Orbital angular momentum at the tight focus of a circularly polarized Gaussian beam

When tightly focusing a circularly polarized optical vortex, a phenomenon of spin-to-orbit conversion has been known to occur. As a rule, the longitudinal component of the spin angular momentum (SAM) vector is assumed to be converted into the longitudinal component of the orbital angular momentum (OAM) vector. In this work, we show that, due to the focusing, the original longitudinal SAM component, averaged over the beam cross-section, is partly converted to the transverse SAM component. In a similar way, the original longitudinal energy flow is partly converted, upon focusing, to the azimuthal component. Meanwhile, the longitudinal component of the OAM vector, averaged over the entire beam cross-section at the focus, increases exactly by the magnitude of the (canonical) averaged azimuthal orbital energy flow at the focus. We show that, upon focusing, the azimuthal energy flow is formed at the focus due to the fact that a right-handed circularly polarized light wave generates two optical vortices: a transverse left-handed circularly polarized vortex with topological charge 2 and a longitudinal vortex with topological charge 1.