The zero helicity and chirality of optical vortices

We show that any uniformly linearly-polarised paraxial vortex mode carrying orbital angular momentum (OAM) has zero spin angular momentum (SAM) density, but exhibits non-zero helicity density distributions. Such a mode then possesses chirality as confirmed by experiment and so can engage with chiral matter. We show that confining the treatment for the general paraxial fields only to leading order leads directly to agreement of our theory with the experimental results, provided we ensure that crucially the paraxial fields obey duality. We find that the space integral of the helicity and chirality densities vanish identically for all such optical vortex modes without specifying the kind of mode. These generally applicable properties of optical vortex modes carrying orbital angular momentum thus assert that without optical spin due to elliptical wave polarisation of index σ, an optical vortex alone cannot possess total helicity, even though it always exhibits non-zero helicity density distributions.