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

N. A. Lopez, I. Y. Dodin

TL;DR

This work develops 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, and proves that the discrete N IMT converges to the discrete MT when iterated, thereby allowing the NIMt to compute MTs that are not necessarily near-identity.

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.

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

TL;DR

This work develops 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, and proves that the discrete N IMT converges to the discrete MT when iterated, thereby allowing the NIMt to compute MTs that are not necessarily near-identity.

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.

Paper Structure

This paper contains 17 sections, 61 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Operator representation of the metaplectic transform
  3. Definition of the MT
  4. Manifestly unitary form of the MT
  5. Discrete metaplectic transform
  6. Derivation
  7. Unitarity verification
  8. Runtime estimate
  9. Discrete near-identity metaplectic transform
  10. Derivation
  11. Unitarity verification
  12. Runtime estimate
  13. Convergence of iterated discrete Near-identity metaplectic transform
  14. Derivation
  15. Local and global error convergence
  16. ...and 2 more sections

Figures (3)

  • Figure 1: (a) Comparing the error $\epsilon$ [Eq. ()] of the 2nd-order, 4th-order, and 6th-order dMTs applied to the first five HG modes [Eq. ()] for the first test case (). (b) Same as (a), but for the change in norm $\eta$ [Eq. ()] rather than $\epsilon$. (c) Comparing the initial and transformed field for the fifth HG mode computed using the 6th-order dMT. For all cases, $q$ is uniformly discretized on the interval $[-20, 20]$ with a step size of $0.1$, so $N = 401$. (d) - (f) Same as (a) - (c), but for the second test case (). (g) - (i) Same as (a) - (c), but for the third test case (). (j) - (l) Same as (a) - (c), but for the fourth test case ().
  • Figure 2: Local (single-step) error convergence of the 2nd-order dNIMT to the dMT for the first five HG modes. The expected convergence rate of $3$ is clearly observed. Although not shown, the 4th-order and the 6th-order dNIMT also exhibit the expected converge rate and in fact, the values of $\epsilon_r$ [Eq. ()] are nearly identical to those of the 2nd-order dNIMT.
  • Figure 3: (a) - (c) Global convergence of the 2nd-order, 4th-order, and 6th-order dNIMT to the respective dMT for the first five HG modes. The dashed line shows the intrinsic error of the corresponding dMT. As can be seen, the convergence rate achieves the expected value of $2$ before asymptoting to match the error of the dMT. (d) Global norm conservation of the 2nd-order, 4th-order, and 6th-order dNIMT for the first five HG modes. In all cases, the norm is conserved to near machine precision.