Spin angular momentum at the sharp focus of a cylindrical vector vortex beam

Sharp focusing of a light field with double (phase and polarization) singularity is studied. Using the Richards-Wolf method, an exact analytical expression for the longitudinal projection of the spin angular momentum (SAM) vector at the focus is obtained. The expression derived suggests that $4(n–1)$ subwavelength regions are formed at the focus, where $n$ is the cylindrical vector beam order, with their centers located on a certain circle centered on the optical axis. Notably, the SAM projections are found to have the opposite sign in the neighboring regions. This means that in the neighboring focal regions, the light has alternating left or right elliptical polarization (manifestation of a spin Hall effect). At the center of the focal spot near the optical axis, the field is right-handed elliptically polarized at $m>0$, or left-handed elliptically polarized at $m<0$, where m is the vortex charge. The total longitudinal spin, i.e., the longitudinal SAM component averaged over the beam-cross section, is zero and preserved upon focusing. Due to the beam containing an optical vortex with charge $m$, the transverse energy flow rotates on a spiral path near the focal plane, rotating on a circle in the focal plane. The rotation direction near the optical axis is counterclockwise for $m>0$, and clockwise for $m<0$.