Statistical Optics JOSEPH W GOODMAN Wiley Classics Library Edition Published 2000 A Wiley-Interscience Publication JOHN WILEY & SONS, INC New York • Chichester • Weinheim • Brisbane • Singapore • Toronto This text is printed on acid-free paper © Copyright © 1985 by John Wiley & Sons, Inc All rights reserved Published simultaneously in Canada Wiley Classics Library Edition published 2000 No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744 Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ @ WILEY.COM For ordering or customer information, please call 1-800-CALL-WILEY Library of Congress Cataloging in Publication Data: Library of Congress Catalog Card Number: 84-13160 ISBN 0-471-01502-4 ISBN 0-471-39916-7 (Classics Edition) Printed in the United States of America 10 To Hon Mai, who has provided the light Preface Since the early 1960s it has gradually become accepted that a modern academic training in optics should include a heavy exposure to the concepts of Fourier analysis and linear systems theory This book is based on the thesis that a similar stage has been reached with respect to the tools of probability and statistics and that some training in the area of statistical optics should be included as a standard part of any advanced optics curriculum In writing this book I have attempted to fill the need for a suitable textbook in this area The subjects covered in this book are very physical but tend to be obscured by mathematics An author of a book on this subject is thus faced with the dilemma of how best to utilize the powerful mathematical tools available without losing sight of the underlying physics Some compromises in mathematical rigor must be made, and to the largest extent possible, a repetitive emphasis of the physical meaning of mathematical quantities is needed Since fringe formation is the most fundamental underlying physical phenomenon involved in most of these subjects, I have tried to stay as close as possible to fringes in dealing with the meaning of the mathematics I would hope that the treatment used here would be particularly appealing to both optical and electrical engineers, and also useful for physicists The treatment is suitable for both self-study and for formal presentation in the classroom Many homework problems are included The material contained in this book covers a great deal of ground An outline is included in Chapter and is not repeated here The course on which this text is based was taught over the 10 weeks of a single academic quarter, but there is sufficient material for a full 15-week semester, or perhaps even two academic quarters The problem is then to decide what material to omit in a single-quarter version If the material is to be covered in one quarter, it is essential that the students have previous exposure to probability theory and stochastic processes as well as a good grasp of Fourier methods Under these conditions, my suggestion to the instructor is viii PREFACE to allow the students to study Chapters 1-3 on their own and to begin the lectures directly with optics in Chapter Later sections that can be omitted or left to optional reading if time is short include Sections 5.6.4, 5.7, 6.1.3, 6.2, 6.3, 7.2.3, 7.5, 8.2.2, 8.6.1, 8.7.2, 8.8.3, 9.4, 9.5, and 9.6 It is perhaps worth mentioning that I have also occasionally used Chapters and as the basis for a full one-quarter course on the fundamentals of probability and stochastic processes The book began in the form of rough notes for a course at Stanford University in 1968 and thus has been a long time in the making In many respects it has been too long in the making (as my patient publisher will surely agree), for over a period of more than 15 years any field undergoes important changes The challenge has thus been to treat the subject matter in a manner that does not become obsolete as time progresses In an attempt to keep the information as up to date as possible, supplementary lists of recent references have been provided at the ends of various chapters The transition from a rough set of notes to a more polished manuscript first began in the academic year 1973-1974, when I was fortunate enough to spend a sabbatical year at the Institute d'Optique, in Orsay, France The hospitality of my immediate host, Professor Serge Lowenthal, as well as the Institute's Director, Professor Andre Marechal, was impeccable Not only did they provide me with all the surroundings needed for productivity, but they were kind enough to relieve me of duties normally accompanying a formal appointment I am most grateful for their support and advice, without which this book would never have had a solid start One benefit from the slowness with which the book progressed was the opportunity over many years to expose the material to a host of graduate students, who have an uncanny ability to spot the weak arguments and the outright errors in such a manuscript To the students