Two-dimensional nonseparable linear canonical transform: sampling theorem and unitary discretization

Liang Zhao; John J. Healy; John T. Sheridan

doi:10.1364/JOSAA.31.002631

Journal of the Optical Society of America A
Vol. 31,
Issue 12,
pp. 2631-2641
(2014)
•https://doi.org/10.1364/JOSAA.31.002631

Two-dimensional nonseparable linear canonical transform: sampling theorem and unitary discretization

Liang Zhao, John J. Healy, and John T. Sheridan

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Liang Zhao, John J. Healy, and John T. Sheridan, "Two-dimensional nonseparable linear canonical transform: sampling theorem and unitary discretization," J. Opt. Soc. Am. A 31, 2631-2641 (2014)

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Table of Contents Category
- Fourier Optics and Signal Processing
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- ABCD transforms (070.2590)
- Discrete optical signal processing (070.2025)
- Numerical approximation and analysis (000.4430)
- Digital holography (090.1995)

About this Article
History
- Original Manuscript: August 4, 2014
- Revised Manuscript: October 5, 2014
- Manuscript Accepted: October 6, 2014
- Published: November 11, 2014

Abstract

The two-dimensional (2D) nonseparable linear canonical transform (NS-LCT) is a unitary, linear integral transform that relates the input and output monochromatic, paraxial scalar wave fields of optical systems characterized by a $4 \times 4$ ray tracing matrix. In addition to the obvious generalizations of the 1D LCT (which are referred to as separable), the 2D-NS-LCT can represent a variety of nonaxially symmetric optical systems including the gyrator transform and image rotation. Unlike the 1D LCT, the numerical approximation of the 2D-NS-LCT has not yet received extensive attention in the literature. In this paper, (1) we develop a sampling theorem for the general 2D-NS-LCT which generalizes previously published sampling theorems for the 1D case and (2) we determine which sampling rates may be chosen to ensure that the obvious discrete transform is unitary.

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2D-Separable		2D-Nonseparable
FT	$(\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \end{matrix})$	Gyrator	$(\begin{matrix} \cos θ & 0 & 0 & \sin θ \\ 0 & \cos θ & \sin θ & 0 \\ 0 & - \sin θ & \cos θ & 0 \\ - \sin θ & 0 & 0 & \cos θ \end{matrix})$
FST	$(\begin{matrix} 1 & 0 & z & 0 \\ 0 & 1 & 0 & z \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$	Coupling	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & τ & 1 & 0 \\ τ & 0 & 0 & 1 \end{matrix})$
FRT	$(\begin{matrix} \begin{matrix} \cos θ_{x} & 0 & \sin θ_{x} & 0 \\ 0 & \cos θ_{y} & 0 & \sin θ_{y} \end{matrix} \\ \begin{matrix} - \sin θ_{x} & 0 & \cos θ_{x} & 0 \\ 0 & - \sin θ_{y} & 0 & \cos θ_{y} \end{matrix} \end{matrix})$	Coordinate (general)	$(\begin{matrix} a_{11} & a_{12} & 0 & 0 \\ a_{21} & a_{22} & 0 & 0 \\ 0 & 0 & \frac{a_{22}}{\det (A)} & \frac{- a_{21}}{\det (A)} \\ 0 & 0 & \frac{- a_{12}}{\det (A)} & \frac{a_{11}}{\det (A)} \end{matrix})$
Chirp	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ - 1 / f_{x} & 0 & 1 & 0 \\ 0 & - 1 / f_{y} & 0 & 1 \end{matrix})$	Rotation	$(\begin{matrix} \cos θ & \sin θ & 0 & 0 \\ - \sin θ & \cos θ & 0 & 0 \\ 0 & 0 & \cos θ & \sin θ \\ 0 & 0 & - \sin θ & \cos θ \end{matrix})$
Scaling- $x$	$(\begin{matrix} 1 / s_{x} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & s_{x} & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$	Shearing- $x$	$(\begin{matrix} 1 & s h_{x} & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & - s h_{x} & 1 \end{matrix})$
Scaling- $y$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 / s_{y} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & s_{y} \end{matrix})$	Shearing- $y$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ s h_{y} & 1 & 0 & 0 \\ 0 & 0 & 1 & - s h_{y} \\ 0 & 0 & 0 & 1 \end{matrix})$

