 Stationary phase-screen power spectrum

For the random phase screen $E(z)=e^{i\beta W(z)}$, the Gaussian phase-screen correlation and spectral measure of a stationary random field giveIf $\rho (\zeta )\to 0$ and $e^{{\beta }^2\rho }-1$ is integrable, this is $e^{-{\beta }^2}\delta (\nu )$ plus the continuous density $(2\pi {)}^{-1}{e}^{-{\beta }^2}\int (e^{{\beta }^2\rho (\zeta )}-1)e^{-i\nu \zeta }d\zeta$. The total mass is one, split into coherent mass $e^{-{\beta }^2}$ and diffuse mass $1-e^{-{\beta }^2}$. Without the decay and integrability assumptions there may be additional spectral atoms or singular components.