass filter, the pulse compression filter, and the phase coded matched filter that the SAW device can perform without using light. A brief description of these functions is given below. A delay line can be realized by taking advantage of the low speed of propagation of the acoustic wave. Figure 15.22 e shows a SAW device on a circular shaped substrate [15.11]. The surface wave circulates around the disc and a total delay time of 400 µs can be achieved, which is about ten times that obtainable in a single pass through a rectangular chip. The delay line becomes a memory device if the output signal is amplified and fed back to the input transducer again, as shown in the figure. A band pass filter in the VHF and UHF bands is fabricated by properly designing the length of the fingers of the interdigital transducer. Figure 15.22 f shows the design principle of the band-pass filter by the delta-function model. The transmitting transducer is a wide-band transducer and the receiver transducer has band-pass characteristics. (The device is reciprocal and the transmitter and receiver can be exchanged). The voltage induced by the piezoelectric effect in each pair of fingers is proportional to its length. The length of the fingers is modulated as sinc z in the z direction. When one impulse is launched from the transmitting transducer, the output voltage from the receiver transducer varies as the impulse propagates through the fingers, where 1/ f 0 is the time for the impulse to pass through only one of the pairs of the teeth. As this impulse goes through the teeth the output voltage Vout varies as sinc(Bt) cos(2π f 0 t). A filter whose impulse response is a sinc function in the time domain possesses a rectangular pass-band in the frequency domain. This

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Fig. 15.22 f. Principle of the SAW bandpass filter

method of design is known as the design by the delta-function model and is widely used because of its simplicity. Figure 15.22 g shows a chirp filter which is different from that in Fig. 15.22 d. The spacing between the pairs of electrodes is made wider towards the end of the substrate. Since the transducer has maximum efficiency when the acoustic wavelength and the spacing between the pair of the electrodes coincide, the lower frequency component is excited predominantly in the outer region and the higherfrequency component, in the inner region. The shape of the receiver transducer is a mirror image of that of the transmitter. With this arrangement, the lower-frequency component has to travel a longer distance than that of the higher-frequency component. When a chirped radar echo, such as shown at the lower right of Fig. 15.22 d, is fed to this SAW device, the lower-frequency component is launched first, but is delayed longer in the SAW device, so that the shorter-frequency component of the echo has a chance to catch up. Thus the output from the SAW device is a compressedin-time version of the input. This device converts the frequency information into a time delay and can be used also as a spectrum analyzer.

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Fig. 15.22 g, h. (g) Geometry of the SAW chirp filter, also used for a spectrum analyzer. (h) Principle of the PSK filter

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The phase-shift keyed (PSK) filter is fabricated by taking advantage of the fact that a phase reversal of the output voltage can be easily obtained by repeating the finger belonging to the same bonding pad. Figure 15.22 h shows the geometry of the fingers of a PSK filter. The fingers of the transducer are spaced by a half wavelength, and are alternately connected to the same bonding pad, like most other interdigital transducers. However, at the locations where the phase is to be inverted, the two adjacent fingers are connected to the same bonding pad. When an impulse acoustic wave passes the particular finger pattern shown at the top right side of Figure 15.22 h, the output voltage across the terminal pad with respect to time becomes such as shown in the middle figure. Notice the sudden π radian phase shifts. The PSK filter is used for both coding and decoding signals. If a PSK coded signal is launched into a PSK filter, the output voltage is the convolution of the input signal and the finger pattern of the filter. As the SAW beam propagates along the fingers, at the instant in which the two patterns match up, the output signal bursts into a large signal. At all other times, the contribution of roughly half the fingers is positive and that of the other half is negative, and no voltage is seen at the output terminal, as shown at the bottom of Fig. 15.22 h. The PSK filter is used for the same purpose as a chirp filter to compress a pulse, but the PSK filter has the advantage of discouraging electronic espionage because the signal can be decoded only by the corresponding PSK mode.

15.4 Bistable Optical Devices The devices so far discussed were strictly linear, i.e., when the input light level is lowered the output light level is lowered accordingly. In this section, devices are presented which give out two different levels of output for the same level of input, depending upon the past history of the input power level. The output of these devices depends upon whether the present input level was achieved by lowering from a higher level or raising from a lower level. This phenomenon of optical hysteresis is known as optical bistability. Such a phenomenon can be used for making optical memories, optically operated switches, optical amplifiers, and optical logic gates.

15.4.1 Optically Switchable Directional Coupler If the light itself is used for changing the switching state of a directional coupler, then the light can steer itself through a switch matrix. Such a switch is materialized by combining a reversed β directional coupler with a feed-back loop (Fig. 15.23). The output from the switch is fed back to control the switching state [15.12–15]. A more specific explanation is given as follows. The input light Pin is incident upon guide 1. A portion of the output light PL1 from guide 1 is taken out by a Tee coupler and detected by a photodiode. The detected signal after amplification is

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Fig. 15.23 Optically switched directional coupler

used to apply an electric field to induce a difference in the propagation constants between the guides. If ∆ is assumed proportional to PL1 , then ∆ = G PL1 ,

(15.94)

where G absorbed such constants as the coupling constant of the Tee coupler, the gain of the amplifier, the electrooptic constant and dimensions of the guides. Insertion of (15.94) into (15.51) gives φ = βb L =

|k12 |2 + (G PL1 )2 L .

(15.95)

The transmission PL1 /Pin of guide 1 is | Ar |2 and rewriting of (15.91) in terms of φ gives 2 |k12 | L 2 PL1 2 φ = 1−2 sin . (15.96) Pin φ 2 The transmission for a given input can be found from the simultaneous solution of (15.95, 96). The solution can be found either by use of a computer or by graphical presentation, and here, the latter means will be chosen. For easier representation, (15.95) is rewritten as 2 φ |k12 | 2 PL1 2 = − . (15.97) Pin G L Pin G Pin Both (15.96, 97) are plotted in the same graph in Fig. 15.24 a. The horizontal axis is the electrical coupling length φ and the vertical is the transmission PL1 /Pin of guide 1. The plot of (15.96) is the curve with the sinc2 characteristic. Equation (15.97)

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435

is plotted with different input powers. The curves for (15.97) are hyperbolas, but in the region of interest they are almost linear with the slopes of (G L Pin )−1 . The slope depends inversely on the incident light power. The movement of the point of intersection is indicated by numbers, and the corresponding points are indicated on the input-output characteristic of the coupler, shown in Fig. 15.24 b. In Fig. 15.24 a, the intersection when the input power is low is at point 1. As the input power is further increased and as the line bends beyond point 2, there are three intersections indicated by the points 3, 3 and 3 . All three points are solutions, but point 3 is unstable. The response to a minute fluctuation in output power in the unstable region between 4 and 2 is considered. An increase in PL1 leads to an increase in φ, according to the sinusoidal curve in Fig. 15.24 a, and this increase in φ in this region of the transmission curve leads to a further increase in PL1 of (15.95). On the

Fig. 15.24 a, b. Optical bistability. (a) Movement of operating points as the input power is varied. (b) Illustration of the graphic method for finding the output-input characteristics of optical bistability. They are with |k12 |L = π and G L = 4π .

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other hand, a minute initial decrease in PL1 leads to a decrease in φ and thus leads to a further decrease in PL1 . Regardless of what the fluctuation at point 3 is, the fluctuation always grows bigger (positive feed back) and the system is unstable. If one makes the same argument with 3 and 3 , one soon discovers that the fluctuation is always suppressed (negative feed back) and the system is stable at these two points. Between the choice of points 3 and 3 , the system operates at point 3 if the input power is moved up from point 2. This is because, as the input power is increased, point 3 is reached first and becomes stable and the output does not grow further to reach point 3 . All of the above discussions on stability were based upon a diagram that does not include any time dependence (static diagram). Strictly speaking, stability is inherently a time dependent phenomenon, and the preceding discussion is not a rigorous argument. The temporal behavior of such devices can be investigated by solving the differential equations with respect to time. An unstable solution of the differential equations grows indefinitely with respect to time [15.16–19]. Returning to Fig. 15.24 a, the behavior of the system beyond point 3 is considered. When the input power exceeds point 4, there is no other alternative but to jump up to 4 and proceed onto point 5. The path of decreasing input power is different from that of increasing power. As the input power is decreased from point 5 and passes beyond point 4 , the triple state is again reached. Between the stable states of point 3 and 3 , point 3 becomes the operating point because, in the process of decreasing the input power, point 3 is first found stable and the operating point stays at 3 , and does not further decrease to point 3. When the input power is decreased to point 2 , the state jumps down to point 2, and it eventually reaches point 1 with a further decrease in the input power. The arrows in Fig. 15.24 b indicate the path of change of the operating point when Pin is cycled. The diagram takes the form of a hysteresis loop. There are higher and lower levels of the light output PL1 for the same value of input Pin , and the choice of one of the two levels depends upon the history of Pin . Such a system is called a bistable system. Figure 15.25 illustrates how such a bistable device is used. An initial triggering light pulse A brings the operating point to point 5 in the upper level, and at the end of the trigger pulse, PL1 stays in the upper level. The output PL1 remains channelled to guide 1, as shown by the solid line. As soon as the triggering light pulse B arrives, the operating point is brought down to point 1 in the lower level. After the B triggering pulse, PL1 stays in the lower level, and the light remains channelled to guide 2 until a further triggering pulse is received. The output from guide 2 is PL2 = Pin − PL1 , and is shown by the dotted line. The light signal is thus channelled through the switch according to a switch pulse contained in the incident light itself.

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Fig. 15.25 Principle of bistable switching triggered by light

15.4.2 Optical Triode Making use of the fact that the output PL1 of the optically controlled directional coupler critically depends upon the input power, an optical triode, such as shown in Fig. 15.26 a, can be assembled [15.14, 17]. The steep slope of the input-output curve of a modified hysteresis is utilized. The loop of the hysteresis shown in Fig. 15.24 b is flattened by making the spacing between points 2 and 4 conform to the shape shown in Fig. 15.26 b. The spacing between points 2 and 4 in Fig. 15.24 b can be reduced by making the two curves in Fig. 15.24 a as parallel as possible between points 2 and 4 by adjusting the values of |k12 |2 , G, or L. Amplitude modulated light properly biased to the center of the hysteresis curve can be amplified as shown in Fig. 15.26 b. It should be recognized that the light energy was not amplified. Rather, the distribution of the input energy between the two guides was changed by the input. The total energy of the output from guides 1 and 2 is constant at all times.

15.4.3 Optical AND and OR Gates Using a modified hysteresis loop, such as shown in Fig. 15.26 b, the operation of AND and OR logic can be performed optically [15.14].

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Fig. 15.26 a, b. Optical triode. (a) Arrangement of the optical triode. (b) Characteristic of operation

When the optical input power is set below the lower knee point of the curve, the output light is off, unless both of the incident lights are on, and the operation of AND logic is achieved. When each input is set near the upper knee, as long as one of the inputs is on, the output light is on, and the logic of OR is performed.

15.4.4 Other Types of Bistable Optical Devices A few other systems that operate on the principle of optical bistability will be covered in this section. First, a bistable optical device using a Mach-Zehnder interferometric modulator is mentioned. The optically switched directional coupler of Sect. 15.4.1 used a reversed β directional coupler, as shown in Fig. 15.23, but a Mach-Zehnder interferometric switch as shown in Fig. 15.19 can be used for the same purpose [15.19].

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Fig. 15.27 Bistable optical device using a Fabry-Perot resonator

This is because the output power PL /Pin of the Mach-Zehnder interferometric modulator varies with the electrical length φ as PL /Pin = cos2 (φ/2) in a manner exactly the same as the reversed ∆β directional coupler. The difference is that the device with the Mach-Zehnder interferometric modulator has only one output port, while that of the reversed ∆β directional coupler has two output ports. It is also possible to replace the reversed ∆β directional coupler by a Fabry-Perot resonator (Fig. 15.27). The cavity of the resonator contains an electrooptic material which is controlled by the output light. The transmission PL /Pin of the Fabry-Perot resonator as a function of φ has a much sharper peak than that of the reversed ∆β directional coupler, as indicated at the upper left corner of the figure, but in other respects they both have the same general shape. Thus, the Fabry-Perot device has a bistable characteristic similar to that of the reversed ∆β directional coupler [15.14]. As seen from the above examples, whether a reversed β directional coupler, a Mach-Zehnder interferometric switch, or a Fabry-Perot resonator is used, what is essential is that the transmission characteristic can be controlled by a positive feedback of the optical output power. Materials such as sodium vapor, carbon disulfide CS2 , and nitrobenzene (C6 H5 NO2 ) have an intensity dependent index of refraction. When such a material is enclosed in the Fabry-Perot resonator, a change in the input power changes the electric length of the resonator, and bistable operation is achieved without the need for an external feedback loop. All one needs is a resonator filled with this intensity dependent material, and one can remove the electrooptic material and the amplifier drawn in dotted lines in Figure 15.27. The range of the bistability of the device with this built-in feedback, however, is much narrower than that with an external feedback loop.

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15 Integrated Optics

15.4.5 Self-Focusing Action and non-linear optics Another noteworthy phenomenon connected with optical bistability is the selffocusing effect [15.20–23]. When a material is illuminated by an intense beam of light, the index of refraction of the medium will change by an amount that depends on the intensity, and consequently the propagation characteristics of the medium will also change. In this situation, doubling the input does not necessarily double the output. Such phenomena have been under intense study in the field of nonlinear optics. The dependence of the refractive index on the light intensity I can be approximately expressed as (15.98) n = n0 + n2 I if higher-order terms of I are ignored. Quite often, the intensity distribution of a light beam in the transverse plane is bell shaped, e.g. a Gaussian distribution. If the intensity of such a beam is high enough to create nonlinear effects, the transverse distribution of the refractive index also becomes bell shaped (if n 2 is positive like that of CS2 ) and the medium starts to behave as a converging lens. This is the basis of self-focusing. Materials such as InSb have a negative value of n 2 and possess properties of self defocusing. The self-focusing effect is useful not only for beam manipulation, but also for producing bistable optical devices. Figure 15.28 shows an example of bistability achieved through the self-focusing phenomenon. A light beam with a Gaussian intensity distribution is focused onto the input face of a cell which contains a nonlinear medium. The output from the cell falls upon a convex lens which focuses the image of the input face of the cell onto a half mirror. The half mirror transmits a portion of the light through as the output, and reflects the rest into the cell again, so that both incident and reflected beams participate in the self-focusing effect.

Fig. 15.28 Bistable optical device based upon self-focusing

15.5 Consideration of Polarization

441

When the input intensity is low, no self-focusing action takes place, and the output power is low (lower level). Once the input-intensity reaches a critical point sufficient to initiate self-focusing, the intensity at the center of the beam increases and the gradient of the refractive index in the cell increases. The increased gradient means better focusing, and better focusing means higher intensity, and higher intensity means increased gradient, and so on. The intensity of the light beam grows rapidly. Once a high intensity is established, the focusing action continues. Even after the input light power is lowered, the output power is still maintained high (higher level) by the self-focusing action, and the input-to-output curve displays a hysteresis loop characterizing the bistability. The device uses the light beam as a feedback loop but needs neither an external circuit nor a resonant cavity, and the switching speed can be as fast as a fraction of a picosecond. In contrast, the switching speed of a bistable device with external feedback is limited by the time constant RC, which is the product of the input resistance R of the amplifier and the capacitance of the electrode. In resonator devices, the build up time of the resonant cavity normally slows down the switching time. In Fig. 15.28, the aperture adjusts the degree of the feedback to reach the condition of bistability.

15.5 Consideration of Polarization So far nothing has been mentioned about the influence of the mode of excitation upon the function of the electrooptic devices. The modes in an optical guide are classified into TE (transverse electric) and TM (transverse magnetic) modes. The former does not have an E component in the direction of propagation, and the latter does not have an H component in the direction of propagation. If the z axis is taken as the direction of propagation, the components present in each mode are E y Hx Hz

(15.99)

TE mode

Hy E x E z

(15.100)

TM mode.

The direction of the E component is especially important for the operation of electrooptic devices. In the coordinate system shown in Fig. 15.29, the E component of the TE0 mode is in the y direction, i.e., parallel to the surface of the layer whereas that of TM0 mode is in the x direction, i.e., in a plane perpendicular to the surface of the layer. The TE0 mode has a lower cut-off frequency than the TM0 mode. The devices that use the electrooptic effect are normally sensitive to the type of mode excited because the electrooptic constant for one direction of polarization (direction of E) is different from that for another direction. For instance, when both the external electric field and the optical axis of LiNbO3 are in the x direction, as shown in Fig. 15.29, the E component of the TM0 mode sees the r33 coefficient, whereas that of the TE0 mode sees the r13 coefficient. The value of r13 is one quarter

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15 Integrated Optics

Fig. 15.29 Direction of the E component of TE0 and TM0 modes. (a) TE0 mode. (b) TM0 mode

Fig. 15.30 a–c. Polarization insensitive Mach-Zehnder interferometric modulator. (a) Plan view. (b) Cross section at a − a . Primarily for TM0 mode. (c) Cross section at b − b . Primarily for TE0 mode

to one third of that of r33 . Some devices have to avoid this difference and others make use of this difference. The solution to eliminating the difference is either to use a mode stripper such as mentioned in connection with Fig. 15.22 b or to design the guide so as to pass only one of the modes. Otherwise, one must use a device which is designed to operate for both TE0 and TM0 modes. Figure 15.30 shows a Mach-Zehnder interferometric modulator which can operate for both TE0 and TM0 modes [15.4]. The electrodes are divided into two

15.5 Consideration of Polarization

443

Fig. 15.31 a, b. TE-TM mode splitter. (a) Plan view. (b) Cross section at a − a

sections, each having a slightly different arrangement. The lower electrode in Section I in the figure is deposited right on the optical guide. This provides an electric field E which is primarily vertical and parallel to the E component of the TM0 mode. On the other hand, the electrodes in Section II are placed symmetrically with respect to the optical guide and the E field is primarily horizontal and parallel to the E component of the TE0 mode. Thus, by adjusting the applied voltages and length of each section, it is possible to find the condition that both TE0 and TM0 modes satisfy the switching condition. Figure 15.31 shows a TE ↔ TM mode splitter using a Y junction [15.6]. As shown in Fig. 15.31 a, the electrodes are displaced from the line of symmetry so that the upper half of the guide in the figure has the external field E parallel to the surface, whereas the lower half has the E field perpendicular to the surface. Moreover, in the lower half of the guide, a buffer layer of Al2 O3 film is deposited between the electrode and the guide to increase the effective refractive index. When the external field is absent, both TE0 and TM0 modes converge into Region II in Fig. 15.31 b where the effective refractive index is higher because of the buffer layer. But once the external field ε is applied, a horizontal field is established in Region I. The direction of polarization of the TE0 mode is horizontal and the associated electrooptic constant r3 3 is the largest (optical axis is horizontal). If the intensity of the external field is raised to such an extent that the effective refractive

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index for the TE0 mode is larger in Region I than in Region II, the TE0 mode converges to Region I. On the other hand, the external electric field E in Region II is predominantly vertical and the TM0 mode is even more confined in this region. With optimum adjustment of the external voltage, TE0 and TM0 modes exit from separate output ports.

15.6 Integrated Optical Lenses and the Spectrum Analyzer Even though the operation of a lens fabricated on the substrate is restricted to one dimension, the lens is useful for such operations as imaging, focusing, spatial filtering, Fourier transforming, and convolving operations. As seen from Fig. 15.32, the physical length of the ray along the axis is the shortest and the rays get longer towards the sides. The focusing requirement is to equalize the optical length of all these rays whose physical lengths are different. The focusing is achieved by either detouring the path or by adjusting the phase velocity of the different rays. Specific geometries will be introduced.

