Characteristics of hollow Gaussian vortex beams in the far field

Based on the methods of vector angular spectrum and stationary phase, the analytical expressions of the TE and TM terms and the energy flux distributions of hollow Gaussian vortex beams in the far field are derived and used to analyze the phase singularities and energy flux distributions. It is shown that the far field characteristics of hollow Gaussian vortex beams depend on the control parameters. By varying the off-axis distance, both the optical vortex and the black nucleus in energy flux distributions will move. The radius of the circular edge dislocation and the dark ring position are related to the waist width, however the symmetrical spatial distribution of energy flux distributions is mainly affected by the off-axis distance.