Open AccessRays, beams and diffraction in a discrete phase space: Wilson bases
Author(s) -
John M. Arnold
Publication year - 2002
Publication title -
optics express
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.394
H-Index - 271
ISSN - 1094-4087
DOI - 10.1364/oe.10.000716
Subject(s) - diffraction , basis (linear algebra) , fourier transform , sampling (signal processing) , discrete fourier transform (general) , phase (matter) , phase space , representation (politics) , space (punctuation) , amplitude , physics , discrete fourier series , optics , phase retrieval , mathematical analysis , mathematics , fourier analysis , quantum mechanics , geometry , computer science , short time fourier transform , detector , politics , political science , law , operating system
For high-frequency fields, which can be separated into superpositions of a few distinct components with rapidly varying phases and slowly varying amplitudes, phase-space representations exhibit a strong localisation in which the coefficients are negligible over most of the phase space. This leads, potentially, to a very large reduction in the computational cost of computing propagators. Using the Windowed Fourier Transform, a number of fundamental problems from diffraction theory are studied using a representation of continuous wavefields by superpositions of beams that are continuously parameterised in phasespace and which propagate along ray trajectories. The existence of noncanonical WFT coefficients is observed, due to the nonuniqeness of the WFT. Numerical evaluations require discrete finite bases. The discrete Wilson basis is generated by a discrete sampling of the windowed Fourier Transform in the phase-space. The sampling is optimal, in the sense that the smallest number of coefficients is generated in an orthogonal basis.