Phase-Space Optics: Fundamentals and Applications: Fundamentals and Applications Book

Preface xiii

1 Wigner Distribution in Optics 1

1.1 Introduction 1

1.2 Elementary Description of Optical Signals and Systems 2

1.2.1 Impulse Response and Coherent Point-Spread Function 3

1.2.2 Mutual Coherence Function and Cross-Spectral Density 3

1.2.3 Some Basic Examples of Optical Signals 4

1.3 Wigner Distribution and Ambiguity Function 5

1.3.1 Definitions 5

1.3.2 Some Basic Examples Again 7

1.3.3 Gaussian Light 9

1.3.4 Local Frequency Spectrum 11

1.4 Some Properties of the Wigner Distribution 12

1.4.1 Inversion Formula 12

1.4.2 Shift Covariance 12

1.4.3 Radiometric Quantities 12

1.4.4 Instantaneous Frequency 14

1.4.5 Moyal's Relationship 15

1.5 One-Dimensional Case and the Fractional Fourier Transformation 15

1.5.1 Fractional Fourier Transformation 15

1.5.2 Rotation in Phase Space 16

1.5.3 Generalized Marginals-Radon Transform 16

1.6 Propagation of the Wigner Distribution 18

1.6.1 First-Order Optical Systems-Ray Transformation Matrix 18

1.6.2 Phase-Space Rotators-More Rotations in Phase Space 19

1.6.3 More General Systems-Ray-Spread Function 21

1.6.4 Geometric-Optical Systems 22

1.6.5 Transport Equations 23

1.7 Wigner Distribution Moments in First-Order Optical Systems 24

1.7.1 Moment Invariants 25

1.7.2 Moment Invariants for Phase-Space Rotators 26

1.7.3 Symplectic Moment Matrix-The Bilinear ABCD Law 28

1.7.4 Measurement of Moments 29

1.8 Coherent Signals and the Cohen Class 29

1.8.1 Multicomponent Signals-Auto-Terms and Cross-Terms 30

1.8.2 One-Dimensional Case and Some Basic Cohen Kernels 32

1.8.3 Rotation of the Kernel 33

1.8.4 Rotated Version of the Smoothed Interferogram 35

1.9 Conclusion 40

References40

2 Ambiguity Function in Optical Imaging 45

2.1 Introduction 45

2.2 Intensity Spectrum of a Fresnel Diffraction Pattern Under Coherent Illumination 47

2.2.1 General Formulation 47

2.2.2 Application to Simple Objects 48

2.2.3 Contrast Transfer Functions 49

2.3 Propagation through a Paraxial Optical System in Terms of AF 49

2.3.1 Propagation in Free Space 49

2.3.2 Transmission through a Thin Object 50

2.3.3 Propagation in a Paraxial Optical System 51

2.4 The AF in Isoplanatic (Space-Invariant) Imaging 52

2.5 The AF of the Image of an Incoherent Source 53

2.5.1 Derivation of the Zernike-Van Cittert Theorem from the Propagation of the AF 53

2.5.2 Partial Coherence Properties in the Image of an Incoherent Source 54

2.5.3 The Pupil-AF as a Generalization of the OTF 54

2.6 Phase-Space Tomography 55

2.7 Another Possible Approach to AF Reconstruction 56

2.8 Propagation-Based Holographic Phase Retrieval from Several Images 58

2.8.1 Fresnel Diffraction Images as In-Line Holograms 58

2.8.2 Application to Phase Retrieval and X-Ray Holotomography 59

2.9 Conclusion 60

References 60

3 Rotations in Phase Space 63

3.1 Introduction 63

3.2 First-Order Optical Systems and Canonical Integral Transforms 64

3.2.1 Canonical Integral Transforms and Ray Transformation Matrix Formalism 64

3.2.2 Modified Iwasawa Decomposition of Ray Transformation Matrix 66

3.3 Canonical Transformations Producing Phase-Space Rotations 67

3.3.1 Matrix and Operator Description 67

3.3.2 Signal Rotator 69

3.3.3 Fractional Fourier Transform 69

3.3.4 Gyrator 73

3.3.5 Other Phase-Space Rotators 74

