Hopf index and the helicity of elliptically polarized twisted light

Here, we describe a systematic derivation of the general form of the optical helicity density of ellipticaly polarized paraxial Laguerre-Caussian modes LG(l,p,sigma).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 sigma and topological charge l, as well as a spin-orbit sigma vertical bar l vertical bar 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 sigma determines the Hopf index, with the optical vortex LG(lp) character featuring only in the action constant. (C) 2022 Optical Society of America