This book discusses statistical methods that are useful fortreating problems in modern optics, and the application of thesemethods to solving a variety of such problems This book covers a variety of statistical problems in optics,including both theory and applications. This book discusses statistical methods that are useful for treating problems in modern optics, and the application of these methods to solving a variety of such problems This book covers a variety of statistical problems in optics, including both theory and applications. 1 Introduction 1 1.1 Deterministic Versus Statistical Phenomena and Models 2 1.2 Statistical Phenomena in Optics 3 1.3 An Outline of the Book 5 2 Random Variables 7 2.1 De nitions of Probability and Random Variables 7 2.2 Distribution Functions and Density Functions 9 2.3 Extension to Two or More Joint Random Variables 13 2.4 Statistical Averages 15 2.4.1 Moments of a Random Variable 16 2.4.2 Joint Moments of Random Variables 17 2.4.3 Characteristic Functions and Moment Generating Functions 19 2.5 Transformations of Random Variables 21 2.5.1 General Transformations. 21 2.5.2 Monotonic Transformations 23 2.5.3 Multivariate Transformations 27 2.6 Sums of Real Random Variables 28 2.6.1 Two Methods for Finding pZ (z) 28 2.6.2 Independent Random Variables 30 2.6.3 The Central Limit Theorem 31 2.7 Gaussian Random Variables 33 2.7.1 De nitions 33 2.7.2 Special Properties of Gaussian Random Variables 35 2.8 Complex-Valued Random Variables 39 2.8.1 General Descriptions 39 2.8.2 Complex Gaussian Random Variables 40 2.8.3 The Complex Gaussian Moment Theorem 43 2.9 Random PhasorSums 43 2.9.1 Initial Assumptions. 44 2.9.2 Calculations of Means, Variances, and the Correlation Co­ef cient 45 2.9.3 Statistics of the Length and Phase 47 2.9.4 Constant PhasorPlus a Random PhasorSum 49 2.9.5 Strong Constant PhasorPlus a Weak Random PhasorSum 53 2.10Poisson Random Variables 54 Problems -Chapter2 55 3 Random Processes 59 3.1 De nition and Description of a Random Process 59 3.2 Stationarity and Ergodicity 62 3.3 Spectral Analysis of Random Processes 68 3.3.1 Spectral Densities of a Known Function 68 3.3.2 Spectral Densities of a Random Process 70 3.3.3 Energy and Power Spectral Densities for Linearly Filtered Random Processes 71 3.4 Autocorrelation Functions, Wiener-Khinchin Theorem 72 3.4.1 De nitions and Properties 72 3.4.2 Relationship to the Power Spectral Density 74 3.4.3 An Example Calculation 75 3.4.4 Auto covariance Functions and Structure Functions 78 3.5 Cross-Correlation and Cross-Spectral Density 78 3.6 Gaussian Random Processes 81 3.6.1 De nition. 81 3.6.2 Linearly Filtered Gaussian Random Processes 82 3.6.3 Wide-Sense Stationarity and Strict Stationarity 83 3.6.4 Fourth-and Higher-Order Moments 83 3.7 Poisson Impulse Processes 83 3.7.1 De nition. 84 3.7.2 Derivation of Poisson Statistics from Fundamental Hy­potheses 87 3.7.3 Derivation of Poisson Statistics from Random Event Times 88 3.7.4 Energy and Power Spectral Densities of Poisson Processes 89 3.7.5 Doubly Stochastic Poisson Processes 92 3.7.6 Spectral Densities of Filtered Processes 95 3.8 Random Processes Derived from Analytic Signals 96 3.8.1 Representation of a Monochromatic Signal by a Complex Signal. 97 3.8.2 Representation of a Nonmonochromatic Signal by a Complex Signal 98 3.8.3 Complex Envelopes or Time-Varying Phasors 101 3.8.4 The Analytic Signal as a Complex-Valued Random Process 102 3.9 The Circular Complex Gaussian Random Process 105 3.10 The Karhunen-Loéve Expansion 106 Problems -Chapter3 108 4 First-Order Statistical Properties 113 4.1 Propagation of Light 114 4.1.1 Monochromatic Light 114 4.1.2 Nonmonochromatic Light 115 4.1.3 Narrow band Light 117 4.1.4 Intensity or Irradiance 117 4.2 Thermal Light. 118 4.2.1 Polarized Thermal Light 119 4.2.2 Unpolarized Thermal Light 122 4.3 Partially Polarized Thermal Light 123 4.3.1 Passage of Narrow band Light Through Polarization-Sensitive Systems. 