Measures of Helicity and Chirality of Optical Vortex Beams

Analytical forms of the optical helicity and optical chirality of monochromatic Laguerre-Gaussian optical vortex beams are derived up to second order in the paraxial parameter $kw_0$. We show that input linearly polarised optical vortices which possess no optical chirality, helicity or spin densities can acquire them at the focal plane for values of a beam waist $w_0 \approx \lambda$ via an OAM-SAM conversion which is manifest through longitudinal (with respect to the direction of propagation) fields. We place the results into context with respect to the intrinsic and extrinsic nature of SAM and OAM, respectively; the continuity equation which relates the densities of helicity and spin; and the newly coined term Kelvins chirality which describes the extrinsic, geometrical chirality of structured laser beams. Finally we compare our work (which agrees with previous studies) to the recent article K\"oksal, et al. Optics Communications 490, 126907 (2021) which shows conflicting results, highlighting the importance of including all relevant terms to a given order in the paraxial parameter.