Polarization Strips in the Focus of a Generalized Poincaré Beam

Abstract

We analyze the tight focusing of a generalized Poincaré beam using a Richards–Wolf formalism. Conventional Poincaré beams are superpositions of two Laguerre–Gaussian beams with orthogonal polarization, while the generalized Poincaré beams are composed of two arbitrary optical vortices with rotationally symmetric amplitudes. Analytical relationships for projections of the electric field in the focal plane are derived. Using the superposition of a right-handed circularly polarized plane wave and an optical vortex with a topological charge of −1 as an example, relationships for the intensity distribution and the longitudinal projection of the spin angular momentum vector are deduced. It is theoretically and numerically shown that the original beam has a topological charge of −1/2 and a C-point of circular polarization, and it is generated at the focal plane center, producing an on-axis C-line with a singularity index of −1/2 (a star). Furthermore, when making a full circle of some radius around the optical axis, the major axis vector of polarization ellipse is theoretically and numerically shown to form a one-sided polarization (Möbius) strip of order −3/2, which has three half-twists and a single ‘patching’ in which two oppositely directed vectors of the major axis of polarization ellipse occur close to each other.