Preface | p. xiii |

Wigner Distribution in Optics | p. 1 |

Introduction | p. 1 |

Elementary Description of Optical Signals and Systems | p. 2 |

Impulse Response and Coherent Point-Spread Function | p. 3 |

Mutual Coherence Function and Cross-Spectral Density | p. 3 |

Some Basic Examples of Optical Signals | p. 4 |

Wigner Distribution and Ambiguity Function | p. 5 |

Definitions | p. 5 |

Some Basic Examples Again | p. 7 |

Gaussian Light | p. 9 |

Local Frequency Spectrum | p. 11 |

Some Properties of the Wigner Distribution | p. 12 |

Inversion Formula | p. 12 |

Shift Covariance | p. 12 |

Radiometric Quantities | p. 12 |

Instantaneous Frequency | p. 14 |

Moyal's Relationship | p. 15 |

One-Dimensional Case and the Fractional Fourier Transformation | p. 15 |

Fractional Fourier Transformation | p. 15 |

Rotation in Phase Space | p. 16 |

Generalized Marginals-Radon Transform | p. 16 |

Propagation of the Wigner Distribution | p. 18 |

First-Order Optical Systems-Ray Transformation Matrix | p. 18 |

Phase-Space Rotators-More Rotations in Phase Space | p. 19 |

More General Systems-Ray-Spread Function | p. 21 |

Geometric-Optical Systems | p. 22 |

Transport Equations | p. 23 |

Wigner Distribution Moments in First-Order Optical Systems | p. 24 |

Moment Invariants | p. 25 |

Moment Invariants for Phase-Space Rotators | p. 26 |

Symplectic Moment Matrix-The Bilinear ABCD Law | p. 28 |

Measurement of Moments | p. 29 |

Coherent Signals and the Cohen Class | p. 29 |

Multicomponent Signals-Auto-Terms and Cross-Terms | p. 30 |

One-Dimensional Case and Some Basic Cohen Kernels | p. 32 |

Rotation of the Kernel | p. 33 |

Rotated Version of the Smoothed Interferogram | p. 35 |

Conclusion | p. 40 |

References | p. 40 |

Ambiguity Function in Optical Imaging | p. 45 |

Introduction | p. 45 |

Intensity Spectrum of a Fresnel Diffraction Pattern Under Coherent Illumination | p. 47 |

General Formulation | p. 47 |

Application to Simple Objects | p. 48 |

Contrast Transfer Functions | p. 49 |

Propagation through a Paraxial Optical System in Terms of AF | p. 49 |

Propagation in Free Space | p. 49 |

Transmission through a Thin Object | p. 50 |

Propagation in a Paraxial Optical System | p. 51 |

The AF in Isoplanatic (Space-Invariant) Imaging | p. 52 |

The AF of the Image of an Incoherent Source | p. 53 |

Derivation of the Zernike-Van Cittert Theorem from the Propagation of the AF | p. 53 |

