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

The phase screen adds the phase accumulated across its thickness, so the reduced field just after the screen isFor a zero-mean random variable $w$ with a normal distribution and variance ${\sigma }^2$, its characteristic function givesThe same expression is approximately valid for a non-Gaussian weak fluctuation: the cumulant expansion begins with $-k^2{\xi }^2{\sigma }^2/2$, while higher cumulants give higher-order corrections.

For $x>0$, every realization obeys the parabolic wave equationLinearity permits ensemble averaging, soThe initial mean is independent of $z$, hence diffraction does not change it and $⟨E(x,z)⟩=\exp (-k^2{\xi }^2{\sigma }^2/2)$ for every $x\ge 0$.

LetApplying the parabolic wave equation to each factor givesIf $C(s)=⟨w(z)w(z+s)⟩$, Gaussian averaging at the screen givesStationarity 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,Writing ${\hat{M}}_0(q)$ for the Fourier transform of the screen value, the solution at arbitrary range is

Choose the overall phase so that the coherent mean $m=⟨E⟩$ is real. With $E_j=m+A_j+i{B}_j$,Because the diffuse field has zero mean, keeping terms through second order givesand, at coincident points,The normalized spatial intensity correlation is thereforeThus weak-scattering intensity fluctuations measure the normalized correlation of the in-phase part of the diffuse field.