Inhomogeneously polarized light fields: polarization singularity indices derived by analogy with the topological charge

In this work, we study several different vector and hybrid light fields, including those with multiple polarization singularities. We derive polarization singularity indices by adopting a well-known M.V. Berry's formula, which is commonly used to obtain the topological charge of scalar vortex light fields. It is shown that fields whose polarization state depends only on the polar angle in the beam cross section can have either polarization singularity lines outgoing from the center, or a single polarization singularity in the center of the beam cross section. If the polarization state of the field depends only on the radial variable, then such fields have no polarization singularities and their index is equal to zero. If the polarization state of a vector field depends on both polar coordinates, then such a field can have several polarization singularities at different locations in the beam cross section. We also investigate a vector field with high-order radial polarization and with a real parameter. At different values of this parameter, such a field has either several polarization singularity lines outgoing from the center, or a single singular point in the center. The polarization singularity index of such a field for different parameters can be either half-integer, or integer, or zero.