Sharp focusing of beams with V-point polarization singularities

It is theoretically and numerically shown that when tightly focusing an n -th order vector light field that has the central V-point (at which the linear polarization direction is undetermined), the polarization singularity index n , and a "flower"-shaped intensity pattern with 2( n - 1) lobes it forms a transverse intensity distribution with 2( n - 1) local maxima. At the same time, a vector light field with the polarization singularity index - n , which has the form of a "web" with 2( n + 1) cells generates at the sharp focus a transverse intensity distribution with 2( n + 1) local maxima. In the focal spot, either 2( n -1) or 2( n + 1) V-point polarization singularities with alternating indices +1 or -1 are formed at the intensity zeros.