Geometric Representation of Vector Vortex Beams: The Total Angular Momentum-Conserving Poincaré Sphere and Its Braid Clusters

Abstract

This paper presents the total angular momentum-conserving Poincaré sphere (TAM-C PS), which offers a novel framework for efficiently characterizing a wide range of vector vortex beams. Unlike other types of Poincaré spheres, the TAM-C PS achieves a better balance between generality and validity, while also providing clearer physical interpretation. By linking the poles of different spheres, the study also introduces two distinct categories of TAM-C PS braid clusters, enabling the representation of various Poincaré spheres within a unified framework. The Poincaré spheres include classical, higher-order, hybrid-order, Poincaré sphere with orbital angular momentum, and TAM-C PS. This is the first clear and unified approach to express multiple Poincaré spheres within a single framework. The TAM-C PS and its braid cluster can be employed to guide the creation of targeted vector vortex light beams, offer a geometric description of optical field evolution, and calculate the geometric phase of optical cyclic evolution.