Transverse and Longitudinal Energy Flows in a Sharp Focus of Vortex and Cylindrical Vector Beams

Abstract

It is shown in this work that, with strong focusing of a beam with optical vortex and circular polarization, three energy flows take place in the focal plane: direct longitudinal, reverse longitudinal and azimuthal transverse flows. Calculations are made analytically using the Richards–Wolf formalism and by numerical simulation. Moreover, the energy rotation at different lengths from the optical axis occurs in different directions. Therefore, the focal plane intersects along the optical axis only part of the initial beam energy per unit time. The same energy part (other things being equal) intersects the focal plane along the positive direction of the optical axis when an optical vortex with cylindrical polarization is focused. The difference is that, if an optical vortex is present, then the transverse energy flux at the focus rotates around the optical axis. If an optical vortex is not present (a beam with only cylindrical polarization), then the average transverse flow in the focal plane is zero, though, in some regions in the focal plane, the flow is directed towards the optical axis and, in other regions, away from it. This behavior of the transverse energy flow at the focus (flow direction towards the optical axis and away from the optical axis) of a cylindrical vector beam can be deemed another kind of Hall effect.