Spin and orbital Hall effects in the tight focus of coaxial superposition of two cylindrical vector beams of different-parity orders

We study tight focusing of coaxial superposition of two cylindrical vector beams (CVB) of different orders. In the initial plane, the polarization singularity index of such superposition equals the half-sum of the orders of the two constituent CVBs. Such superposition has neither spin angular momentum (SAM) nor transverse energy flow in the initial plane. We demonstrate that if two constituent CVBs are of different-parity orders, then, in the focal plane, there occur regions with nonzero longitudinal SAM components of alternating sign, alongside regions where opposite-handed transverse energy flows are rotating along closed paths (clockwise and counterclockwise). This means that the longitudinal spin and orbital Hall effects arise in the focal plane. On the contrary, if the two CVBs are of same-parity orders, polarization in the focal plane is inhomogeneous linear and the energy flow (Umov-Poynting vector) only has an on-axis component.