Spin angular momentum and angular harmonics spectrum of two-order polarization vortices at the tight focus

We investigate the spin angular momentum (SAM) of two-order cylindrical vector beams at the tight focus. Such beams are a generalization of the conventional cylindrical vector beams since the transverse field components on the Cartesian axes have different polarization orders. Using the Richards-Wolf approximation, we derive an expression for the longitudinal component of the SAM density distribution. We show that if the polarization indices are of different parity then, an optical spin Hall effect arises at the tight focus, meaning that alternating areas with positive and negative SAM are originated although the initial light field is linearly polarized. We study the angular harmonics spectrum of all the components of the focused light field and determine the predominant angular harmonics. Then, neglecting the insignificant harmonics, we define the shape of the longitudinal component of the SAM density distribution and demonstrate the possibility of generating a focal SAM distribution where the areas with positive and negative SAM reside on a circle as alternating pairs or separated into different half-circles.