= Quadratic refractive-index shift in the coherent field
{title2=$\partial_xm\supset ik\mu^2m/2$}
If the physical <refractive index> is $n=1+\mu W$, then $n^2-1=2\mu W+\mu^2W^2$. In the <parabolic wave equation>, the last term produces $ik\mu^2\mathbb E[W^2E]/2$ in the mean equation. For unit-variance, finite-correlation fluctuations, $\mathbb E[W^2E]=m+O(\mu)$ in a weak-fluctuation expansion, giving a coherent phase drift $ik\mu^2m/2$ to second order. This term is absent if the model deliberately linearizes $n^2-1$ to $2\mu W$. It is not obtained by squaring ideal <Gaussian white noise>, whose pointwise square is not the finite-variance random variable $W^2$.
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