= Solution
To first order in the weak fluctuation, $n^2-1=2\mu W+O(\mu^2)$, so
$$
E_x=\frac{i}{2k}E_{zz}+ik\mu WE.
$$
For
$$
F=E_1E_2^*E_3E_4^*,
\qquad E_j=E(x,z_j),
$$
introduce the signs $s=(1,-1,1,-1)$. The <product rule> gives
$$
\partial_x\langle F\rangle
=\frac{i}{2k}
\left(\partial_{z_1}^2-\partial_{z_2}^2
+\partial_{z_3}^2-\partial_{z_4}^2\right)m_4
+ik\mu\sum_{j=1}^4s_j\langle W(x,z_j)F\rangle.
$$
Assume that $W$ is a zero-mean <stationary Gaussian random field>, that its longitudinal <correlation length> is short compared with the envelope's evolution scale, and that the propagation distance is long compared with that correlation length. The forward <Markov approximation> then neglects diffraction during one correlation length. Applying the <Furutsu–Novikov formula> closes the last average at second order in $\mu$. Define the integrated longitudinal <autocorrelation function of a random field>
$$
C(\zeta)=\int_{-\infty}^{\infty}
\rho(\xi,\zeta)\,d\xi.
$$
Then
$$
\boxed{
\partial_xm_4
=\frac{i}{2k}
\left(\partial_{z_1}^2-\partial_{z_2}^2
+\partial_{z_3}^2-\partial_{z_4}^2\right)m_4
-Q_4m_4},
$$
where
$$
\boxed{
Q_4=\frac{k^2\mu^2}{2}
\sum_{i,j=1}^4s_is_jC(z_i-z_j)}.
$$
Writing $C_{ij}=C(z_i-z_j)$ and using the evenness of the covariance gives the equivalent expression
$$
Q_4=k^2\mu^2
\left(2C(0)-C_{12}+C_{13}-C_{14}
-C_{23}+C_{24}-C_{34}\right).
$$
The derivation also assumes <paraxial propagation>, weak scattering, sufficient regularity to interchange differentiation and <expected value>[expectation], and statistical homogeneity in both coordinates. Without the short-correlation approximation, the Gaussian identity produces a nonlocal longitudinal memory integral rather than this local closed equation.
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