3.4 Properties of the Phase-Space Rotators 74

3.4.1 Some Useful Relations for Phase-Space Rotators 75

3.4.2 Similarity to the Fractional Fourier Transform 76

3.4.3 Shift Theorem 77

3.4.4 Convolution Theorem 77

3.4.5 Scaling Theorem 77

3.4.6 Phase-Space Rotations of Selected Functions 78

3.5 Eigenfunctions for Phase-Space Rotators 80

3.5.1 Some Relations for the Eigenfunctions 80

3.5.2 Mode Presentation on Orbital Poincaré Sphere 82

3.6 Optical Setups for Basic Phase-Space Rotators 84

3.6.1 Flexible Optical Setups for Fractional FT and Gyrator 85

3.6.2 Flexible Optical Setup for Image Rotator 87

3.7 Applications of Phase-Space Rotators 88

3.7.1 Generalized Convolution 88

3.7.2 Pattern Recognition 90

3.7.3 Chirp Signal Analysis 94

3.7.4 Signal Encryption 94

3.7.5 Mode Converters 95

3.7.6 Beam Characterization 96

3.7.7 Gouy Phase Accumulation 100

3.8 Conclusions 101

Acknowledgments 102

References 102

4 The Radon-Wigner Transform in Analysis, Design, and Processing of Optical Signals 107

4.1 Introduction 107

4.2 Projections of the Wigner Distribution Function in Phase Space: The Radon-Wigner Transform (RWT) 108

4.2.1 Definition and Basic Properties 108

4.2.2 Optical Implementation of the RWT: The Radon-Wigner Display 117

4.3 Analysis of Optical Signals and Systems by Means of the RWT 122

4.3.1 Analysis of Diffraction Phenomena 122

4.3.1.1 Computation of Irradiance Distribution along Different Paths in Image Space 122

4.3.1.2 Parallel Optical Display of Diffraction Patterns 132

4.3.2 Inverting RWT: Phase-Space Tomographic Reconstruction of Optical Fields 134

4.3.3 Merit Functions of Imaging Systems in Terms of the RWT 138

4.3.3.1 Axial Point-Spread Function (PSF) and Optical Transfer Function (OTF) 138

4.3.3.2 Polychromatic OTF 143

4.3.3.3 Polychromatic Axial PSF 146

4.4 Design of Imaging Systems and Optical Signal Processing by Means of RWT 151

4.4.1 Optimization of Optical Systems: Achromatic Design 151

4.4.2 Controlling the Axial Response: Synthesis of Pupil Masks by RWT Inversion 156

4.4.3 Signal Processing through RWT 157

Acknowledgments 162

References 162

5 Imaging Systems: Phase-Space Representations 165

5.1 Introduction 165

5.2 The Product-Space Representation and Product Spectrum Representation 166

5.3 Optical Imaging Systems 170

5.4 Bilinear Optical Systems 173

5.5 Noncoherent Imaging Systems 176

5.6 Tolerance to Focus Errors and to Spherical Aberration 178

5.7 Phase Conjugate Plates 183

References 189

6 Super Resolved Imaging in Wigner-Based Phase Space 193

6.1 Introduction 193

6.2 General Definitions 195

6.3 Description of SR 197

6.3.1 Code Division Multiplexing 200

6.3.2 Time Multiplexing 201

6.3.3 Polarization Multiplexing 202

6.3.4 Wavelength Multiplexing 203

6.3.5 Gray-Level Multiplexing 203

6.3.6 Description in the Phase-Space Domain 205

6.4 Conclusions 213

References 214

7 Radiometry, Wave Optics, and Spatial Coherence 217

7.1 Introduction 217

7.2 Conventional Radiometry 218

7.3 Lambertian Sources 221

7.4 Mutual Coherence Function 221

7.5 Stationary Phase Approximation 224

7.6 Radiometry and Wave Optics 226

7.7 Examples 231

7.7.1 Blackbody Radiation 231

7.7.2 Noncoherent Source 232

7.7.3 Coherent Wave Fields 233

7.7.4 Quasi-Homogeneous Wave Field 234

Acknowledgments 235

References 235

8 Rays and Waves 237

8.1 Introduction 237

