  The topological charge of a photonic vortex is an essential quantity in
singular optics and the critical parameter to characterize the vorticity of
twisted light. However, the definition of the photonic topological charge
remains elusive. Here we put forth a theoretical formalism to provide a
comprehensive treatment of photonic vortices. We introduce quantum operators
for the photon current density and helicity current density based on the
continuity equations from the paraxial Helmholtz equation. Our formalism allows
us to introduce flow velocity and circulation for photonic currents in parallel
to their counterparts in superfluids. The quantized circulation of the photonic
currents is conserved during propagation and it gives an explicit definition of
the photonic topological charge as the winding number of a photonic vortex. In
particular, we predict helicity current generated pure helicity vortices, in
which the photon current vanishes. Finally, we show an interesting effect that
the quantum statistics of twisted photon pairs are essentially determined by
their spin states.
