Spin-orbit conversion at the sharp focus of a circularly polarized light beam

It is known that when tightly focusing a circularly polarized optical vortex, spin-to-orbital angular momentum conversion occurs. It is usually said that the longitudinal component of the spin angular momentum (SAM) vector is transformed into the longitudinal component of the orbital angular momentum (OAM) vector. In this paper, it is shown that the initial longitudinal SAM, averaged over the beam cross-section, is preserved during focusing, and part of the longitudinal SAM component is transformed into the transverse SAM component. Likewise, the initial longitudinal energy flux averaged over the beam cross-section is preserved during focusing, and part of its longitudinal component is transformed into the azimuthal component. In this case, the longitudinal component of the OAM vector, averaged over the beam cross-section at the focus, increases by exactly the amount equal to the average azimuthal orbital (canonical) energy flux at the focus. The azimuthal energy flux at the focus is formed due to the fact that when focused, light with right-handed circular polarization forms two optical vortices, namely, a left-handed circular polarized transverse vortex with topological charge 2 and a longitudinal one with topological charge 1.