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

In a manner commensurate to 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. To address this, we recently demonstrated, by modifying Euler-angle parameterization, that an optical gadget consisting of two quarter-wave $q$-plates and one half-wave $q$-plate, each endowed with the same topological charge $q$, operates as a viable SU(2) gadget for the higher-order Poincaré sphere, contingent upon the fulfillment of the holonomy condition. This work presents the controlled navigation on the higher-order Poincaré sphere through the proposed architecture, formulated under the concept of an effective waveplate and thus can be referred to as an effective SU(2) gadget. The notion of an effective waveplate refers to a coaxial arrangement of multiple waveplates that, under specific constraints, can act as a single waveplate. 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 bear direct relevance to the deterministic control and engineering of structured light, encompassing polarization singularities, vector vortex beams and topological optical fields. Moreover, these results holds potential applications in the contemporary research frontiers in singular optics, spin-orbit photonics and quantum communication.