Phase and polarization singularities of nonparaxial Gaussian vortex beams

The analytical expression for nonparaxial Gaussian vortex beams propagating in free space is derived, which enables us to study phase and polarization singularities in nonparaxial vector wavefields. Differing from the polarization singularities formed by two transverse electric-field components in the paraxial regime, the polarization singularities can be formed by the transverse and longitudinal electric-field components of nonparaxial beams, and there exist C-points and L-lines. The variation of the beam parameters and propagation distance will result in a shift of phase and polarization singularities, but their position relation remains unchanged and the topological relationship holds true.