Effective SU(2) gadget: holonomic walk on higher-order Poincaré sphere

In a manner commensurate with the SU(2) gadget for the Poincaré sphere, which involves a combination of two quarter-wave plates and one half-wave plate, regardless of their sequential order, an analogous construct for the higher-order Poincaré sphere had long remained elusive. This work presents controlled navigation on the higher-order Poincaré sphere through the proposed architecture, a combination of two q^Q-plates and one q^H-plate in any order, formulated under the concept of an effective waveplate, and thus can be referred to as an effective SU(2) gadget. In this gadget, the relative alignment of the offset angles of the constituent q-plates emerges as the decisive parameter governing systematic navigation on the higher-order Poincaré sphere. This study bears direct relevance to the deterministic control and engineering of structured light, encompassing polarization singularities, vector vortex beams, and topological optical fields.