Energy backflow in a focal spot of the cylindrical vector beam

Using Richards-Wolf formulae it is shown that a tightly focused azimuthally-radially polarized m-th order laser beam with an arbitrary apodization function produces a reverse energy flow in the focal plane ($m=2$). If $m=3$, the reverse energy flow on the axis is equal to zero, increasing in the axis vicinity as the square of the distance to the axis. The azimuthally-radially polarized beam of the m-th order is an example of a polarization vortex. Previously, the reverse energy flow in the focus was obtained only for circularly polarized vortex beams with the topological charge m. Using the FDTD method and the Richards-Wolf formulae, we show numerically that in the focus of a zone plate such laser beams produce regions where the Poynting vector is opposite to the direction of the beam propagation.