15.6.1 Mode Index Lens A mode index lens profiles its effective refractive index by varying the thickness of the guiding layer [15.24]. Figure 15.33 illustrates how the effective index of refraction, N = n 1 cos α = β/k, of the TM0 mode is increased with the thickness of the guiding layer. Here β is the propagation constant in the direction of propagation z, and k is that of free space. As the thickness is increased, the angle of inclination

Fig. 15.32 Imaging condition of a lens

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Fig. 15.33 Comparison of the angle α for the TM0 mode in the thinner and thicker guiding layer. The thicker the layer is, the smaller α is, hence the larger the effective refractive index N = n 2 cos α is

Fig. 15.34 a, b. Luneberg lens. (a) Plan view. (b) Side view

α decreases so that the zero lines still match the guide boundary and thus, N is increased with the thickness. In order to equalize the optical path in the center of the lens with that on the sides, the effective index of refraction is raised towards the center. For this purpose, the thickness of the guiding layer is increased by depositing the same material as the guiding layer as shown in Fig. 15.34. A larger increase in effective refractive index is achieved if a material with a higher index of refraction n 3 than n 1 is used for depositing. The light ray goes into the deposited region and the amount of increase in effective refractive index has to be analyzed as a four-layer problem. The focusing power, however, is limited by the availability of usable transparent materials with a high refractive index. In the case of a spherical lens, Maxwell found that a fish-eye lens whose refractive index distribution is n(r ) = 2/(1 + r 2 ) can make the image of an object on the sphere at a diametrically √ opposite point on the same sphere. Luneberg found that a sphere with n(r ) = 2 − r 2 can converge a parallel beam into a point. Among the refractive index distribution of mode-index lenses, the one-dimensional Luneberg type distribution is most often employed.

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Fig. 15.35 a, b. Geodesic lens. (a) Plan view. (b) Side view

15.6.2 Geodesic Lens Figure 15.35 shows the geometry of a geodesic lens [15.25]. The thickness of the guiding layer is kept constant, but the contour length near the center is made longer by indentation. A protrusion would also give the same effect. For the lens to be aberration free, the contour of the indentation has to follow an aspheric complex profile. The difficulty in doing this is to polish the ground profile without destroying the original profile. The edge of the indentation has to be ground to prevent unwanted scattering. The ground edge should be taken into consideration in designing the profile to achieve a diffraction limited lens.

15.6.3 Fresnel Zone Lens The principle of the Fresnel zone plate can be incorporated into an optical guide to produce a lens [15.26]. The phase distribution that a lens has to set up so as to converge a parallel beam to a point is, from (6.9), φ = −k

y2 + φ0 , 2f

(15.101)

where f is the focal length of the lens. Figure 15.36 shows a method for approximating this quadratic expression. The curve is approximated by a staircase function. The phase φ is quantized into an integral number of π radians in the upper graph. Since subtraction of an integral multiple of 2π radians (represented by the hatched sections) does not change anything, the required phase distribution for one half of the lens becomes like that shown in the lower graph. By depositing π radian phase shift pads in accordance with the pattern shown in the lower graph, a Fresnel lens is fabricated. Figure 15.37 illustrates this Fresnel lens. It is also possible to replace these phase pads by absorption pads thereby forming a zone plate of the absorption type, as mentioned in Sect. 4.5, with the sacrifice

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447

Fig. 15.36 Binary approximation of quadratic phase distribution (with φ0 = 0)

Fig. 15.37 a, b. Fresnel lens. (a) Plan view. (b) Side view

of half of the incident power. This type of lens generates background noise. The noise is due to higher-order beams which are inherent to the Fresnel zone plate lens discussed in Sect. 4.5. A modified Fresnel lens is shown in Fig. 15.38 in which the phase pads have been tilted inwards so that the beams of specular reflection from the phase pads are directed toward the focus [15.24].

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Fig. 15.38 a, b. Grating lens. (a) Plan view. (b) Side view

Fig. 15.39 Integrated optical spectrum analyzer

15.6.4 Integrated Optical Spectrum Analyzer Two geodesic lenses and a surface acoustic device can be combined to build an integrated optical spectrum analyzer [15.27]. Such a device can display the frequency spectrum distribution of a radio frequency electrical signal. Figure 15.39 shows a sketch of an integrated optical spectrum analyzer. The first geodesic lens L1 collimates the beam from a laser diode. The collimated beam

15.7 Methods of Fabrication

449

intercepts the surface acoustic wave launched from the acoustic transducer. The surface acoustic wave introduces a periodic strain and spatially modulates the refractive index of the guiding layer. The light wave is diffracted in the direction that satisfies the Bragg condition, i.e. in the direction proportional to the acoustic frequency. The diffracted light is focused, or equivalently, is Fourier transformed onto an array of detectors at the end of the substrate. The light intensity distribution across the detector array represents the frequency spectral distribution of the acoustic wave. The electrical output from the detector is stored on the comb shaped electrodes of a charge coupled device (CCD) [15.28]. The CCD sequentially shifts the charge on the comb electrodes to the output, in accordance with clocked advance pulses, in a manner similar to a shift register that clocks out a parallel signal as a serial signal. The orientation of the acoustic transducer is slightly tilted so that the Bragg condition is satisfied at the center of the band of acoustic frequencies. The required time of operation, physical dimensions and weight of the integrated optical spectrum analyzer are substantially less than the electronic counterpart of the device.

15.7 Methods of Fabrication A localized increase in refractive index forms an optical guide. In this section, we describe methods of fabrication for optical guides, followed by those of forming the patterns for the guides.

15.7.1 Fabrication of Optical Guides There are two major approaches to fabricating optical guides. One approach involves the deposition of a higher-index material onto the lower-index substrate. The other approach is to introduce a local increase in the refractive index of the substrate. Methods that belong to the former approach [15.29] are (1) epitaxial growth and (2) sputtering; and those in the latter are (3) ion-implantation, (4) out-diffusion, (5) in-diffusion, (6) ion migration and (7) photochromic film. The method of epitaxial growth is a method that uses the growth of one crystalline substance on another having a similar crystal structure [15.30]. It is used to build layers of different optical characteristics, for example, the growth of LiNbO3 onto LiTaO3 is used to build a layer of higher refractive index. InGaAs is grown over GaAs to build a layer with a lower energy band gap for an integrated Schottky barrier photo-detector. Epitaxy, however, requires careful control of the growth process before good results can be obtained. The sputtering method [15.31] is a widely used technique. Figure 15.40 shows an apparatus for sputtering. A material (metallic or non-metallic) such as Ti is held onto the cathode. The substrate to be coated, such as LiNbO3 , is placed on the anode.

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15 Integrated Optics

Fig. 15.40 Apparatus for sputtering

An inert gas such as argon is enclosed at a constant gas pressure between 10−2 and 10−1 torr by constant pumping. A few thousand volts is applied between the cathode (−) and the anode (+) in order to accelerate the ions of the enclosed gas. The bombardment of accelerated Ar ions into the cathode ejects the atoms of the material (Ti). The ejected material, which is either neutral or slightly negatively charged, travels to the anode and is deposited onto the surface of the substrate (LiNbO3 ) to form a film. Some of the materials used for sputtering are LiNbO3 , Ta, and Nb. Corning 7059 glass is one of the most common substrate materials used. The next method to be mentioned is ion implantation [15.32]. When a crystalline substrate like fused silica (SiO2 ) is bombarded by highly accelerated (a few hundred KeV) heavy ions like Li, Ne, Ar, Kr or Xe, the index of refraction is changed due to implantation. For instance, the implantation of Li into a fused silica substrate reduces the index of refraction by 0.8%. Another method is out-diffusion [15.33, 34]. When a substrate is heated to a high temperature in vacuum, some of the elements in the compound literally boil out. The method of out-diffusion uses this fact. Crystallized LiNbO3 can take the form of (Li2 O)ν (Nb2 O5 )1−ν . (This is called nonstoichio-metric form. With ν = 0.5, it becomes LiNbO3 .) The value of ν ranges from 0.48 to 0.5, and the refractive index of the crystal depends upon the value of ν. A crystal with a smaller value of ν has a higher index of refraction. When (Li2 O)ν (Nb2 O5 )1−ν is heated in vacuum, more (Li2 O) boils out than (Nb2 O5 ) because Li is not as strongly bound to the lattice as Nb. The refractive index of the out-diffused part becomes higher. For example, an optical guide is made by heating a lithium niobate substrate to 1, 100◦ C for 21 h in a 6 × 10−6 torr vacuum.

15.7 Methods of Fabrication

451

As a matter of fact, when LiNbO3 is heated for the purpose of some other treatment to the substrate, guiding layers are formed all over the surface because of the out-diffusion. This out-diffusion, however, can be prevented by maintaining an equilibrium vapor pressure by externally providing Li2 O vapor [15.34]. The method of in-diffusion [15.35] diffuses the material into the surface of the substrate to change the refractive index. In-diffusion of such transition elements as Ti, V or Ni into either a LiNbO3 or LiTaO3 substrate raises its refractive index. As ˚ of a metal such as those mentioned a matter of procedure, a thin layer (200–800 A) above is evaporated on the surface of a LiNbO3 substrate. This prepared substrate is heated at 850 ∼ 1, 000◦ C in a vacuum chamber containing Ar gas. After several hours of heating, a layer of higher refractive index a few microns thick is formed, due to the in-diffusion process. Ion migration [15.36] uses the migration of metal ions into a glass substrate. When AgNO3 is heated, it changes from solid form into liquid form. If a soda lime glass is immersed in the molten AgNO3 , a migration of Ag into the glass takes place, and Na in the glass is replaced by Ag. The refractive index of the soda lime glass is raised in the areas where Na has been replaced by Ag. A technique which is quite different from all the others is the photochromic technique [15.37]. A normally transparent acrylic sheet coated with photochromic material changes its colour to dark blue when it is exposed to ultraviolet light. With the change in colour, the refractive index of the film is also changed. The refractive index of the colored state is higher than that of the uncolored state for wavelengths longer than the photochromic’s absorption peak. The pattern can be fabricated at room temperature. The lines, however, are subject to fading, but this can be an advantage or a disadvantage depending on whether permanence or erasability is desired.

15.7.2 Fabrication of Patterns The methods of fabricating the guiding layers were the subject of the previous subsections. In order to make a useful device out of these guiding layers, the guides have to have a pattern [15.38–40]. The sputtering technique can be used for this purpose. If the substrate is placed on the cathode rather than on the anode, the bombardment of ions erodes the substrate material. This process can be used for pattern fabrication, and is called sputter etching. Photoresist is one of the most commonly used techniques for pattern fabrication. Figure 15.41 illustrates the photoresist process. First (1), the substrate is coated evenly by a photoresist such as poly-methyl-methacry-late (PMM) by spinning the substrate, and then (2) it is exposed to light through a mask which has the desired pattern. The substrate is then (3) immersed into a developer bath where the portions exposed to light are dissolved. After development, a thin film of a metal like Ti is (4) deposited by evaporation. It is then put into an acetone bath where all photoresist

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Fig. 15.41 (1–6). Procedure for fabricating a strip of optical guide with photoresist. (1) Photoresist (PMM) is poured and the excess resist is spun off. (2) Pattern mask is laid over the photoresist and exposed to light. (3) Photoresist is developed and exposed part is dissolved (positive photoresist). (4) Ti is evaporated over the entire top surface. (5) Photoresist is dissolved by acetone. Ti on the photoresist layer is removed. (6) Heated at 970 ◦ C for several hours to allow Ti film to diffuse into LiNbO3

15.7 Methods of Fabrication

453

is dissolved (5), including the photoresist under the thin metal film. Now, only the pattern of thin Ti film is left intact on the substrate. The substrate is then (6) put into an oven for several hours at 970◦ C so that in-diffusion of Ti takes place. Thus, the process of fabricating the desired pattern of the optical guide is completed. Finer resolution of the pattern can be achieved by writing in a pattern directly onto the photoresist rather than using a mask. A fine laser beam, electron beam, or even an x-ray beam is used for the writing [15.41, 42].

15.7.3 Summary of Strip Guides Figure 15.42 summarizes the configurations of strip guides used in integrated optics [15.43–45]. The ridged guide shown in Fig. 15.42 a is bordered by air on three sides. Since the dielectric discontinuity is so large, there is practically no evanescent wave, hence no cross-talk. A variation of this configuration is the rib guide shown in Fig. 15.42 a . The raised portion uses the same material as that of the guiding layer. The light energy is still confined in the raised portion, but a larger evanescent wave is present at the bottom. By adding another rib guide, as indicated by the dotted lines, a coupler can be made. The amount of coupling is adjustable by the amount of filling between the guides. Again, because of the large dielectric discontinuity on the surface, the surface roughness of the guide critically influences the transmission loss of both the ridge and rib guides. The bulge guide in Fig. 15.42 a is a more realistic shape for fabrication. This type of guide is produced by removing the unmasked

Fig. 15.42 a–c. Various configurations of channel guides. (a) Ridge guide. (a ) Rib guide. (a ) Bulge guide. (b) Imbedded guide. (c) Strip loaded guide

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15 Integrated Optics

parts by either chemical means or sputter etching. Alternatively epitaxial growth can be used to make a bulge guide. Figure 15.42 b shows an imbedded guide. Only one surface is bordered by air, and all other sides are bordered by a medium with only a slightly lower refractive index. The evanescent waves are present in all three sides. This is the most widely used guide. The amount of isolation as well as the coupling is easily controlled by the geometry and the refractive index. This type of guide is made by the methods of in- or out-diffusion. Figure 15.42 c shows a strip loaded guide. A strip of dielectric of higher refractive index is placed over the guiding layer. The light energy is confined in the guiding layer under the strip, and the tolerance in fabricating the loading strip line is not so severe as for ridge or rib guides. If the same material as the guiding layer is used for the loading strip, it becomes a rib guide. Sputtering or epitaxial growth is used as the method of fabrication. Sometimes a metal is used as the loading strip. In this case, the strip gives a means of applying an external electric field, as well as confining the light, because the metal at the light frequency acts as a lossy dielectric, even though its degree of influence is very minute [15.29]. Sometimes a SiO2 or Al2 O3 layer is placed between the metal strip and the guiding layer as a buffer layer to minimize the transmission loss.

15.7.4 Summary of Geometries of Electrodes Figure 15.43 summarizes the configurations of the electrodes so far described. The configuration of the electrode is determined in accordance with the requirement of

Fig. 15.43 a–d. Configurations for applying an external electric field in various directions. (a) Horizontal field in imbedded guide. (b) Horizontal field in rib guide. (c) Vertical field in rib guide. (d) Vertical field in imbedded guide

Problems

455

the direction of the applied field. In Fig. 15.43 a, b the direction of electric line of force is horizontal. Figure 15.43 a is for an imbedded guide, and Fig. 15.43 b for a rib guide. The effectiveness of the applied electric field in the latter configuration is much larger than the former, because of the smaller area of divergence of the electric line of force. When the electric field has to be vertical, the configurations shown in Fig. 15.43 c, d are used. The fact that the electric line of force always terminates perpendicular to the surface of a perfect conductor is used. With the configuration in Fig. 15.43 d, the field intensity can be doubled by placing another electrode, as shown in dotted lines.

Problems 15.1. Based upon the delta-function model, design an interdigital transducer of a SAW bandpass filter. The frequency of the pass band is from 80 to 120 MHz. The velocity of the surface acoustic wave in the lithium niobate substrate is 3,485 m/s. 15.2. Show that the loops of the ⊗ modes of the reversed ∆ directional coupler (Fig. 15.16) are all confined in the sector bounded by the vertical axis and the 45 degree line (a line bisecting the horizontal and vertical axis). 15.3. An optical directional coupler such as shown in Fig. 15.44 is to be deposited on a LiNbO3 substrate with the following parameters by locally raising index of refraction using titenium diffusion. The electrical coupling length of the ⊗ switching mode is selected as π2 rad. a = 1.0 µm, n 1 = 1.1 n 2 ,

b = 0.5 µm,

c = 1.5 µm,

α = 0.3 (Electrode efficiency)

λ = 1.3 µm

Fig. 15.44 Geometries of the optical directional coupler

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x modes. (——) E x coupling for n /n = 1.5; (–·–·–) E x Fig. 15.45 Coupling coefficient for E 1q 1 2 1q 1q coupling for n 1 /n 2 = 1.1

Fig. 15.46 Operating points of the optical bistability

Problems

457

Assume total length d of electric line of force of the dc field is 10 µm and the ratio of propagation constants k z /k1 is unity for the dominant mode. a) What is the orientation of the crystal and the electrode for minimizing the control voltage? b) What is the control voltage to achieve the = switching mode? Only excitation of the dominant mode (TE mode) is desired. The condition that only the dominant mode be excited is approximately a<

0.8λ n 21 − n 22

.

The coupling constant for the given dimensions and the refractive indices are found from the universal curves prepared by Marcatili [15.46] in Fig. 15.45. 15.4. Derive the expressions which would be useful for drawing the cross bar diagram of a two-bay reversed ∆β directional coupler (two reversed ∆β directional couplers in tandem), and make a rough sketch of the cross bar diagram for the twobay reversed ∆β directional coupler. Also, verify that the required control voltage for the two-bay coupler is less than that of the one-bay coupler. 15.5. In Fig. 15.24a only the range of 0 < φ < 3π/2 of the curve was considered. Complete the input-output characteristic curve like the one in Fig. 14.34b taking a wider range of 3π/2 φ < 3π as shown in Fig. 15.46 into consideration. Assume |k12 |L = π, and G L = 4π.

Chapter 16

3D Imaging

The chapter begins with a brief history [16.1–5] of 3D imaging followed by a discussion of physiological factors contributing to 3D vision. The principles underlying various means for realizing the 3D effect are then described in detail.

16.1 Historical Development of 3D Displays Human beings have long been curious about how to create 3D images out of 2D images. As early as 280 BC, Euclid of Greece had already recognized that a simultaneous look at slightly different scenes of the same object viewed from the left and right produced the sensation of seeing a 3D image. It is not certain whether or not he made any further progress beyond this recognition [16.1].

Euclid

459

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Around 1600, which predated the invention of photography by about two hundred years, the Italian scientist Giovanni Battista della Porta tried to create a 3D image by arranging side-by-side stereoscopic pictures that consisted of accurately hand drawn views from the left and the right. In 1838 before an audience of the Royal Scottish Society of Arts, Sir Charles Wheatstone of England demonstrated a 3D display using a pair of mirrors that combined a stereoscopic pair of pictures viewed from the left and the right. The stereoscope was well received, but the cumbersome structure inhibited wider acceptance. In 1849, Sir David Brewster of Scotland combined a pair of stereoscopic images using a set of prisms, which replaced the mirrors in Wheatstone’s stereoscope. This model is popular even now as Brewster’s stereoscope.