124 4.3.2 The Coherency Matrix 127 4.3.3 The Degree of Polarization 130 4.3.4 First-Order Statistics of the Instantaneous Intensity 133 4.4 Single-Mode Laser Light 134 4.4.1 An Ideal Oscillation 136 4.4.2 Oscillation with a Random Instantaneous Frequency 137 4.4.3 The Vander Pol Oscillator Model 138 4.4.4 A More Complete Solution for Laser Output Intensity Statistics 146 4.5 Multimode Laser Light 149 4.5.1 Amplitude Statistics 150 4.6 Pseudo thermal light. 155 Problems -Chapter4 156 5 Coherence of Optical Waves 159 5.1 Temporal Coherence 160 5.1.1 Interferometers that Measure Temporal Coherence 160 5.1.2 The Role of the Autocorrelation Function in Predicting the Interferogram. 163 5.1.3 Relationship Between the Interferogram and the Power Spectral Density of the Light 166 5.1.4 Fourier Transform Spectroscopy 170 5.1.5 Optical Coherence Tomography 173 5.1.6 Coherence Multiplexing 179 5.2 Spatial Coherence 181 5.2.1 Young s Experiment 181 5.2.2 Mathematical Description of the Experiment 182 5.2.3 Some Geometrical Considerations 187 5.2.4 Interference Under Quasimonochromatic Conditions 190 5.2.5 Cross-Spectral Density and the Spectral Degree of Coherence. 192 5.2.6 Summary of the Various Measures of Coherence 195 5.2.7 Effects of Finite Pinhole Size 197 5.3 Separability of Coherence effects 197 5.4 Propagation of Mutual Coherence 202 5.4.1 Solution Based on the Huygens-Fresnel Principle 202 5.4.2 Wave Equations Governing Propagation of Mutual Coherence. 205 5.4.3 Propagation of Cross-Spectral Density 206 5.5 Special Forms of the Mutual Coherence Function 207 5.5.1 A Coherent Field 207 5.5.2 An Incoherent Field 210 5.5.3 A Schell-Model Field 211 5.5.4 A Quasihomogeneous Field 212 5.5.5 Expansion of the Mutual Intensity Function in Coherent Modes 212 5.6 Diffraction of Partially Coherent Light 213 5.6.1 Effect of aThinTransmittingStructureonMutualIntensity214 5.6.2 Calculation of the Observed Intensity Pattern 214 5.6.3 Discussion 217 5.6.4 An Example 218 5.7 The Van Cittert Zernike Theorem 220 5.7.1 Mathematical Derivation of the Theorem 220 5.7.2 Discussion 222 5.7.3 An Example 224 5.8 A Generalized Van Cittert Zernike Theorem 226 5.9 Ensemble Average Coherence 230 Problems -Chapter5 232 6 Higher-Order Coherence 239 6.1 Statistical Properties of Integrated Intensity 240 6.1.1 Mean and Variance of the Integrated Intensity 241 6.1.2 Approximate Form of the Probability Density Function of Integrated Intensity. 244 6.1.3 Exact Solution for the Probability Density Function of Integrated Intensity 250 6.2 Mutual Intensity with Finite Measurement Time 256 6.2.1 Moments of the Real and Imaginary Parts of J12(T ) 257 6.3 Analysis of the Intensity Interferometer 262 6.3.1 Amplitude versus Intensity Interferometry 263 6.3.2 Ideal Output of the Intensity Interferometer 265 6.3.3 Noise at the Interferometer Output 268 Problems -Chapter6 272 7 Partial Coherence in Imaging Systems 275 7.1 Preliminaries 276 7.1.1 PassageofPartiallyCoherentLightthroughaThinTrans­mittingStructure 276 7.1.2 Hopkins Formula 278 7.1.3 Focal Plane to Focal Plane Coherence Relationships 280 7.1.4 A Generic Optical Imaging System 281 7.2 Space-Domain Calculation of Image Intensity 283 7.2.1 An Approach to Calculating the Mutual Intensity Incident on the Object 284 7.2.2 Zernike s Approximation 284 7.2.3 Critical Illumination and Köhler s Illumination 286 7.3 Frequency Domain Calculation. 287 7.3.1 Mutual Intensity Relations in the Frequency Domain 288 7.3.2 The Transmission Cross-Coef cient 290 7.4 The Incoherent and Coherent Limits 294 7.4.1 The Incoherent Case 294 7.4.2 The Coherent Case 296 7.4.3 When is an Optical Imaging System Fully Coherent or Fully Incoherent? 297 7.5 Some Examples. 300 7.5.1 The Image of Two Closely Spaced Points 300 7.5.2 The Image of an Amplitude Step 303 7.5.3 The Image of a -Radian Phase Step 305 7.5.4 The Image of a Sinusoidal Amplitude Object 306 7.6 Image Formation as an Interferometric Process 308 7.6.1 An Imaging System as an Interferometer 308 7.6.2 The Case of an In coherent Object 312 7.6.3 Gathering Image Information with Interferometers 314 7.6.4 The Michelson Stellar Interferometer 316 7.6.5 The Importance of Phase Information 318 7.6.6 Phase Retrieval in One Dimension 322 7.6.7 Phase Retrieval in Two Dimensions Iterative Phase Retrieval 324 7.7 The Speckle Effect In Imaging 326 7.7.1 The Origin and First-Order Statistics of Speckle. 