Partial Coherence Properties in the Image of an Incoherent Source | p. 54 |

The Pupil-AF as a Generalization of the OTF | p. 54 |

Phase-Space Tomography | p. 55 |

Another Possible Approach to AF Reconstruction | p. 56 |

Propagation-Based Holographic Phase Retrieval from Several Images | p. 58 |

Fresnel Diffraction Images as In-Line Holograms | p. 58 |

Application to Phase Retrieval and X-Ray Holotomography | p. 59 |

Conclusion | p. 60 |

References | p. 60 |

Rotations in Phase Space | p. 63 |

Introduction | p. 63 |

First-Order Optical Systems and Canonical Integral Transforms | p. 64 |

Canonical Integral Transforms and Ray Transformation Matrix Formalism | p. 64 |

Modified Iwasawa Decomposition of Ray Transformation Matrix | p. 66 |

Canonical Transformations Producing Phase-Space Rotations | p. 67 |

Matrix and Operator Description | p. 67 |

Signal Rotator | p. 69 |

Fractional Fourier Transform | p. 69 |

Gyrator | p. 73 |

Other Phase-Space Rotators | p. 74 |

Properties of the Phase-Space Rotators | p. 74 |

Some Useful Relations for Phase-Space Rotators | p. 75 |

Similarity to the Fractional Fourier Transform | p. 76 |

Shift Theorem | p. 77 |

Convolution Theorem | p. 77 |

Scaling Theorem | p. 77 |

Phase-Space Rotations of Selected Functions | p. 78 |

Eigenfunctions for Phase-Space Rotators | p. 80 |

Some Relations for the Eigenfunctions | p. 80 |

Mode Presentation on Orbital Poincaré Sphere | p. 82 |

Optical Setups for Basic Phase-Space Rotators | p. 84 |

Flexible Optical Setups for Fractional FT and Gyrator | p. 85 |

Flexible Optical Setup for Image Rotator | p. 87 |

Applications of Phase-Space Rotators | p. 88 |

Generalized Convolution | p. 88 |

Pattern Recognition | p. 90 |

Chirp Signal Analysis | p. 94 |

Signal Encryption | p. 94 |

Mode Converters | p. 95 |

Beam Characterization | p. 96 |

Gouy Phase Accumulation | p. 100 |

Conclusions | p. 101 |

Acknowledgments | p. 102 |

References | p. 102 |

The Radon-Wigner Transform in Analysis, Design, and Processing of Optical Signals | p. 107 |

Introduction | p. 107 |

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

Definition and Basic Properties | p. 108 |

Optical Implementation of the RWT: The Radon-Wigner Display | p. 117 |

Analysis of Optical Signals and Systems by Means of the RWT | p. 122 |

Analysis of Diffraction Phenomena | p. 122 |

Computation of Irradiance Distribution along Different Paths in Image Space | p. 122 |

Parallel Optical Display of Diffraction Patterns | p. 132 |

Inverting RWT: Phase-Space Tomographic Reconstruction of Optical Fields | p. 134 |

Merit Functions of Imaging Systems in Terms of the RWT | p. 138 |

Axial Point-Spread Function (PSF) and Optical Transfer Function (OTF) | p. 138 |

Polychromatic OTF | p. 143 |

Polychromatic Axial PSF | p. 146 |

Design of Imaging Systems and Optical Signal Processing by Means of RWT | p. 151 |

Optimization of Optical Systems: Achromatic Design | p. 151 |

Controlling the Axial Response: Synthesis of Pupil Masks by RWT Inversion | p. 156 |

Signal Processing through RWT | p. 157 |

Acknowledgments | p. 162 |

References | p. 162 |

Imaging Systems: Phase-Space Representations | p. 165 |

Introduction | p. 165 |

The Product-Space Representation and Product Spectrum Representation | p. 166 |

Optical Imaging Systems | p. 170 |

Bilinear Optical Systems | p. 173 |

Noncoherent Imaging Systems | p. 176 |

Tolerance to Focus Errors and to Spherical Aberration | p. 178 |

Phase Conjugate Plates | p. 183 |

References | p. 189 |

Super Resolved Imaging in Wigner-Based Phase Space | p. 193 |

Introduction | p. 193 |

General Definitions | p. 195 |

Description of SR | p. 197 |

Code Division Multiplexing | p. 200 |

Time Multiplexing | p. 201 |

Polarization Multiplexing | p. 202 |

Wavelength Multiplexing | p. 203 |

Gray-Level Multiplexing | p. 203 |

Description in the Phase-Space Domain | p. 205 |

Conclusions | p. 213 |

References | p. 214 |

Radiometry, Wave Optics, and Spatial Coherence | p. 217 |

Introduction | p. 217 |

Conventional Radiometry | p. 218 |

Lambertian Sources | p. 221 |

Mutual Coherence Function | p. 221 |

Stationary Phase Approximation | p. 224 |

Radiometry and Wave Optics | p. 226 |

Examples | p. 231 |

Blackbody Radiation | p. 231 |

Noncoherent Source | p. 232 |

Coherent Wave Fields | p. 233 |

Quasi-Homogeneous Wave Field | p. 234 |

Acknowledgments | p. 235 |

References | p. 235 |

Rays and Waves | p. 237 |

Introduction | p. 237 |

Small-Wavelength Limit in the Position Representation I: Geometrical Optics | p. 238 |

