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 closely related, but their connection is hidden by use four different vector fields for representation, namely, electric magnetic $\mathbf{E}$ $\mathbf{B}$, two transverse potential vectors $\mathbf{C}^\perp$ $\mathbf{A}^\perp$. Helmholtz's decomposition theorem restricted to solenoidal fields, entails introduction a bona fide inverse curl operator, which permits one express above terms observable only. This yields clear expressions $H, C$, $\mathbf{S}$, automatically gauge-invariant display electric-magnetic democracy.