Spin angular momentum of Gaussian beams with several polarization singularities

We study a paraxial vector Gaussian beam with several polarization singularities located on a circle. Such a beam is superposition of a cylindrically polarized Laguerre-Gaussian beam and a linearly polarized Gaussian beam. It is found that although polarization in the initial plane is linear, alternating regions with the different-sign spin angular momentum density are generated upon free-space propagation, showing that a spin Hall effect arises. For an arbitrary transverse plane, it is shown that the spin angular momentum magnitude is maximal on a certain-radius circle. We obtain an approximate expression for the distance to the transverse plane where the spin angular momentum density is maximal. Besides, we derive an optimal radius of the singularity-containing circle in the initial plane for which the maximal spin angular momentum density can be achieved upon propagation. It is revealed that in this case, the energies of the Laguerre-Gaussian beam and the Gaussian beam are equal to each other. We also obtain an expression for the orbital angular momentum density and find it to be defined by the spin angular momentum density, multiplied by - m /2, with m being the upper index of the Laguerre-Gaussian beam, equal to the number of the polarization singularities. An analogy with plane waves reveals that the spin Hall effect arises due to different divergence rates of the linearly polarized Gaussian beam and the cylindrically polarized Laguerre-Gaussian beam.