Holography and Coherent Optics

Holography and Coherent Optics L.M.Soroko Joint Institute for Nuclear Research Dubna, USSR

Translated from Russian by Albin Tybulewicz Editor, Soviet Journal of Quantum Electronics

With a Foreword by George W. Stroke Translation Editor State University of New York at Stony Brook

PLENUM. PRESS· NEW YORK AND LONDON

Library of Congress Cataloging in Publication Data

Soroko, Lev Markovich. Holography and coherent optics.

Translation of Osnovy golograiJi i kogerentnoloptiki. Bibliography: p. Includes index. 1. Holography. 2. Infonnation theory in optics. 3. Coherence (Optics) I. Title.

QC449.S6713 535'.4 784479

ISBN-13: 978-1-4684-3422-4 e-ISBN-13: 978-1-4684-3420-0 DOT: 10.1007/978-1-4684-3420-0

The original Russian text, published by Nauka Press in Moscow in 1971, has been corrected by the author for the present edition. This translation is published under an agreement with the Copyright Agency of the USSR (V AAP).

OCHOBbI fOJIOfPAIDHH H KOfEPEHTHOH OnTHKH JI.M.COPORO

OSNOVY GOLOGRAFII I KOGERENTNOI OPTIKl L. M. Soroko

© 1980 Plenum Press, New York Softcover reprint of the hardcover I st edition 1980

A Division of Plenum Publishing Corporation 227 West 17th Street, New York, N.Y. 10011

All righ ts reserved

No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any fonn or by any means, electronic, mechanical, photocopying, microfllming, recording, or otherwise, without written pennission from the Publisher

Foreword

Now that holography has matured (wIth Dennis Gabor having received the Nobel Prize In 1971) and that coherent optics has proved to be even more powerful than originally imagined, in open­ing the new field of "optical computing," it is particularlyappropri­ate to make available to English-speaking readers the translation of a uniquely outstanding text which presents the general and per­manent foundations of the field.

L. M. Soroko's book has proved its usefulness, since It ap­peared in Russian In 1971, following the basic pattern set in the world's first monograph In the field, "An Introduction to Coherent Optics and Holography," written by G. w. Stroke in 1966, and since also available in Its second US edition (1969) from Academic Press, following the publication of its Russian edition by Mlr In 1967.

To further enhance the usefulness of L. M. Soroko's book, and to assure that Its fundamental nature be completely updated, the Translation Editor, In agreement with the author, felt that this could best be assured by including an "updating appendix" In the form of three recent papers In the general field of "opto-dlgltal holographic Image processing and 3-D reconstructions," especially -also since their bibliographies Include numerous further refer­ences to a great many authors and workers throughout the world. In addition, the reader's attention Is drawn to two books, also re­cently pubUshed by Plenum Press, under ihe title "Optical Infor­mation Processing," the first In 1976 (edited by Yu. E. Nesterikhln, G. w. Stroke, and w. E. Kock), and the second In 1977 (edited by E. S. Barrekette, G. w. Stroke, Yu. E. Nesterlkhln, and W. E. Kock).

v

vi FOREWORD

These books present the complete set of papers from two US-USSR science cooperation seminars. organized and co-chaired by Aca­demician Yu. E. Nesterikhin. for the USSR. and Prof. G. w. Stroke. for the USA. and held respectively in June 1975 at the National Academy of Sciences in Washington. D. C. under sponsorship from the National Science Foundation. and in July 1976 in the Soviet science city of Akademgorodok (Novosibirsk). under sponsorship of the Academy of Sciences of the USSR. Among the US participants in these seminars. there were W. E. Kook. David Casasent. J. W. Goodman. A. Korpel. Rolf Landauer. Brian J. Thompson. A. Vander Lugt. E. S. Barrekette. W. K. Pratt. M. P. Zampino. H. Kogelnik. S. H. Lee. A. Kozma. and J. C. Urbach. in addition to G. W. Stroke. Papers from all these participants. and a counterpart series of papers from the USSR participants. are included in the two "Optical Information Processing· volumes. In addition. seminar reports have also appeared in Applied Optics. which. in the recent years. together with the IEEE Proceedings. have been perhaps most rep­resentative of the progress and ramifications of the work in th~ field.

