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.

In this work, the sharp focusing of a laser beam whose initial polarization pattern is formed by superposition of a cylindrical mth-order vector beam and a homogeneous linearly polarized beam is considered theoretically and numerically. Although in the source plane of such a beam both the angular spin momentum and the third Stokes parameter are equal to zero, we reveal that given odd m, subwavelength local regions are formed in the focal plane, where transverse vortex energy flows occur and the third Stokes parameter (the on-axis component of the angular spin momentum) is non-zero. Thus, at odd m, at the focus of such a beam there are – sub-regions with elliptical polarization of light with alternating handedness in the adjacent sub-regions (clockwise and counterclockwise). This phenomenon can be interpreted as a variant of an optical Hall effect. We note that at even m, the field at the focus is linearly polarized at every point and no transverse energy flow is observed.