Study of the Longitudinal Component of the Poynting Vector of Hybrid-Polarized Optical Vortices

In this paper, we study the sharp focusing of hybrid-polarized optical vortices using the Richards-Wolf formalism. The defining feature of these beams is a polarization state that combines azimuthal and circular properties—in the cross section of such a beam, depending on the azimuthal angle, the polarization changes from circular to azimuthal and back to circular. It is demonstrated theoretically and numerically that, although the total intensity forms a symmetric ring, the longitudinal component of the Poynting vector (Pz) exhibits a pronounced azimuthal asymmetry, indicating an asymmetric energy flow in the focal region. Central symmetry in the energy flux is a direct consequence of the hybrid polarization under sharp focusing, contrasting sharply with symmetric Pz distributions of uniformly or cylindrically polarized vortices.