Stony Brook. N. Y .• 18 January 1978

George W. Stroke Translation Editor

Preface

This monograph on the fundamentals of holography and co­herent optics is a considerably revised and expanded version of a course of lectures first presented by the author to senior under­graduates at the Moscow Physicotechnical Institute and to par­ticipants of a continuous seminar on holography in 1966-67.

The science of holography was founded about 30 years ago but it has grown only since the appearance of lasers in pliysicallab­oratories. It is now rapidly and widely applied in seemingly most unlikely branches of science and technology. This process is ac­companied by the continuous growth of the number of specialists who are developing optical systems suitable for their applications. These systems are based on the modern coherent optics and holog­raphy. This wide range of specialists needs a modern and com­prehensive treatment of the theory and physical principles under­lying holography. 'The present book is intended to fill this need, and the first step in this direction is a discussion of the theory of holography. Many of the subjects discussed in the present mono­graph represent the essence or development of numerous discus­sions in which the author has participated.

During the writing of this book the first and only monograph on holography in the world literature was G. W. Stroke'S "An In­troduction to Coherent Optics and Holography," published by Aca­demic Press in 1966. The Russian translation of this book was published by Mir in 1967. Stroke's book is used widely throughout the world and a greatly enlarged second edition has been published in 1969.

vii

vUi PREFACE

The reviews on holography published in various journals have also been necessarily brief. Therefore, even if they were collected together, they would not provide a systematic treatment of the sub­ject which is needed during a period of rapid spread of holography to various branches of science and technology.

The intention of the present author has been to present the fundamentals of holography in a sufficiently comprehensive form and from the same point of view. The present monograph is effec­tively an attempt to provide such a systematic treatment on the basis of physical optics, statistical radiophysics, and information theory, which are leading branches of modern science.

Holography is a relatively new subject and, therefore, the monograph begins with an introductory and descriptive chapter in which the physical basis of holography is presented in a clear but somewhat simplified manner. In spite of that, all the aspects of holography are introduced rigorously and this applies also to the concept of coherence of light which is of basic importance in holog­raphy.

The second chapter introduces the concept of an optical signal and deals in detail with the properties of integral linear transforma­tions which can be applied to an optical signal. These transforma­tions are of direct relevance to holography, Fourier spectroscopy, synthesis (computer generation) of holograms, and preparation of two-dimensional translators and filters for coherent optical systems.

The discussion is illustrated by several systems in which in­coherent illumination is used. A detailed discussion of correla­tors and complex two-dimensional filters is deferred to the seventh chapter.

The general properties of integral transformations with Fou­rier-type kernels are formulated at the end of the second chapter. These transformations have the property of invertlbility and they can be used to build multistage optical information processing sys­tems in which there are hardly any losses of any aspects of in­formation.

The third chapter deals with the main properties of a random optical signal. 'The material in this chapter is used later in deal­ing with the coherence of light (in the fourth chapter), the funda­mentals of holography (in the sixth chapter), and the optical in-

PREFACE

formation processing method and filtering of optical signals (in the seventh chapter).

The fourth chapter is devoted to the classical (nonquantum) theory of coherence. '!be latest experiments on the coherence of light are described and methods for measuring coherence are dis­cussed. The theory of coherence of light is basic to holography.

The fifth chapter is devoted to the information structure of an optical signal, particularly those information invariants which determine the resolution in optics and holography, the informa­tion capacity of the message being transmitted, and the influence of the degree of coherence of Ught on the amount of information which can be transmitted without distortion by a partially coherent light beam.

The sixth chapter deals with the following top ics: the prin­ciple of holography as a method for complete recording of light waves; the relationship between an object and the optical signal recorded in a hologram of this object; the classification of holo­grams and the resolution requirements which must be satisfied by recording materials; the various holographic systems, including those employing coherent illumination and spatially coherent il­lumination, ,achromatic systems. volume and stroboscopic holo­grams, and synthesized (computer-generated) holograms. Gen­eralized holography is discussed at the end of the chapter and possi­ble ways of bull ding a holographic x-ray microscope are discussed.

