Focusing of a vector beam with C-lines of polarization singularity

We discuss the sharp focusing of an initial hybrid vector light field of the n-th order that has n C-lines (lines along which the polarization is circular). Using a complex Stokes field, it is shown that the polarization singularity index of such a field is equal to n/2. The initial field is shown to retain it singularity index in the focal plane. Analytical expressions are obtained for the intensity and Stokes vector components in the focal plane. It is shown theoretically and numerically that for an even-order field with n=2p, the intensity pattern at the focus has symmetry and instead of C-lines, C-points are formed, with the axes of polarization ellipses rotating around them. For n=4, there are C-points with singularity indices 1/2 and with a “lemon” topology. Around such points, a surface that is formed by the polarization ellipses in a three-dimensional space has a Möbius strip topology. For an odd-order initial field with n=2p+1, the intensity pattern at the focus is shown to have no symmetry, with the field becoming pure vectorial (no elliptical polarization) and the linear polarization vectors rotating around V-points.