Spin angular momentum at the tight focus of a cylindrical vector beam with an imbedded optical vortex

We discuss sharp focusing of a light field that possess a double (phase + polarization) singularity. An exact analytical expression for the longitudinal projection of the spin angular momentum (SAM) vector at the focus is deduced based on a Richards-Wolf approach. The expression deduced suggests that at the focus, 4(n-1) subwavelength regions are generated (n denotes the cylindrical beam order), which are arranged on a circle centered on the optical axis. In the adjacent regions, the SAM is revealed to change sign, meaning that the light has alternating left- and right-handed elliptic polarization (an optical spin Hall effect) in these regions. At the near-axis focus center, the light is shown to have right-handed elliptical polarization at m>0 or left-handed elliptic polarization at m<0, where m denotes the topological charge of the optical vortex. The total longitudinal spin, i.e. the cross-section-averaged longitudinal SAM component, is shown to be zero and conserve upon focusing. Thanks to the presence of an optical vortex with topological charge m, the near-focus transverse energy flow is found to rotate on a spiral path, rotating on a circle at the focus. In the optical axis vicinity, the rotation occurs anticlockwise at m>0 or clockwise at m<0. No data was used for the research described in the article.