Spin Angular Momentum at the Focus of a Superposition of an Optical Vortex and a Plane Wave with Linear Polarizations

Abstract

In this paper, tight focusing of a superposition of a vortex laser beam with topological charge n with linear polarization and a plane wave with the same linear polarization directed along the horizontal axis is considered. Using the Richards–Wolf formalism, analytical expressions are obtained for the intensity distribution and longitudinal projection of the spin angular momentum in the focal plane. It is shown that for even and odd numbers n, the intensity and the spin angular momentum have different symmetries: for even n they are symmetric about both Cartesian axes, and for odd n they are symmetric only about the vertical axis. The intensity distribution has n local maxima at the focus, and it is nonzero on the optical axis for any n. The distribution of the longitudinal spin angular momentum (spin density) in the focal plane has (n + 2) subwavelength regions with a positive spin angular momentum and (n + 2) regions with a negative spin angular momentum, the centers of which alternately lie on a circle of a certain radius with a center on the optical axis. This spin distribution with different signs demonstrates the spin Hall effect at the focus. Negative and positive spins are mutually compensated, and the total spin is equal to zero at the focus. We have shown that by changing the topological charge of the optical vortex, it is possible to control the spin Hall effect at the focus, that is, to change the number of regions with spins of different signs.

Keywords

• spin angular momentum
• Richards–Wolf formulas
• spin Hall effect
• optical vortex
• plane wave
• linear polarization