	LCTs	Sampling Theorems	Unitary Rate Chosen
S-LCT when $B \neq 0$	2D-FT	$L_{x^{'}} \leq 1 / T_{x}$ , and $L_{y^{'}} \leq 1 / T_{y}$	$T_{x^{'}} = 1 / L_{x}$ , and $T_{y^{'}} = 1 / L_{y}$
	2D-FRT	$L_{x^{'}} \leq \sin θ_{x} / T_{x}$ , and $L_{y^{'}} \leq \sin θ_{y} / T_{y}$	$T_{x^{'}} = \sin θ_{x} / L_{x}$ , and $T_{y^{'}} = \sin θ_{y} / L_{y}$
	2D-FST	$L_{x^{'}} \leq λ z / T_{x}$ , and $L_{y^{'}} \leq λ z / T_{y}$	$T_{x^{'}} = λ z / L_{x}$ , and $T_{y^{'}} = λ z / L_{y}$
	2D-S-LCT	$L_{x^{'}} \leq b_{11} / T_{x}$ , and $L_{y^{'}} \leq b_{22} / T_{y}$	$T_{x^{'}} = b_{11} / L_{x}$ , and $T_{y^{'}} = b_{22} / L_{y}$
S-LCT when $B = 0$	2D Chirp	$L_{x^{'}} \leq L_{x}$ , and $L_{y^{'}} \leq L_{y}$	$T_{x^{'}} = T_{x}$ , and $T_{y^{'}} = T_{y}$
	Scaling- $x$	$L_{x^{'}} \leq L_{x} / S_{x}$ , and $L_{y^{'}} \leq L_{y}$	$T_{x^{'}} = T_{x} / S_{x}$ , and $T_{y^{'}} = T_{y}$
	Scaling- $y$	$L_{x^{'}} \leq L_{x}$ , and $L_{y^{'}} \leq L_{y} / S_{y}$	$T_{x^{'}} = T_{x}$ , and $T_{y^{'}} = T_{y} / S_{y}$
NS-LCT when $B \neq 0$	Gyrator	$L_{x^{'}} \leq \sin θ / T_{y}$ , and $L_{y^{'}} \leq \sin θ / T_{x}$	$T_{x^{'}} = \sin θ / L_{y}$ , and $T_{y^{'}} = \sin θ / L_{x}$
NS-LCT when $B = 0$	Coordinate	$L_{x^{'}} \leq L_{x} \sqrt{a_{11}^{2} + a_{21}^{2}}$ , and $L_{y^{'}} \leq L_{y} \sqrt{a_{12}^{2} + a_{22}^{2}}$	$T_{x^{'}} = T_{x} \sqrt{a_{11}^{2} + a_{21}^{2}}$ , and $T_{y^{'}} = T_{y} \sqrt{a_{12}^{2} + a_{22}^{2}}$
	Coupling	$L_{x^{'}} \leq L_{x}$ , and $L_{y^{'}} \leq L_{y}$	$T_{x^{'}} = T_{x}$ , and $T_{y^{'}} = T_{y}$
	Shearing- $x$	$L_{x^{'}} \leq L_{x}$ , and $L_{y^{'}} \leq L_{y} \sqrt{1 + s h_{x}^{2}}$	$T_{x^{'}} = T_{x}$ , and $T_{y^{'}} = T_{y} \sqrt{1 + s h_{x}^{2}}$
	Shearing- $y$	$L_{x^{'}} \leq L_{x} \sqrt{1 + s h_{y}^{2}}$ , and $L_{y^{'}} \leq L_{y}$	$T_{x^{'}} = T_{x} \sqrt{1 + s h_{y}^{2}}$ , and $T_{y^{'}} = T_{y}$

2D-Separable		2D-Nonseparable
FT	$(\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \end{matrix})$	Gyrator	$(\begin{matrix} \cos θ & 0 & 0 & \sin θ \\ 0 & \cos θ & \sin θ & 0 \\ 0 & - \sin θ & \cos θ & 0 \\ - \sin θ & 0 & 0 & \cos θ \end{matrix})$
FST	$(\begin{matrix} 1 & 0 & z & 0 \\ 0 & 1 & 0 & z \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$	Coupling	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & τ & 1 & 0 \\ τ & 0 & 0 & 1 \end{matrix})$
FRT	$(\begin{matrix} \begin{matrix} \cos θ_{x} & 0 & \sin θ_{x} & 0 \\ 0 & \cos θ_{y} & 0 & \sin θ_{y} \end{matrix} \\ \begin{matrix} - \sin θ_{x} & 0 & \cos θ_{x} & 0 \\ 0 & - \sin θ_{y} & 0 & \cos θ_{y} \end{matrix} \end{matrix})$	Coordinate (general)	$(\begin{matrix} a_{11} & a_{12} & 0 & 0 \\ a_{21} & a_{22} & 0 & 0 \\ 0 & 0 & \frac{a_{22}}{\det (A)} & \frac{- a_{21}}{\det (A)} \\ 0 & 0 & \frac{- a_{12}}{\det (A)} & \frac{a_{11}}{\det (A)} \end{matrix})$
Chirp	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ - 1 / f_{x} & 0 & 1 & 0 \\ 0 & - 1 / f_{y} & 0 & 1 \end{matrix})$	Rotation	$(\begin{matrix} \cos θ & \sin θ & 0 & 0 \\ - \sin θ & \cos θ & 0 & 0 \\ 0 & 0 & \cos θ & \sin θ \\ 0 & 0 & - \sin θ & \cos θ \end{matrix})$
Scaling- $x$	$(\begin{matrix} 1 / s_{x} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & s_{x} & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$	Shearing- $x$	$(\begin{matrix} 1 & s h_{x} & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & - s h_{x} & 1 \end{matrix})$
Scaling- $y$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 / s_{y} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & s_{y} \end{matrix})$	Shearing- $y$	$(\begin{matrix} 1 & 0 & 0 & 0 \\ s h_{y} & 1 & 0 & 0 \\ 0 & 0 & 1 & - s h_{y} \\ 0 & 0 & 0 & 1 \end{matrix})$

