= Solution
Set $\psi=e^{ikx+\mu\chi_1+O(\mu^2)}$. Substitution into the <Helmholtz equation> gives the first <Rytov approximation> equation
$$
(\Delta+2ik\partial_x)\chi_1=-2k^2W.
$$
For the forward parabolic approximation, omit $\partial_x^2\chi_1$. With zero perturbation at $x=0$, the <Fresnel propagator> then gives
$$
\boxed{\chi_1(x,\mathbf s)=ik\int_0^x
\exp\!\left[\frac{i(x-u)\Delta_\perp}{2k}\right]W(u,\mathbf s)\,du,
\qquad \psi_R=e^{ikx}e^{\mu\chi_1}.}
$$
A jointly <Gaussian random field> makes $\chi_1$ a centered complex Gaussian variable. The <complex Gaussian exponential moment> therefore gives the <first-Rytov coherent mean>
$$
\boxed{\mathbb E[\psi_R]=e^{ikx}\exp\left(\frac{\mu^2}{2}\mathbb E[\chi_1^2]\right).}
$$
The second moment here has no complex conjugation. It cannot be replaced by the positive intensity variance $\mathbb E[|\chi_1|^2]$.
For example, let $\widehat C(u,v;\boldsymbol\kappa)$ be the transverse spectrum of the covariance between depths $u$ and $v$. The Fourier multiplier of the <Fresnel propagator> is $e^{-i(x-u)|\boldsymbol\kappa|^2/(2k)}$, so
$$
\mathbb E[\chi_1^2]=-k^2\int_0^x\int_0^x du\,dv
\int\frac{d^2\boldsymbol\kappa}{(2\pi)^2}\widehat C(u,v;\boldsymbol\kappa)
\exp\left[-\frac{i|\boldsymbol\kappa|^2(2x-u-v)}{2k}\right].
$$
This supplies the requested mean in terms of medium statistics. For the longitudinal white-noise model, $\widehat C=\delta(u-v)\Phi$, which reduces the double longitudinal integral to one.
The exponential is the mean of the first-order logarithmic model. It is not a complete second-order perturbation expansion of the true mean: the omitted logarithmic term $\mu^2\chi_2$ can itself have a nonzero mean. Strictly at first order, $\mathbb E[\psi]=e^{ikx}+O(\mu^2)$. As an independent check, the closed white-noise parabolic model gives exactly
$$
\mathbb E[E]=\exp[-k^2\mu^2C(0)x/2]
$$
for plane-wave input. A truncated first-Rytov covariance expression with transverse diffraction need not equal this exact mean; omitted logarithmic terms account for the difference.
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