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–Gaussian modes LG`; p;  . The treatment incorporates the contributions of the longitudinal field components for both the paraxial electric E and magnetic B fields, which satisfyMaxwell’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 .k2w2 0/1 (where k is the axial wavenumber and w0 the beam waist) include terms proportional to optical spin   and topological charge `, aswell as a spin-orbit  j`j term.However, evaluations of the space integrals leading to the total helicity confirmthat the space integral of the `-dependent termin the density (which is due entirely to the longitudinal fields) vanishes identically for all ` and p, so that, in general, only   determines theHopf index, with the optical vortex LG` p character featuring only in the action constant.