Skyrmion number for a generalized vector Poincaré beam

We investigate two initial vector Poincaré-type light fields that can be treated as optical skyrmions – topological quasiparticles. For such fields, explicit expressions are obtained for the components of a 3D-vector skyrmionic field in the initial plane as well as for the skyrmion numbers, proportional to the topological charges of the optical vortices constituting the Poincaré beams. A new constructive formula is derived for effective calculation of the skyrmion number via the components of the normalized Stokes vector, rather than via the components of the vector skyrmion field. The skyrmion numbers computed by the well known and new formulae coincide. We also show that each component of the 3D-vector skyrmion field has a number equal to one third of the full skyrmion number. The numerical simulation results are consistent with the theoretical conclusions.