Fast and accurate computation of normalized Bargmann transform

TL;DR

This paper introduces a normalized version of the Bargmann transform and develops fast, accurate computational methods for it, leveraging its relationships with other transforms like Gabor and Hermite-Gaussian functions.

Contribution

The paper presents a normalized Bargmann transform and novel computational algorithms, enhancing accuracy and efficiency in optical system analysis.

Findings

Normalized Bargmann transform is bounded near infinity.

Derived relationships with Gabor, Hermite-Gaussian, gyrator, and 2D LCT.

Proposed fast algorithms for computation and inversion.

Abstract

The linear canonical transform (LCT) was extended to complex-valued parameters, called complex LCT, to describe the complex amplitude propagation through lossy or lossless optical systems. Bargmann transform is a special case of the complex LCT. In this paper, we normalize the Bargmann transform such that it can be bounded near infinity. We derive the relationships of the normalized Bargmann transform to Gabor transform, Hermite-Gaussian functions, gyrator transform, and 2D nonseparable LCT. Several kinds of fast and accurate computational methods of the normalized Bargmann transform and its inverse are proposed based on these relationships.