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Optical coherence and quantum optics LEONARD MANDEL Lee DuBridge Professor it Physics and Optics University of Rochester, Rochester, N. Y., USA AND EMIL WOLF Wilson Pro和ssor of Optical Physics University of Rochester, Rochester, N. Y., USA

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 lRP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia ©Cambridge University Press 1995 First Published 1995 Printed in the United States of America A catalogue record for this book is available from the British Library Library of Congress cataloguing in publication data Includes bibliographical references and index. ISBN O 521 41711 2 1. Coherence (Optics) 2. Quantum optics. I. Wolf, Emil. II. Title. Optical coherence and quantum optics/ by Leonard Mandel and Emil Wolf. Mandel, Leonard. p. cm. QC403.M34 1995 535'.2-dc20 93-48873 CIP ISBN O 521 41711 2 hardback KT

Dedicated to our wives Jeanne and Marlies in appreciation of their patience, understanding and help

Contents Preface 1 Elements of probability theory 1.1 Definitions 1.2 Properties of probabilities 1.2.1 Joint probabilities 1.2.2 Conditional probabilities 1.2.3 Bayes'theorem on inverse probabilities 1.3 Random variables and prob动ility distributions 1.3.1 Transformations of variates 1.3.2 ECxhpebecytsahtieov nis n and moments 1.3.3 equality 1.4 Generating functions 1.4.1 Moment generating function 1.4.2 Characteristic function 1.4.3 Cumulants 1.5 Some examples of probability distributions 1.5.1 Bernoulli or binomial-distriooti硕 1.5.2 Poisson distribution 1.5.3 Bose-Einstein distribution 1.5.4 The weak law of large numbers 1.5.5 Normal or Gaussian distribution 1.5.6 The central limit theorem 1.5. 7 Gamma distribution 1.6 Multivariate Gaussian distribution 1.6.1 The Gaussian moment theorem 1.6.2 Moment generating function and characteristic function 1.6.3 Multiple complex Gaussian variates Problems 2 Random (or stochastic) processes 2.1 Introduction to statistical ensembles 2.1.1 The ensemble average 2.1.2 Joint probabilities and correlations 2.1.3 The probability functional XXV 1 1 3 3 5 6 7 9 11 15 16 16 17 19 21 21 23 25 26 27 30 30 33 36 37 38 39 41 41 41 43 44 ix

x Contents 2.2 Stationarity and ergodicity f . e time average o a stationary process 2.2.1 Th 2.2.2 Ergodicity 2.2.3 Examples of random processes 2.3 Properties of th e autocorrelation function 2 .4 Spectral prop erties of a stat10nary random process 2.4.1 Spectral density and the Wiener-Khintchine theorem 2.4.2 Singularities of the spectral density 2.4.3 Normalized correlations and normalized spectral densities 2.4.4 Cross-correlations and cross-spectral densities 2.5 Orthogonal representation of a random process 2.5.1 The Karhunen-Loeve expansion 2.5.2 The limit T- co; an alternative approach to the Wiener- Khintchine theorem 2.6 Time development and class1ficat1on of random processes 2.6.1 Conditional probability densities 2.6.2 Completely random or separable process 2.6.3 First-order Markov process 2.6.4 Higher-order Markov process 2 7 M . aster equat10ns 1n mtegro-differentrnl form 2.8 M aster equations in differential form 2.8.1 The Kramers-Moyal differential equation 2.8.2 Vector random process 2.8.3 The order of the Kramers-Moyal differential equation 2.9 Langevin equation and Fokker-Planck equation . Transition moments for the Langevin process 2.9 1 2.9.2 Steady-state solution of the Fokker-Planck equation 2.9.3 Time-dependent solution of the Fokker-Planck equation 2.10 The Wiener process (or one-dimensional random walk) 2.10.1 The random walk problem 2.10.2 Joint probabilities and autocorrelation 2.10.3 Equation of motion of the Wiener process Problems 3 Some useful mathematical techniques 3.1 The complex analytic signal 3. 1. 1 Definition and basic properties of analytic signals 3.1.2 Quasi-monochromatic signals and their envelopes 3.1.3 Relationships between correlation functions of real and associated complex analytic random processes 3 .1.4 Statistical properties of the analytic signal associated with a real Gaussian random process 3.2 The angular spectrum representation of wavefields 3.2.1 The angular spectrum of a wavefield in a slab geometry 3.2.2 The angular spectrum of a wavefield in a half-space 3.2.3 An example: diffraction by a semi-transparent object 3.2.4 The Weyl representation of a spherical wave 3.2.5 The Rayleigh diffraction formulas 45 47 48 50 52 56 56 60 61 62 65 66 68 70 70 71 72 73 74 75 76 78 79 79 80 81 83 84 84 86 87 88 92 92 92 97 102 106 109 109 112 118 120 125

Contents 3.3 The method of stationary phase 3.3.1 Definition of an asymptotic expansion 3.3.2 Method of stationary phase for single integrals 3.3.3 Method of stationary phase for double integrals 3.3.4 An example: far也zone behavior of the angular spectrum representation of wavefields Problems 4 Second-order coherence theory of scalar wavefields 4.1 Introduction 4.2 Some elementary concepts and definitions 4.2.1 Temporal coherence and the coherence time 4.2.2 Spatial coherence and the coherence area 4.2.3 Coherence volume and the degeneracy parameter 4.3 Interference of two stationary light beams as a second-order correlation phenomenon 4.3.1 The laws of interference. The mutual coherence function and the complex degree of coherence 4.3.2 Second-order correlations in the space-frequency domain. The cross-spectral density and the spectral degree of coherence 4.3.3 Coherence time and bandwidth 44 . Propagation of correlations 4.4.1 Differential equations for the propagation of the mutual coherence and of the cross-spectral density in free space 4.4.2 Propagation of correlations from a plane 4.4.3 Propagation of correlations from finite surfaces 4.4.4 The van Cittert-Zemike theorem 4 4 5 . . Propagation o correlations from pnmary sources . f 4.5 Fields of special types 4.5.1 Cross-spectrally pure light 4.5.2 Coherent light in the space-time domain 4.5.3 Coherent light in the space-frequency domain 4.6 Free fields of any state of coherence 4.6.1 Sudarshan's equations for the propagation of second-order correlation functions of free fields 4.6.2 Time evolution of the second-order correlation functions of free 4. 7 .2 Rigorous representation of the cross-spectral density as a space correlation function 4.7.3 Natural modes of oscillations of partially coherent primary sources and a representation of their cross-spectral density as a correlation function fields of free fields 4.6.3 A relationship between temporal and spatial coherence properties 4.6.4 A relationship between spectral properties and spatial coherence properties of free fiel~s 4.7 Coherent-mode representation and ensemble representation of sources and fields in the space-frequency domain 4.7.1 Coherent-mode representation of partially coherent fields in free xi 128 128 128 133 141 144 147 147 148 148 150 155 159 160 170 176 180 181 183 186 188 193 196 196 200 205 207 208 211 212 213 213 214 218 220

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