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

TL;DR

This paper introduces a new discrete metaplectic transform that is exactly unitary and can be computed efficiently in linear time, improving numerical stability and convergence in optical and geometrical optics applications.

Contribution

The authors develop a discrete, exactly unitary metaplectic transform and a linear-time approximate version that converges to the true transform, enhancing stability and computational efficiency.

Findings

The discrete NIMT converges to the discrete MT when iterated.

The new algorithms are demonstrated with practical examples.

The discrete NIMT is exactly unitary, unlike previous approximations.

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.