Tight Focusing Properties of Gaussian Beams with Asymmetric Off-Axis Bi-Phase Singularities

The center of a vortex beam is a phase singularity, and in general, the vortex axis (i.e., the phase singularity) of the vortex beam is on the optical axis of the beam. Due to consideration errors, however, a generated vortex beam is off-axis to some extent, and therefore, it is of great application value to investigate the characteristics of beams with off-axis phase singularities. In this paper, the tight focusing properties of radially polarized Gaussian beams with asymmetric off-axis bi-phase singularities upon passing through a lens with a large numerical aperture are investigated on the basis of the Richards-Wolf vector diffraction theory. In particular, the influences of off-axis distance and the vortex topological charge number on the focusing field are analyzed. The results reveal that the relative quantities of two off-axis distances and the corresponding two topological charge numbers have a consistent influence on the offset direction of the maximum light intensity of the focusing field. In addition, it is found that dark core splitting occurs in the focusing field of high-order vortex beams. Thus, multiple dark cores appear, and the number of dark cores is equal to the sum of two vortex topological charge numbers subtracted by 1.