SU(2) Poincaré sphere: A generalized representation for multidimensional structured light

Structured light, as a general term for arbitrary states of amplitude, phase, and polarization in optical fields, is highly topical because of a myriad of applications it has fostered. A geometric description to graphically group classes of structured light has obvious benefits, with some notable advances in analogous Poincar\'e sphere (PS) mapping for both spin and orbital angular momentum (OAM), as well as ray-optical PS approaches for propagation-invariant fields, but all limited in dimensionality they can describe. Here we propose a generalized SU(2) PS for arbitrary dimensional structured light. The states on it represent extended families of beams with multidimensional ray-wave structures, accurately described by SU(2) symmetry groups. We outline how to construct this mapping theoretically, revealing insights into mode transformations involving OAM and geometric phase, and fully verify its efficacy experimentally. The generality of our approach is evident by the reduction to prior PS representations as special cases. We also demonstrate an extension of our approach to explain a more general high-dimensional vector beam. This construction naturally accounts for the salient topology of the classical PSs while bringing to more new degrees of freedom and dimensions for tailoring a larger variety of quantum-to-classical structured beams for a variety of applications.