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Exactly unitary discrete representations of the metaplectic transform for linear-time algorithms

Journal Article · · Journal of the Optical Society of America. A, Optics, Image Science, and Vision
DOI:https://doi.org/10.1364/josaa.417412· OSTI ID:1804649
 [1];  [2]
  1. Princeton Univ., NJ (United States)
  2. Princeton Univ., NJ (United States); Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
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The metaplectic transform (MT), a generalization of the Fourier transform sometimes called the linear canonical transform, is a tool used ubiquitously in modern optics, for example, when calculating the transformations of light beams in paraxial optical systems. The MT is also an essential ingredient of the geometrical-optics modeling of caustics that we recently proposed. In particular, this application relies on the near-identity MT (NIMT); however, the NIMT approximation used so far is not exactly unitary and leads to numerical instability. Here, we develop a discrete MT that is exactly unitary, and approximate it to obtain a discrete NIMT that is also unitary and can be computed in linear time. We prove that the discrete NIMT converges to the discrete MT when iterated, thereby allowing the NIMT to compute MTs that are not necessarily near-identity. Finally, we then demonstrate the new algorithms with a series of examples.
Research Organization:
Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States)
Sponsoring Organization:
USDOE
Grant/Contract Number:
AC02-09CH11466
OSTI ID:
1804649
Alternate ID(s):
OSTI ID: 1777629
Journal Information:
Journal of the Optical Society of America. A, Optics, Image Science, and Vision, Journal Name: Journal of the Optical Society of America. A, Optics, Image Science, and Vision Journal Issue: 5 Vol. 38; ISSN 1084-7529
Publisher:
Optical Society of America (OSA)Copyright Statement
Country of Publication:
United States
Language:
English

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