Focusing of linearly polarized optical vortex and a hall effect

Polarization of a higher-order cylindrical vector beam (CVB) is known to be locally linear. The higher the beam order, the larger number of full circles the local linear polarization vector makes around the optical axis. It is also known that the CVB with radially symmetric amplitude has zero spin angular momentum (SAM) and zero orbital angular momentum (OAM) both in the initial plane and in the focal plane (because in both Cartesian components of the vector field, the angular derivative of phase is zero). We show here that near the focal plane of the CVB (i.e. before and beyond the focus), an even number of local subwavelength areas with rotating polarization vectors are generated. In addition, in the neighboring areas, the polarization vectors are rotating in the opposite directions. Thus, the longitudinal components of the SAM vector in such neighboring areas are of different sign. After passing through the focal plane, the rotation direction of the polarization vector at each point of the beam cross-section changes to the opposite one. Such a spatial separation of the left and right rotation of the polarization vectors is a manifestation of the optical spin Hall effect.