Energy Backflow in Tightly Focused Fractional Order Vector Vortex Beams with Binary Topological Charges

Using the Richards-Wolf diffraction integral, the longitudinal energy evolution on the focal plane of the fractional order vector vortex (FOVV) beams was studied. These beams possessed a vortex topological charge n and a polarization topological charge m, and were subjected to tight focusing through a larger numerical aperture. Our investigation revealed the existence of backflow energy when the binary topological charges n and m satisfied the conditions of n + m = 2 or n - m = -2. The component circularly polarized vortex beams of e(-i2f)e(<^>)+ (i.e., the minus second-order vortex right circularly polarized beam) and e(i2f)e(<^>)- (i.e., the second-order vortex left circularly polarized beam) played significant roles in the generation of reverse energy flux at the focal region. For FOVV beams with binary topological charges n and m, whose sum and difference were integers, the longitudinal energy on the focal plane exhibited axial symmetry. If the sum or the difference of the topological charges n and m was not an integer, the axisymmetric longitudinal energy on the focal plane was disrupted.