Effect of Diffraction on Wigner Distributions of Optical Fields and how to Use It in Optical Resonator Theory. II -- Unstable Resonators

TL;DR

This paper investigates how diffraction affects Wigner distributions in optical fields, focusing on complex fractional Fourier transforms and their application to unstable resonator theory, revealing hyperbolic rotation effects in phase-space.

Contribution

It introduces a novel analysis of diffraction effects using complex fractional Fourier transforms and applies this to improve understanding of unstable resonator behavior.

Findings

Diffraction induces hyperbolic rotations in phase-space representations.

Complex fractional Fourier transforms can model field transfer in unstable resonators.

The matrix decomposition reveals new insights into optical field transformations.

Abstract

The second part of the article is devoted to field transfers by diffraction that are represented by fractional Fourier transformations whose orders are complex numbers. The corresponding effects on the Wigner distributions associated with optical fields are still represented by $4\times4$ matrices operating on the scaled phase-space, but unlike matrices involved in the first part, those matrices decompose into two matrices that essentially represent 2--dimensional hyperbolic rotations, not elliptical rotations. The result is applied to the theory of unstable resonators.