Spin Hall Effect While Focusing an Optical Vortex and a Plane Wave with Linear Polarisations

The sharp focusing of the superposition of a plane wave with the linear polarisation (LP) and an optical vortex with topological charge (TC) m and with the same LP directed along the horizontal axis is considered. Applying the Richards–Wolf formalism, analytical formulas are derived for the intensity and the spin angular momentum (SAM) lengthwise component in the focal plane. It is demonstated that for odd and even TCs m the SAM and the intensity have different symmetries: for even m they are symmetric about both Cartesian axes, and for odd m they are symmetric only about the vertical axis. The intensity pattern has 2m local maxima in the focal plane. The intensity on the optical axis for any TC m is nonzero. The pattern of the lengthwise SAM (spin density) in the focal plane has 2(m + 2) subwavelength areas in the half of which the SAM has positive value while in others it has negative value. The centers of these areas alternately lie on a circle of a certain radius with a center on the optical axis. This spin pattern with different signs shows the spin Hall effect in the focal plane. The total negative and positive spin in the focal plane are mutually compensated and equal to zero.