Sir David Brewster

In the 1853, Wilhelm Rollman of Germany demonstrated the principle of the anaglyph, which was further developed by Joseph D’Almeida of France in 1858. The anaglyph takes advantage of complementary colors to separate the pictures viewed from the left and right [16.2]. The pictures representing the left and right views are drawn in complementary colored inks, one over the other, in the same plane. The viewer wears glasses made of these complementary colors to separate the left and right views in order to achieve the stereoscopic effect. In 1895, John Anderton, a British subject, obtained an US patent on “Method by which pictures projected upon screens by magic lanterns are seen in relief” [16.6]. In his magic lanterns, a polarizer such as a pile-of-plate polarizer or a Nichol prism was installed to project polarized light onto a screen. Viewers had to wear bulky glasses that were also installed with pile-of-plate polarizers or Nichol prisms. Unlike anaglyphs, polarization-based techniques are capable of producing true color 3D images. In 1903, F. E. Ives of the USA invented the parallax barrier method that does not require wearing glasses or any other viewing aid [16.7]. A parallax barrier is an opaque screen that has an array of strategically spaced fine slits. The parallax barrier is placed over a stereoscopic pair of pictures drawn on a plane and aligned such that a particular slit matches the line of sight of only one of the viewer’s eyes but not the other. The resolution of the picture, however, is halved and the viewer’s eyes have to be parallel to the axis of the slit array.

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In 1908, Gabriel Lippmann of France [16.8] proposed the lens sheet method, which was further developed by H. E. Ives (the son of F. E. Ives) of the USA in 1930. An array of cylindrical lenses, which is known as a lenticular sheet [16.9], is placed over an array of slits. The lenticular sheet slits are significantly wider than the parallax barrier slits. The basic principle of the lens sheet method is the same as that of the parallax barrier, but the lens sheet method has finer resolution, a wider field of view, and abundant light for viewing compared to the parallax barrier method. Lippmann also invented Integral Photography (IP), which uses an array of circular lenses. Unlike the lenticular sheet, IP does not restrict the orientation of the viewer’s eyes.

Gabriel Lippmann

H. E. Ives

In 1928, Edwin H. Land of the USA (founder of the Polaroid Corporation) invented the first synthetic sheet polarizer. This invention was significant because it

Edwin H. Land

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meant that the more bulky, costly polarizers used in systems such as Anderton’s magic lanterns could be replaced with inexpensive, lightweight synthetic polarizers. Thanks to Land’s pioneering work, synthetic thin film polarizers are used today in a myriad of applications ranging from polarizing sunglasses to liquid crystal displays. By their very nature, liquid crystal displays are convenient sources of polarized light, which make them attractive for polarization-based 3D imaging [16.11]. In 1940, Land introduced the vectrograph process for embedding orthogonally polarized images in a medium, which produce a 3D effect when viewed through glasses with orthogonal polarizer sheets. While the vectograph was originally designed for black and white, the process was later extended to color. The principle of the vectograph technique is similar to anaglyphs, except that it is based on polarization instead of complementary colors. In 1948, Dennis Gabor of Hungary conceived the idea of the hologram [8.13]. The hologram became the prima donna of the 3D imaging world, especially after the laser became available in the early 1960’s. Gabor won the Nobel Prize in 1971 for his invention.

Dennis Gabor

In 1967, Alan C. Traub invented the varifocal mirror [16.12, 16.13]. The mirror material is evaporated over a thin membrane to form a concave mirror. The membrane is then stretched over a cup. The radius of curvature is varied by controlling the degree of vacuum in the cup. An image is synthesized by relaying image components at varying focal lengths. In 1968, the white light hologram was invented by Stephen A. Benton while working at the Polaroid Corporation [8.11]. The white light hologram does not require a laser for viewing and the 3D image can be seen in ordinary light, which further increased the popularity of holograms. In 1968, Ivan Sutherland of the USA developed the head-mounted display (HMD) [16.14]. Stereoscopic pairs of moving pictures are displayed on a small liquid crystal screen directly over each eye. The viewer not only sees a 3D movie, but a wide field of view is possible by using a set of lenses.

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Ivan Sutherland Table 16.1 Classification of 3D imaging according to the method ⎛ C. Wheatstone, 1838 Looking through Images combined by mirrors ⎜ Images combined by prisms D. Brewster, 1849 eyepieces ⎜ ⎝ Images directed to the eyes I. Sutherland, 1968 via head-mounted displays Wearing glasses

⎛

No viewing aids

⎛

Colored glasses: anaglyphs ⎜ Projection of polarized light ⎜ ⎜ Polarizer glasses ⎜ ⎜ ⎝ Time sharing display method Horse blinder glasses Parallax barrier ⎜ Lenticular lens ⎜ ⎜ Integral photography ⎝ Holography Varifocal Mirror

W. Rollman, 1853, and J. D’ Almeida, 1858 J. Anderton, 1895 (US patent) E. Land, 1928 (first synthetic polarizer sheet) and 1940 (vectograph) JVC, 1985 K. Iizuka, 2006 F. E. Ives, 1903 G. Lippmann, 1908, and H. E. Ives, 1930 G. Lippmann, 1908 D. Gabor, 1948 A. Traub, 1968

Around 1985, JVC of Japan marketed a 3D display system employing a time sharing method. The left and right images are displayed alternately on the same monitor and the observer sees the respective views by wearing glasses whose left and right views are switched by liquid crystal optical switches. 3D imaging schemes classified by method are tabulated in Table 16.1. This book has already described many aspects of holography in Chapter 8 Holography, Chapter 9 Laboratory Procedures for Fabricating Holograms, and Chapter 12 Applications of Microwave Holography. Discussions of holography are not repeated in this chapter, but one should not forget that among the methods listed in Table 16.1, holography is one of the champions in 3D imaging. Holograms produce realistic 3D images without eye fatigue even for prolonged viewing times, as long as slight speckle patterns can be tolerated when a coherent light source is used.

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16.2 Physiological Factors Contributing to 3D Vision The basic factors that make the images three dimensional are summarized as follows [16.1–16.3]: The physiological factors are: (1) Parallax. Due to the separation of the left and right eyes, the two eyes see slightly different scenes. This is called parallax. This parallax assists the viewer in obtaining information about the location of the object. (2) Movement Parallax. A picture of scenery painted on a flat panel does not change even if the observer moves his or her head sideways. In the case of 3D objects, however, the side-to-side movement of the head creates changes in visibility or occlusion (some hidden portions of the object become visible and some visible portions become hidden) as well as differential movement speed (the foreground appears to move faster than the background). This is called movement parallax. Movement parallax can be experienced even with one eye. (3) Accommodation. A neurophysiological process adjusts the radius of curvature of the eye lens to focus the image onto the retina. Figure 16.1 is a cross section of the eyes viewed from above. The ciliary muscle squeezes the circumference of the eye lens so as to bulge the lens to focus to a closer object. The suspensory ligament flattens the lens so as to focus to a farther object. This control of the eye muscles is called accommodation. The brain acquires distance information from the amount of the required accommodation. (5) Convergence. Both eye balls are continually moving so that the image is made at the fovea of the respective eye where the density of the photoreceptor cells

Fig. 16.1 Cross section of the eyes viewed from the above

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(5)

(6) (7) (8)

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is the highest. Convergence refers to the fact that the angle of the visual axes subtended by an object to the fovea decreases with the distance to the object. (The visual axis does not exactly coincide with the optical axis of the crystalline lens. There is 1.5◦ to 5◦ discrepancy between them as shown in Fig. 16.1.) The amount of necessary convergence of the visual axis is manipulated by a combination of the medial rectus and lateral rectus muscles that control the orientation of the eye ball. The amount of convergence is used by the brain to interpret the distance to the object. The psychological factors are: Remembered geometry of the object and linear perspective. The observer’s memories about the shape and the size of the object play an important role. A picture properly drawn on a 2D canvas looks as if the objects in the picture are three dimensional. Artists scale closer objects larger than distant ones to create linear perspective, thus, making a picture on a 2D canvas look three dimensional. Occlusion. The occluded object is always located farther away. Shading. The addition of appropriate shading or relative brightness to fine art significantly augments the 3D effect. Resolution. Lower resolution or blurry objects are farther away. Figure 16.2 explains various 3D effects using examples.

Fig. 16.2 Explanation of various 3D effects

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16.3 How Parallax and Convergence Generate a 3D Effect Even though all of the above mentioned physiological and psychological factors are important, parallax and convergence effects are the two most effective means for achieving a 3D effect and are sufficient for most people to perceive a 3D effect. This chapter will focus on methods based on these two effects. Generating the 3D effect based on parallax and convergence can be categorized into two major types. One is the projection type, and the other, the interception type.

16.3.1 Projection Type Here is a simple explanation for how parallax combined with convergence generates a 3D effect. Figure 16.3a shows what a viewer’s left and right eyes see when a ball is in front of the viewer. The left eye sees the ball to its right, while the right eye sees the same object to its left. This is the parallax effect because the same object is seen differently by the left and right eyes. This is also the convergence effect because the angle of the visual axes subtended by an object to the fovea depends on the distance to the object. The viewer’s brain judges the distance to the ball using this information as well as the other factors mentioned above. Figure 16.3b is an attempt to fool the brain using pictures on a projection plane. Two pictures of the ball are drawn on the projection plane located behind the ball by extending the lines of sight onto the plane. The view seen from the left eye is projected on the right-hand side of the projection plane and the view seen from the right eye is projected on the left-hand side. As far as the light paths between the center-crossing point and both eyes are concerned, they are the same as the light path that would have been created by the actual ball. To the eyes, whether the light path

Fig. 16.3 a, b, c. Parallax effect. (a) Object. (b) Projected views of the object. (c) Transposed images

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started from the object or from the projection plane does not make any difference as long as the light looks as if it is starting from the object. The observer’s eyes perceive the object as if it were at the original location. Do these two ball pictures on the back plane alone in Fig. 16.3c give the viewer the illusion that the ball exists off the plane? The answer is emphatically “No!” The viewer sees a picture of two balls side by side simply because each eye sees both pictures of the ball on the projection plane as indicated by the solid-line traces and the dotted-line traces in Fig. 16.3c. In order to create the illusion that the ball exists off the plane, a way must be found to ensure that each eye is permitted to see only one picture of the ball as indicated by the crisscross solid-line traces while blocking the parallel dotted-line traces. Try to block only the dotted-line traces and pass the solid-line traces by extending your hands as illustrated in Fig. 16.4a. With the hands in the proper position, the picture becomes instantly three dimensional. If the dotted-line traces from the pictures in Fig. 16.4a are eliminated, the light paths shown in Fig. 16.4a become identical to the light paths shown in Fig. 16.3b and the illusion that the ball exists outside the plane is created. A hands-free way to accomplish this is by wearing a pair of horse blinder glasses made out of cardboard as shown in Fig. 16.4 b. The blind over each eye is opened just wide enough to block the dotted line trace but pass the solid line trace, so that each eye sees only the picture corresponding to the solid-line traces. The goal of achieving 3D displays by the parallax effect essentially boils down to finding ways to pass the crisscross path while blocking the parallel path.

Fig. 16.4 a, b. Methods for blocking the dotted line path. (a) Blocking with the hands. (b) Blocking with horse blinders

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16.3.1.1 Amount of Image Pop-Out The sensation that the ball pops out of the projection plane in Fig. 16.4 when the 3D effect is perceived will be referred to as the image “pop-out.” Figure 16.5 shows a simple experiment using a laptop computer for exploring the degree of image popout. Draw two red dots, one stationary at point a and the other movable by dragging the computer mouse at point b. Wear a pair of horse blinder glasses so that the crisscross path can be established while blocking the straight parallel path. When this is achieved, the red dot will be perceived as popping out of the screen to point c. By increasing the separation of the two dots a and b , the amount of pop-out increases.

16.3.1.2 Fabrication of a Stereoscopic Pair of Images This section explores various ways of fabricating pairs of stereoscopic images. It is important that the left and right views are transposed when they are displayed in projection-based methods. (a) Drawing the images freehand Figure. 16.6 shows an example of a pair of stereoscopic images drawn freehand of a pitcher throwing a baseball toward home plate. The positions of the home plate and the ball in the left figure are shifted toward the left with respect the pitcher, while those in the right figure are shifted toward the right. The amount of the desired pop-out can be controlled by the amount of the shift. (b) Taking photographs with a camera Establish a reference point, pick a scene, and frame the scene in the camera’s view finder. Step about 5 to 10 cm to the right of the reference point (5 cm if the scene is a close-up and 10 cm if the scene contains more distant objects) and take a snapshot. This snapshot is the picture viewed from the right, which will be placed on the left-hand side of the projection plane. Then, step about 5–10 cm to the left of the reference point and take another snapshot of the same

Fig. 16.5 Pop-out experiment

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Fig. 16.6 Creating a 3D cartoon from a 2D cartoon.

scene. This is the picture viewed from the left, which will be placed on the right hand-hand side of the projection plane. To minimize distortion, it is best not to rotate or tilt the camera when stepping to each side of the reference point, unless the distortion is desired for artistic effect. (c) Using a camera adapter for the real time display In the case of the real time displays such as a 3D camcorder, the left and right view images have to be fabricated and transposed in real time. A special adapter can be used as shown in Fig. 16.7, which is the top view of the adapter [16.10]. It has two mirrors and a 60◦ equilateral prism. The scene viewed from the right is drawn in green. The incident light is first reflected by the right mirror and directed toward the 60 ◦ equilateral prism. Total internal reflection takes place at the left side of the prism, and the light beam is finally funneled into the left half of the camcorder objective lens. In a similar fashion, the scene viewed from the left is drawn in red. The incident light is likewise is reflected by the left mirror and directed toward the prism. Total internal reflection takes place at the right side of the prism, and the light beam is finally funneled into the right half of the camcorder objective lens. The 60 ◦ equilateral prism serves two important purposes: the first is to direct the light paths so that the left and right scene images are transposed in real time, and the second is to prevent rays coming directly from the object from entering the camcorder objective. The direct rays bypassing the side mirrors, which are shown in yellow in Fig. 16.7, are deflected from the sides of the prism.

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Fig. 16.7 Ray path of the 3D camcorder

Fig. 16.8 Interception type of 3D display

16.3.2 Interception Type Interception types such as the one shown in Fig. 16.8 are also possible. There are important differences between the projection type and the interception type. In the interception type, the following points apply: (1) The 3D image is located at the intersection of the lines projected from the eyes toward the images on the screen, hence the wider the separation of the pair of the stereoscopic images, the deeper the generated 3D image is. (2) The view from the left eye should be placed on the left-hand side of the picture plane, and that from the right eye, on the right-hand side. There is no transpose of the image in the interception type. While the crisscross light path shown in

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green lines in Fig. 16.5 is prevented by the optical fence, the parallel light path is permitted to pass. (3) The entire constructed 3D image appears behind the interception screen. The projection method usually produces a more dramatic 3D effect simply because the image is pushed out of the screen toward the viewer. The interception plane is not necessarily between the viewer’s eyes and the 3D image. Often, the plane of the interception is placed almost in the same location as the 3D image plane. This placement lessens eye fatigue and improves binocular fusion because there is less displacement between the points of accommodation and convergence. The remainder of the discussions in this chapter is limited to the parallax effect and convergence, which are sufficient for the majority of people to see a 3D image. However, a small percentage of people have poor binocular fusion and cannot fuse the stereoscopic pairs of images as one image but see them as two images, and they cannot see the 3D effect from the parallax and convergence effects alone. The situation is analogous to color blindness in which some people cannot see certain colors.

16.4 Details of the Means Used for Realizing the 3D Effect Here, more detailed explanations of the various methods for creating 3D displays are provided.

16.4.1 Methods based on Polarized Light The light from the liquid crystal display of a laptop computer screen or camera phone is polarized, which makes it a convenient source of polarized light for polarization-based methods [16.10–11, 16.15–16]. Figure 16.9 shows how a liquid crystal display is used to create a 3D display. The polarization direction depends on the manufacturer. For the arrangement shown in Fig. 16.9, the light from the monitor is assumed to be vertically polarized. A cellophane sheet of the proper thickness (25 micron thick) has the property of rotating the direction of polarization of the light [16.11]. The right half of the monitor screen is covered with the cellophane sheet to rotate the direction of polarization by 90◦ . Thus, the image on the left half is polarized vertically, while that on the right half, horizontally. If the viewer wears a pair of orthogonally polarized glasses with a horizontally polarized eyepiece over the left eye and a vertically polarized eyepiece over the right eye, the viewer’s left eye can see only the right half of the monitor screen, and his right eye sees only the left half of the screen. Thus, the parallel paths represented by the dotted lines in Fig. 16.3c are blocked by the glasses, while the desired crisscross paths represented by the solid lines reach the viewer’s eyes.

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Fig. 16.9 Method by polarized light

Eye fatigue is a common concern with all 3D displays. It has been clinically proven that if the point of accommodation is located at a place different from that of the convergence, the viewing causes eye fatigue [16.2]. With the projection type, the point of convergence is at the intersection, which is located at a point between the projection plane and the eyes, whereas the point of accommodation is located on the projection plane. A set of prisms with a 3◦ deviation angle as shown in Fig. 16.9 translates the formed 3D image to the plane of the projection without distorting the image. The points of convergence and accommodation now coincide, thereby relieving eye fatigue. In addition to relieving eye fatigue, the set of prisms often aids binocular fusion.

16.4.2 3D Movies Producing 3D images by means of polarized light has been applied to 3D movies. As shown in Fig. 16.10, two movie projectors are used. One is for the picture in the vertical polarization, and the other for the horizontal polarization. Projected light is polarized by means of a polarizing beam splitter. The position of the projection screen almost coincides with the location of the 3D image. The audience wears polarizer glasses with one eyepiece polarized vertically and the other eyepiece polarized horizontally so that each eye sees only the picture of the specific polarization. The crisscross path is established as in the case of the liquid crystal displays. Aside from the requirement of wearing glasses, the movie provides a dramatic effect.

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Fig. 16.10 3D movie

Fig. 16.11 Wheatstone’s stereoscope

16.4.3 Wheatstone’s Stereoscope With Wheatstone’s stereoscope, views from the left and right eyes are posted on left and right vertical walls, and a set of vertical mirrors creates the superimposed stereoscopic images in the center as illustrated in Fig. 16.11. It is important to notice that the mirrors upset the orientation of the left and right images. The pictures on the

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walls look as if one is a 180◦ rotation of the other. The pictures have to be oriented so that the images Lˆ and Rˆ reflected by the 45◦ mirror are orientated properly. Observe the movement of a particular point of the image by the mirrors. As a specific example, consider the point where the color bands of the ball converge, which will be referred to as the “navel” of the ball. The navel of the right image R is denoted by < r , and the corresponding navel of the left image L is denoted by < l. The combined image in the center reveals exactly how these two particular points move. The points represented by r and l are connected to the respective eyes as shown by the red lines in Fig. 16.11. The point r and point l are converged to make a pop-out image at point c, which is lifted above the plane of the pictures.

16.4.4 Brewster’s Stereoscope Brewster replaced the cumbersome mirrors in Wheatstone’s stereoscope with a set of prisms, and Brewster’s stereoscope has become very popular. As shown in Fig. 16.12, Brewster’s stereoscope uses a pair of prisms to shift a side-by-side stereoscopic pair of images toward the center where the two images meet. The picture viewed from the left, which is labeled L, is placed on the left. Similarly, the picture viewed from the right, which is labeled as R, is placed on the right.

Fig. 16.12 Brewster’s stereoscope

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The plane of the projection for this method almost coincides with the 3D image plane. This arrangement belongs to the interception-type 3D display. In a manner similar to the illustration of Wheatstone’s stereoscope, the red line connecting point r to the right eye and the red line connecting point l to the left eye form the crisscross light path making a pop-out image at point c. The merits of Brewster’s method are not only simplicity of construction, but also a lesser degree of eye fatigue because the position of convergence is situated close to that of the accommodation of the eyes, as elaborated earlier in Sect. (16.4.1).