327 7.7.2 Ensemble-Average VanCittert Zernike Theorem 329 7.7.3 The Power Spectral Density of Image Speckle 331 7.7.4 Speckle Suppression 334 Problems -Chapter7 337 8 Randomly Inhomogeneous Media 341 8.1 Effects of Thin Random Screens 342 8.1.1 Assumptions and Simpli cations. 342 8.1.2 The Average Optical Transfer Function 343 8.1.3 The Average Point-Spread Function 346 8.2 Random Phase Screens 346 8.2.1 General Formulation 347 8.2.2 The Gaussian Random-Phase Screen 348 8.2.3 Limiting Forms. 352 8.3 The Earth s Atmosphere as a Thick Phase Screen 355 8.3.1 De nitions and Notation. 357 8.3.2 Atmospheric Model 360 8.4 Wave Propagation Through the Atmosphere 364 8.4.1 WaveEquationinanInhomogeneousTransparentMedium364 8.4.2 The Born Approximation. 365 8.4.3 The Rytov Approximation 367 8.4.4 Intensity Statistics 369 8.5 The Long-Exposure OTF 371 8.5.1 Long-Exposure OTF in Terms of the Wave Structure Function 372 8.5.2 Near-Field Calculation of the Wave Structure Function 377 8.5.3 Effects of Smooth Variations of the Refractive Index Structure Constant C2 383 8.5.4 The Atmospheric Coherence Diameter r0 387 8.5.5 Structure Function for a Spherical Wave 389 8.5.6 Extension to Longer Paths 390 8.6 The Short-Exposure OTF 397 8.6.1 Long versus Short Exposures. 397 8.6.2 Calculation of the Average Short-Exposure OTF 399 8.7 Stellar Speckle Interferometry 405 8.7.1 Principles of the Method 405 8.7.2 Heuristic Analysis of the Method 408 8.7.3 Simulation 412 8.7.4 A More Complete Analysis 413 8.8 The Cross-Spectrum or Knox-Thompson Technique 416 8.8.1 The Cross-Spectrum Transfer Function 417 8.8.2 Constraints on | | 418 8.8.3 Simulation 419 8.8.4 Recovering Object Spectral Phase Information from the Cross-Spectrum 419 8.9 The Bispectrum Technique 423 8.9.1 The Bispectrum Transfer Function 423 8.9.2 Recovering Full Object Information from the Bispectrum 424 8.10AdaptiveOptics. 426 8.11Generalityof the Theoretical Results 430 8.12Laserilliminated Objects 431 Problems -Chapter8 435 9 Photoelectric Detection of Light 441 9.1 The Semiclassical Model for Photoelectric Detection 442 9.2 Fluctuations of Classical Intensity 443 9.2.1 Photocount Statistics for Well-Stabilized, Single-Mode Laser Light 445 9.2.2 Photocount Statistics for Polarized Thermal Light 446 9.2.3 Polarization Effects. 451 9.2.4 Effects of Incomplete Spatial Coherence. 453 9.3 The Degeneracy Parameter 455 9.3.1 Fluctuations of Photocounts 456 9.3.2 The Degeneracy Parameter for Blackbody Radiation 460 9.3.3 Read Noise 463 9.4 Noise Limitations of the Amplitude Interferometer 465 9.4.1 The Measurement System and the Quantities to Be Measured 467 9.4.2 Statistical Properties of the Count Vector 469 9.4.3 The Discrete Fourier Transform as an Estimation Tool 470 9.4.4 Accuracy of the Visibility and Phase Estimates 472 9.4.5 Amplitude Interferometer Example 476 9.5 Noise Limitations of the Intensity Interferometer 477 9.5.1 The Counting Version of the Intensity Interferometer 477 9.5.2 Expected Value of the Count-Fluctuation Product 478 9.5.3 The Signal-to-Noise Ratio Associated with the Visibility Estimate 481 9.5.4 Intensity Interferometer Example 483 9.6 Noise Limitations in Stellar Speckle Interferometry 485 9.6.1 A Continuous Model for the Detection Process 486 9.6.2 The Spectral Density of the Detected Image 487 9.6.3 Fluctuations of the Estimate of Image Spectral Density 491 9.6.4 Signal-to-Noise Ratio for Stellar Speckle Interferometry 493 9.6.5 Discussion of the Results 495 Problems -Chapter9 497 A The Fourier Transform 501 A.1 Fourier Transform De nitions 501 A.2 Basic Properties of the Fourier Transform 502 A.3 Tables of Fourier Transforms 505 B Random PhasorSums 507 C The Atmospheric Filter Functions 513 D Speckle Interferometry 517 E Fourth-Order Moment of Speckle Spectrum 521 Bibliography 524 Index 537