= Solution
Put $\beta=k\xi\mu$. The <random phase screen> produces
$$
E(0,z)=e^{i\beta W(z)},\qquad
\widehat E(x,\nu)=e^{-i\nu^2x/(2k)}\widehat E(0,\nu).
$$
Interpret the normality assumption as a jointly <Gaussian process>, not merely Gaussian one-point <marginal distributions>. Write its normalized <autocorrelation> as $\rho(\zeta)=\langle W(z+\zeta)W(z)\rangle$. The difference $W(z+\zeta)-W(z)$ is a zero-mean <Gaussian random variable> with <variance> $2(1-\rho(\zeta))$. Its <characteristic function> therefore gives the <Gaussian phase-screen correlation>
$$
\boxed{C_E(\zeta):=\langle E(0,z+\zeta)\overline{E(0,z)}\rangle
=\exp\{-\beta^2[1-\rho(\zeta)]\}.}
$$
Without joint Gaussianity the <autocorrelation> alone does not determine this expectation.
For an acoustic pressure amplitude $\psi$, with background mass density $\rho_0$, the time-averaged <acoustic energy flux> is
$$
I_x=\frac{1}{2\rho_0\omega}\operatorname{Im}(\overline\psi\,\partial_x\psi).
$$
An acoustic potential convention changes the dimensional prefactor. The physical field includes the carrier $e^{ikx}$: a transverse <Fourier transform> mode has axial <wavenumber> $\kappa_\nu=k-\nu^2/(2k)$ in the <parabolic wave equation>. Thus its <spectral acoustic flux> is proportional to $\kappa_\nu|\widehat E(x,\nu)|^2$, or to $k|\widehat E(x,\nu)|^2$ at leading paraxial order. Differentiating the reduced field alone would omit the carrier contribution.
The infinite stationary illumination has a <spectral measure of a stationary random field>, rather than a finite ordinary value of $\langle|\widehat E|^2\rangle$ at each $\nu$. To specify the normalization, truncate the screen to an aperture of length $L$, propagate this truncated initial field, and set
$$
\widehat E_L(0,\nu)=\frac1{2\pi}\int_{-L/2}^{L/2}E(0,z)e^{-i\nu z}\,dz.
$$
The propagation multiplier has modulus one. Using the <Gaussian phase-screen correlation> and the overlap of the two aperture intervals gives
$$
\boxed{\langle|\widehat E_L(x,\nu)|^2\rangle
=\frac1{4\pi^2}\int_{-L}^{L}(L-|\zeta|)
e^{-\beta^2[1-\rho(\zeta)]}e^{-i\nu\zeta}\,d\zeta,\qquad x>0.}
$$
Multiplication by $\kappa_\nu/(2\rho_0\omega)$ gives the ensemble-averaged pressure-normalized modal <acoustic energy flux> for this finite-aperture convention.
The <infinite-aperture spectral normalization> consistent with the specified <Fourier transform> is
$$
\boxed{S_E(\nu)=\lim_{L\to\infty}\frac{2\pi}{L}
\langle|\widehat E_L(x,\nu)|^2\rangle
=\frac1{2\pi}\int_{\mathbb R}e^{-\beta^2[1-\rho(\zeta)]}e^{-i\nu\zeta}\,d\zeta.}
$$
The integral and limit are understood as <spectral measures of a stationary random field> when necessary. By <Parseval's identity>, this is power per unit transverse length: $\int_{\mathbb R}S_E(d\nu)=C_E(0)=1$. \b[The ensemble-averaged axial <spectral acoustic flux> is independent of $x$:]
$$
\boxed{\frac{d\langle I_x\rangle}{d\nu}
=\frac{\kappa_\nu}{2\rho_0\omega}S_E(\nu)
\simeq I_{\rm inc}S_E(\nu),\qquad I_{\rm inc}=\frac{k}{2\rho_0\omega}.}
$$
Here the notation includes <Dirac delta distributions> in the <spectral measure of a stationary random field>; the leading-order total flux is $I_{\rm inc}$. The $\kappa_\nu$ correction is meaningful within $|\nu|\ll k$, the range of the <paraxial approximation>.
If $\rho(\zeta)\to0$ and $e^{\beta^2\rho(\zeta)}-1$ is integrable, the <stationary phase-screen power spectrum> separates into <coherent and diffuse wave fields>:
$$
S_E(\nu)=e^{-\beta^2}\delta(\nu)
+\frac{e^{-\beta^2}}{2\pi}\int_{\mathbb R}
[e^{\beta^2\rho(\zeta)}-1]e^{-i\nu\zeta}\,d\zeta.
$$
The coherent fraction is $e^{-\beta^2}$ and the diffuse fraction is $1-e^{-\beta^2}$. If additionally $|\beta|\ll1$, the <stationary phase-screen power spectrum> is $(1-\beta^2)\delta+\beta^2\widehat\rho+O(\beta^4)$ with the same <Fourier transform> convention. Small $\mu$ alone does not imply small accumulated phase $k\xi\mu$.
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