16.4.5 Anaglyph The anaglyph uses complementary colors to separate the views from the left and right. The dual-color anaglyph uses only two of the three primary colors red, green, and blue; and in the tri-color anaglyph, all three primary colors are used.

16.4.5.1 Dual-Color Anaglyph For example, if the left and right views are converted into a red and green colored stereoscopic pair of images, then the image is viewed by wearing a pair of glasses with red and green colored filters. Figure 16.13 shows a schematic of an anaglyph. Focus your attention on the viewer’s right eye, which is covered with a red filter. When green light from the green stereoscopic image is incident onto this eye, the red filter attenuates the green light and green light does not reach the eye. As a result, the green lines appear as dark lines to the viewer. On the other hand, when the red light from red stereoscopic image is incident to the red filter it certainly passes into the eye as red lines, but at the same time, the white light coming from the white background space of the stereogram also becomes red after passing through the red filter. The red lines are no longer distinguishable from the background. The red lines disappear into the background. Thus, the right eye covered with the red filter can recognize only the green image, which represents the view from the right eye, as a dark line-sketch. In a similar manner, the left eye covered with the green filter can recognize only the red lines, which represent the view from the left eye, as a dark line-sketch. In the figure, the navel of the ball viewed from the left is denoted by l and is connected to the left eye of the viewer. The same point viewed from the right is denoted by r and is connected the right eye. As with both Brewster’s and Wheatstone’s stereogram, a pop-out image of the navel is formed at point c.

16.4.5.2 Tri-Color Anaglyph If the intersections in the drawing in Fig. 16.14a are left as red, they will appear black when viewed through green filter, but will disappear for the red filter. As

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Fig. 16.13 Anaglyph

Fig. 16.14 a, b. Comparison between dualcolor and tri-color anaglyphs. (a) Dual-color. (b) Tri-color

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Fig. 16.15 a, b. Simple tri-color anaglyphs drawn freehand. (a) Chair. (b) Face

shown in Fig. 16.14b, the 3D perception can be improved by coloring the intersections of the red and green lines with a third color, blue, that appears dark when viewed through both the red and green filters [16.17]. The difference between the dual-color anaglyph and tri-color anaglyph becomes increasingly noticeable as the size of the Figure is magnified. Figure 16.15a shows a very simple tri-color anaglyph. It was drawn freehand with three colored pencils. The chair viewed from the right is drawn in green to the left, and the chair viewed from the left is drawn in red to the right. The intersections of the two colored pictures are drawn in blue. By wearing a pair of colored glasses as shown in Fig. 16.13, the picture of the chair looks as if it is standing on the page of the book. Figure 16.15b shows a cartoon similarly drawn by three colored pencils, but the nose was shifted. The position of the green nose was shifted to the left and the red nose to the right. The intersections are again drawn with blue pencil. Thus, when viewed with red and green 3D glasses, only the nose will pop out. The merit of the anaglyph is that, like Brewster’s method, the locations of convergence and accommodation of the eye coincide resulting in much less eye fatigue. The demerit is the loss of color information of the scene.

16.4.5.3 Vectographs The principle of the vectograph invented by Edwin H. Land has a lot in common with anaglyphs [16.18]. The difference is that anaglyphs are based on complementary colors whereas the vectograph is polarization based. Figure 16.16 shows the principle of operation of the vectograph. The left view is drawn by the vertically-polarized polarizer molecules while the right view is drawn

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Fig. 16.16 Principle of the vectograph

by the horizontally-polarized polarizer molecules on a transparent film. The film is backlit with ordinary white light. The 3D image is viewed by wearing a pair of orthogonally polarized glasses. The viewer’s right eye is covered with the vertically polarized eyepiece. When the horizontally polarized light from the image made of horizontally-polarized polarizer molecules is incident onto this eye, the vertically polarized eyepiece attenuates the horizontally polarized light and this light does not reach the eye. As a result, the horizontally polarized image appears as a dark line image to the viewer. On the other hand, when the vertically polarized light from the image made of vertically-polarized polarizer molecules is incident onto this eye, the vertically polarized eyepiece certainly allows this light to pass into the eye. But at the same time, white light coming from the white background space of the stereogram where there are no polarizer molecules also passes into the eye

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and the vertically polarized image is no longer distinguishable from the background and disappears into the background. Thus, the right eye covered with the vertically polarized eyepiece can recognize only the horizontally polarized image, which represents the view from the right eye, as a dark line-sketch. In a similar manner, the left eye covered with the horizontally polarized eyepiece can recognize only the vertically polarized image, which represents the view from the left eye, as a dark line-sketch. Where the two images overlap, the directions of the light polarization are orthogonal, and the back light can no longer pass through the film. This, however, does not cause any harm but does good. Recall that the overlapped intersections have to be seen as dark for both eyes in the tri-color anaglyph. In the case of the vectograph, light is not transmitted from these intersections and will appear dark to both eyes of the observer. Fabrication of the vectograph makes use of a polarized filter sheet in which exposed areas become more strongly polarized in response to a photographic image projected on it. When the sheet is processed, it appears relatively clear in normal light, but an image can be seen when viewed through a correctly oriented polarizer. In the vectograph film, the orthogonally polarized left and right stereographic images are placed on opposite sides of the same transparent base. Vectographs had limited commercial success because of the intricate and painstaking fabrication process, but they produced excellent quality 3D displays when viewed through polarizer glasses. The StereoJet invented by Julius Scarpetti [16.19] is a modern version of the vectograph that replaces the complicated dye transfer process of the vectograph with a digital printing process. The StereoJet uses polarizable inks on special transparency film.

16.4.6 Time Sharing Method In the time sharing method shown in Fig. 16.17, the left and right pictures each occupy the entire space of the monitor, but they are displayed alternately in time. The liquid crystal optical switch on the viewer’s glasses turns on and off to separate the two pictures. When the monitor is showing the left view, the right eyepiece of the glasses shuts off. Similarly, when the monitor is showing the right view, the left eyepiece shuts off. A typical switching cycle is 120 Hertz in order to avoid the flickering of the image. This method has the advantage of using the whole screen at the cost of the complexity of the scheme and the necessity of wearing the viewing glasses.

16.4.7 Head Mounted Display (HMD) Figure 16.18 shows a sketch of a head-mounted display (HMD). Two liquid crystal displays for each eye are installed onto the goggles so that the stereoscopic pictures

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Fig. 16.17 Time sharing method

Fig. 16.18 Head mounted display

are directly displayed to each eye. A set of lenses are added to create the illusion of viewing a wide scene [16.14]. The electrical signal from the head tracker changes the display in accordance with the movement of the head. The primary applications of the HMD are for various kinds of simulators such as those used for aviation, medical operations, and military maneuvers. It is also used for architectural displays and entertainment in general. The demerit is that prolonged use may create strain on the head and neck muscles. All the methods mentioned so far require the viewer to wear glasses. The following methods do not require glasses.

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16.4.8 Volumetric Methods Volumetric methods are 3D display methods based on the superposition of images displayed at different distances. They are called volumetric 3D displays simply because the devices are capable of emitting, scattering or reflecting light at any point in a volume [16.5].

16.4.8.1 Longitudinal Translation of the Reflector Plane As shown in Fig. 16.19a, the location of a vertical reflector plane is mechanically translated in the longitudinal direction with a repetition time much faster than response time of normal eye vision [16.2]. The image is drawn by a fine intensity modulated light beam steered by an x-y light scanner onto the reflector plane. With the appropriate synchronization among the light intensity modulator, the x-y scanner controller, and the z direction stage movement controller, a steady 3D image is displayed. An inherent drawback is the bulky mechanical system.

Fig. 16.19 a, b. Methods of moving the reflector screen (a) Translation. (b) Rotation

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16.4.8.2 Spinning Screen Method As shown in Fig. 16.19b the reflector screen is spun around a shaft. The x-y light scanner and the light intensity modulator draw the image on the rotating reflector screen. The reflector screen is enclosed inside a transparent dome. Air pressure inside the dome is lowered in order to avoid air friction and noise. A computer controls the timing of the overall system. The rotating scheme has the advantage of easier movement of the reflector screen and a much wider viewing angle. The drawbacks are the cumbersome mechanical system and the dead zone region near the shaft where the depth of the image cannot be changed.

16.4.8.3 3D Image Out of Series of Cross-Sectional Images A series of sliced images are superimposed using mirrors and half mirrors. As shown in Fig. 16.20a, a series of image slices was generated by switching each pixel of the output of the color camera in accordance with the distance information obtained from an Axi-Vision Camera (depth mapping camera) [16.20]. As shown in Fig. 16.20b, the monitors are arranged so that the distances to the observer are increased with a constant increment d. Seven half mirrors and four additional surface reflecting mirrors are used to superimpose the images. The four additional mirrors are necessary so that the left and right parity of all images are kept the same [16.21]. Fig. 16.20c shows the results of the 3D image. The object is a wedge whose two sides are colored red and green. When the observer’s head moves left and right, the 3D scene changes accordingly.

16.4.9 Varifocal Mirror A 3D display produced by means of a varifocal mirror is shown in Fig. 16.21. A varifocal mirror is made by stretching a thin aluminized Mylar film over a loud speaker [16.13]. The radius of curvature of the Mylar film is varied by changing the air pressure behind the film with the movement of the speaker cone. The aluminized Mylar film plays the role of a variable focus concave mirror. Thus, the size and position of the virtual image formed by the varifocal mirror is electronically controllable. With the arrangement shown in Fig. 16.21, a picture of an arrow AB is displayed on the cathode ray tube. At the instant that the mirror surface is flat as indicated by P0Q, the virtual image A B of the arrow is made at the location of the mirror symmetry of AB. When the surface of the mirror becomes concave as indicated by PRQ, the location of the virtual image A B moves further away from the observer. The shorter the focal length of the mirror surface is made, the further the location of the virtual image moves away from the observer. If the arrow is replaced by a series of two-dimensional sectional pictures of the object, and the focus is scanned faster than the response time of the eye, a threedimensional image can be viewed. It should be cautioned that the size of the virtual

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Fig. 16.20 a, b, c. 3D image constructed from a series of cross-sectional images. (a) Pixel switch controlled by the distance information. (b) Arrangement of the monitors. (c) 3D image made out of cross sections.

image A B , which is located further from the observer than A B , is larger in size than A B . The proper size relationship is not preserved. It has to be compensated for by a computer or by some other means when it is utilized for generating a 3D display. Fig. 16.22 shows an example of a 3D display using the varifocal mirror [16.13].

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Fig. 16.21 Stereoscopic display using a varifocal mirror

Fig. 16.22 a, b. 3D images using the varifocal mirror method. (a) Object. (b) Photograph of the 3D image [16.13]

16.4 Details of the Means Used for Realizing the 3D Effect

485

16.4.10 Parallax Barrier The stereogram for the parallax barrier method is made by interspersing the left and right views [16.1–2,7]. The left and right views are sliced into columns. Only even numbered columns in the left view and only odd numbered columns in the right view are used and the rest of the columns are discarded as shown in Fig. 16.23a. The parallax barrier stereogram is fabricated from the columns of the two views by interspersing in an orderly manner into one sheet as indicated in Fig. 16.23b. A parallax barrier sheet is used for separating the left and right views in the parallax barrier stereogram. The parallax barrier is an array of narrow slits on an opaque sheet. As shown in Fig. 16.23c, in the vicinity of any slit, there is always a small region that only the right eye ray can reach (right column) and a small region that only the left eye ray can reach (left column). In other words, the same small region cannot be seen by both eyes. The light ray indicated by the thick red line from a small region just below letter l can reach the left eye, but the light ray indicated by the short red dotted line from the same small region cannot reach the right eye because it is blocked by the barrier.

Fig. 16.23 a, b, c. Parallax barrier stereogram. (a) Sampled view. (b) Front view. (c) Top view

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The even numbered columns can be seen only by the left eye and the odd numbered columns can be seen only by the right eye. In other words, a particular column that matches the line of sight of one of the viewer’s eyes does not match the line of sight of the viewer’s other eye. The approximate point of the navel r of the ball is connected by the heavier green line in Fig. 16.23c to the right eye, and the corresponding point l is connected by the heavier red line to the left eye. The intersection c represents the pop-out point of the ball as in the cases discussed earlier. Next, the size of the slits for the masking barrier will be calculated. The proper dimensions of the barrier are important for obtaining a satisfactory result. All lines extended from a column of the stereogram through the aperture of the barrier have to be converged to the respective eyes. The only sighting device is the barrier apertures. Unless each aperture properly aims at the viewer’s respective eye, the system certainly does not work. The separation S between the barrier and the stereogram for a given column size of the stereogram will be calculated. In Fig. 16.24, notice that the two triangles bordered by the heavy black line are connected tip to tip. The goal of the calculation is to determine how far to place the barrier from the stereogram so that the extended lines enter the eyes. The eyes are separated by a distance E, where E is 6.5 cm for an average human being. Using the similarity of the two triangles, D−S S = P E

(16.1)

The required S is calculated as S=

D 1+

E P

(16.2)

Next, the aperture size Q of the barrier is calculated. A small error in the aperture size accumulates as the edge of the stereogram is approached from the center. The

Fig. 16.24 Determining the location and aperture size of the barrier

16.4 Details of the Means Used for Realizing the 3D Effect

487

aperture size Q is calculated using the proportionality associated with the red-shaded triangle in Fig. 16.24. P Q = (16.3) D D−S Solving (16.3) for Q gives S (16.4) Q = P 1− D The value of Q has to be smaller than P, but not by very much. For instance, with P = 0.25 cm, D = 30 cm, the result is Q = 0.24 cm which is 96% of P. Even though this method has the merit of not requiring glasses, it has the following drawbacks. Not only does the resolution of the 3D image become one half of the original, but also the brightness of the image is reduced to one half. Backlit illumination such as that from a computer monitor is preferable. An additional drawback is that the tolerance in positioning the viewer’s eyes relative to the barrier sheet and stereogram is rather limited.

16.4.11 Horse Blinder Barrier Method The horse blinder barrier method uses slanted blades. On top of the stereoscopic pair of images, two sets of slanted blinds are installed. Each set of blinds are slanted so that each eye can see only the appropriate image of the stereogram as shown in Fig. 16.25. In this scheme, the screen effectively wears the horse blinder glasses instead of the viewer. An advantage of this method is its simplicity. The size and spacing of the blades are not as critical as those of the parallax barrier and can be adjusted as the 3D image is viewed.

Fig. 16.25 Horse blinder barrier

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16 3D Imaging

16.4.12 Lenticular Sheet Method The lenticular sheet method evolved from the parallax barrier method [16.9, 16.22– 23] as shown in Fig. 16.26. The parallax barrier method uses the line of sight of the barrier sheet to separate the left and right views so that the observer’s eyes see only the respective views of the stereogram. In the lenticular method, the slits of the barrier sheet are filled by convex cylindrical lenses as shown in Fig. 16.26a. With the lenticular sheet, the directivity of the light converged by the convex lenses is used to direct the path of light. The lenses deliver the images indicated by the red stripes only to the left eye but not to the right eye. Since this directivity is preserved regardless of the lens width, the lens width is expanded to the fullest extent in order to increase the brightness of the 3D image, and the lenses are connected as a sheet as shown in Fig. 16.26b. Such an array of cylindrical lenses is known as a lenticular sheet. Beside the advantage of the improved brightness of the image, multiplexing of the images is possible with the lenticular sheet method. In the columns of the ordinary parallax barrier method shown in Fig. 16.23, the left and right views themselves were directly copied from the original pictures, and there is no compression of the pictures. With the lenticular method, the left and right views are first reduced to smaller images and the observer sees the images magnified by the convex lens in the lenticular sheet. In the new spaces made available by reducing the size of the images as shown in Fig. 16.26a, more than one image can be imbedded in each column. As shown in Fig. 16.26c, a new set of stereoscopic images of the scenery viewed from a slightly different angle is imbedded in each column. When the observer moves his or her head to the left in Fig. 16.26c, a new set of stereographic images drawn in yellow and blue lines are seen. Even though this example has only one additional set of stereoscopic images, an even greater number

Fig. 16.26 a, b, c. Evolution of the parallax barrier method into the lenticular sheet method. (a) Parallax barrier covered with cylindrical lenses. (b) Lenticular sheet with one image. (c) Lenticular sheet with two images

16.4 Details of the Means Used for Realizing the 3D Effect

489

of sets can be imbedded as long as the space is available. With the parallax barrier method, the image does not change to reveal hidden objects when the observer’s head moves, resulting in an unnatural look to the 3D image. The 3D image of the lenticular method helps to correct these deficits by imbedding more stereoscopic images of scenes taken at different angles. The advantages of the lenticular sheet method are not only that the viewer sees a 3D image without wearing glasses but also that the 3D image appears more natural looking and brighter. The demerits are not only the complexity of fabricating the stereogram, but also the orientation of the observer’s eyes has to be horizontal (parallel to the array axis of the lenticular lenses).

16.4.13 Integral Photography (IP) The scheme of 3D imaging by integral photography (IP) has many points in common with the method of the lenticular sheet. The lenticular system uses a sheet of cylindrical lenses but integral photography uses an array of fly’s eye lenslets [16.24–28].

16.4.13.1 IP Imagelet Formation and Image Reconstruction Integral photography involves two-step processing of the imagelets. As the first step, the prime IP stereogram is taken using a fly’s eye lens sheet as shown in Fig. 16.26a. This primary IP stereogram is an array of the real images of the object that differ from the adjacent imagelet by a perspective shift of the lenslet diameter. Note that each real image is inverted. In the second step, each imagelet of the primary IP stereogram is individually inverted by a 180◦ rotation in the plane of the paper in order to make the secondary IP stereogram. The necessity of individual inversion of the imagelet rather than whole inversion of the primary IP stereogram will be explained in the next section. The same fly’s eye lens sheet is placed in front of the secondary IP stereogram as shown in Fig. 16.27b to view the 3D image. How do such imagelets, which superficially look quite redundant, converge into one big 3D image? To answer this question, consider the ray paths of the zeroth and third imagelets of the particular row of the IP stereogram shown in Fig. 16.28a. This figure demonstrates how the virtual images of these two imagelets grow into one big 3D image as the distance between the stereogram and the lens sheet is increased. Note that not only does the distance x to the virtual images increase, but also the size of the virtual images expands. As the size of the images expands, the adjacent images start overlapping each other. The correct amount of overlap can be determined by observing the overlap of a particular point in both images. For instance, in the case of Fig. 16.28a, the images of the eye in the two imagelets overlap each other. This is the desired position x = b of the virtual image. Fig. 16.28b shows the case when the virtual imagelets of one

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16 3D Imaging

Fig. 16.27 a, b. Integral Photography. (a) Fabrication of the primary IP stereogram. (b) Construction of the 3D image from the secondary IP stereogram

Fig. 16.28 a, b. Small imagelets forming into one big 3D image. (a) Two of the imagelets. (b) All of the imagelets

entire row are amalgamated. The merits of the integral photograph are practically the same as those of the lenticular sheet, but the most significant advantage is that the 3D image can be viewed without the restriction of the horizontal orientation of eyes.

