= Colored Gaussian paraxial mean propagator
{title2=$m(x)=\langle U_W(x,0)\rangle E_0$}
For $E_x=i\Delta_\perp E/(2k)+ik\mu WE$, a jointly Gaussian colored medium has covariance $C(s,\mathbf r)$. Averaging the equation gives a term $\langle WE\rangle$; the <Furutsu–Novikov formula> expresses it through the random propagator and the functional derivative of $E$. It does not generally close in the mean field alone. A phase-screen product followed by Gaussian averaging gives a formal exact mean propagator, with factor $\exp[-k^2\mu^2\Delta x^2\sum_{j,l}C(x_j-x_l,\mathbf r_j-\mathbf r_l)/2]$ for each transverse path. In the longitudinal white-noise limit the covariance collapses to equal slices, and the mean closes as $m_x=i\Delta_\perp m/(2k)-k^2\mu^2A(0)m/2$.
Back to article page
