Optical Helicity of Light in the Tight Focus

Using the Richards–Wolf formalism, we obtain explicit analytical expressions for the optical helicity density at the tight focus of four different light beams: a linearly polarized optical vortex, an optical vortex with right-handed circular polarization, superposition of a cylindrical vector beam and a linearly polarized beam, and a beam with hybrid circular-azimuthal polarization. We show that, in all four cases, the helicity density at the focus is nonzero and has different signs in different focal plane areas. If the helicity density changes sign, then the full helicity of the beam (averaged over the beam cross section at the focus) is zero and is conserved upon propagation. We reveal that the full helicity is zero when the full longitudinal component of the spin angular momentum is zero. If the helicity density does not change sign at the focus, such as in a circularly polarized optical vortex, then it is equal to the intensity in the focus, with the full helicity being equal to the beam power and conserving upon propagation. Although the helicity is related to the polarization state distribution across the beam at the focus, the expressions for the helicity density are found to be different from those for the longitudinal component of the spin angular momentum for the beams of interest.