Direction of energy flows at the focus of cylindrical vortex vector beams

In this work, we show analytically using the Richards-Wolf formalism and through the numerical simulation that at the sharp focus of a circularly polarized optical vortex, three energy flows occur: the direct longitudinal, reverse longitudinal, and azimuthal transverse ones. Moreover, the rotation of energy at different distances from the optical axis is different-handed. Therefore, only a part of the initial energy of the beam intersects the focal plane along the optical axis per unit time. The same portion (all other things being equal) intersects the focal plane along the positive direction of the optical axis if an optical vortex with cylindrical polarization is focused. The difference is that in the presence of an optical vortex, the transverse energy flow at the focus rotates around the optical axis, and if an optical vortex is absent (a beam with only cylindrical polarization), then the transverse flow is, on average, zero in the focal plane. But in some areas in the focal plane the flow is directed toward the optical axis, and in some areas – away from the optical axis.