Page 8 - Phase Space Optics Fundamentals and Applications

Contents vii

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 Poincar´e 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