Superposition of an optical vortex and a plane wave with linear polarization states at the tight focus

We analyze the sharp focusing of the superposition of a vortex laser beam with topological charge n and linear polarization and a plane wave with the same linear polarization directed along the horizontal axis. Using the Richards-Wolf formalism, analytical expressions are obtained for the intensity distribution and longitudinal projection of the spin angular momentum (SAM) in the focal plane. It is shown that for even and odd numbers n the intensity and SAM have different symmetries: for even n they are symmetric with respect to both Cartesian axes, and for odd n they are symmetric only with respect to the vertical axis. The intensity distribution has 2n local maxima at the focus, and the intensity on the optical axis is nonzero for any n. The distribution of the longitudinal SAM (spin density) in the focal plane has (n+2) subwavelength regions with a positive SAM and (n+2) regions with a negative SAM, the centers of which alternately locate on a circle of a certain radius centered at the optical axis. Such an alternating-spin pattern demonstrates the spin Hall effect at the focus. At the focus, the negative and positive spins are mutually compensated, meaning that the total spin is zero.