Helicity, chirality, and spin of optical fields without vector potentials

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 one to express the above three quantities in terms of the observable electric and magnetic fields only. This yields clear expressions for $H,C$, and $\mathbf{S}$, which are automatically gauge invariant and display electric-magnetic democracy.