Focusing a vortex beam with circular polarization: angular momentum

Based on the Richards–Wolf formalism, we obtain two different exact expressions for the angular momentum (AM) density in the focus of a vortex beam with the topological charge $n$ and with right circular polarization. One expression for the AM density is derived as the cross product of the position vector and the Poynting vector and has a nonzero value at the focus for an arbitrary integer number $n$. The other expression for the AM density is deduced as a sum of the orbital angular momentum (OAM) and the spin angular momentum (SAM). We reveal that at the focus of the light field under analysis, the latter turns zero at $n=–1$. While both these expressions are not equal to each other at each point of space, 3D integrals thereof are equal. Thus, exact expressions are obtained for densities of AM, SAM and OAM at the focus of a vortex beam with right-hand circular polarization and the identity for the densities AM = SAM + OAM is shown to be violated. Besides, it is shown that the expressions for the strength vectors of the electric and magnetic fields near the sharp focus, obtained by adopting the Richards–Wolf formalism, are exact solutions of the Maxwell's equations. Thus, Richards–Wolf theory exactly describes the behavior of light near the sharp focus in free space.