Spin-Orbital Transformation in a Tight Focus of an Optical Vortex with Circular Polarization

In the framework of the Richards-Wolf formalism, the spin-orbit conversion upon tight focusing of an optical vortex with circular polarization is studied. We obtain exact formulas which show what part of the total (averaged over the beam cross-section) longitudinal spin angular momentum is transferred to the total longitudinal orbital angular momentum in the focus. It is shown that the maximum part of the total longitudinal angular momentum that can be transformed into the total longitudinal orbital angular momentum is equal to half the beam power, and this maximum is reached at the maximum numerical aperture equal to one. We prove that the part of the spin angular momentum that transforms into the orbital angular momentum does not depend on the optical vortex topological charge. It is also shown that by virtue of spin-orbital conversion upon focusing, the total longitudinal energy flux decreases and partially transforms into the whole transversal (azimuthal) energy flow in the focus. Moreover, the longitudinal energy flux decreases by exactly the same amount that the total longitudinal spin angular momentum decreases.