Angular and Orbital Angular Momenta in the Tight Focus of a Circularly Polarized Optical Vortex

Abstract

Based on the Richards-Wolf (RW) formalism, we obtain two different exact expressions for the angular momentum (AM) density of light in the focus of an optical vortex with a topological charge n and 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 in the focus for an arbitrary integer n. Another expression for the AM density is equal to a sum of the orbital angular momentum (OAM) and the spin angular momentum (SAM) and, in the focus of a considered light field, is equal to zero at n = −1. These expressions are not equal at each point in space, but their 3D integrals are equal. Thus, we derive exact expressions for the AM, SAM and OAM densities in the focus of an optical vortex with right circular polarization and demonstrate that the identity for the densities AM = SAM + OAM is not valid. In addition, we show that the expressions for the strength vectors of the electric and magnetic field near the tight focus, obtained on the basis of the RW formalism, are exact solutions of Maxwell’s equations. Thus, the RW theory exactly describes the behavior of light near the tight focus in free space.