Higher topological index spaces in polarization optics

Substantial research in polarization optics has been devoted to topological phenomena since the seminal introduction of the Pancharatnam phase. The review begins by outlining several representative topological phenomena in optics. The topological phases and Pancharatnam–Berry phase-mediated interactions between the spin and orbital angular momenta, referred to as spin–orbit interactions, are discussed. We then narrate how these topological characteristics, formulated in various parameter spaces such as momentum, Stokes, and modal spaces, engender spatially structured topologies in optical fields. Furthermore, various topological constructs for structured light are discussed. Optical elements for higher-order Poincaré sphere beams operating under holonomy conditions enable the extension of conventional polarization optics into higher topological index spaces, forming the central theme of this review. In this context, the SU(2) description of polarization transformations is presented, along with holonomic and non-holonomic evolutions and their realization through structured optical elements such as q -plates. Structured light has also enabled the exploration of topological field configurations such as optical skyrmions, merons, and three-dimensional topological structures, including hopfions and torons. Meta-optics now enables the realization of previously unattainable element topologies in fabrication, allowing access to higher-index spaces.