= Solution
The <phase screen> adds the phase accumulated across its thickness, so the reduced field just after the screen is
$$
E(0,z)=e^{ik\xi w(z)}.
$$
For a zero-mean random variable $w$ with a <normal distribution> and variance $\sigma^2$, its <characteristic function> gives
$$
\boxed{\langle E(0,z)\rangle
=\exp\left(-\frac12k^2\xi^2\sigma^2\right).}
$$
The same expression is approximately valid for a non-Gaussian weak fluctuation: the <cumulant expansion> begins with $-k^2\xi^2\sigma^2/2$, while higher cumulants give higher-order corrections.
For $x>0$, every realization obeys the <parabolic wave equation>
$$
2ikE_x+E_{zz}=0.
$$
Linearity permits ensemble averaging, so
$$
\boxed{2ik\partial_x\langle E\rangle
+\partial_z^2\langle E\rangle=0.}
$$
The initial mean is independent of $z$, hence diffraction does not change it and $\langle E(x,z)\rangle=\exp(-k^2\xi^2\sigma^2/2)$ for every $x\geq0$.
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