Laguerre-Gaussian Modes and the Wigner Transform

TL;DR

This paper explores the mathematical relationship between Laguerre-Gaussian modes and Hermite-Gaussian modes through the Wigner transform, revealing how these modes are interconnected in laser physics and quantum optics.

Contribution

It provides a new closed-form expression for the Wigner transform of Laguerre-Gaussian modes, offering an alternative derivation and deeper insight into their structure.

Findings

Wigner transform of LG modes is derived in closed form.

LG modes are shown to be Wigner transforms of Hermite-Gaussian modes.

The Wigner transform interchanges creation and annihilation operators.

Abstract

Recent developments in laser physics have called renewed attention to Laguerre-Gaussian (LG) beams of paraxial light. In this paper we consider the corresponding LG modes for the two-dimensional harmonic oscillator, which appear in the transversal plane at the laser beam's waist. We see how they arise as Wigner transforms of Hermite-Gaussian modes, and we proceed to find a closed form for their own Wigner transforms, providing an alternative to the methods of Simon and Agarwal. Our main observation is that the Wigner transform intertwines the creation and annihilation operators for the two classes of modes.