The seventh chapter· deals with optical information processing methods, primarily those based on the holographic principle. Co­herent methods for image (pattern) recognition, spatial differen­tiation of optical signals, and methods for preparing two-dimen­sional translators are discussed.

The monograph deliberately restricts itself to the fundamental theory of holography, and therefore does not deal with recording materials or with the theory of aberrations in holography (in par­ticular. the aberration theory of the holographic microscope). Applications of holography in various branches of science and technology are also not considered, in order to maintain the size of the volume within customary limits.

In the writing of the present monograph it has been necessary to partly rephrase some of the nomenclature used in those disci-

x PREFACE

plines which make essential contributions to holography. This rephrasing and the very compact notation used in the basic rela­tionships simplify the amalgamation of these disciplines and make it possible to develop a new language suitable for the description of the fundamental prinCiples of holography and radio-frequency optics which are essentially interdisciplinary subjects and are among the youngest branches of modern science.

The monograph is written for a wide range of researchers, including physiCiSts, radiophysicists, biophysicists, engineers, and mathematicians working in image recognition and other branches of cybernetics. The monograph can also be used as a textbook for undergraduate and graduate students.

Naturally, it has been difficult to avoid shortcomings in writ­ing the first monograph on a young branch of applied physics such as holography. The author will be very grateful to the readers who will respond to his request for critical comments and suggestions.

The author is deeply grateful to Professor Ya. A. Smorodin­skii,who has stimulated the author's interest in holography, and to the Corresponding Member of the USSR Academy of Sciences V. P. Dzelepov for his constant encouragement of the author's research in holography. The author is also grateful to Academician B. M. Pontekorvo (pontecorvo), Corresponding Member of the USSR Academy of Sciences I. I. Gurevich, Doctor of Physicomathe-mati cal Sciences G. I. Kopylov, Candidate of Physicomathematical Sciences A. F. Pisarev, and Engineer I. P. Nalimov for interesting discussions of holography, and to Professors S. L. Mandel'shtam and G. V. Skrotskii, who have responded in a lively manner to the progress in optics and quantum electronics, have introduced im­portant modifications into senior undergraduate course~, and have greatly encouraged the author.

The author is particularly grateful to V. M. Soroko without whose help in many matters this book would not have been written.

Thanks are also due to Professor G. W. Stroke of the State UniverSity of New York at Stony Brook for scientific contacts which have helped the author to gain a fuller understanding of the latest achievements in holography.

L. Soroko

.Dubna, 1969

Contents

Chapter 1 Introduction

§ 1.1

§1.2

§1.3 §1.4 § 1.5 §1.6 §1.7

Chapter 2

Principle of Holography. Interference of Light ..............••.........•

Three-Dimensional Nature of an Image. Parallax ......•................•.

Plane Hologram ..••............•...•. Diffraction of Light in a Plane Hologram . . . . • . Coherence of Light • . . . . . . . . . . . . . . • . . . . Light Sources for Holography ..•••.. . . . . • . History of Holography . . . • . . . . . . . . . . . . . .

1

3

11 13 16 20 26 30

Optical Signal and Its Transformations. ••• 43

§2.1 §2.2 §2.3 §2.4

§ 2.5

§ 2.6 §2.7 §2.8 § 2.9

§2.10

Complex Signal in Optics .••••....••••... Fourier Transformation • • . . . • . . . • • • • • • . . Properties of Fourier Transformation ••..... Typical Fourier Transformations and

Standard Notation for Commonly Occurring

43 48 49

Functions. • • • • . . . . . . . . . . . . . . . . . . • . 57 Optical Systems Performing Fourier

Transformation. . . • • . . . . . . • . . . . . . . • . Convolution and Its Properties .........•.•. Correlation and Its Properties. • • • . • . . ..... Convolution and Correlation in a Plane ...•... Optical System for Convolution and Cross-

Correlation Operations .•..•.•..••...•• Scalar Products of Functions. • . • . • • . . . . . . •

xl

65 68 72 73

74 78

xli

§2.11 §2.12 §2.13

§2.14 §2.15

§ 2.16 §2.17

§2.18 §2.19 § 2.20 §2.21 §2.22 §2.23

§2.24

§ 2.25 § 2.26 §2.27

§2.28 §2.29 §2.30 §2.31 §2.32

Chapter 3 Random

§3.1 §3.2 § 3.3 § 3.4

CONTENTS

Generalized Functions or Distributions . • • . . • • 81 Examples of Generalized Functions ...• . . . . . 84 Fourier Transformation of Generalized

