= Gaussian phase-screen correlation
{c}
{title2=$C_E(\zeta)=e^{-\beta^2(1-\rho(\zeta))}$}
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.
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