Geometric Hybrid Poincar\'e Sphere with Variable Poles

Abstract

We propose a geometric hybrid Poincar\'e sphere (GHPS) as a unified geometrical framework for describing structured photon states with independently controllable spin angular momentum (SAM) and orbital angular momentum (OAM). Unlike the conventional higher-order Poincar\'e sphere, in which the SAM and OAM are intrinsically coupled through fixed basis states, the GHPS is constructed by defining its poles as direct products of arbitrary orthogonal bases on the Poincar\'e sphere (PS) and orbital Poincar\'e sphere (OPS) and by superposing these pole states. Using numerical simulations, we analyze representative GHPS states and show that the GHPS spherical coordinates govern the amplitude ratio and relative phase between the pole bases. This framework enables spatially inhomogeneous polarization distributions and intensity patterns, including nonseparable structures in which polarization and intensity are intrinsically intertwined, and provides a systematic state-space description for the coherent geometrical control of advanced structured light fields.