On-axis energy backflow in the focal fields of vector vortex beams with arbitrary initial phase

Energy backflow is an intriguing counterintuitive phenomenon, which has been reported in the focal region of light beams with phase or polarization singularities. Although the scenarios involving vector vortex beams with both polarization and phase singularities have been discussed, the impact of the light beam's initial phase on energy backflow remains not fully clear. Here, we theoretically prove and numerically demonstrate that the longitudinal component of the Poynting vector in the focal plane is independent of the initial phase of incident light beams. We further reveal the general condition for the emergence of on-axis energy backflow near the focus of such light beams with arbitrary initial phase: specifically, the polarization order <i>l</i> and the phase topological charge <i>m</i> need to satisfy <i>l</i> ± <i>m</i> = 2. And we unveil the existing Poynting vector singularities associated with on-axis energy backflow. More remarkably, the exceptional cases are uncovered, where <i>l</i> = 1 and <i>m</i> = ±1. And we find that it is possible to construct on-axis energy backflow by appropriately modulating the amplitude of the incident light beam. Furthermore, we propose a general method to achieve a strong longitudinal electric field on the optical axis by utilizing light beams satisfying the condition with <i>l</i> ± <i>m</i> = 1. The results enrich the toolkit for constructing and modulating light fields as well as energy flow distributions in the focal region.