= Solution
For a general colored medium and unit plane-wave input $E(0,\mathbf s)=1$, let $U_W(x,0)$ be the propagator of $\partial_x=i\Delta_\perp/(2k)+ik\mu W(x,\mathbf s)$. Then the solution is
$$
\Gamma(x;\mathbf s_1,\mathbf s_2)=\mathbb E\left[(U_W(x,0)1)(\mathbf s_1)(U_W(x,0)1)(\mathbf s_2)^*\right].
$$
A time-ordered operator exponential defines $U_W$. Its expectation depends on the full longitudinal covariance; the initial Gaussian and transverse-stationarity assumptions do not specify a universal closed expression.
With the stated longitudinal <Gaussian white noise> and transverse <statistical homogeneity>, the moment depends only on $\boldsymbol\rho=\mathbf s_1-\mathbf s_2$. Hence $(\Delta_1-\Delta_2)\Gamma=0$. The closed equation in the previous part and initial value $\Gamma(0)=1$ give
$$
\boxed{\Gamma(x,\boldsymbol\rho)=\exp\{-k^2\mu^2x[C(0)-C(\boldsymbol\rho)]\}.}
$$
In particular $\Gamma(x,0)=1$: the ensemble mean intensity remains constant for plane-wave input, even though <coherent attenuation in a white-noise random medium> reduces the mean complex field.
Use a consistent transverse <power spectrum> convention,
$$
C(\boldsymbol\rho)=\int\frac{d^2\boldsymbol\kappa}{(2\pi)^2}\Phi(\boldsymbol\kappa)e^{i\boldsymbol\kappa\cdot\boldsymbol\rho}.
$$
For an isotropic spectrum, angular integration gives the <isotropic transverse covariance spectrum>, $C(\rho)=(2\pi)^{-1}\int_0^\infty\Phi(\kappa)J_0(\kappa\rho)\kappa\,d\kappa$. Therefore
$$
\boxed{\Gamma(x,\rho)=\exp\left[-\frac{k^2\mu^2x}{2\pi}
\int_0^\infty\Phi(\kappa)(1-J_0(\kappa\rho))\kappa\,d\kappa\right].}
$$
The supplied two-dimensional <Fourier transform> definition has angular factor $2\pi$, whereas the displayed radial transform pair omits it. The latter is a normalized <Hankel transform> pair. If its radial spectrum is denoted $\Phi_H=\Phi/(2\pi)$, use $\Phi_H$ in the integral and omit the prefactor $1/(2\pi)$. These are equivalent conventions; mixing them changes the attenuation rate.
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