Sharp focusing of on-axis superposition of a high-order cylindrical vector beam and a beam with linear polarization

In this work, the sharp focusing of a laser beam whose initial polarization pattern is formed by superposition of a cylindrical $m$th-order vector beam and a homogeneous linearly polarized beam is considered theoretically and numerically. Although in the source plane of such a beam both the angular spin momentum and the third Stokes parameter are equal to zero, we reveal that given odd $m$, subwavelength local regions are formed in the focal plane, where transverse vortex energy flows occur and the third Stokes parameter (the on-axis component of the angular spin momentum) is non-zero. Thus, at odd $m$, at the focus of such a beam there are –sub-regions with elliptical polarization of light with alternating handedness in the adjacent sub-regions (clockwise and counterclockwise). This phenomenon can be interpreted as a variant of an optical Hall effect. We note that at even $m$, the field at the focus is linearly polarized at every point and no transverse energy flow is observed.