## Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms

JOSA A, Vol. 22, Issue 5, pp. 917-927 (2005)

http://dx.doi.org/10.1364/JOSAA.22.000917

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### Abstract

By use of matrix-based techniques it is shown how the space–bandwidth product (SBP) of a signal, as indicated by the location of the signal energy in the Wigner distribution function, can be tracked through any quadratic-phase optical system whose operation is described by the linear canonical transform. Then, applying the regular uniform sampling criteria imposed by the SBP and linking the criteria explicitly to a decomposition of the optical matrix of the system, it is shown how numerical algorithms (employing interpolation and decimation), which exhibit both invertibility and additivity, can be implemented. Algorithms appearing in the literature for a variety of transforms (Fresnel, fractional Fourier) are shown to be special cases of our general approach. The method is shown to allow the existing algorithms to be optimized and is also shown to permit the invention of many new algorithms.

© 2005 Optical Society of America

**OCIS Codes**

(070.4560) Fourier optics and signal processing : Data processing by optical means

(080.2730) Geometric optics : Matrix methods in paraxial optics

(100.2000) Image processing : Digital image processing

(200.2610) Optics in computing : Free-space digital optics

(200.3050) Optics in computing : Information processing

(200.4560) Optics in computing : Optical data processing

(200.4740) Optics in computing : Optical processing

**History**

Original Manuscript: August 16, 2004

Manuscript Accepted: October 11, 2004

Published: May 1, 2005

**Citation**

Bryan M. Hennelly and John T. Sheridan, "Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms," J. Opt. Soc. Am. A **22**, 917-927 (2005)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-22-5-917

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