SU(2) gadget for higher-order Poincaré sphere

Abstract: The combination of two quarter-wave plates and one half-wave plate, regardless of their sequential arrangement, constitutes a well-established universal SU(2) gadget capable of implementing all polarization transformations on the standard Poincar\'{e} sphere. However, there is no analogous system for realizing all polarization transformations on the higher-order Poincar\'{e} sphere, a member of a higher topological index space. This work demonstrates that an optical gadget, comprising two quarter-wave $q$-plates and one half-wave $q$-plate, arranged in any order, is an SU(2) gadget to realize arbitrary polarization evolution on the higher-order Poincar\'{e} sphere.