A note on helicity, chirality and spin of optical fields

Helicity $H$, chirality $C$ and spin angular momentum $\mathbf{S}$ are three physical observables that play an important role in the study of optical fields. These quantities are closely related, but their connection is hidden by the use of four different vector fields for their representation, namely the electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$, and the two transverse potential vectors $\mathbf{C}^\perp$ and $\mathbf{A}^\perp$. Helmholtz's decomposition theorem restricted to solenoidal vector fields, entails the introduction of a bona fide inverse curl operator, which permits to express the above three quantities in terms of the electric and magnetic fields only. This gives clear expressions for $H, C$ and $\mathbf{S}$, which are automatically gauge-invariant and show electric-magnetic democracy.