8.2 Small-Wavelength Limit in the Position Representation I: Geometrical Optics 238

8.2.1 The Eikonal and Geometrical Optics 239

8.2.2 Choosing z as the Parameter 242

8.2.3 Ray-Optical Phase Space and the Lagrange Manifold 243

8.3 Small-Wavelength Limit in the Position Representation II: The Transport Equation and the Field Estimate 245

8.3.1 The Debye Series Expansion 245

8.3.2 The Transport Equation and Its Solution 245

8.3.3 The Field Estimate and Its Problems at Caustics 247

8.4 Flux Lines versus Rays 249

8.5 Analogy with Quantum Mechanics 250

8.5.1 Semiclassical Mechanics 251

8.5.2 Bohmian Mechanics and the Hydrodynamic Model 253

8.6 Small-Wavelength Limit in the Momentum Representation 254

8.6.1 The Helmholtz Equation in the Momentum Representation 254

8.6.2 Asymptotic Treatment and Ray Equations 256

8.6.3 Transport Equation in the Momentum Representation 258

8.6.4 Field Estimate 259

8.7 Maslov's Canonical Operator Method 260

8.8 Gaussian Beams and Their Sums 261

8.8.1 Parabasal Gaussian Beams 261

8.8.2 Sums of Gaussian Beams 264

8.9 Stable Aggregates of Flexible Elements 266

8.9.1 Derivation of the Estimate 266

8.9.2 Insensitivity to γ 269

8.9.3 Phase-Space Interpretation 270

8.10 A Simple Example 271

8.11 Concluding Remarks 275

References 275

9 Self-Imaging in Phase Space 279

9.1 Introduction 279

9.2 Phase-Space Optics Minimum Tool Kit 280

9.3 Self-Imaging of Paraxial Wavefronts 284

9.4 The Talbot Effect 285

9.5 The "Walk-off" Effect 289

9.6 The Fractional Talbot Effect 290

9.7 Matrix Formulation of the Fractional Talbot Effect 295

9.8 Point Source Illumination 298

9.9 Another Path to Self-Imaging 301

9.10 Self-Imaging and Incoherent Illumination 302

9.11 Summary 305

References 306

10 Sampling and Phase Space 309

10.1 Introduction 309

10.2 Notation and Some Initial Concepts 312

10.2.1 The Wigner Distribution Function and Properties 312

10.2.2 The Linear Canonical Transform and the WDF 314

10.2.3 The Phase-Space Diagram 314

10.2.4 Harmonics and Chirps and Convolutions 316

10.2.5 The Comb Function and Rect Function 318

10.2.5.1 Comb Functions 318

10.2.5.2 Rect Functions 320

10.3 Finite Supports 321

10.3.1 Band-limitedness in Fourier Domain 321

10.3.2 Band-limitedness and the LCT 322

10.3.3 Finite Space-Bandwidth Product-Compact Support in x and k 324

10.4 Sampling a Signal 325

10.4.1 Nyquist-Shannon Sampling 325

10.4.2 Generalized Sampling 328

10.5 Simulating an Optical System: Sampling at the Input and Output 329

10.6 Conclusion 332

References 332

11 Phase Space in, Ultrafast Optics 337

11.1 Introduction 337

11.2 Phase-Space Representations for Short Optical Pulses 338

11.2.1 Representation of Pulsed Fields 338

11.2.2 Pulse Ensembles and Correlation Functions 340

11.2.3 The Time-Frequency Phase Space 343

11.2.4 Phase-Space Representation of Paraxial Optical Systems 349

11.2.5 Temporal Paraxiality and the Chronocyclic Phase Space 353

11.3 Metrology of Short Optical Pulses 357

11.3.1 Measurement Strategies 357

11.3.2 Pulse Characterization Apparatuses as Linear Systems 358

11.3.3 Phase-Space Methods 361

11.3.3.1 Spectrographic Techniques 362

11.3.3.2 Tomographic Techniques 366

11.3.4 Interferornetric or Direct Techniques 369

11.3.4.1 Two-Pulse Double-Slit Interferometry 370

11.3.4.2 Shearing Interferometry 374

11.4 Conclusions 378

References 379

Index 385