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

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      Author
      Koç A.
      Haldun M. Özaktaş
      Hesselink, L.
      Date
      2010-05-12
      Source Title
      Journal of the Optical Society of America A: Optics and Image Science, and Vision
      Print ISSN
      1084-7529
      Publisher
      Optical Society of America
      Volume
      27
      Issue
      6
      Pages
      1288 - 1302
      Language
      English
      Type
      Article
      Item Usage Stats
      137
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      113
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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.
      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
      Permalink
      http://hdl.handle.net/11693/22309
      Published Version (Please cite this version)
      http://dx.doi.org/10.1364/JOSAA.27.001288
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      • Department of Electrical and Electronics Engineering 3601
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