Poincare Skyrmion Number

We discuss two source vector fields of Poincare-beam type that can be looked upon as optical skyrmions, i.e. topological quasiparticles. We derive explicit analytical relationships that describe projections of a three-dimensional (3D) skyrmion vector field in the source plane and skyrmion numbers, which are shown to be pro-portional to the topological charges of constituent optical vortices of the Poincare beams. We also propose a new constructive formula as an effective tool for calculating the skyrmion number via normalized Stokes vector projections rather than skyrmion vector field projections. The skyrmion numbers calculated using the familiar and newly proposed formulae coincide. Numbers of each projection of a 3D skyrmion vector field are shown to comprise a third of the full skyrmion number. The theoretical conclusions are validated by a numerical simulation. The non-uniform linear polarization in the skyrmion cross-section depends on the azimuthal angle and can be used to form a spiral microrelief due to the mass transfer of molecules on the surface of the material.