Energy backflow in the focus of an optical vortex

By employing Richards-Wolf formulae for an aplanatic system, we show that when tightly focusing a left-hand circularly polarized optical vortex with the topological charge 2 and an arbitrary apodization function, an on-axis energy backflow occurs in the focus plane, which is verified by a negative optical-axis projection of the Poynting vector. This result is numerically reconfirmed using the rigorous FDTD-based analysis of diffraction of a left-hand circularly polarized plane wave by a spiral zone plate with the topological charge 2 and a numerical aperture of similar to 1. Significantly, the back and direct flows of energy are comparable in magnitude. While the on-axis energy backflow is shown to take place on the entire axis, its maximum is attained in the focal plane, quickly decaying with distance from the focus. The on-axis depth of backflow on which it drops by a factor of two is nearly the same as the depth of focus, while the transverse spot in which the energy backflow occurs is approximately equal to the Airy disk.