Functions. • • . . . . . . . . . . . • • . • . • • • . . . 91 Convolution of Generalized Functions. .• • . . . •• 100 Fourier Transformation and Convolution

of Functions in Multidimensional Spaces . . .. 107 Delta Functions in a Plane ••••.. . . • . • • . •. 111 Fourier Transformation of Radial Functions

Defined in a Plane . . . . . . . • • . . . . . • . . •• 115 Hankel Transformation. . . . . . . . . . . . . . . . .. 119 Truncated Fourier Transformation. • • • . . . . .. 121 Hilbert Transformation ... . . . . . . • . • . . . •. 128 Linear Filter . . . . . . . . . . . . . . • . • • . . . • •• 131 Linear Two-Dimensional Filter. . • • . . . . . . .. 137 Dispersion Relationships and Causality

Principle • • . . . . . • • • . • . . . . . • .. • . . • .• 143 Relationship Between Fourier, Hilbert, and

Hankel Transformations • . . • • . • • . . . . . •• 148 Modulation of a Signal ...•. . . . . . • . . . . . •• 154 Fresnel Transformation. • • . . . . • . . . . . • • .• 167 Relationship Between Fresnel and Fourier

Transformations • • • • • . • • • • • • • • . • • • •• 172 Fresnel Sandwich . • • • • . . • • . • • . • • • • • • . . Bickel- Bowman System ..•...•.•••..••.• Auxiliary Parameter in Fresnel Transformation. Dirac and Sampling Transformations ••....•. Comparison of Integral Transformations Used

in OptiCS •........................

Signal •••••••••••••••••••••••••••

Random Function and Random Signal ••••••••• Stationary Random Signal • • • • • • • • • • • • • • • • Correlation Functions •••••••••••••••••• Fourier Transformation of Cross-Correlation

175 177 182 183

184

193

193 196 198

Functions • • • • • • • • • • . • • • • • • • . . • . • • 206 § 3.5 Average Power. • • • • • • • • • • • • • • • • • • • • • • 207 § 3.6 Energy Spectrum. and lts Physical Meaning •••• 216

13.7 13.8

13.9 13.10 13.11

Chapter 4

CONTENTS xW

RBDdom Noise BDd Interference. • • • • • • • • • •• 219 Correlative Methods for Suppression of

Noise and Interference. • • • • • • • • • • • • • • • 221 Detection of a Signal of Known Waveform. • • • • • 223 Indeterminacy Function ••••••••••••••••• 226 Reconstruction of a Function From Its

Autocorrelation •••••••••••••••••••• 233

Coherence of Light....................... 239

14.1 Mutual Coherence Function. • • • • • • • • • • • • •• 239 14.2 Properties of an Analytic Signal and of

Mutual Coherence Function • • • • • • • • • • • • • 247 §4.3 Emission of Light by Atoms •••••••••••••• 249 §4.4 Temporal and Spatial Coherence ••••••••••• 256 14.5 Temporal Coherence • • • • • • • • • • • • • • • • • • • 258 §4.6 Interference of Waves of Different

Frequencies • • • • • • • • • • • • • • • • • • • • • • • 261 14.7 Generalization of Mutual Coherence Function

to Interference of Light of Different Frequencies • • • • • • • • • • • • • • • • • • • • • •• 270

§ 4.8 Transverse and Longitudinal Spatial

§4.9

14.10

14.11

14.12

14.13

14.14

14.15

14.16

Coherence • • • • • • • • • • • • • • • • • • • • • • • 274 Incoherent Extended Source Generating

Spatially Coherent Light • • • • • • • • • • • • • •• 282 ReJationship Between Coherence and

Diffraction of Light • • • • • • • • • • • • • • • • •• 284 Coherence Determination and Measurement

Methods. • • • • • • • • • • • • • • • • • • • • • • • • • 300 Relationship Between Coherence and

Emission Spectrum • • • • • • • • • • • • • • • • • • 316 Wave Equation of Mutual Coherence