of my statistical optics courses at Stanford, therefore, I owe an enormous debt The evolving notes were also used at a number of other universities, and I am grateful to both William Rhodes (Georgia Institute of Technology) and Timothy Strand (University of Southern California) for providing me with feedback that improved the presentation The relationship between author and publisher is often a distant one and sometimes not even pleasant Nothing could be further from the truth in this case Beatrice Shube, the editor at John Wiley & Sons who encouraged me to begin this book 15 years ago, has not only been exceedingly patient and understanding, but has also supplied much encouragement and has become a good personal friend It has been the greatest of pleasures to work with her I owe special debts to K.-C Chin, of Beijing University, for his enormous investment of time in reading the manuscript and suggesting improvements, PREFACE ix and to Judith Clark, who typed the manuscript, including all the difficult mathematics, in an extremely professional way Finally, I am unable to express adequate thanks to my wife, Hon Mai, and my daughter Michele, not only for their encouragement, but also for the many hours they accepted being without me while I labored at writing JOSEPH W GOODMAN Stanford, California October 1984 Contents Introduction 1.1 1.2 1.3 Deterministic versus Statistical Phenomena and Models Statistical Phenomena in Optics An Outline of the Book Random Variables 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Definitions of Probability and Random Variables Distribution Functions and Density Functions Extension to Two or More Joint Random Variables Statistical Averages 2.4.1 Moments of a Random Variable 2.4.2 Joint Moments of Random Variables 2.4.3 Characteristic Functions Transformations of Random Variables 2.5.1 General Transformation 2.5.2 Monotonic Functions 2.5.3 Multivariate Probability Transformations Sums of Real Random Variables 2.6.1 Two Methods for Finding pz(z) 2.6.2 Independent Random Variables 2.6.3 The Central Limit Theorem Gaussian Random Variables 2.7.1 Definitions 2.7.2 Special Properties of Gaussian Random Variables Complex-Valued Random Variables 2.8.1 General Descriptions 2.8.2 Complex Gaussian Random Variables 7 12 15 16 17 19 21 21 23 27 29 29 31 31 33 34 37 40 40 41 536 RANDOM PHASOR SUMS means from ri, we obtain the variances and the covariance, (B-9) For the special case of a probability density function for the phases that is even about the origin, we obtain the simpler expressions cov(r,/) = (B-10) Finally, when the phases are uniformly distributed, we have M,(l) = M,(2) = 0, (B-ll) and the variances and covariances reduce to = =— cov(r,/) = 0, which is identical with the results obtained in Section 2.9 (B-12) RANDOM PHASOR SUMS 537 There is one further subtlety regarding the random walk problem that we wish to clarify here It was argued in Section 2.9 that, when the number of terms in the sums of Eq (B-2) grows large, the central limit theorem implies that the statistics of the real and imaginary parts of the sum tend asymptotically toward Gaussian Such an argument is valid regardless of whether the phases associated with the individual contributions have uniform statistics However, the assumption has been made that the real and imaginary parts are asymptotically jointly Gaussian random variables; that is, they satisfy together a second-order Gaussian probability density function [cf Eq (2.9-5)] Whereas the Gaussian character of their marginal densities is implied by the central limit theorem, their joint Gaussian character is less obvious To prove joint Gaussianity, we make the simplifying assumption that the phases k are uniformly distributed, independent random variables We retain the assumption that the amplitudes ak are independent of the phases and of each other The joint characteristic function of the real and imaginary parts r and / is given by We define polar coordinate variables in the (cov co2) plane, through o)x —ftcos x w m = r^ p * q , /t ^ m 4- P f, n 4k m =5* P U n4- m i* q K terms K(K- 1) terms K(K- 1)(K — 2) terms K(K- 1) terms K(K~ 1)CAT - 2) terms K(K- 1) terms K(K - l)(K - 2) terms K(K- 1) terms K(K- 1) terms K(K- 1) terms K(K- 1) terms K(K- 1)(A: - 2)(K - 3) terms K(K- 1)(# - 2) terms K(K- l)(K - 2) terms K(K - l)(K- 2) terms 539 540 SPECTRUM OF A DETECTED SPECKLE IMAGE For the moment, condition the statistics by a known rate function X(JC, y)\ we shall later average over the statistics of X Thus we first average over the 2K + random variables (x l9 yx), (x , y2), , (xK, yK), K Noting that for a Poisson random variable K, the following expectation over K, conditioned on a known X, is given by E[K(K - 1) • • • (K - k +1)] = [^(A)] \ (C-2) where # ( X ) represents the conditional mean of K The contributions of the 15 sets of terms identified above can now be written as follows: (1) *