Energy backflow in the focus of a light beam with phase or polarization singularity

We show that if a nonzero E-field intensity is found in the focus of a circularly polarized optical vortex with phase singularity or a higher-order azimuthally polarized beam with polarization singularity, an energy backflow will occur at the center of the focal plane. Using the Richards-Wolf formulas, which describe the electromagnetic field near the tight focus of an aplanatic system containing an ideal spherical lens, we prove that the negative on-axis propagation of light occurs when focusing (i) a left-hand circularly polarized optical vortex with the topological charge $m=2$ or (ii) a conventional second-order azimuthally polarized radially symmetric laser beam. Notably, in the former case, the negative energy flow rotates on a spiral about the optical axis, while in the latter case, the Poynting vector is perpendicular to the focal plane at any point. The theoretical conclusions are confirmed by numerical simulation.