Fresnel and Fraunhofer diffraction of a Gaussian beam with several polarization singularities

Alongside phase singularities (optical vortices), there may be light fields with polarization singularities (PS), i.e. isolated intensity s with radial, azimuthal, or radial-azimuthal polarization around them. Here, we study Gaussian beams with several arbitrarily located PS. An analytic expression is obtained for their complex amplitude. A partial case is studied when the PS are at the vertices of a regular polygon. If the beam has one or two PS, then these are points with radial polarization. If there are four PS, then two of the points will have azimuthal polarization. It is shown that while propagating in free space, the PS can appear only in a discrete set of planes, in contrast to the phase singularities, which exist in any transverse plane. In the case of two PS, it is shown that their polarization transforms from radial in the initial plane to azimuthal in the far field.