Statistical Optics

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 6

2.1 Definitions of Probability and Random Variables 6

2.2 Distribution Functions and Density Functions 8

2.3 Extension to Two or More Joint Random Variables 12

2.4 Statistical Averages 14

2.5 Transformations of Random Variables 20

2.6 Sums of Real Random Variables 27

2.7 Gaussian Random Variables 32

2.8 Complex-Valued Random Variables 37

2.9 Random Phasor Sums 42

2.10 Poisson Random Variables 52

3 Random Processes 56

3.1 Definition and Description of a Random Process 56

3.2 Stationarity and Ergodicity 59

3.3 Spectral Analysis of Random Processes 64

3.4 Autocorrelation Functions and the Wiener-Khinchin Theorem 69

3.5 Cross-Correlation Functions and Cross-Spectral Densities 75

3.6 Gaussian Random Processes 78

3.7 Poisson Impulse Processes 80

3.8 Random Processes Derived from Analytic Signals 93

3.9 The Circular Complex Gaussian Random Process 101

3.10 The Karhunen-Loeve Expansion 102

4 Some First-Order Statistical Properties of Light 109

4.1 Propagation of Light 110

4.2 Thermal Light 114

4.3 Partially Polarized Thermal Light 119

4.4 Single-Mode Laser Light 130

4.5 Multimode Laser Light 143

4.6 Pseudothermal Light Produced by Passing Laser Light Through a Changing Diffuser 148

5 Temporal and Spatial Coherence of Optical Waves 152

5.1 Temporal Coherence 153

5.2 Spatial Coherence 172

5.3 Separability of Spatial and Temporal Coherence Effects 188

5.4 Propagation of Mutual Coherence 191

5.5 Special Forms of the Mutual Coherence Function 196

5.6 Diffraction of Partially Coherent Light by a Transmitting Structure 202

5.7 The Van Cittert-Zernike Theorem 208

5.8 A Generalized Van Cittert-Zernike Theorem 214

5.9 Ensemble-Average Coherence 218

6 Some Problems Involving Higher-Order Coherence 227

6.1 Statistical Properties of the Integrated Intensity of Thermal or Pseudothermal Light 228

6.2 Statistical Properties of Mutual Intensity with Finite Measurement Time 243

6.3 Classical Analysis of the Intensity Interferometer 249

7 Effects of Partial Coherence in Imaging Systems 262

7.1 Preliminaries 263

7.2 Space-Domain Calculation of Image Intensity 269

7.3 Frequency Domain Calculation of the Image Intensity Spectrum 274

7.4 The Incoherent and Coherent Limits 280

7.5 Some Examples 286

7.6 Image Formation as an Interferometric Process 293

7.7 The Speckle Effect in Imaging 308

8 Imaging Through Randomly Inhomogeneous Media 323

8.1 Effects of Thin Random Screens on Image Quality 324

8.2 Random-Phase Screens 328

8.3 The Earth's Atmosphere as a Thick Phase Screen 336

8.4 Electromagnetic Wave Propagation Through the Inhomogeneous Atmosphere 344

8.5 The Long-Exposure OTF 352

8.6 The Short-Exposure OTF 375

8.7 Stellar Speckle Interferometry 382

8.8 The Cross-Spectrum or Knox-Thompson Technique 392

8.9 The Bispectrum Technique 398

8.10 Adaptive Optics 401

8.11 Generality of the Theoretical Results 404

8.12 Imaging Laser-Illuminated Objects through a Turbulent Atmosphere 406

9 Fundamental Limits in Photoelectric Detection of Light 415

9.1 The Semiclassical Model for Photoelectric Detection 416

9.2 Effects of Random Fluctuations of the Classical Intensity 417

9.3 The Degeneracy Parameter 429

9.4 Noise Limitations of the Amplitude Interferometer at Low Light Levels 439

9.5 Noise Limitations of the Intensity Interferometer at Low Light Levels 449

9.6 Noise Limitations in Stellar Speckle Interferometry 456

Appendix A The Fourier Transform 471

A.1 Fourier Transform Definitions 471

A.2 Basic Properties of the Fourier Transform 473

A.3 Tables of Fourier Transforms 476

Appendix B Random Phasor Sums 478

Appendix C The Atmospheric Filter Functions 484

Appendix D Analysis of Stellar Speckle Interferometry 489

Appendix E Fourth-Order Moment of the Spectrum of a Detected Speckle Image 493

Bibliography 496

Index 509