Abstract

Here, we describe a systematic derivation of the general form of the optical helicity density of ellipticaly polarized paraxial Laguerre–Gaussian modes ${{\rm LG}_{\ell, p,\sigma}}$. The treatment incorporates the contributions of the longitudinal field components for both the paraxial electric ${\textbf E}$ and magnetic ${\textbf B}$ fields, which satisfy Maxwell’s self-consistency condition in the sense that ${\textbf E}$ is derivable from ${\textbf B}$ 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 $\ell$, as well as a spin-orbit $\sigma |\ell |$ term. However, evaluations of the space integrals leading to the total helicity confirm that the space integral of the $\ell$-dependent term in the density (which is due entirely to the longitudinal fields) vanishes identically for all $\ell$ and $p$, so that, in general, only $\sigma$ determines the Hopf index, with the optical vortex ${{\rm LG}_{\ell p}}$ character featuring only in the action constant.

© 2022 Optical Society of America

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