## Fast numerical algorithm for the linear canonical transform

JOSA A, Vol. 22, Issue 5, pp. 928-937 (2005)

http://dx.doi.org/10.1364/JOSAA.22.000928

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### Abstract

The linear canonical transform (LCT) describes the effect of any quadratic phase system (QPS) on an input optical wave field. Special cases of the LCT include the fractional Fourier transform (FRT), the Fourier transform (FT), and the Fresnel transform (FST) describing free-space propagation. Currently there are numerous efficient algorithms used (for purposes of numerical simulation in the area of optical signal processing) to calculate the discrete FT, FRT, and FST. All of these algorithms are based on the use of the fast Fourier transform (FFT). In this paper we develop theory for the discrete linear canonical transform (DLCT), which is to the LCT what the discrete Fourier transform (DFT) is to the FT. We then derive the fast linear canonical transform (FLCT), an NlogN algorithm for its numerical implementation by an approach similar to that used in deriving the FFT from the DFT. Our algorithm is significantly different from the FFT, is based purely on the properties of the LCT, and can be used for FFT, FRT, and FST calculations and, in the most general case, for the rapid calculation of the effect of any QPS.

© 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

**Citation**

Bryan M. Hennelly and John T. Sheridan, "Fast numerical algorithm for the linear canonical transform," J. Opt. Soc. Am. A **22**, 928-937 (2005)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-22-5-928

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