The Eikonal and Geometrical Optics | p. 239 |

Choosing z as the Parameter | p. 242 |

Ray-Optical Phase Space and the Lagrange Manifold | p. 243 |

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

The Debye Series Expansion | p. 245 |

The Transport Equation and Its Solution | p. 245 |

The Field Estimate and Its Problems at Caustics | p. 247 |

Flux Lines versus Rays | p. 249 |

Analogy with Quantum Mechanics | p. 250 |

Semiclassical Mechanics | p. 251 |

Bohmian Mechanics and the Hydrodynamic Model | p. 253 |

Small-Wavelength Limit in the Momentum Representation | p. 254 |

The Helmholtz Equation in the Momentum Representation | p. 254 |

Asymptotic Treatment and Ray Equations | p. 256 |

Transport Equation in the Momentum Representation | p. 258 |

Field Estimate | p. 259 |

Maslov's Canonical Operator Method | p. 260 |

Gaussian Beams and Their Sums | p. 261 |

Parabasal Gaussian Beams | p. 261 |

Sums of Gaussian Beams | p. 264 |

Stable Aggregates of Flexible Elements | p. 266 |

Derivation of the Estimate | p. 266 |

Insensitivity to ¿ | p. 269 |

Phase-Space Interpretation | p. 270 |

A Simple Example | p. 271 |

Concluding Remarks | p. 275 |

References | p. 275 |

Self-Imaging in Phase Space | p. 279 |

Introduction | p. 279 |

Phase-Space Optics Minimum Tool Kit | p. 280 |

Self-Imaging of Paraxial Wavefronts | p. 284 |

The Talbot Effect | p. 285 |

The "Walk-off" Effect | p. 289 |

The Fractional Talbot Effect | p. 290 |

Matrix Formulation of the Fractional Talbot Effect | p. 295 |

Point Source Illumination | p. 298 |

Another Path to Self-Imaging | p. 301 |

Self-Imaging and Incoherent Illumination | p. 302 |

Summary | p. 305 |

References | p. 306 |

Sampling and Phase Space | p. 309 |

Introduction | p. 309 |

Notation and Some Initial Concepts | p. 312 |

The Wigner Distribution Function and Properties | p. 312 |

The Linear Canonical Transform and the WDF | p. 314 |

The Phase-Space Diagram | p. 314 |

Harmonics and Chirps and Convolutions | p. 316 |

The Comb Function and Rect Function | p. 318 |

Comb Functions | p. 318 |

Rect Functions | p. 320 |

Finite Supports | p. 321 |

Band-limitedness in Fourier Domain | p. 321 |

Band-limitedness and the LCT | p. 322 |

Finite Space-Bandwidth Product-Compact Support in x and k | p. 324 |

Sampling a Signal | p. 325 |

Nyquist-Shannon Sampling | p. 325 |

Generalized Sampling | p. 328 |

Simulating an Optical System: Sampling at the Input and Output | p. 329 |

Conclusion | p. 332 |

References | p. 332 |

Phase Space in, Ultrafast Optics | p. 337 |

Introduction | p. 337 |

Phase-Space Representations for Short Optical Pulses | p. 338 |

Representation of Pulsed Fields | p. 338 |

Pulse Ensembles and Correlation Functions | p. 340 |

The Time-Frequency Phase Space | p. 343 |

Phase-Space Representation of Paraxial Optical Systems | p. 349 |

Temporal Paraxiality and the Chronocyclic Phase Space | p. 353 |

Metrology of Short Optical Pulses | p. 357 |

Measurement Strategies | p. 357 |

Pulse Characterization Apparatuses as Linear Systems | p. 358 |

Phase-Space Methods | p. 361 |

Spectrographic Techniques | p. 362 |

Tomographic Techniques | p. 366 |

Interferornetric or Direct Techniques | p. 369 |

Two-Pulse Double-Slit Interferometry | p. 370 |

Shearing Interferometry | p. 374 |

Conclusions | p. 378 |

References | p. 379 |

Index | p. 385 |

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