Function ••••••••••••••••••••••••• 318 Integral Representation of Mutual Coherence

Function •••••••••••••••••••••.••• 320 Measurement of Coherence of a Laser

Beam. ••••••••••••••••••••••••••• 324 Coherence and Diffusing Screens •••• • • • • • • • 328

xiv CONTENTS

Chapter 5 Information Structure of Optical

Signals •...•..........•.••..•..•.••.•. 335

§ 5.1 Introduction • • • • • • • • • • • • • • • • • • • • • • • • • 335 § 5.2 Indeterminacy Principle • • • • • • • • • • • • • • • • • 336 § 5.3 Classical Representation of a Signal in

Information Theory • • • • • • • • • • • • • • • • • • 338 § 5.4 Elementary Signal. • • • • • • • • • • • • • • • • • • • • 342 § 5.5 Determinate Function With a Bounded

Spectrum . • • . • • . . • • . • • . • • • . . • . • • .• 344 § 5.6 Properties of Sampling Function ••••••••••• 350 §5.7 Determinate Function of Finite Duration. • • • •• 351 § 5.8 One-Dimensional Functions With

Finite Spectra. • • • • • • • • • • • • • • • • • • • • • 353 § 5.9 Two-Dimensional Functions With

§ 5.10 §5.11 § 5.12 §5.13 § 5.14

Chapter 6

Finite Spectra. • • • • • • • • • • • • • • • • • • • • • 362 Sampling Theorem in Optics • • • • • • • • • • • • • • 366 lliumination Matrix • • • • • • • • • • • • • • • • • • • • 371 Intensity Matrix •••••••••••••••••••••• 380 Image and Detector Matrices. • • • • • • • • • • • •• 383 Temporal Sampling Points ••••••••••••••• 392

Holography ..•....•.••.• 0......... .. .. ... 395

§ 6.1 Introduction • • • • • • • • • • • • • • • • • • • • • • • •• 395 § 6.2 Basic Holographic System. • • • • • • • • • • • • • •• 399 § 6.3 Development of Photographic Emulsions • • • • •• 404 § 6.4 Relationship Between an Object and a Light

Wave Reaching Entrance Pupil. • • • • • • • • •• 410 § 6.5 Holograms of Very Simple Objects • • • • • • • • •• 417 § 6.6 Holographic Microscope. • • • • • • • • • • • • • • • • 421 § 6.7 Relation Between an Object and a Light

Wave Reaching Entrance Pupil for Arbitrary lliumination • • • • • • • • • • • • • • •• 424

§ 6.8 Theory of Most General Holographic System 428 § 6.9 Alternative Representations of Spherical

§ 6.10 §6.11

Fresnel Transform and Its Fourier Transform. • . . • • • • • • • • • . • • • • • • . • • •. 434

Rigorous Classification of Holograms. • • • • • • • 440 Most Typical Systems for Obtaining Different

Types of Hologram •••••••••••••••••• 447

§6.12 §6.13 §6.14

§6.15

§ 6.16 § 6.17

§ 6.18

§ 6.19 § 6.20 § 6.21

§ 6.22 § 6.23 § 6.24 § 6.25

§6.26 § 6.27 §6.28 § 6.29

§6.30

Chapter 7

CONTENTS xv

Absorption and Relief-Phase Hologr;oams • • • • •• 452 Amplitude Synthesis of Images. • • • • • • • • • • •• 457 Correlative Compensation of Extended

Reference Source Effects • • • • • • • • • • • • •• 459 Theory of Correlative Compensation of Extended

Source Effects ••••••••••••••••••••• 462 Holograms of Phase Objects • • • • • • • • • • • • •• 465 Information Capacity of a Photographic

Emulsion and Its Use in Holography. • • • • •• 480 Some Features of Amplitude Synthesis of

Images by Holography •••••••••••••••• 485 Synthesized Holograms. • • • • • • • • • • • • • • • • • 489 Coherence in Holography •••••••••••••••• 505 Intensity Holograms Obtained Under Incoherent

Illumination Conditions. • • • • • •.• • • • • • • •• 519 Holograms Formed with a Fresnel Zone Plate. • 532 Achromatic Holography ••••••••••••••••• 535 Volume Holograms •••••••••••••••••••• 545 Vector Field Holograms of Two Polarization

