Gaussian phase-screen correlation ($C_E(\zeta )=e^{-{\beta }^2(1-\rho (\zeta ))}$)

Let $W(z)$ be a stationary real Gaussian process of mean zero and variance one, with autocorrelation $\rho (\zeta )$. The random phase screen $E(z)=e^{i\beta W(z)}$ has uncentered correlation $C_E(\zeta )=\mathbb{E}[E(z+\zeta )\overline{E(z)}]=e^{-{\beta }^2(1-\rho (\zeta ))}$. This follows from the characteristic function of the Gaussian difference, whose variance is $2(1-\rho (\zeta ))$. Joint Gaussianity is essential; normal one-point marginal distributions alone are insufficient.