A ray-optical Poincar\'e sphere for structured Gaussian beams

TL;DR

This paper introduces a ray-optical framework using a Poincaré sphere to describe structured Gaussian beams, unifying various beam families and providing new insights and approximation methods for their propagation.

Contribution

It develops a general ray-based model on a Poincaré sphere for structured Gaussian beams, extending beyond traditional beam families and enabling amplitude approximations without diffraction calculations.

Findings

Unified description of Gaussian beam families

Incorporation of caustics via Poincaré sphere mapping

Effective amplitude approximation method

Abstract

A general family of structured Gaussian beams naturally emerges from a consideration of families of rays. These ray families, with the property that their transverse profile is invariant upon propagation (except for cycling of the rays and a global rescaling), have two parameters, the first giving a position on an ellipse naturally represented by a point on the Poincar\'e sphere (familiar from polarization optics), and the other determining the position of a curve traced out on this Poincar\'e sphere. This construction naturally accounts for the familiar families of Gaussian beams, including Hermite-Gauss, Laguerre-Gauss and Generalized Hermite-Laguerre-Gauss beams, but is far more general. The conformal mapping between a projection of the Poincar\'e sphere and the physical space of the transverse plane of a Gaussian beam naturally involves caustics. In addition to providing new insight into the physics of propagating Gaussian beams, the ray-based approach allows effective approximation of the propagating amplitude without explicit diffraction calculations.