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

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      Author
      Koç A.
      Haldun M. Özaktaş
      Hesselink, L.
      Date
      2010-08-05
      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
      9
      Pages
      1896 - 1908
      Language
      English
      Type
      Article
      Item Usage Stats
      128
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      97
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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.
      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
      Permalink
      http://hdl.handle.net/11693/22227
      Published Version (Please cite this version)
      http://dx.doi.org/10.1364/JOSAA.27.001896
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      • Department of Electrical and Electronics Engineering 3601
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