Far-field properties of nonparaxial Gaussian beams with a pair of vortices

The analytical far-field expressions for the TE and TM terms and energy flux distributions of nonparaxial Gaussian beams with a pair of vortices are derived by using vector angular spectrum representation and stationary phase method. The far-field properties including phase singularities and energy flux distributions of the TE and TM terms and whole beam are studied in detail. It is shown that there exist a saddle point and phase singularities that depend on the off-axis distance, waist width, and propagation distance. By suitably varying the off-axis distance, the motion and variation of topological charge of phase singularities may take place. The results are interpreted and compared with the previous work.