Let
$$
M(x;z_1,z_2)=\langle E(x,z_1)E(x,z_2)\rangle.
$$
Applying the <parabolic wave equation> to each factor gives
$$
\boxed{2ikM_x+(\partial_{z_1}^2+\partial_{z_2}^2)M=0.}
$$
If $C(s)=\langle w(z)w(z+s)\rangle$, Gaussian averaging at the screen gives
$$
M(0;z_1,z_2)
=\boxed{\exp\{-k^2\xi^2[\sigma^2+C(z_1-z_2)]\}.}
$$
Stationarity makes this a function only of $s=z_1-z_2$. Since $\partial_{z_1}^2+\partial_{z_2}^2=2\partial_s^2$ on such functions,
$$
M_x=\frac{i}{k}M_{ss}.
$$
Writing $\widehat M_0(q)$ for the <Fourier transform> of the screen value, the solution at arbitrary range is
$$
\boxed{
M(x;s)=\frac1{2\pi}\int_{\mathbb R}
\widehat M_0(q)e^{iqs-iq^2x/k}\,dq.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 335 / 1 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-335/1/ii/solution