Rough-surface far-field intensity per illuminated length ($\frac{⟨|{\psi }_s{|}^2⟩}{D}\sim \frac{2k{q}_i^2|A_i{|}^2{\cos }^2{\theta }_s}{\pi r}{S}_h(k\sin {\theta }_s-p_i)$)

First-order Dirichlet roughness has mean-plane trace $2i{q}_i{A}_{i}h(x)e^{i{p}_{i}x}$, where $p_i^2+q_i^2=k^2$ and $q_i>0$. Stationary phase applied to the outgoing angular spectrum evaluates the far-field amplitude at tangential wavenumber $k\sin {\theta }_s$. For a stationary surface illuminated over length $D$, the mean squared height transform divided by $D$ tends to its power spectrum of surface height. Thus the diffuse intensity samples the spectrum at the tangential momentum transfer. An infinite stationary surface has no finite total far-field amplitude without a window or an intensity-density convention.