Symmetry of electric spin angular momentum density in the tight focusing of linearly polarized vortex beams

Symmetry of electric spin angular momentum (SAM) density in the tight focusing of linearly polarized vortex beams is investigated in this paper. By calculating analytic expressions of SAM density, reflection symmetry and rotation symmetry of SAM density are found in the focal region. The vortex phase can break some symmetry, leading to non-zero longitudinal SAM. When the vortex phase vanishes, more symmetry will occur. It is proved that longitudinal SAM is zero when the topological charge is zero, and that transverse SAM is always zero. This paper is expected to advance our understanding of distribution of SAM density and optical spin-orbit coupling.