Fast and accurate algorithm for the computation of complex linear canonical transforms

Date
2010-08-05
Advisor
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Source Title
Journal of the Optical Society of America A: Optics and Image Science, and Vision
Print ISSN
1084-7529
Electronic ISSN
Publisher
Optical Society of America
Volume
27
Issue
9
Pages
1896 - 1908
Language
English
Type
Article
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Abstract

A fast and accurate algorithm is developed for the numerical computation of the family of complex linear canonical transforms (CLCTs), which represent the input-output relationship of complex quadratic-phase systems. Allowing the linear canonical transform parameters to be complex numbers makes it possible to represent paraxial optical systems that involve complex parameters. These include lossy systems such as Gaussian apertures, Gaussian ducts, or complex graded-index media, as well as lossless thin lenses and sections of free space and any arbitrary combinations of them. Complex-ordered fractional Fourier transforms (CFRTs) are a special case of CLCTs, and therefore a fast and accurate algorithm to compute CFRTs is included as a special case of the presented algorithm. The algorithm is based on decomposition of an arbitrary CLCT matrix into real and complex chirp multiplications and Fourier transforms. The samples of the output are obtained from the samples of the input in ∼N log N time, where N is the number of input samples. A space-bandwidth product tracking formalism is developed to ensure that the number of samples is information-theoretically sufficient to reconstruct the continuous transform, but not unnecessarily redundant.

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Keywords
Algorithms, Bandwidth, Eigenvalues and eigenfunctions, Fast Fourier transforms, Optical systems, Complex number, Complex parameter, Fractional Fourier transforms, Free space, Gaussian apertures, Gaussians, Graded index, Input sample, Input-output, Linear canonical transform, Lossless, Lossy systems, matrix, Number of samples, Numerical computations, Paraxial optical systems, Phase systems, Space-bandwidth product, Thin lens, Mathematical transformations
Citation
Published Version (Please cite this version)