Properties of a strongly focused Gaussian beam with an off-axis vortex

The intensity distribution and the phase properties, especially the Gouy phase and the phase singularities are studied in a strongly focused Gaussian beam with an off-axis vortex. The symmetry relation of the focused field is also derived. It is found that the off-axis vortex induces a rotation of the field pattern, the transverse focal shift, and the asymmetric distribution of the phase singularities. Our results also show that the initial position of the off-axis vortex in the incident beam strongly influences the distance of the transverse focal shift, but does not have an effect on the Gouy phase along the central axis.