16.4 Details of the Means Used for Realizing the 3D Effect

491

16.4.13.2 IP Pseudoscopic Image The necessity of individual inversion of the imagelet of the primary stereogram in the process of making the secondary IP stereogram will be explained. Without the individual inversion, the reconstructed 3D image becomes a pseudoscopic image, which is an image whose depth information is reversed. Fig. 16.29 compares two secondary IP stereograms made by flipping the IP stereogram in different ways. The top center figure in Fig. 16.29 is the primary IP stereogram, which is the array of the real images of the object obtained by using the fly’s eye lenslets in the first step. Fig. 16.29 a is the secondary IP stereogram made by just rotating the whole primary IP stereogram 180◦ in the plane of the paper. Fig. 16.29b is the secondary IP stereogram made by rotating each individual imagelet of the primary IP stereogram by 180◦ in the plane of the paper. There is a difference between (a) and (b). While the boys are facing away from the center of the IP in (a), the boys are facing toward the center of the IP in (b). In order to clearly demonstrate the difference in the depth information of the reconstructed 3D image from these secondary IP stereograms, the plan views of the boy’s head are used in Fig. 16.30. Only one pair of images is considered for simplicity. The two boys are facing away from the center in Fig. 16.30a representing Fig. 16.29a, while the boys are facing toward the center in Fig. 16.30b representing Fig 29b. Each lens delivers the image to the respective eye. The depth information of the reconstructed 3D image in Fig. 16.30a is reversed from what is expected. Such an image is categorized as a pseudoscopic image. The depth information of the reconstructed image in Fig. 16.30b however is in the correct orientation and such an image is categorized as an orthoscopic image. Thus, unless the imagelets of the array of real images obtained in the first step are individually

Fig. 16.29 a, b. Comparison between two ways of flipping the IP stereogram. (a) Primary IP stereogram is rotated by 180◦ as a whole unit. (b) Imagelets of the primary IP stereogram are rotated by 180◦ individually

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16 3D Imaging

Fig. 16.30 a, b. IP 3D images. (a) Pseudoscopic image. (b) Orthoscopic image

rotated to make the secondary IP stereogram, the reconstructed 3D image becomes a distance reversed pseudoscopic image rather than a desired orthoscopic image.

16.5 Concluding Remarks The era of High Definition Television (HDTV) has come, and it is the author’s belief that the era of 3D television is not far behind. The major principles used for generating the 3D effects have been presented in this chapter together with simple explanations and a glimpse into the many applications of this field. It is the author’s hope that the ideas presented in this chapter and throughout the book have sparked the reader’s imagination and inspired a desire to take on the reins of the optics of the future. It is with this spirit that this book was written.

Problems 16.1. The center of Fig. 16.31 shows sketches of a pencil viewed from the left and from the right. The sketches are used as a stereoscopic pair of images in a projectiontype 3D imaging system. What kind of 3D image will the viewer see if the sketches are not transposed? Hint Make pop-up’s of both tip and bottom of the pencil by drawing crisscross lines. 16.2. The range of vibration of the varifocal mirror is extended as shown in Fig. 16.32. What effect does this have on the range of image displacement compared to the varifocal mirror in Fig. 16.21?

Problems

493

Fig. 16.31 A non-transposed stereoscopic pair of pencil images in a projection-type 3D imaging system

Fig. 16.32 Varifocal mirror with an extended range of vibration

16.3. (a) With the parallax barrier method, the annoyance due to the barrier decreases as the pitch of the barrier is reduced. Referring to the parameters shown in Fig. 16.33, the condition that the pitch of the barrier becomes unnoticeable [16.1] is Pitch = 2Q D/3500 (in mm) (16.5) Calculate the values of S and P/Q that satisfy this condition for D = 1 meter and E = 6.5 cm. (b) As the size of slits Q is decreased, the diffraction effect of the slits becomes prominent. The diffraction effect not only harms the quality of the image, but it

494

16 3D Imaging

Fig. 16.33 Parallax barrier method

Fig. 16.34 Calculating the condition for seeing a single amalgamated 3D image from the secondary IP stereogram

Problems

495

also has a detrimental effect on the ability of the barrier to block the unwanted views due to a spreading of the viewing angles subtended by the viewer’s eyes. Assume that the diffraction pattern of a slit in the barrier is similar to that shown in Fig. 4.2b in Chapter 4. What is the value of slit size Q such that the zero-tozero width of the main beam of the diffraction is less than 10−2 mm for a light wavelength λ = 555 nm and D = 1 meter? 16.4. For the given parameters in Fig. 16.34, determine the spacing a between the secondary IP stereogram and the IP lens sheet in order to reconstruct one big 3D image from the secondary IP stereogram. The spacing between the 0th and 3rd imagelets is 67 mm. The left eye of the 0th imagelet is shifted by 9 mm to the right from the center line of the 0th imagelet. The same left eye of the 3rd imagelet is shifted by 7 mm to the left of the center line of the 3rd imagelet. The focal length of the lens sheet is 25.4 mm. Hint Both the 0th and 3rd imagelets are magnified by the same amount. When properly magnified, the magnification factor x should fill the 67 mm space and 9x + 7x = 67.

References

Chapter 1 1.1 R. J. Unstead: Egypt and Mesopotamia (Adam and Charles Black, London 1977) 1.2 S. N. Kramer: From the Tablets of Sumer (The Falcon’s Wing Press, Indian Hills, CO 1956) 1.3 S. N. Kramer: History Begins at Sumer (The University of Pennsylvania Press, Philadelphia 1981) 1.4 C. G. Fraser: Half-hours with Great Scientists; The Story of Physics (The University of Toronto Press, Toronto 1948) 1.5 L. Gorelick, A. J. Gwinnett: “Close Work without Magnifying Lenses?”, Expedition 23, No. 2, pp. 27–34 (Winter 1981) 1.6 T. Whittle: The World of Classical Greece (William Heinemann, London 1971) 1.7 G. Gamow: Biography of Physics (Hutchinson, London 1962) 1.8 V. Ronchi: The Nature of Light, an Historical Survey, translated into English by V. Barocas (William Heinemann, London 1970) 1.9 K. Iizuka, H. Ogura, J. L. Yen, V. K. Nguyen, J. Weedmark: A hologram matrix radar, Proc. IEEE 64, 1493–1504 (1976) 1.10 D. C. Lindberg: Theories of Vision from Al-Kindi to Kepler (The University of Chicago Press, Chicago 1976) 1.11 A. I. Sabra: Theories of Light from Descartes to Newton (Cambridge University Press, Cambridge 1967) 1.12 M. Born, E. Wolf: Principles of Optics (Pergamon, Oxford 1964) 1.13 News column, Nobel Physics Prize to Bloembergen, Schawlow and Siegbahn, Physics Today, pp. 17–20 (Dec. 1981)

497

498

References

Chapter 2 2.1 C. L. Andrews: Optics of the Electromagnetic Spectrum (Prentice-Hall, Englewood Cliffs, NJ 1960) 2.2 J. W. Goodman: Introduction to Fourier Optics (McGraw-Hill, New York 1968) 2.3 R. Bracewell: The Fourier Transform and its Applications (McGraw-Hill, New York 1965) 2.4 M. L. Boas: Mathematical Methods in the Physical Sciences (Wiley, New York 1966) 2.5 A. Papoulis: Systems and Transforms with Applications in Optics (McGrawHill, New York 1968)

Additional Reading Barakat, R.: In The Computer in Optical Research, ed. by R. Frieden, Topics Appl. Phys., Vol. 41 (Springer, Berlin, Heidelberg 1980) Churchill, R. V.: Operational Mathematics (McGraw-Hill, New York 1958) Erdelyi, A., Magnus, W., Oberhattinger, F., Tricomi, F.: Tables of Integral Transforms, Vol. 1 and 2 (McGraw-Hill, New York 1954) Lighthill, M. J.: Introduction to Fourier Analysis and Generalized Functions (Cambridge University Press, Cambridge, London 1958) Sneddon, I. N.: Fourier Transforms (McGraw-Hill, New York 1951)

Chapter 3 3.1 J. A. Stratton: Electromagnetic Theory (McGraw-Hill, New York 1941) 3.2 J. W. Goodman: Introduction to Fourier Optics (McGraw-Hill, New York 1968) 3.3 A. Papoulis: Systems and Transforms with Applications in Optics (McGrawHill, New York 1968) 3.4 A. Sommerfeld: Partial Differential Equation in Physics (Academic, New York 1949) 3.5 D. C. Stinson: Intermediate Mathematics of Electromagnetics (Prentice-Hall, Englewood Cliffs, NJ 1976)

Additional Reading Babiˇc, V. M., Kirpiˇcnikova, N. Y.: The Boundary-Layer Method in Diffraction Problems, Springer Ser. Electrophys., Vol. 3 (Springer, Berlin, Heidelberg 1979)

References

499

Born, M., Wolf, E.: Principles of Optics (Pergamon, New York 1964) Economou, E. N.: Green’s Functions in Quantum Physics, 2nd ed., Springer Ser. Solid-State Sci., Vol. 7 (Springer, Berlin, Heidelberg 1983) Fowles, G. R.: Introduction to Modern Optics (Holt Rinehart and Winston, New York 1975) Lotsch, H. K. V.: Multiple-Diffraction Formula, Optik 25, 598–605 (1967) Sommerfeld, A.: Optics (Academic, New York 1954) Stone, J. M.: Radiation and Optics (McGraw-Hill, New York 1963) Young M.: Optics and Lasers, 2nd ed. Springer Ser. Opt. Sci., Vol. 5 (Springer, Berlin, Heidelberg 1984)

Chapter 4 4.1 M. Rousseau, J. P. Mathieu, J. W. Blaker: Problems in Optics (Pergamon, New York 1973) 4.2 A. Papoulis: Systems and Transforms with Applications in Optics (McGrawHill, New York 1968) 4.3 E. Hecht, A. Zajac: Optics (Addison-Wesley, Reading, MA 1974) 4.4 S. Miyaoka: Digital audio compact and rugged, IEEE Spectrum, 35–39, (March 1984) 4.5 D. J. Broer, L. Vriens: Laser-induced optical recording in thin films, Appl. Phys. A 32, 107–123 (1983) 4.6 A. Engel, J. Steffen, G. Herziger: Laser machining with modulated zone plates, Appl. Opt. 13, 269–273 (1974)

Additional Reading Bracewell, R.: The Fourier Transform and its Applications (McGraw-Hill, New York 1965) Feynman, R. P., Leighton, R. B., Sands, M.: The Feynman Lectures on Physics Vol. I (Addison-Wesley, Reading, MA 1964) Goodman, J. W.: Introduction to Fourier Optics (McGraw-Hill, New York 1968) Nussbaum, A., Phillips, R. A.: Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, NJ 1975) Petit, R. (ed): Electromagnetic Theory of Gratings, Topics Current Phys., Vol. 22 (Springer, Berlin, Heidelberg 1980)

Chapter 5 5.1 E. Kreyszig: Advanced Engineering Mathematics (Wiley, New York 1972) 5.2 A. A. Khinchin: A Course of Mathematical Analysis (translated from Russian) (Hindustan Publishing, Delhi 1960)

500

References

5.3 M. Kline, I. W. Kay: Electromagnetic Theory and Geometrical Optics (Interscience, New York 1965) 5.4 R. K. Luneburg: Mathematical Theory of Optics (University of California Press, Berkeley 1964) 5.5 K. Maeda, I. Kimura: Electromagnetic Theory (Ohm-Sha, Tokyo 1970) 5.6 A. Sommerfeld: Optics (Academic, New York 1954) 5.7 G. Petit Bois: Table of Indefinite Integrals (Dover, New York 1961) 5.8 L. Jacomme: A model for ray propagation in a multimode graded index-fiber, Optics Commun. 14, 134–138 (1975) 5.9 S. Shimada: Optical Fiber Communication Techniques (Ohm-Sha, Tokyo 1980) 5.10 M. Ikeda: Propagation characteristics of multimode fibers with graded core index, IEEE J. QE-10, 362–371 (1974) 5.11 T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura: Optical characteristics of a light-focusing fiber guide and its applications, IEEE J. QE-6, 606–612 (1970) 5.12 S. E. Miller, E. A. J. Marcatili, T. Li: Research toward optical-fiber transmission systems, Proc. IEEE 61, 1703–1751 (1973) 5.13 H. K. V. Lotsch: Physical-optics theory of planar dielectric waveguides, Optik 27, 239–254 (1968) 5.14 H. K. V. Lotsch: Beam displacement at total reflection: the Goos-H¨anchen effect, Optik 32, 116–137, 189–204, 299–319 and 553–569 (1970/71)

Additional Reading Adams, M. J.: An Introduction to Optical Waveguides (Wiley, New York 1981) Born, M., Wolf, E.: Principles of Optics (Pergamon, New York 1964) Burrus, C. A., Standley, R. D.: Viewing refractive-index profiles and small-scale inhomogeneities in glass optical fibers: some techniques, Appl. Opt. 13, 2365– 2369 (1974) Gloge, D.: Optical fibers for communication, Appl. Opt. 13, 249–254 (1974) Kaiser, P., Marcatili, E. A. J., Miller, S. E.: A new optical fiber, Bell Syst. Tech. J. 52, 265–269 (1973) Kumagai, N., Sawa, S.: Propagation of TE01 mode in a lens-like medium, J. Inst. Electronics and Commun. Eng. Jpn. 54, 39–44 (1971) Marcuse, D.: Light Transmission Optics (Van Nostrand Reinhold, New York 1972) Sharma, A. B., Halme, S. J., Butusov, M. M.: Optical Fiber Systems and Their Components, Springer Ser. Opt. Sci., Vol. 24 (Springer, Berlin, Heidelberg 1981) Synge, J. L.: Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge University Press, London 1937)

References

501

Chapter 6 6.1 J. W. Goodman: Introduction to Fourier Optics (McGraw Hill, New York 1968) 6.2 A. Nussbaum, R. A. Phillips: Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, NJ 1976) 6.3 S. H. Lee (ed.): Optical Information Processing, Fundamentals, Topics Appl. Phys., Vol. 48 (Springer, Berlin, Heidelberg 1981) Chap. 1 6.4 L. P. Yaroslovsky: Digital Picture Processing, Springer Ser. Info. Sci., Vol. 9 (Springer, Berlin, Heidelberg, in preparation) 6.5 A. Papoulis: Systems and Transforms with Applications in Optics (McGraw Hill, New York 1968)

Additional Reading Casasent, D. (ed.): Optical Data Processing, Applications, Topics Appl. Phys., Vol. 23 (Springer, Berlin, Heidelberg 1978) Frieden, B. R.: Probability, Statistical Optics and Data Analysis, Springer Ser. Infor. Sci., Vol. 10 (Springer, Berlin, Heidelberg 1982) Hecht, E., Zajac, A.: Optics (Addison-Wesley, Reading, MA 1974) Huang, T. S. (ed.): Picture Processing and Digital Filtering, 2nd ed., Topics Appl. Phys., Vol. 6 (Springer, Berlin, Heidelberg 1979) Meyer-Arendt, J. R.: Introduction to Classical and Modern Optics (Prentice-Hall, Engle-wood Cliffs, NJ 1972) O’Neal, E. L.: Introduction to Statistical Optics (Addison-Wesley, Reading, MA 1963) Stavroudis, O. N.: Modular Optical Design, Springer Ser. Opt. Sci., Vol. 28 (Springer, Berlin Heidelberg 1982)

Chapter 7 7.1 J. W. Cooley, J. W. Tukey: An algorithm for the machine calculation of complex Fourier series, Math. Comput. 19, 297–301 (1965) 7.2 H. J. Nussbaumer: In Two-Dimensional Digital Signal Processing II, ed. by T. S. Huang, Topics Appl. Phys., Vol. 43 (Springer, Berlin, Heidelberg 1981) Chap. 3 7.3 H. J. Nussbaumer: Fourier Transform and Convolution Algorithms, 2nd ed. Springer Ser. Inform. Sci., Vol. 2 (Springer, Berlin, Heidelberg 1982) 7.4 W. T. Cochran, J. W. Cooley, D. L. Favin, H. D. Helms, R. A. Kaenel, W. W. Lang, G. C. Maling Jr., D. E. Nelson, C. M. Rader, P. D. Welch: What is the Fast Fourier Transform?, Proc. IEEE 55, 1664–1674 (1967)

502

References

Additional Reading Cooley, J. W., Lewis, P. A. W., Welch, P. D.: Historical notes on the Fast Fourier Transform, Proc. IEEE 55, 1675–1677 (1967) Gold, B., Rader, C. M.: Digital Processing of Signals (McGraw Hill, New York 1969) Bergland, G. D.: A guided tour of the Fast Fourier Transform, IEEE Spectrum 41–52 (July 1969) Brigham, E. O.: The Fast Fourier Transform (Prentice Hall, Englewood Cliffs, NJ 1974)

Chapter 8 8.1 J. W. Goodman: Introduction to Fourier Optics (McGraw-Hill, New York 1968) 8.2 W. T. Cathey: Optical Information Processing and Holography (Wiley, New York 1974) 8.3 M. Akagi, T. Kaneko, T. Ishiba: Electron micrographs of hologram cross sections, Appl. Phys. Lett. 21, 93–95 (1972) 8.4 H. M. Smith (ed.): Holographic Recording Materials, Topics in Appl. Phys., Vol. 20 (Springer, Berlin, Heidelberg 1977) 8.5 R. J. Collier, C. B. Buckhardt, L. H. Lin: Optical Holography (Academic, New York 1971) 8.6 T. S. Huang: Digital holography, Proc. IEEE, 59, 1335–1346 (1971) 8.7 W. H. Lee: Sampled Fourier transform hologram generated by computer, Appl. Opt. 9, 639–643 (1970) 8.8 M. C. King, A. M. Noll, D. H. Berry: A new approach to computer-generated holography, Appl. Opt. 9, 471–475 (1970) 8.9 W. J. Dallas: In The Computer in Optical Research, ed. by B. R. Frieden, Topics in Appl. Phys., Vol. 41 (Springer, Berlin, Heidelberg 1980) 8.10 S. A. Benton: In Applications of Holography and Optical Data Processing, ed. by E. Marom, A. A. Friesem, W. Wiener-Avenear (Pergamon, New York 1977) pp. 401–409 8.11 S. A. Benton: Hologram reconstruction with expanded light sources, J. Opt. Soc. Am. 59, 1545 A (1969) 8.12 J. C. Dainty (ed.): Laser Speckle and Related Phenomena, 2nd ed., Topics in Appl. Phys., Vol. 9 (Springer, Berlin, Heidelberg 1984) 8.13 D. Gabor: Light and information, Prog. Opt., 1, 109–153 (1961) 8.14 B. R. Frieden: Probability, Statistical Optics, and Data Testing, Springer Ser. Inform. Sci., Vol. 10 (Springer, Berlin, Heidelberg 1983) 8.15 W. Schumann, M. Dubas: Holographic Interferometry, Springer Ser. Opt. Sci., Vol. 16 (Springer, Berlin, Heidelberg 1979)

