Tight focusing properties of hybrid-order Poincaré sphere beams

The hybrid-order Poincaré sphere (HPS) was proposed to describe high-order vector vortex beams. In this paper, we theoretically study the intensity and longitudinal component phase distribution of tightly focusing HPS beams within a homogeneous and isotropic medium. We analyse the phase and demonstrate the orbital angular momentum (OAM) state distribution of the longitudinal component. Arbitrary OAM states can be obtained by manipulating the order of HPS beams, which comprise two orthogonally circularly polarised vortex beams with topological charges m and n. Meanwhile, the OAM states of the longitudinal component are destroyed when m=−n; otherwise, they split into two parts with topological charges m+1 and n−1. The focus intensity distribution is petal-shaped if the two topological charges have opposite signs, where the number of intensity lobes in the focal plane is m−n+2. Moreover, the ratio of the maximum intensities of the longitudinal and total fields depends on the sign and value of the two topological charges, m and n. Finally, we investigate a generalised case of HPS beams, and demonstrate that the focal intensity and phase distribution are also influenced by the coefficients. The results obtained in this paper may facilitate a new application of HPS beams in various fields.