States • • • • • • • • • • • • • • • • • • • • • • • • • •• 560 Ghost Images • • • • • • • • • • • • • • • • • • • • • • •• 562 Polychromatic Images • • • • • • • • • • • • • • • • • • 563 Aberrations in Holography ••••••••••••••• 567 Generalized Hologram and Principle

of a Reference Object. • • • • • • • • • • • • • • • • 572 Stroboscopic Holograms. • • • • • • • • • • • • • • •• 575

Optical Information Processing Methods.... 579

§ 7.1 Increasing Importance of Optics in Computing Technology • • • • • • • • • • • • • • • • 579

§ 7.2 Fourier Transformation in a Coherent Optical System • • • • • • • • • • • • • • • • • • • • • 580

~ 7.3 Linear Optical Filter • • • • • • • • • • • • • • • • • • • 585 § 7.4 Principle of Spatial Filtering of Images ••••••• 587 § 7.5 Optical Correlator ••••••••••••••••••••• 587 § 7.6 Convolution and Correlation in Coherent

Optical Systems . . . . . . . . . . . . . . . . . . . . 593 § 7.7 Image Recognition • • • • • • • • • • • • • • • • • • • • • 596 § 7.8 Binary Matched Filter • • • • • • • • • • • • • • • • •• 599

xvi

§ 7.9 §7.10 §7.11 § 7.12 § 7.13 § 7.14 § 7.15 § 7.16 § 7.17 § 7.18

§ 7.19 § 7.20

§ 7.21 ~ 7.22 § 7.23

§ 7.24 § 7.25 § 7.26 § 7.27

§7.28

§ 7.29 §7.30 §7.31 §7.32

CONTENTS

Main Components of a Coherent Optical System. 601 Generalized Fresnel Function • • • • • • • • • • • • • 604 Optical stage •••••••••••••••••••••••• 606 Im.aging Condition • • • • • • • • • • • • • • • • • • • • • 608 Condition for Obtaining a Fourier Transform. • • 610 Change of Scale in a Fourier Transform • • • • • • 612 Multistage Optical Systems • • • • • • • • • • • • • • • 614 Multichannel Systems. • • • • • • • • • • • • • • • • • • 616 Spatial Invariance of Linear Optical Filter •••• 618 Suppression of Noise by Spatial Differentiation

of Optical Signals ••••••••••••••••••• 622 Permissible Displacements of a Filter. • • • • • • 627 Some Optical Systems Used in Preparation

of Matched Filters. • • • • • • • • • • • • • • • • • • 635 File of Matched Filters. • • • • • • • • • • • • • • • • • 639 Capacity of Matched-Filter File. . . . • . . . . . .• 642 Incoherent Optical Information Processing

Systems. . • . • . . • • • . • . . . . • . . . . . • • .. 650 Two-Dimensional Translator. • • • • • • • • • • • • • 658 Temporal Holographic Filter. • • • • • • • • • • • • • 662 Correlative Comparison of Two Signals. • • • • • • 666 Redistribution of Information Degrees of

Freedom . • • • • • • • • • • • • • • • • • • • • • • • • 671 Operation of a Coherent Side-Looking

Radar Station • • • • • • • • • • • • • • • • • • • • • • 683 Theory of Coherent Side-Looking Radar •••••• 687 Analysis of Data in a Coherent Radar Frame. • • 695 A Posteriori Image Processing Apodization 699 Removal of Spherical Aberration of Lenses • • • • 702

A p pen d i x . • • • • . . . . . . . . . . . . . . . . . • • . . . . . • . . 703

Optical Computing, by George W. Stroke. . . . . . . . . • • • • 705 Retrieval of Good Images from ACCidentally

Blurred Photographs, by George W. stroke, Maurice Halioua, Venugopal Srinivasan, and Morimasa Shinoda. • • . . . • • • • • • . • • • . . • . . • . 749

Image Improvement and Three-Dimensional Reconstruction Using Holographic Image Processing by George W. Stroke, Maurice Halioua, Friedrich Thon, and Dieter H. Willasch • • • • • • • • • . . . • . • • • • 755

Holography and Coherent Optics