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 mth-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.