Focused polarization ellipse field singularities: interaction of spin-orbital angular momentum and the formation of optical Möbius strips

We study here the intensity distribution and formation of optical polarization Möbius strips by tightly focusing of C-point singularity beams. These beams are characterized by a central circular polarization point (C-point) surrounded by a spatially varying elliptic polarization. Under tight focusing conditions, the different polarization components of the beam interfere and exhibit clear difference between left-handed and right handed input beams. The transverse polarization distribution at the focal plane is similar to the input distribution for left-handed lemon beam, but exhibits 180° rotation for right handed lemon beam. Moreover, the longitudinal polarization component exhibits spiral phase distribution, owing to spin-orbit angular momentum conversion at the focal plane, with opposite winding directions for the left-handed and right-handed input beams. We show that the shape of the resulting Möbius strip is determined by the helicity of the C-point and by the polarization singularity index, which is the contour integral of polarization ellipse angle around the singularity. It is found that inverting the helicity leads to 180° rotation in the focal plane intensity distribution, accompanied by handedness inversion for the polarization ellipses. The number of separatrices in the input polarization distribution is equivalent to the number of twist points of the Möbius strip in the focal plane, as well as to the number of intensity zeros in the z-component of the focused field. These phenomena are observed for beams with a bright C-point, but also for dark C-point, in which the electric field is zero at the center of the beam.