The near-axis backflow of energy in a tightly focused optical vortex with circular polarization

Using the Richards-Wolf formulae for a diffractive lens, we show that in the focal plane of a sharply focused left-hand circularly polarized optical vortex with the topological charge 2 there is an on-axis backflow of energy (as testified by the negative axial projection of the Poynting vector). The result is corroborated by the FDTD-aided rigorous calculation of the diffraction of a left-hand circularly polarized plane wave by a vortex zone plate with the topological charge 2 and the NA about 1. Moreover, the back- and direct flows of energy are comparable in magnitude. We have also shown that while the backflow of energy takes place on the entire optical axis, it has a maximum value in the focal plane, rapidly decreasing with distance from the focus. The length of a segment along the optical axis at which the modulus of the backflow drops by half (the depth of backflow) almost coincides with the depth of focus, and the transverse circle in which the energy flow is reversed roughly coincides with the Airy disk.