 Single-polarization two-dimensional photon density of states

For an area $A$ with periodic in-plane boundary conditions, one polarization contributes $A2\pi kdk/(2\pi {)}^2$ normal modes of the electromagnetic field. For linear dispersion relation $\omega =ck$ and effective physical volume $Ad$, this gives the density of states in the title. No extra factor two for polarization is allowed. Opposite in-plane directions are already counted in the full circle of wavevectors.