Measures of helicity and chirality of optical vortex beams

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