We have studied diverse vector and hybrid light fields, including those with multiple polarization singularities, and have derived relationships for polarization singularity indices based on the familiar Berry formula, which is normally utilized to find the topological charge of a scalar vortex light field. The fields with pure polar-angle-dependent polarization in the beam cross-section are shown to feature either polarization singularity lines outgoing from the center or a single polarization singularity point at the beam center. The fields with pure radial-variable-dependent polarization are shown to have no polarization singularities and zero polarization index. The vector fields with both polar-angle- and radial-variable-dependent polarization are shown to have multiple polarization singularity points that are scattered across the cross-section. A vector field with higher-order radial polarization and a real parameter was also studied and was shown to feature either several polarization singularity lines outgoing from the center or a central singular point, depending on the parameter value. Notably, at different parameter values, the polarization singularity index of such a field can take half-integer, integer, or zero values.

By simulating tight focusing of vector beams with azimuthal polarization of fractional order 0 <2022