Quadratic refractive-index shift in the coherent field (${\partial }_{x}m\supset ik{\mu }^{2}m/2$)

If the physical refractive index is $n=1+\mu W$, then $n^2-1=2\mu W+{\mu }^2{W}^2$. In the parabolic wave equation, the last term produces $ik{\mu }^2\mathbb{E}[W^{2}E]/2$ in the mean equation. For unit-variance, finite-correlation fluctuations, $\mathbb{E}[W^{2}E]=m+O(\mu )$ in a weak-fluctuation expansion, giving a coherent phase drift $ik{\mu }^{2}m/2$ 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 $W^2$.