Page 12 - Phase Space Optics Fundamentals and Applications

Contents xi

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