Fig. 1

Definition of Skyrmions and their interaction with various polarization aberrations. a The various symmetry properties of the field on its boundary that allow for compactification and the corresponding compactified manifold. The symmetry properties are shown here via polygons, with arrows indicating pairwise identification of different edges oriented in the direction of the arrow. Notice that certain symmetries allow the field to be tesselated, which gives topological character to periodic structured fields. As mentioned in the main text, due to the topological nature of compactification, one can freely deform this polygon and, in the case of unbounded fields, one can consider the boundary at infinity. Possible Stokes fields satisfying the relevant symmetries are also shown. It is worth noting that a field with a constant value on its boundary satisfies every symmetry property simultaneously, however, there exist fields, non-uniform on their boundary, that can also be compactified, as shown in the figure. Throughout this paper, Stokes fields are depicted using hue to signify azimuthal angle \(\tan \theta ={s}_{2}/{s}_{1}\) and saturation to represent height s₃ (similar to ref. ¹⁵). b Schematic depicting the effects of a complex spatially varying medium on an incident optical Skyrmion field. In this paper, systems exhibiting spatially varying retardance, diattenuation, depolarization, and combinations of the aforementioned are carefully considered. Note that the boundary of the incident field and the corresponding aberrations are of key importance in guaranteeing topological protection. Throughout this work, fields with a constant value on their boundaries are considered for physical reasons expanded on in the main text