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

Abstract

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 was recently proposed by the authors. 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. We then demonstrate the new algorithms with a series of examples.

10 pages, 3 figures, 1 appendix

References in corpus (5)

• Restoring geometrical optics near caustics using sequenced metaplectic transforms
• Two-dimensional nonseparable discrete linear canonical transform based on CM-CC-CM-CC decomposition
• Metaplectic geometrical optics for modeling caustics in uniform and nonuniform media
• Simulating Pitch Angle Scattering Using An Explicitly Solvable Energy Conserving Algorithm
• An Explicitly Solvable Energy-Conserving Algorithm for Pitch-Angle Scattering in Magnetized Plasmas

Cited by in corpus (4)

• Steepest-descent algorithm for simulating plasma-wave caustics via metaplectic geometrical optics
• Metaplectic geometrical optics for ray-based modeling of caustics: Theory and algorithms
• Quasioptical modeling of wave beams with and without mode conversion: IV. Numerical simulations of waves in dissipative media
• Demonstration of Metaplectic Geometrical Optics for Reduced Modeling of Plasma Waves