Hopf index and the helicity of elliptically polarized twisted light

Abstract

Here, we describe a systematic derivation of the general form of the optical helicity density of ellipticaly polarized paraxial Laguerre–Gaussian modes L G ℓ , p , σ . The treatment incorporates the contributions of the longitudinal field components for both the paraxial electric E and magnetic B fields, which satisfy Maxwell’s self-consistency condition in the sense that E is derivable from B and vice versa. Contributions to the helicity density to leading order in ( k 2 w 0 2 ) − 1 (where k is the axial wavenumber and w 0 the beam waist) include terms proportional to optical spin σ and topological charge ℓ , as well as a spin-orbit σ | ℓ | term. However, evaluations of the space integrals leading to the total helicity confirm that the space integral of the ℓ -dependent term in the density (which is due entirely to the longitudinal fields) vanishes identically for all ℓ and p , so that, in general, only σ determines the Hopf index, with the optical vortex L G ℓ p character featuring only in the action constant.