References

503

8.16 Y. I. Ostrovsky, M. M. Butusov, G. V. Ostrovskaya: Interferometry by Holography, Springer Ser. Opt. Sci., Vol. 10 (Springer, Berlin, Heidelberg 1980) 8.17 C. Knox, R. E. Brooks: Holographic motion picture microscopy, Proc. Roy. Soc. (London) B174, 115–121 (1969) 8.18 D. H. McMahon: Holographic ultrafiche, Appl. Opt. 11, 798–806 (1972) 8.19 Y. Takeda, Y. Oshida, Y. Miyamura: Random phase shifters for Fourier transformed holograms, Appl. Opt. 11, No. 4, 818–822 (1972) 8.20 A. Ioka, K. Kurahashi: Grating modulated phase plates for holographic storage, Appl. Opt. 14, 2267–2273 (1975) 8.21 Y. Tsunoda, Y. Takeda: High density image-storage holograms by a random phase sampling method, Appl. Opt. 13, 2046–2051 (1974) 8.22 L. K. Anderson: Holographic Optical Memory for Bulk Data Storage, Bell Laboratories Records, 319–325 (Nov. 1968) 8.23 K. Kubota, Y. Ono, M. Kondo, S. Sugama, N. Nishida, M. Sakaguchi: Holographic disk with high data transfer rate; its application to an audio response memory, Appl. Opt. 19, 944–951 (1980) 8.24 D. J. Channin, A. Sussman: In Display Devices, ed. by J. I. Pankove, Topics Appl. Phys., Vol 40 (Springer, Berlin, Heidelberg 1980) Chap. 4 8.25 A. Engel, J. Steffen, G. Herziger: Laser machining with modulated zone plates, Appl. Opt. 13, 269–273 (1974) 8.26 S. Amadesi, F. Gori, R. Grella, G. Guattari: Holographic methods for painting diagnostics, Appl. Opt. 13, 2009–2013 (1974) 8.27 G. von Bally (ed.): Holography in Medicine and Biology, Springer Ser. Opt. Sci., Vol. 18 (Springer, Berlin, Heidelberg 1979) 8.28 A. J. MacGovern, J. C. Wyant: Computer generated holograms for testing optical elements, Appl. Opt. 10, 619–624 (1971) 8.29 K. Bromley, M. A. Monahan, J. F. Bryant, B. J. Thompson: Holographic subtraction, Appl. Opt. 10, 174–181 (1971) 8.30 C. Agren, K. A. Stetson: Measuring the wood resonance of treble viol plates by hologram interferometry, J. Acoust. Soc. Am. 46, 120 (1969); N.-E. Molin: Private communication 8.31 W. Schmidt, A. Vogel, D. Preusser: Holographic contour mapping using a dye laser, Appl. Phys. 1, 103–109 (1973) 8.32 L. O. Heflinger, R. F. Wuerker: Holographic contouring via multifrequency lasers, Appl. Phys. Lett., 15, 28–30 (1969)

Chapter 9 9.1 Basic Photographic Sensitometry Workbook, (Kodak Publication, No. Z-22Ed 1968) 9.2 T. H. Jeong: Gaertner-Jeong Holography Manual, Gaertner Scientific Corporation (1968)

504

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9.3 M. Lehmann: Holography, Technique and Practice (The Focal Press, London 1970) 9.4 G. T. Eaton: Photographic Chemistry (Morgan and Morgan Inc., Hastings on Hudson, NY 1957) 9.5 J. Upatnieks, C. Leonard: Diffraction efficiency of bleached photographically recorded interference patterns, Appl. Opt. 8, 85–89 (1969)

Additional Reading Chathey, T. W.: Optical Information Processing and Holography (Wiley, New York 1974) Fujinami, S.: Photography, Kyoritsu series No. 62 (Kyoritsu Shuppan, Tokyo 1963) Fujinami, S.: Advanced Photography, Kyoritsu series No. 118 (Kyoritsu Shuppan, Tokyo 1966) Smith, H. M. (ed.): Holographic Recording Materials, Topics Appl. Phys., Vol. 20 (Springer, Berlin, Heidelberg 1977)

Chapter 10 10.1 J. W. Goodman: Introduction to Fourier Optics (McGraw-Hill, New York 1968) 10.2 M. Born, E. Wolf: Principles of Optics (Pergamon, New York 1964) 10.3 W. T. Cathey: Optical Information Processing and Holography (Wiley, New York 1974) 10.4 J. R. Meyer-Arendt: Introduction to Classical and Modern Optics (PrenticeHall, Inglewood Cliffs, NJ 1972)

Additional Reading Lee, S. H. (ed.): Optical Information Processing, Fundamentals, Topics Appl. Phys., Vol. 48 (Springer, Berlin, Heidelberg 1981) Levi, L.: Applied Optics (Wiley, New York 1968) Suzuki, T.: Optical signal processing, Proc. of Institute of Electronics and Communication Engineers of Japan 57, 1278–1284 (1974) Troup, G. J.: Optical Coherence Theory, Recent Developments (Methuen and Co. Ltd., London 1967)

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505

Chapter 11 11.1 Basic Photographic Sensitometry Workbook (Kodak Publication, No. Z-22Ed 1968) 11.2 H. M. Smith (ed.): Holographic Recording Materials, Topics Appl. Phys., Vol. 20 (Springer, Berlin, Heidelberg 1977) 11.3 Y. Taki, M. Aoki, K. Hiwatashi: Picture Processing (Corona-sha, Tokyo 1972) 11.4 S. H. Lee (ed.): Optical Information Processing, Fundamentals, Topics Appl. Phys., Vol. 48 (Springer, Berlin, Heidelberg 1981) 11.5 L. Gregoris, K. Iizuka: Additive and subtractive microwave holography, Appl. Opt. 12, 2641–2648 (1973) 11.6 R. G. Eguchi, F. P. Carlson: Linear vector operations in coherent optical data processing systems, Appl. Opt. 9, 687–694 (1970) 11.7 A. Vander Lugt: Coherent optical processing, Proc. IEEE 62, 1300–1319 (1974) 11.8 I. McCausland: Introduction to Optimal Control (Wiley, New York 1979) 11.9 J. L. Horner: Optical restoration of images blurred by atmospheric turbulence using optimum filter theory, Appl. Opt. 9, 167–171 (1970) 11.10 D. Casasent (ed.): Optical Data Processing, Applications, Topics Appl. Phys., Vol. 23 (Springer, Berlin, Heidelberg 1978) 11.11 J. W. Goodman: Introduction to Fourier Optics (McGraw-Hill, New York 1968) 11.12 F. T. S. Yu, T. H. Chao: Color signal correlation detection by matched spatial filtering, Appl. Phys. B 32, 1–6 (1983) 11.13 D. P. Jablonowski, S. H. Lee: Restoration of degraded images by composite gratings in a coherent optical processor, Appl. Opt. 12, 1703–1712 (1973) 11.14 A. Wouters, K. M. Simon, J. G. Hirschberg: Direct method of decoding multiple images, Appl. Opt. 12, 1871–1873 (1973) 11.15 K. A. Frey, D. M. Wieland, L. E. Brown, W. L. Rogers, B. W. Agranoff: Development of a tomographic myelin scan, Ann. Neurology 10, 214– 221 (1981) 11.16 A. W. Lohmann, H. W. Werlich: Incoherent matched filtering with Fourier holograms, Appl. Opt. 10, 670–672 (1971) 11.17 H. Kato, J. W. Goodman: Nonlinear filtering in coherent optical systems through halftone screen processes, Appl. Opt. 14, 1813–1824 (1975) 11.18 G. T. Herman (ed.): Image Reconstruction from Projections, Topics Appl. Phys., Vol. 32 (Springer, Berlin, Heidelberg 1979) 11.19 W. J. Meredith, J. B. Massey: Fundamental Physics of Radiology, 2nd ed. (Wright, Bristol, UK 1972) 11.20 R. A. Brooks, G. DiChiro: Principles of computer assisted tomography (CAT) in radiographic and radioscopic imaging, Phys. Med. Biol. Vol. 21, 689–732 (1976)

506

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Additional Reading Berrett, A., Br¨unner, S., Valvassori, G. E. (eds): Modern Thin-Section Tomography (Charles C. Thomas Publisher, Springfield, IL 1973) Franc¸on, M.: Modern Applications of Physical Optics (Wiley, New York 1963) Kak, A. C.: Computerized tomography with x-ray, emission, and ultrasound sources, Proc. IEEE 67, 1245–1272 (1979) Yaroslavsky, L. P.: Digital Picture Processing, Springer Ser. Inform. Sci., Vol. 9 (Springer, Berlin, Heidelberg 1984)

Chapter 12 12.1 R. P. Dooley: X-band holography, Proc. IEEE 53, 1733–1735 (1965) 12.2 Y. Aoki: Microwave hologram and optical reconstruction, Appl. Opt. 6, 1943–1946 (1967) 12.3 H. Shigesawa, K. Takiyama, T. Toyonaga, M. Nishimura: Microwave holography by spherical scanning, Trans. Inst. Electr. Commun. Engs. Jpn., 56-B, No. 3, 99–106 (1973) 12.4 N. H. Farhat, A. H. Farhat: Double circular scanning in microwave holography, Proc. IEEE 61, 509–510 (1973) 12.5 T. Hasegawa: A new method of observing electromagnetic fields at high frequencies by use of test paper, Bull. Yamagata Univ., Yamagata, Jpn., IV, 1 (1955) 12.6 K. Iizuka: Mapping of electromagnetic fields by photochemical reaction, Electr. Lett. 4, 68 (1968) 12.7 C. F. Augustine, C. Deutsch, D. Fritzler, E. Marom: Microwave holography using liquid crystal area detectors, Proc. IEEE 57, 1333–1334 (1969) 12.8 K. Iizuka: In situ microwave holography, Appl. Opt. 12, 147–149 (1973) 12.9 L. Gregoris: Aspects of Microwave Holography, Ph.D. Thesis, University of Toronto (1975) 12.10 K. Iizuka: Microwave holography by means of optical interference holography, Appl. Phys. Lett. 17, 99–101 (1970) 12.11 L. Gregoris, K. Iizuka: Moir´e microwave holography, Appl. Opt. 16, 418– 426 (1977) 12.12 L. G Gregoris, K. Iizuka: Visualization of internal structure by microwave holography, Proc. IEEE 58, 791–792 (1970) 12.13 N. H. Farhat, W. R. Guard: Millimeter wave holographic imaging of concealed weapons, Proc. IEEE, 59, 1383–1384 (1971) 12.14 K. Iizuka: Microwave hologram by photoengraving, Proc. IEEE 57, 813– 814 (1969) 12.15 K. Iizuka: Use of microwave holography in the study of electromagnetic phenomena, Intern. Symp. on Antenna and Propagation, Japan 163–164 (1971)

References

507

12.16 K. Iizuka: Subtractive microwave holography and its application to plasma studies, Appl. Phys. Lett. 20, 27–29 (1972) 12.17 P. F. Checcacci, G. Papi, V. Russo, A. M. Scheggi: Holographic VHF antennas, IEEE Trans. AP-19, 278–279 (1971) 12.18 K. Iizuka, M. Mizusawa, S. Urasaki, H. Ushigome: Volume type holographic antennas, IEEE Trans. AP-23, 807–810 (1975) 12.19 E. B. Joy, C. P. Burns, G. P. Rodrigue: Accuracy of far field patterns based on near field measurements, 1973 IEEE G-AP Symp., 57 (1973) 12.20 J. C. Bennett, A. P. Anderson, P. A. McInnes, A. J. T. Whitaker: Investigation of the characteristics of a large reflector antenna using microwave holography, 1973 IEEE G-AP Symp. 298–301 (1973) 12.21 M. W. Brown, L. J. Porcello: An introduction to synthetic aperture radar, IEEE Spectrum, 52–62 (Sept. 1969) 12.22 L. J. Cutrona, E. N. Leith, L. J. Porcello, W. E. Vivian: On the application of coherent optical processing techniques to synthetic aperture radar, Proc. IEEE 54, 1026–1032 (1966) 12.23 E. N. Leith: In Optical Data Processing, Applications, ed. by D. Casasent, Topics Appl. Phys., Vol. 23 (Springer, Berlin, Heidelberg 1978) Chap. 4 12.24 K. Iizuka, H. Ogura, J. L. Yen, V. K. Nguyen, J. Weedmark: A hologram matrix radar, Proc. IEEE 64, 1493–1504 (1976)

Chapter 13 13.1 A. B. Sharma, S. J. Halme, M. M. Butusov: Optical Fiber Systems and Their Components, Springer Ser. Opt. Sci., Vol. 24 (Springer, Berlin, Heidelberg 1981) 13.2 Special issue of J. Inst. Electr. Commun. Jpn. 63, (Nov. 1980) 13.3 Y. Suematsu, K. Iga: Introduction to Optical Fiber Communications (Ohm-Sha, Toky 1976) 13.4 H. Takashima, N. Uchida, S. Takashima: Overview on techniques of optical fiber cables, Special Issue on fiber optical communication, J. Shisetsu 31, 28–43 (1979) 13.5 S. D. Personick: Optical Fiber Transmission Systems (Plenum, New York 1981) 13.6 H. Takanashi, M. Kunita: Photodetectors, J. Inst. of Electr. Commun. Engs. Jpn. 63, 1178–1182 (1980) 13.7 R. H. Kingston: Detection of Optical and Infrared Radiation, Springer Ser. Opt. Sci., Vol. 10 (Springer, Berlin, Heidelberg 1978) 13.8 H. Kressel (ed.): Semiconductor Devices for Optical Communication, 2nd ed., Topics Appl. Phys., Vol. 39 (Springer, Berlin, Heidelberg 1982) 13.9 J. Hamasaki, Y. Fuji: Photoelectric detectors, J. Inst. Electr. Commun. Engs. Jpn. 56, 524–529 (1973)

508

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13.10 S. D. Personick: Receiver design for optical fiber systems, Proc. IEEE 65, 1670–1678 (1977) 13.11 J. Gowar: Optical Communication Systems (Prentice-Hall, Englewood Cliffs, NJ 1984) 13.12 R. J. Keyes (ed.): Optical and Infrared Detectors, 2nd ed., Topics Appl. Phys., Vol. 19 (Springer, Berlin, Heidelberg 1980) 13.13 Y. Suematsu: Electrooptics sources, J. Inst. Electr. Commun. Engs. Jpn. 56, 535–540 (1973) 13.14 W. Koechner: Solid-State Laser Engineering, Springer Ser. in Opt. Sci., Vol. 1 (Springer, Berlin, Heidelberg 1976) 13.15 K. Seeger: Semiconductor Physics, 2nd ed., Springer Ser. Solid-State Sci., Vol. 40 (Springer, Berlin, Heidelberg 1982) 13.16 C. H. Gooch: Injection Electroluminescent Devices (Wiley, London 1973) 13.17 Optical Fiber Communications Link Design (ITT, Technical Note R-1, 1978) 13.18 U. Haller, W. Herold, H. Ohnsorge: Problems arising in the development of optical communication systems, Appl. Phys. 17, 115 (1978) 13.19 J. Millman, C. C. Halkias: Integrated Electronics: Analog and Digital Circuits and Systems (McGraw-Hill, New York 1972)

Additional Reading Bendow, B., Mitra, S. S.: Fiber Optics (Plenum, New York 1979) Giallorenzi, T.: Optical communications research and technology: fiber optics, Proc. IEEE 66, 745–780 (1978) Howes, M. J., Morgan, D. V.: Optical Fiber Communications (Wiley, New York 1980) Iizuka, K.: Fundamentals of Engineering Optics (Ohm-Sha, Tokyo 1980) Midwinter, J. E.: Optical Fibers for Transmission (Wiley, New York 1979) Nonaka, S.: Fiber optic components, J. Inst. Electr. Commun. Engs. Jpn. 63, 1183– 1191 (1980) Shimada, S.: Fiber Optical Communication Techniques (Ohm-Sha, Tokyo 1980) Sodha, M. S., Ghatak, A. K.: Inhomogeneous Optical Waveguides (Plenum, New York 1977) Tanaka, M., Izawa, T.: Fundamental of optical fiber communications (III), J. Inst. Electr. Commun. Engs. Jpn. 60, 783–794 (1977) Young, M.: Optics and Lasers, 2nd ed., Springer Ser. Opt. Sci., Vol. 5 (Springer, Berlin, Heidelberg 1984)

Chapter 14 14.1 I. P. Kaminow, E. H. Turner: Linear Electrooptical Materials, in Handbook of Lasers with Selected Data on Optical Technology, ed. R. J. Pressley (Chemical Rubber Company, Cleveland, OH 1971) pp. 447–459

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14.2 T. Okada (ed.): Crystal Optics (Morikita Publishing Company, Tokyo 1975) 14.3 J. Koyama, H. Nishihara: Optoelectronics (Corona-Sha, Tokyo 1978) pp. 298–300 14.4 C. L. M. Ireland: A-20 ps resolution crystal streak camera, Opt. Commun. 30, 99–103 (1979) 14.5 T. Sueta, M. Izutsu: Optical guide modulators, in: Editorial committee of the special project research on optical guided-wave electronics, Tokyo (1981) pp. 397–418 14.6 B. D. Cullity: Elements of X-ray Diffraction (Addison-Wesley, Reading MA 1956) 14.7 B. K. Vainshtein: Modern Crystallography I, Springer Ser. Solid-State Sci., Vol. 15 (Springer, Berlin, Heidelberg 1981) 14.8 A. A. Chernov (ed.): Modern Crystallography III, Springer Ser. Solid-State Sci. Vol. 36 (Springer, Berlin, Heidelberg 1984) 14.9 J. F. Nye: Physical Properties of Crystals (Oxford at the Clarendon Press, Oxford 1979) 14.10 K. Kudo: Fundamentals of Optical Properties of Matter (Ohm-Sha, Tokyo 1977) 14.11 L. V. Keldysh: Behavior of non-metallic crystals in strong electric fields, Sov. Phys. JETP 6 (33), 763–770 (1958) 14.12 J. C. Campbell, J. C. Dewinter, M. A. Pollack, R. E. Nahory: Buried heterojunction electroabsorption modulator, Appl. Phys. Lett. 32, 471–473 (1978)

Chapter 15 15.1 R. V. Schmidt, R. C. Alferness: Directional coupler switches, modulators, and filters using alternating ∆β Techniques, IEEE Trans. CAS-26, 1099– 1108 (1979) 15.2 W. H. Louisell: Coupled Mode and Parametric Electronics (Wiley, New York 1960) 15.3 H. Kogelnik, R. V. Schmidt: Switched directional couplers with alternating ∆β, IEEE J. QE-12, 396–401 (1976) 15.4 R. C. Alferness: Guided wave devices for optical communication, IEEE J. QE-17, 946–959 (1981) and T. G. Giallorenzi, R. A. Steinberg: US Patent. 4291, 939 15.5 R. C. Alferness, R. V. Schmidt: Tunable optical waveguide directional coupler filter, Appl. Phys. Lett. 33, 161–163 (1978) 15.6 J. Koyama, H. Nishihara, M. Haruna, T. Seihara: Optical guide switches and their applications, in: “Editorial committee of the special project research on optical guided-wave electronics”, Tokyo (1981) pp. 419–431 15.7 T. R. Ranganath, S. Wang: Ti-diffused LiNbO3 branched-waveguide modulators; performance and design, IEEE J. QE-13, 290–295 (1977) ¯

