The Geometry of Paraxial Vector Beams

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.