Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals

Date
2010-05-12
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Source Title
Journal of the Optical Society of America A: Optics and Image Science, and Vision
Print ISSN
1084-7529
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Publisher
Optical Society of America
Volume
27
Issue
6
Pages
1288 - 1302
Language
English
Type
Article
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Abstract

We report a fast and accurate algorithm for numerical computation of two-dimensional non-separable linear canonical transforms (2D-NS-LCTs). Also known as quadratic-phase integrals, this class of integral transforms represents a broad class of optical systems including Fresnel propagation in free space, propagation in gradedindex media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic/astigmatic/non- orthogonal cases. The general two-dimensional non-separable case poses several challenges which do not exist in the one-dimensional case and the separable two-dimensional case. The algorithm takes ∼ñ log ñ time, where ñ is the two-dimensional space-bandwidth product of the signal. Our method properly tracks and controls the space-bandwidth products in two dimensions, in order to achieve information theoretically sufficient, but not wastefully redundant, sampling required for the reconstruction of the underlying continuous functions at any stage of the algorithm. Additionally, we provide an alternative definition of general 2D-NS-LCTs that shows its kernel explicitly in terms of its ten parameters, and relate these parameters bidirectionally to conventional ABCD matrix parameters.

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Keywords
Algorithms, Bandwidth, Integral equations, Optical systems, ABCD matrix, Continuous functions, Free space, Fresnel propagation, Graded index, Integral transform, Linear canonical transform, Numerical computations, Space-bandwidth product, Thin lens, Two dimensional spaces, Two-dimension, Two dimensional
Citation
Published Version (Please cite this version)