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15.8 F. J. Leonberger: High-speed operation of LiNbO3 electrooptic interferometric waveguide modulators, Opt. Lett. 5, 312–314 (1980) 15.9 W. D. Bomberger, T. Findakly, B. Chen: Integrated optical logic devices, Proc. of SPIE, Vol. 321, Integrated Optics II, 38–46 (1982) 15.10 A. A. Oliner (ed.): Acoustic Surface Waves, Topics Appl. Phys., Vol. 24 (Springer, Berlin, Heidelberg 1978) 15.11 I. M. Mason, E. Papadofrangakis, J. Chambers: Acoustic-surface-wave disc delay lines, Electr. Lett. 10, 63–65 (1974) 15.12 A. Schnapper, M. Papuchon, C. Puech: Remotely controlled integrated directional coupler switch, IEEE J. QE-17, 332–335 (1981) 15.13 R. S. Cross, R. V. Schmidt, R. L. Thornton, P. W. Smith: Optically controlled two channel integrated-optical switch, IEEE, J. QE-14, 577–580 (1978) 15.14 P. W. Smith, W. J. Tomlinson: Bistable optical devices promise subpicosecond switching, IEEE Spectrum, 18, 26–33 (1981) 15.15 D. Vincent and G. Otis: Hybrid bistable devices at 10.6 µm, IEEE J. QE-17, 318–320 (1981) 15.16 H. J. Reich, J. G. Skalnik, H. L. Krauss: Theory and Applications of Active Devices (Van Nostrand, Princeton, N.J. 1966) 15.17 S. Tarucha, M. Minakata, J. Noda: Complementary optical bistable switching and triode operation using LiNbO3 directional coupler, IEEE J. QE-17, 321– 324 (1981) 15.18 M. Okada, K. Takizawa: Instability of an electrooptic bistable device with a delayed feedback, IEEE J. QE-17, 2135–2140 (1981) 15.19 H. Ito, Y. Ogawa, H. Inaba: Analyses and experiments on integrated optical multivibrators using electrooptically controlled bistable optical devices, IEEE J. QE-17, 325–331 (1981) 15.20 J. E. Bjorkholm, P. W. Smith, W. J. Tomlinson, A. E. Kaplan: Optical bistability based on self-focusing, Opt. Lett. 6, 345–347 (1981) 15.21 A. E. Kaplan: Optical bistability that is due to mutual self-action of counterpropagating beams of light, Opt. Lett. 6, 360–362 (1981) 15.22 S. A. Akhmanov, R. V. Khokhlov, A. P. Sukhorukov: Self-focusing, selfdefocusing and self-modulation of laser beams, in Laser Handbook, ed. by F. T. Arecchi and E. O. Schulz-DuBois (North-Holland, Amsterdam 1982) pp. 1151–1228 15.23 J. E. Bjorkholm, P. W. Smith, W. J. Tomlinson: Optical bistability based on self-focusing; an approximate analysis, IEEE J. QE-18, 2016–2022 (1982) 15.24 S. Tanaka: Planar optical components, in: “Editorial committee of the special project research on optical guided-wave electronics”, Tokyo (1981) pp. 449– 462 15.25 B. Chen, O. G. Ramer: Diffraction-limited geodesic lens for integrated optic circuit, IEEE J. QE-15, 853–860 (1979) 15.26 W. C. Chang, P. R. Ashley: Fresnel lenses in optical waveguides, IEEE J. QE-16, 744–753 (1980) 15.27 T. R. Joseph, T. R. Ranganath, J. Y. Lee, M. Pedioff: Performance of the integrated optic spectrum analyzer, SPIE 321, 134–140 (1982)

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511

15.28 D. F. Barbe (ed.): Charge-Coupled Devices, Topics Appl. Phys., Vol. 38 (Springer, Berlin, Heidelberg 1980) 15.29 J. Koyama, H. Nishihara: Optoelectronics (Corona-Sha, Tokyo 1978) 15.30 K. Ploog, K. Graf: Molecular Beam Epitaxy of III-V Compounds (Springer, Berlin, Heidelberg 1984) 15.31 R. Behrisch (ed.): Sputtering by Particle Bombardment I and II, Topics Appl. Phys., Vols. 47 and 52 (Springer, Berlin, Heidelberg 1981 and 1983) 15.32 H. Ryssel, H. Glawischnig (eds.): Ion Implantation Techniques, Springer Ser. Electrophys., Vol. 10 (Springer, Berlin, Heidelberg 1982) 15.33 I. P. Kaminow, J. R. Carruthers: Optical waveguiding layers in LiNbO3 and LiTaO3 , Appl. Phys. Lett. 22, 326–328 (1973) 15.34 T. R. Ranganath, S. Wang: Suppression of Li2 O out-diffusion from Tidiffused LiNbO3 optical waveguides, Appl. Phys. Lett. 30, 376–379 (1977) 15.35 R. V. Schmidt, I. P. Kaminow: Metal-diffused optical waveguides in LiNbO3 , Appl. Phys. Lett. 25, 458–460 (1974) 15.36 G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, R. M. DeLa Rue: Planar optical waveguides formed by silver-ion migration in glass, IEEE, J. QE-13, 192–200 (1977) 15.37 A. G. Hallam, I. Bennion, W. J. Stewart: Photochromic stripe waveguides for integrated optics, 1st Europ. Conf. on Integrated Optics, London, 26–28 (1981) 15.38 T. Sueta, M. Izutsu: Optical guide modulators, in: “Editorial committee of the special project research on optical guided-wave electronics”, Tokyo (1981) pp. 397–418 15.39 M. K. Barnoski: Introduction to Integrated Optics (Plenum, New York 1973) pp. 127–165 15.40 T. Tamir (ed.): Integrated Optics 2nd ed., Topics Appl. Phys., Vol. 7 (Springer, Berlin, Heidelberg 1979) pp. 199–241 15.41 K. E. Wilson, C. T. Mueller, E. Garmire: Integrated optical circuit fabricated with laser-written masks, SPIE 327, 29–36 (1982) 15.42 D. F. Barbe (ed.): Very Large Scale Integration VLSI, 2nd ed., Springer Ser. Electrophys., Vol. 5 (Springer, Berlin, Heidelberg 1982) 15.43 H. G. Unger: Planar Optical Waveguides and Fibers (Oxford, London 1977) and Fibers and Integrated Optics, ed. by D. B. Ostrowsky (Plenum, New York 1978) pp. 183–222 15.44 R. Hunsperger: Integrated Optics Theory and Technology, Springer Ser. Opt. Sci. Vol. 33, (Springer, Berlin, Heidelberg 1982) 15.45 A. B. Sharma, S. J. Halme, M. M. Bustusov: Optical Fiber Systems and Their Components, Springer Ser. in Opt. Sci., Vol. 24 (Springer, Berlin, Heidelberg 1981) 15.46 E. A. J. Marcatili: Dielectric rectangular waveguide and dielectric coupler for integrated optics, Bell Syst. Tech. J. 48, 2071–2102 (1969)

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Chapter 16 16.1 T. Okoshi: Three-Dimensional Imaging Techniques (Academic Press, New York 1976) 16.2 T. Izumi (ed.): The Basics of a 3-Dimensional Image (In Japanese) (Ohmsha, Tokyo 1995) 16.3 K. Iizuka: Welcome to the wonderful world of 3D: Introduction, principles and history, Opt. Photonics News, 17, 43–51 (July/August 2006) 16.4 T. Okoshi: Three-dimensional displays, Proc. IEEE 68, 548–564 (1980) 16.5 A. R. L. Travis: The display of three-dimensional video images, Proc. IEEE 85, 1817–1832 (1997) 16.6 J. Anderton: Method by which pictures projected upon screens by magic lanterns are seen in relief, Patent US 542321 (July 9, 1895) 16.7 K. Iizuka: Welcome to the wonderful world of 3D: Parallax barrier experiments, Opt. Photonics News, 18, 28–37 (April 2007) ´ 16.8 G. Lippmann: Epreuves r´eversibles donnant la sensation du relief, J. de Phys., 4th series 7, 821–825 (1908) 16.9 Herbert E. Ives: Reflecting screens for relief picture projection, J. OSA 21, 109–118 (1931) 16.10 K. Iizuka “Three dimensional camcorder”. Optical Engineering 46(10), 103203–8 (2007) 16.11 K. Iizuka: Cellophane as a half-wave plate and its use for converting a laptop computer screen into a three dimensional display, Rev. Sci. Instrum. 74, 3636–3639 (2003) 16.12 A.C. Traub: Stereoscopic display using rapid varifocal mirror oscillations, Appl. Opt. 6, 1085–1087 (1967) 16.13 H. Higuchi, J. Hamasaki, M. Okada: Real-time optical sectioning having high SNR by using frequency interleaving, Appl. Opt. 16, 1777–1779 (1977) 16.14 H. Hua, A. Girardot, C. Gao, J. P. Rolland: Engineering of head-mounted projective displays, Appl. Opt. 39, 3814–3824 (2000) 16.15 K. Iizuka: Three-dimensional camera phone, Appl. Opt. 43, 6285–6292 (2004) 16.16 K. Iizuka: Welcome to the wonderful world of 3D: How to create the parallax effect, Opt. Photonics News 17, 40–47 (October 2006) 16.17 K. Iizuka: Welcome to the wonderful world of 3D: Anaglyph experiments, Opt. Photonics News 18, 24–29 (February 2007) 16.18 E. H. Land: Vectographs: Images in terms of vectorial inequality and their application in three-dimensional representation, J. Opt. Soc. Am. 30, 230– 238 (1940) 16.19 J. J. Scarpetti: Coating methods and compositions for production of digitized stereoscopic polarizing images, Patent US 5591508 (January 7, 1997) 16.20 M. Kawakita, K. Iizuka, H. Nakamura, I. Mizuno, T. Kurita, T. Aida, Y. Yamanouchi, H. Mitsumine, T. Fukaya, H. Kikuchi, F. Sato: Highdefinition real-time depth-mapping TV camera: HDTV Axi-Vision Camera, Optics Express 12, 2781–2794 (2004)

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16.21 M. Kawakita, K. Iizuka, Y. Tsuchiya, Y. Iino, H. Kikuchi, H. Fujikake, H. Sato: 3D display generated from a series of cross sections sliced by the depth mapping camera, Proceedings of the 3D Image Conference (Japan, July 5, 2001) pp. 141–144 16.22 H. Higuchi, J. Hamasaki: Real-time transmission of 3-D images formed by parallax panoramagrams, Appl. Opt. 17, 3895–3902 (1978) 16.23 T. Okoshi: Optimum design and depth resolution of lens-sheet and projection-type three-dimensional displays, Appl. Opt. 10, 2284–2291 (1971) 16.24 C. B. Burckhardt: Optimum parameters and resolution limitation of integral photography, J. OSA 58, 71–76 (1968) 16.25 C. B. Burckhardt, E. T. Doherty: Beaded plate recording of integral photographys, Appl. Opt. 8, 2329–2331 (1969) 16.26 J.-S. Jang, B. Javidi, “Three-dimensional projection integral imaging using micro-convex-mirror arrays, Optics Express 12, 1077–1083 (2004) 16.27 J. Arai, F. Okano, H. Hoshino, I. Yuyama: Gradient-index lens-array method based on real-time integral photography for three-dimensional images, Appl. Opt. 37, 2034–2045 (1998). 16.28 F. Okano, H. Hoshino, J. Arai, M. Yamada, I. Yuyama, Integral threedimensional video system, in Three-Dimensional Video and Display: Devices and Systems, eds. Bahram Javidi and Fumio Okano, (SPIE Optical Engineering Press, Bellingham, Washington, 2001) pp. 90–116.

Index

3D camcorder 469, 470 camera phone 471 diagram of effects of 466 history of 459 imaging 459 laptop computer 468, 471 longitudinal translation method 481 movie 462, 472 physiological factors 464 psychological factors 465 remembered geometry 465 shading 465 spinning screen method 482 superimposed cross-sectional images 482 television 492 volumetric method 481 Aberration 8 free 157 Absorption edge 396 Absorption region, inherent (see also Fiber) 340 Abu Ali Al-Hasen ibn al-Hasan ibn Al-Haytham 6 Accommodation 464 Acetic acid 230 Acoustooptic deflector 388 effect 386 modulator 388 ADP 372, 375, 377 Aether 19 wind 19 Agfa 8E75 227 Algebraic reconstruction technique (see also Tomography) 292, 293

Alhazen 6, 9, 10 Alhazen’s law of reflection 7 Aluminized Mylar film 482 Ammonium dihydrogen phosphate 375 Anaglyph 460, 475, 477 dual-color 475, 477 tri-color 475, 477, 479 Anderton, John 460, 462, 463 Angstrom 349 Anisotropic media 371 Anisotropy 136 Antenna array 84 APD 346, 347, 348 Aperture diffraction 78 Arabian Empire 6 Arago, Dominique Francois Jean 17–19 Archimedes 3 Aristotle 4 Arsenic 343 ART (see also Tomography) 293 Astronomy 157 Attenuator continuously varying 364 discretely varying 364 Avalanche multiplication 347 Avalanche photodiode 346–348 Axi-Vision Camera 482 Babinet’s principle 72 Back projection method (see also Tomography) 289 Bar state 417, 418 Beam director 220 splitter 221 Benton, Stephen A. 462 BER 348, 365, 367, 368

515

516 Bi-directional transmission 362 Biaxial crystal 372, 383 Binocular fusion 471, 472 Birefringent 371 crystal 371 negative 372 positive 372 Bit Error Rate 348 Bit rate 348, 365 Bit reversal 171 Bleach 231 Bloembergen, Nicolaas 24 Blurred picture 273 Bohr, Niels 16 Boltzmann, Ludwig 20 Bombardment 450 Born, Max 23 Boron 340 Boundary conditions 402 Bragg condition 387 diffraction 387 Branching coupler 361 Brewster’s stereoscope 460, 463, 474, 475, 477 Brightness of the reconstructed image 188 Bulge guide 454 Bulk effect 383 Calcite 17, 18, 372 Camera obscura 7 Carbon disulfide 439 Cauchy’s infinitesimal strain tensor 390 CCD 449 Cellophane half-wave plate 471, 472 Characteristic equation of the optical guide 403 Charge coupled device 449 Chirp filter 431 modulation 429 radar 431 Ciliary muscle 464 Circularly polarized wave 374 Cladding 129 glass 335 Coherence length 238 light 241 time 238 transfer function 235, 242, 243, 263 Coin 312, 314 Collision-ionization process 346 Color blindness 471

Index coma 8 complementary 460, 475 Comb function 41 Comparison between LED and LD 357 between microwave holography and X-ray 311 between PIN photodiode and APD 347 Compton, Arthur Holly 22 Computed hologram 192, 212 Computed tomography 287 Conjugate image 187, 216 Connectors 358 C-type 358 FA-type 358 Continuously variable attenuator 364 Contour lines 215 of an object 214, 215 Convergence 464–466 Convolution 270 filter 268 Convolver 429 Cooley-Turkey algorithm 161 Core glass 129, 335 Cornu’s spiral 70 Corpuscular theory of light 16 Cotton-Mouton effect 396 Coupled-mode theory 409 arbitrary excitation 414 balanced excitation 413, 420 unbalanced excitation 412, 420 Coupler (see also Directional coupler) 358, 453 beam splitter type 362 branching 361 electronically tunable 423 fiber optic 360 grating 361 planar multiple 361 prism 361, 362 reversed ∆β 419, 457 Selfoc micro lens 361 star directional 362 tapered 361 Coupling condition 411 Cross bar switching diagram 419, 422 Cross state 417, 418 Crystal biaxial 372, 383 birefringent 371 intrinsic 344 optically active 372 uniaxial 372 Ctesibiu’s experiment 6

Index CTF 235, 242, 243 Cuneiform tablets 1 Cut-off 134 condition 134 D’Almeida, Joseph 460 Da Vinci, Leonardo 8 Dark-room procedures 229 bleaching 231 drying 231 stop bath 230 water rinsing 231 Davisson, Clinton Joseph 23 De Broglie, Prince Louis-Victor 23 De Waard 8 Decoding by Fourier transform 259 Deflector acoustooptic 388 electro optic 383 light 383, 386 prism 383 quadrupole 384 Delay in the start of lasing 356 line 430 Delta function 40, 276 Democritus 3, 4 Density, optical 251 Density, photographic 230 Derivative operation 258 in an arbitrary direction 105 Descartes globules 11 Descartes’ vat 9 Descartes, Rene 9, 10, 19 Design, optical communication systems 365 Developing time 229 Devices in integrated optics 416 DFT 161, 329 Diffraction aperture 78 edge 78 limited 157, 248 of light 11 one-dimensional formula of 66 Raman-Nath 388 Diffusion 345 Dipole, half wave 314 Dirac, Paul Adrien Maurice 23 Directional coupler 362, 363, 455 beam splitter type 362 electronically tunable 423 optically switchable 433 reversed ∆β 419 star type 362, 363

517 switch 416 tapered 361 Directional derivative 106 Directrix 121 Discrete Fourier transform 161, 329 Dispersion material 338 mode 136, 338 of the optical fiber 336 waveguide 338 Dominant mode (see also Mode) 405 Dopants 340 Double DFT 296, 330 Doubly refractive 371 Drying 231 e wave 372, 373, 375 Edge diffraction 78 Effective index of refraction 408, 444 Egyptian mummy-case 3 priests 3 Eigenvalue method 391 Eikonal 109 Einstein, Albert 21 Einstein’s hypothesis 21 Elastooptic constants 390 effect 386 effect in an anisotropic medium 388 Electrodes, geometries of 454 Electronic espionage 433 Electron-hole pair production 345 Electrooptic 371, 375 coefficients 378 constants 376, 377 digital light deflector 383 effect 376, 417 material 371 modulator 388 Elliptic helix 129, 132 Emmetropic 2 Emulsion 227 Epitaxial growth 449, 454 Euclid 3 Euler-Lagrange formula 116 Evanescent wave 405, 409, 453 Even mode (see also Mode) 402 Excitation arbitrary 414 balanced 413, 419 unbalanced 412, 419 Exhaustion of the fixer 231 Exposure time 227, 228

518 Extraordinary index 372 Extraordinary wave 372, 375 Eye fatigue 463, 471 Fabrication hologram 182, 222 optical guides 449 pattern 451 Fabry-Perot resonator 439 Far field obtaining from near field pattern 316 pattern 317 region (see also Fraunhofer region) 61 Faraday effect 394 Fast Fourier transform 161 Fermat’s principle 11, 23 Fermat, Pierre 11 Feynman, Richard P. 23 FFT 161 FFT by the method of decimation in frequency 164 in time 172 Fiber optic couplers 360 Fiber dispersion of 336 graded index 337 multimode 135 optical 340 Selfoc 125 single mode 134, 136 step index 335 transmission loss characteristics 339 Fiber optical communication 333, 342, 365 receivers for 343 Field in an electrooptic medium 375 Field of view, hologram 206 Film 221 Filter chirp 431 convolution 268 for image recovery 263 inverse 260 logarithmic 283 low pass 278 matched 265, 280 SAW bandpass 455 spatial 220 Wiener 262 Filtered back projection (see also Tomography) 301 Fish-eye lens 445 Fit theory 16 Fixer 321

Index Fizeau, Armand H. Louis 19 Fly’s eye lens 489 Fog of photographic film 252 Foucault, Leon 19 Fourier optics 36 Fourier transform 34, 36, 50, 142, 258, 264, 265, 283, 294, 296, 297 by a lens 142 domain 234 hologram 190, 216 in polar coordinates and filtered back projection (see also Tomography) 298 method of tomography 294 Fovea 464 Franz-Keldysh effect 396 Fraunhofer approximation 63 region 63 Frenet-Serret formula 104, 105, 116 Fresnel cosine integral 70 hologram 190 integral 68, 69 Kirchhoff diffraction formula 57, 59, 60, 63, 67 region 62 sine integral 70 transform 318 zone lens 446 zone plate 91, 95 Fresnel, Augustin Jean 11, 17–19, 53 Front vertex 158 Fust, Johann 8 Gabor, Dennis 24, 202, 462, 463 Galilei, Galileo 8 Gallium 344 arsenide 350, 375, 379, 396 Gating function 36 GaAs 350, 375, 379, 396 Gelatin 230, 231 Geodesic lens (see also Lens) 446 Geometrical optics 101, 407 Germanium 340 Germer, Lester 23 Glasses complementary colors 475, 477 horse blinders 467 crossed polarizers 471, 472 Globules 10, 11 Goos-H¨anchen phase shift 130 Graded-index fiber (see also Fiber) 337, 342 Gamma-ray images 97

