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 $e^{-i2\varphi }{\hat{e}}_{+}$ (i.e., the minus second-order vortex right circularly polarized beam) and $e^{i2\varphi }{\hat{e}}_{-}$ (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.

In this study, we demonstrate both theoretically and experimentally second harmonic generation (SHG) of a twisted vector vortex optical field (TVVOF) using a nonlinear type-II phase-matched β-BaB₂O₄ (BBO) crystal. Our study introduces a novel method to manipulate SHG by independently modulating the two orthogonal polarization components of a TVVOF. This flexibility in controlling SHG can be achieved through accurate experimental adjustments of the polarization components. Furthermore, we reveal that the SHG can be dynamically tuned by varying the angle between the polarization direction of the optical field and the principal axis of the BBO crystal via rotation. These findings provide a new approach for the flexible manipulation of SHG in structured vector optical fields, which have potential applications in optical communication, quantum optics, and photonic device engineering.

Based on the two-dimensional (2D) nonlinear Schrödinger equation, we investigate the collapse dynamics of a vector vortex optical field (VVOF) in nonlinear Kerr media with parity–time (PT)-symmetric modulation. The critical power for the collapse of a VVOF in a Kerr-ROLP medium (Kerr medium with a real optical lattice potential) is derived. Numerical simulations indicate that the number, position, propagation distance, and collapse profile of the collapse of a VVOF in sine and cosine parity–time-symmetric potential (SCPT) Kerr media are closely related to the modulation depth, initial powers, and the topological charge number of a VVOF. The VVOF collapses into symmetric shapes during propagation in a Kerr-ROLP medium, and collapse shapes are sensitively related to the density of the PT-symmetric optical lattice potential. In addition, due to gain–loss, the VVOF will be distorted during propagation in the Kerr-SCPT medium, forming an asymmetric shape of collapse. The power evolution of the VVOF in a Kerr-SCPT medium as a function of the transmission distance with different modulating parameters and topological numbers is analyzed in detail. The introduction of PT-symmetric optical lattice potentials into nonlinear Kerr materials may provide a new approach to manipulate the collapse of the VVOF.