Covering spheres and hyperspheres: skyrmionic-like optical fields

The term “skyrmion”, borrowed from particle theory, denotes a physical field that can be locally described by a real, normalized vector constrained to a sphere, such that over a physical region this vector covers completely this sphere, yielding a configuration that is topologically stable. Recently the term “Stokes skyrmion” has been used to denote types of paraxial optical fields with spatial polarization variations that cover the complete Poincaré sphere (a 2-sphere). Here we describe families of periodic Stokes skyrmionic fields in which the sense of this coverage (i.e. the sign of the Jacobian between a spatial plane and the sphere) does not change. This includes mappings that are conformal, and others that are invariant under propagation. We also discuss short pulses that cover the sphere over a longitudinal one. Finally, we extend the concept from two to four dimensions by considering non-paraxial polarization states, which are described by either a 4-sphere or a complex projective plane, and show that simple optical fields composed of 5 plane waves span over 4D space-time region this parameter space completely.