Swings and roundabouts: Optical Poincar\'e spheres for polarization and Gaussian beams

TL;DR

This paper explores the relationship between Poincaré spheres and Gaussian beams, interpreting structured modes as classical oscillators and their quantization, providing a new perspective on polarization and beam structure.

Contribution

It introduces a novel interpretation of Gaussian modes as eigenfunctions of classical oscillator operators, linking polarization and beam structure through Hamiltonian mechanics.

Findings

Structured Gaussian modes correspond to classical constants of motion.

Elliptic polarization interpreted via isotropic 2D harmonic oscillator.

Semiclassical quantization reveals classical ellipse families.

Abstract

The connection between Poincar\'e spheres for polariz-ation and Gaussian beams is explored, focusing on the interpretation of elliptic polarization in terms of the isotropic 2-dimensional harmonic oscillator in Hamiltonian mechanics, its canonical quantization and semiclassical interpretation. This leads to the interpretation of structured Gaussian modes, the Hermite-Gaussian, Laguerre-Gaussian and Generalized Hermite-Laguerre Gaussian modes as eigenfunctions of operators corresponding to the classical constants of motion of the 2-dimensional oscillator, which acquire an extra significance as families of classical ellipses upon semiclassical quantization.