Index Grating 85 coupler 361 lens 448 reflection 85 Greek Empire 2 Grimaldi, Father Francesco Maria 11, 12 Group delay 135, 337 Group velocity 134, 136 Guide 454 Gutenberg, Johannes 8 Half-wave dipole 314 H-D curve 252 Half mirror 482 Hankel function 67 transform 44, 47 Head mounted display (HMD) 462, 479 Head tracker 480 Hero(n) 4 Hertz, Heinrich 19, 20 High Definition Television (HDTV) 492 High-density recording 205 HISS radar 325, 332 HMD 462, 479 Hologram 3D displays 462, 463 computed 192, 212 enhanced view of 206 experimental procedures for fabrication 222 fabrication of 182, 192, 217, 222 Fourier transform 190, 216 Fresnel 190 image 190, 197 lensless Fourier transform 190, 191 matrix 327 rainbow 197, 200 reflection 198 viewing 232 wave front classification of 190 white light 197, 462 Holographic antenna 316 disk 209 interferometry 210, 211 memory 208 Holography 181 analytical description 183 applications of 204 microwave 305, 327 pictorial illustration of 181, 222 Hooke, Robert 11 Horizontally-polarized polarizer molecules 478

519 Horse blinder barrier 487 glasses 467 Huygens’ principle 11, 17, 53, 60 Huygens, Christiaan 11, 16, 53 Iceland spar 17, 18 Iizuka, Keigo 463 Image conjugate 187, 216 of hologram 186, 190, 197 orthoscopic 187 pop-out 468 pseudoscopic 187 quality, effect of finite lens size 153 real 187 true 187 virtual 186 Imagelets 489 Imbedded guide 454 Impulse response function 233–236, 239 Impurity absorption loss (see also Fiber) 340 In-diffusion 451 Incoherent light 241 matched filter 280 Index ellipsoid 372 Index of refraction 125, 339 effective 408, 444 extraordinary 372 inhomogeneity of 116 ordinary 372 parabolic distribution of 129 Indicatrix 372, 378 Inhomogeneity of the refractive index 116 Inhomogeneous media 101 Integral Photography (IP) 461, 489 Integrated micro-optic circuits 110 optical devices 416 optical lenses 444 optical spectrum analyzer 448 Integrated optical lens 444 Fresnel zone type 446 geodesic type 446 grating type 448 mode index type 444 Integrated optics 399 Interception type 3D effect 470 Interdigital transducer 430, 455 Interferometric hologram 210, 211 holography 309 Interferometry, speckle 203 Intrinsic crystal 343

520 Intrinsic region 343 Inverse filters 260 Ion implantation 450 Ion migration 451 Isotropic 107 Ives, F. E. 460, 463 Ives, H. E. 461, 463 Jupiter 8 JVC (Japan Victor Company) 463 Kao, Charles Kuen 24 KDP 372, 375, 380 Kepler, Johann 9, 10 Kerr electrooptic effect 375 magnetooptic effect 396 Kinoform 193 Kirchhoff’s integral theorem 53 Kirchhoff, Gustav 53 Kodak 649F 227 D-165 developer 229 Land, Edwin H. 461, 463, 477 Laser action 352 cavity 352 diode 339, 348, 351–355, 357 machining 209 mirror 49 Lateral rectus muscle 465 Layard, Austen 1 LED 348–350, 352, 355, 357 LD 339, 348, 351–355, 357 Leith, E. N. 24 Lenses (see also Integrated optical lens) 139 effect of finite size of 152–154 effect of lateral translation of input 148 formation of image by 149 Fourier transformable properties of 142 gating 448 meniscus 158 plano convex 139 rock crystal 1 Lensless Fourier transform hologram (see also Holography) 190, 191 Lenticular sheet method 461, 488 Level surface 105 Light deflector 383, 386 acoustooptic 388 analog type 383 digital type 383 electrooptic 383

Index prism type 383 quadruple type 383 Light emitting diode 348–350, 352, 355, 357 Linear perspective 465 Linearly polarized wave 374 Lippershey, Hans 8 Lippmann, Gabriel 461, 463 Liquid crystal display (LCD) 462, 472, 479, 480 Liquid crystal optical switch 463, 479 Lithium gas 396 niobate (LiNO3 ) 372, 375, 378, 379 tantalite 372, 375 Logarithmic filtering 283 Low-pass filter 278 Lucite 386 Luneberg 445 Mach-Zehnder interferometric modulator 425 Magic lanterns 460, 462 Maiman, Theodore 24 Matched filter 265 Material dispersion 338 Maxwell’s equation 23 Maxwell, James Clerk 19 Medial rectus muscle 465 Meniscus lens 158 Meridional ray 124, 126, 129, 130, 138 Metallic mirrors 3 Michelson’s interferometer 19 Michelson, Albert Abraham 19 Microfiche 205 Micromachining 210 Microwave holography 305, 327 as means of seeing through 311 Microwave mapping methods 305 by changes in color 307 by measuring surface expansion 309 by scanning probe 306 by thermal vision 308 Millikan, Robert Andrews 21 Mirror metallic 3 of a laser 49 Mobile electron 352 Modes 133–135, 400, 401 dispersion 136, 338 dominant 406 even 402 filter 428 index lens 444 odd 402 pattern 134

Index TE 401, 441 TM 400, 441 Modulation, chirp 429 Modulator 423, 439 acoustooptic 427 interferometric type 439 waveguide 427 Modulated zone plate 96, 99 Modulation transfer function 246, 248 and optical transfer function 246 Moir´e technique 309 Morley, Edward Williams 19 MTF 246 Multiple exposures 272 Multiple pinhole camera 274 Multiplexing, wavelength division 362, 363 Myopic 1 MZP 96, 99 NA 335–338 Near field region 62 Neurophysiological process of focusing 464 Newton rings 12, 14, 17 corpuscular theory 12 cross prism experiment 14 reflector telescope 15 two prism experiment 13 Newton, Isaac 11, 12, 16 Newton’s error 14 Nichol prism 460 Nitrobenzene 396, 439 Noise, thermally generated 347 Non-radiative recombination 349 Non-return to zero code 367, 368 Normal 107 Normalized thickness of the guide 404 NRZ 367, 368 Numerical aperture 335–338 Nyquist’s criteria 265 o wave 372, 373, 375 Obliquity factor 59 Occlusion 464, 465 Odd mode (see also Modes) 402 OH− ion absorption (see also Fiber) 340 Object beam (see also Holography) 183 On-off ratio 357 Operation of addition and subtraction (see also Optical signal processing) 253 averaging 255 differentiation 257, 258 division 255

521 Fourier transform 259 multiplication 254 Optical activity 393 AND gate 437 attenuators 364 axis of the crystal 372, 382 axis of a crystalline lens 465 bench 217, 218 bistability 433, 435, 456 connector (see also Connector) 358 density 251 directional coupler 362, 455 exclusive OR 426 fence 471, 474 fiber (see also Fiber optic) 335 hysteresis 433, 436 indicatrix 372, 378 memory 205, 208, 430, 433 OR gate 437 path 109, 116 power, average 348, 367 power, output of LD and LED 356 power, peak 367 switch 384, 423, 426, transfer function 241–244, 246, 248, 249 triode 437 Optical signal processing (see also Operation of) 251 using incoherent light 273 Optically active crystal 372 Opticks 13 Optics geometrical (see also Geometrical optics) 101 physical 101 ray 101 wave 105, 141, 149 Ordinary index 372 wave 372, 373, 375 Orthogonality condition 392 Orthoscopic image 187, 188, 491 OTF 242–244, 246, 248, 249 Out-diffusion 451 Outer-shell electrons 343 Oxidation 230 Parabolic refractive index distribution 129 Parallax 464 barrier 485 barrier method 460, 485 effect 464, 466, 467, 471 movement 464

522 Path of light crisscross from an object 467, 470–472 expression for 101 in a cylindrically symmetric medium 122 in a spherically symmetric medium 117 in an inhomogeneous medium 111 in a uniaxal crystal 371 in Selfoc fiber 126 parallel from an object 467, 471, 472 Pauli, Wolfgang 23 Peak power 367 Pelissier, G. 285 Phantom 289 Phase shift keyed 433 Phase shifters 328 Phase transfer function 248 Phosphorus 340 Photochromic technique 451 Photodiode avalanche 346, 347, 478, 479 PIN 343, 346, 347, 368 Photographic film 251 density 230 plate holder 221 Photography 460, 461 Photometer 226 Photoreceptor cells 464 Photoresist 451, 453 Physical optics 101 Picture element 288 Piezoelectric material 427, 430 Pile-of-plate polarizer 460 PIN photodiode 343, 347, 368 Pinhole camera 274 multiple 274 time modulated 276 Pixel 288, 290 switch 482, 483 Planar multiple coupler (see also Coupler) 361 Planck, Max Karl Ernst Ludwig 20 Plane waves 28 Planigraphic tomography 285 Plasma discharge 314 Plato 5 Pockels effect 375 Point source transfer function 64, 144 Poisson’s spot 18 Poisson, Simeon 18 Polarizable inks 479 Polarization 441 allowed directions 373 circular 374 linear 374 rotation of 393, 394

Index Polarization-based 3D imaging 460, 471 Polaroid Corporation 461, 462 Porta, Giovanni Battista della 460 Position vector 28, 101, 103,104 Potassium dihydrogen phosphate 372, 375 Power average optical 348, 367 limited 157 output of LD and LED 356 peak optical 367 Power spectral density of the signal 262 Principal strain 389 Prism 3˚ flat 472, 474 60˚ equilateral 469 coupler 361, 362 Projection-type 3D effect 466 Pseudoscopic image (see also Image) 186, 187, 491, 492 PSK 433 PTF 248 Ptolemy, Claudius 5, 9 Pulse spreading 337, 338 Pupil function 153 Quadrupole type deflector (see also Deflector) 383, 384 Quantization condition 408 Quantized propagation constant 129 in a slab guide 129 in optical fiber 131 Quantum efficiency 347, 368 Quartz 372 Quasi-coherent source 12 Radar chirp 431 HISS 325, 332 side-looking synthetic aperture 318, 320 Radiative recombination 349 Radio astronomy 36 Radius of curvature 104, 116 Rainbow hologram (see also Holography) 197, 200, 201 Raman-Nath condition 388 Raman-Nath diffraction 386, 388 Raster 322 Ray optics 101 Ray sum 289 Rayleigh scattering loss (see also Fiber) 339 Real image (see also Image) 186, 187 Receivers for fiber optical communication 343 Recombination time 349 Reconstruction image from hologram 182, 188

Index Rectangle function 36 Reference beam 183 Reflection 11 grating 85 hologram 197 laws of 7 total 9, 114 Refraction 12, 13 Refractive index 125, 340 effective 408, 444 extraordinary 372 inhomogeneity of 116 ordinary 372 parabolic distribution of 129 Relationship between MTF and OTF 246 Relativity, theory of 19 Relaxation oscillation 356 Renaissance 6, 7 Resolution 465 limit of 157 Resonant wavelength 354 Resonator, Fabry-Perot 439 Return to zero code 367 Reversed ∆β directional coupler 419, 457 Rib guide 453 Rise time of the optical fiber 368 Rollman, Wilhelm 460, 463 Rotary power 394 Rutile 386 RZ 367 Saccharimeter 394 Sampling theorem 263 SAW bandpass filter 455 devices 427 Scalar wave 30 Scarpetti, Julius 479 Schr¨odinger’s equation 23 Schr¨odinger, Erwin 23 Schwinger, Julian 23 Scrambler, for light signals 362 Self-focusing action of optical bistability 440 Selfoc 115 fiber 116, 125, 133, 137, 138, 336, 337 micro lens 129, 361 slab 116 with half mirror coupler 363 Shearing strain 390 Shot noise 347 Shoulder (see also Photographic film) 252 Shutter 227, 249 Side-looking synthetic aperture radar 318, 320 Sign function 38

523 Signal to noise ratio 347 Silicon 343 Silver bromide 251 chloride 251 grains 229, 230 halide 229, 231, 251 halide compounds 229 iodide 251 Single mode fiber (see also Fiber) 134, 136, 338, 342, 343 Skew ray 124, 127, 129, 131, 132, 138 in Selfoc fiber 127 Slab optical guide 399 Selfoc 116 step index 116 SLAR 318, 320 Snell, Willebrod, 10 Snell’s law 6, 10, 11, 113 Sodium chloride 394 thiosulfate 231 vapor 439 Solarization (see also Photographic film) 252 Sommerfeld absorption condition 58 radiation condition 58 Space domain 233, 235 Spatial coherence 237 coherence width 237 filter 220 frequency 33 frequency domain 233 frequency response 233 standing wave 181, 182, 213, 214 Spatially incoherent light 241 Speckle interferometry 203 pattern 202 objective 202, 203 subjective 202, 203 Spectrum analyzer 431, 448 Splices 358 Splicing fiber 359 by cylindrical rod 359, 360 by fusion 360 by sleeve 359 by V groove method 359 Splitter, TE-TM mode 443 Spontaneous emission 351, 352 Sputter etching 451, 454

524

Index

Sputtering method 449, 450 technique 451 Standing wave (see also Spatial standing wave) 213, 226 pattern 134 Star directional coupler 362 Stefan, J. 20 Step function 39 Step index guide 129 slab 116 multimode guide (see also Fiber) 336–338, 342 Stereogram for Brewster’s stereoscope 474 for head mounted display 479 for horse blinder barrier method 487 for integral photography (IP) 489 for lenticular sheet method 488 for parallax barrier method 485, 486 for polarized light method 468 for primary IP 489, 490 for secondary IP 489, 490 for vectograph 478 for Wheatstone’s stereoscope 473 StereoJet 479 Stereoscopic effect 460 pair of images 468, 479 Stimulated emission 351, 354 Stop bath 230 Straight through state 417 Strain tensor 390 Strain-optic constants 386 Strip guides, summary of 453 Strip loaded guide 454 Structure related losses (see also Fiber) 340, 341 Subtractive microwave holography 314 Surface acoustic wave devices 427 Suspensory ligament 464 Sutherland, Ivan 462, 463 Switch 384, 423, 426 Switching operation 417 Synthetic sheet polarizer 461, 462

Temperature dependence of the laser diode 357 Temporal coherence 237 frequency 33 Tensor, strain 390 Thales 3 Thermal vision 308 Thermally generated noise 347 Threshold gain 354 Time sharing method 479 TM modes 406 TM0 mode 442 Toe (see also Photographic film) 252 Tomography 285 algebraic reconstruction technique (ART) 292, 293 back projection method 289 computed 287 filtered back projection method 301 Fourier transform method 294 planigraphic 285 Tomonaga, Sin-Itiro 23 Townes, Charles 24 Transfer function for coherent light 233, 235, 242, 243, 263 for incoherent light 241 modulation 246, 248 optical 242–244, 246, 248, 249, phase 248 point source 64, 144 Transfer length 418 Transposed images 466, 468, 469 Transition probability 351 Transmission, bi-directional 362 Transmitters for fiber optical communications 348 Traub, Alan C. 462, 463 Triangle function 38 True image (see also Image) 186, 187 Tyndall, John 333

Tangent 102 line 101 vector 102, 104 Taper coupler 361 TE-TM mode splitter 443 TE mode (see also Mode) 401, 441, 457 Tellurium dioxide 388

V groove method (see also Splicing) 359 Valence electrons 344 Varifocal mirror 462, 482, 483 Vectograph 462, 477 Vector position 28, 101, 103, 104 tangent 104

Uniaxial crystals 372 Unit vector 102 University of Alexandria 6 Upatnieks, J. 24

Index

525

unit 102 wave 30 Verdet constant 394 Vertically-polarized polarizer molecules 478 Vibrations of holographic plate 228 Video disk system 85 Viewing the hologram 232 Virtual image 186 of IP 489 Visual axis of a crystalline lens 465 Visualization of microwave phenomena (see also Microwave holography) 313 of the plasma discharge 314, 315 Voigt effect 396 Von Lenard, Philipp Eduard Anton 20

optics 101, 141, 149 vector 29 Waveguide dispersion 338 Wavelength division multiplexing 362, 363 Wheatstone’s stereoscope 463, 473–475 White light hologram 197, 462 Wien, Wilhelm 20 Wiener filter 262

Water 386 Wave optics, differential equations of 400 Wave normal 109

Zinc sulphide 394 Zone plate 91 Fresnel 91, 95 modulated 96

X-rays 181 x-y light scanner 481 Y junction 423, 424 Young, Thomas 11, 12, 17, 19 Yttrium iron garnet 394

Springer Series in

OPTICAL SCIENCES Volume 1 1 Solid-State Laser Engineering By W. Koechner, 5th revised and updated ed. 1999, 472 figs., 55 tabs., XII, 746 pages

Published titles since Volume 91 91 Optical Super Resolution By Z. Zalevsky, D. Mendlovic, 2004, 164 figs., XVIII, 232 pages 92 UV-Visible Reflection Spectroscopy of Liquids By J.A. Räty, K.-E. Peiponen, T. Asakura, 2004, 131 figs., XII, 219 pages 93 Fundamentals of Semiconductor Lasers By T. Numai, 2004, 166 figs., XII, 264 pages 94 Photonic Crystals Physics, Fabrication and Applications By K. Inoue, K. Ohtaka (Eds.), 2004, 209 figs., XV, 320 pages 95 Ultrafast Optics IV Selected Contributions to the 4th International Conference on Ultrafast Optics, Vienna, Austria By F. Krausz, G. Korn, P. Corkum, I.A. Walmsley (Eds.), 2004, 281 figs., XIV, 506 pages 96 Progress in Nano-Electro Optics III Industrial Applications and Dynamics of the Nano-Optical System By M. Ohtsu (Ed.), 2004, 186 figs., 8 tabs., XIV, 224 pages 97 Microoptics From Technology to Applications By J. Jahns, K.-H. Brenner, 2004, 303 figs., XI, 335 pages 98 X-Ray Optics High-Energy-Resolution Applications By Y. Shvyd’ko, 2004, 181 figs., XIV, 404 pages 99 Mono-Cycle Photonics and Optical Scanning Tunneling Microscopy Route to Femtosecond Ångstrom Technology By M. Yamashita, H. Shigekawa, R. Morita (Eds.) 2005, 241 figs., XX, 393 pages 100 Quantum Interference and Coherence Theory and Experiments By Z. Ficek and S. Swain, 2005, 178 figs., XV, 418 pages 101 Polarization Optics in Telecommunications By J. Damask, 2005, 110 figs., XVI, 528 pages 102 Lidar Range-Resolved Optical Remote Sensing of the Atmosphere By C. Weitkamp (Ed.), 161 figs., XX, 416 pages 103 Optical Fiber Fusion Splicing By A.D. Yablon, 2005, 137 figs., XIII, 306 pages 104 Optoelectronics of Molecules and Polymers By A. Moliton, 2005, 229 figs., 592 pages 105 Solid-State Random Lasers By M. Noginov, 2005, 131 figs., XII, 238 pages

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