The Geometry of Paraxial Vector Beams

TL;DR

This paper reveals a fundamental connection between structured light in paraxial vector beams and topological field theory, describing their geometry with a non-Abelian gauge theory and proposing experimental detection methods.

Contribution

It introduces a novel geometrical framework for vector beams using non-Abelian Chern-Simons theory, extending beyond traditional models and linking topological concepts to optical phenomena.

Findings

Vector beams are described by an $SU(2)$ principal bundle over spacetime.

Non-Abelian Chern-Simons gauge theory models the polarization and spatial mode structure.

Proposes experimental detection of non-Abelian phases via Wilson lines.

Abstract

This work unveils a novel and fundamental connection between structured light and topological field theory by showing how the natural geometrical setting for paraxial vector beams is that of a $SU(2)$ principal bundle over $\mathbb{R}^{2+1}$. Going beyond the usual high-order Poincar\'e sphere approach, we show how the nonseparable structure of polarisation and spatial modes in vector beams is naturally described by a non-Abelian Chern-Simons gauge theory. In this framework, we link the Chern-Simons charge to spin-orbit coupling, and we propose a simple way to experimentally detect the presence of non-Abelian phases through Wilson lines. This new insight on vector beams opens new possibilities for realising and probing topological quantum field theories using classical optics, as well as it lays the foundation for implementing topologically protected classical and quantum information protocols with structured light.