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

Aykut Koç; Haldun M. Ozaktas; Lambertus Hesselink

doi:10.1364/JOSAA.27.001896

Journal of the Optical Society of America A
Vol. 27,
Issue 9,
pp. 1896-1908
(2010)
•https://doi.org/10.1364/JOSAA.27.001896

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

Aykut Koç, Haldun M. Ozaktas, and Lambertus Hesselink

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Aykut Koç, Haldun M. Ozaktas, and Lambertus Hesselink, "Fast and accurate algorithm for the computation of complex linear canonical transforms," J. Opt. Soc. Am. A 27, 1896-1908 (2010)

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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 $\sim N log N$ 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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Equations (66)

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Case 1	Case 2	Case 3
$b = 0$	$b \neq 0$ and $a = 0$	$b \neq 0$ and $a \neq 0$
$a_{r} \neq 0$	$b_{c} = 0$	$b_{c} \leq 0$
$d = 1 / a$	$b_{r} d_{c} \geq 0$	$b_{c} a_{r} - b_{r} a_{c} \leq b_{c}$
$a_{c} = 0$		$b_{c} d_{r} - b_{r} d_{c} \leq b_{c}$
$a_{r} c_{c} \geq 0$

	$T 1$	$T 2$	$T 3$	CFRT	DFT
$F 1$	$4.12 \times 10^{- 6}$	$2.19 \times 10^{- 6}$	$2.8 \times 10^{- 3}$	$1.24 \times 10^{- 5}$	$2.0 \times 10^{- 21}$
$F 2$	$3.73 \times 10^{- 4}$	$7.1 \times 10^{- 3}$	$1.4 \times 10^{- 3}$	$1.2 \times 10^{- 3}$	$6.2 \times 10^{- 4}$
$F 3$	0.53	0.35	0.26	0.22	1.2
$F 4$	$1.2 \times 10^{- 3}$	$4.96 \times 10^{- 2}$	$2.0 \times 10^{- 3}$	$2.2 \times 10^{- 3}$	$7.1 \times 10^{- 2}$
$F 5$	0.11	0.2	$8.0 \times 10^{- 3}$	$6.4 \times 10^{- 3}$	1.7

Case 1	Case 2	Case 3
$b = 0$	$b \neq 0$ and $a = 0$	$b \neq 0$ and $a \neq 0$
$a_{r} \neq 0$	$b_{c} = 0$	$b_{c} \leq 0$
$d = 1 / a$	$b_{r} d_{c} \geq 0$	$b_{c} a_{r} - b_{r} a_{c} \leq b_{c}$
$a_{c} = 0$		$b_{c} d_{r} - b_{r} d_{c} \leq b_{c}$
$a_{r} c_{c} \geq 0$

	$T 1$	$T 2$	$T 3$	CFRT	DFT
$F 1$	$4.12 \times 10^{- 6}$	$2.19 \times 10^{- 6}$	$2.8 \times 10^{- 3}$	$1.24 \times 10^{- 5}$	$2.0 \times 10^{- 21}$
$F 2$	$3.73 \times 10^{- 4}$	$7.1 \times 10^{- 3}$	$1.4 \times 10^{- 3}$	$1.2 \times 10^{- 3}$	$6.2 \times 10^{- 4}$
$F 3$	0.53	0.35	0.26	0.22	1.2
$F 4$	$1.2 \times 10^{- 3}$	$4.96 \times 10^{- 2}$	$2.0 \times 10^{- 3}$	$2.2 \times 10^{- 3}$	$7.1 \times 10^{- 2}$
$F 5$	0.11	0.2	$8.0 \times 10^{- 3}$	$6.4 \times 10^{- 3}$	1.7

Abstract

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Equations (66)

Journal of the Optical Society of America A