	LCTs	Sampling Theorems	Unitary Rate Chosen
S-LCT when $B \neq 0$	2D-FT	$L_{x^{'}} \leq 1 / T_{x}$ , and $L_{y^{'}} \leq 1 / T_{y}$	$T_{x^{'}} = 1 / L_{x}$ , and $T_{y^{'}} = 1 / L_{y}$
	2D-FRT	$L_{x^{'}} \leq \sin θ_{x} / T_{x}$ , and $L_{y^{'}} \leq \sin θ_{y} / T_{y}$	$T_{x^{'}} = \sin θ_{x} / L_{x}$ , and $T_{y^{'}} = \sin θ_{y} / L_{y}$
	2D-FST	$L_{x^{'}} \leq λ z / T_{x}$ , and $L_{y^{'}} \leq λ z / T_{y}$	$T_{x^{'}} = λ z / L_{x}$ , and $T_{y^{'}} = λ z / L_{y}$
	2D-S-LCT	$L_{x^{'}} \leq b_{11} / T_{x}$ , and $L_{y^{'}} \leq b_{22} / T_{y}$	$T_{x^{'}} = b_{11} / L_{x}$ , and $T_{y^{'}} = b_{22} / L_{y}$
S-LCT when $B = 0$	2D Chirp	$L_{x^{'}} \leq L_{x}$ , and $L_{y^{'}} \leq L_{y}$	$T_{x^{'}} = T_{x}$ , and $T_{y^{'}} = T_{y}$
	Scaling- $x$	$L_{x^{'}} \leq L_{x} / S_{x}$ , and $L_{y^{'}} \leq L_{y}$	$T_{x^{'}} = T_{x} / S_{x}$ , and $T_{y^{'}} = T_{y}$
	Scaling- $y$	$L_{x^{'}} \leq L_{x}$ , and $L_{y^{'}} \leq L_{y} / S_{y}$	$T_{x^{'}} = T_{x}$ , and $T_{y^{'}} = T_{y} / S_{y}$
NS-LCT when $B \neq 0$	Gyrator	$L_{x^{'}} \leq \sin θ / T_{y}$ , and $L_{y^{'}} \leq \sin θ / T_{x}$	$T_{x^{'}} = \sin θ / L_{y}$ , and $T_{y^{'}} = \sin θ / L_{x}$
NS-LCT when $B = 0$	Coordinate	$L_{x^{'}} \leq L_{x} \sqrt{a_{11}^{2} + a_{21}^{2}}$ , and $L_{y^{'}} \leq L_{y} \sqrt{a_{12}^{2} + a_{22}^{2}}$	$T_{x^{'}} = T_{x} \sqrt{a_{11}^{2} + a_{21}^{2}}$ , and $T_{y^{'}} = T_{y} \sqrt{a_{12}^{2} + a_{22}^{2}}$
	Coupling	$L_{x^{'}} \leq L_{x}$ , and $L_{y^{'}} \leq L_{y}$	$T_{x^{'}} = T_{x}$ , and $T_{y^{'}} = T_{y}$
	Shearing- $x$	$L_{x^{'}} \leq L_{x}$ , and $L_{y^{'}} \leq L_{y} \sqrt{1 + s h_{x}^{2}}$	$T_{x^{'}} = T_{x}$ , and $T_{y^{'}} = T_{y} \sqrt{1 + s h_{x}^{2}}$
	Shearing- $y$	$L_{x^{'}} \leq L_{x} \sqrt{1 + s h_{y}^{2}}$ , and $L_{y^{'}} \leq L_{y}$	$T_{x^{'}} = T_{x} \sqrt{1 + s h_{y}^{2}}$ , and $T_{y^{'}} = T_{y}$

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Journal of the Optical Society of America A

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