Energy backflow in tightly focused hybrid vector beams with noncircular polarization-mapping tracks on the Poincaré sphere

We report the formation of on-axis energy backflow in the focus of tightly focused hybrid vector beams featuring noncircular mapping tracks on the Poincar\'e sphere for azimuthal polarization states. Using the Richards-Wolf vectorial diffraction integral, we give the general expressions for all components of the electric and magnetic fields, as well as the expression for the longitudinal component of the Poynting vector in the focal plane under $xy$ basis vectors. We reveal that the on-axis energy backflow will appear when the amplitude modulation factor $l$ and the topological charge $m$ of the vortex phase carried by basis vectors satisfy the condition $|l|\ensuremath{-}|m|=\ifmmode\pm\else\textpm\fi{}2$. And, we show that the longitudinal Poynting vector is not circularly symmetric but related to the azimuthal angle in the focal plane for the hybrid vector beams with nonzero $l$ and $m$. Furthermore, it is found that the on-axis energy backflow can also be observed in the focus of the hybrid vector beams under other basis vectors beyond the $xy$ case. More intriguingly, we additionally uncover that there is on-axis energy backflow in the focus of a linearly polarized optical vortex with the topological charge $|m|=2$, where the near-axis energy flows spirally along the optical axis. The numerical simulation results are in good agreement with the theoretical analyses.