Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 72 / 1 / c / i

Assume a real refractive index and write the scattering potential as $V=⟨V⟩+W$, so $⟨W⟩=0$ by construction. With the real kernel components in the question, the logarithmic Rytov approximation separates intoInclude the incident wave phase and the mean-potential wave phase in the deterministic reference ${\varphi }_0$: at each observation point it is $\arg {\psi }_i-\int b⟨V⟩$. It need not be spatially constant. The amplitude is $|{\psi }_i|\exp [-\int aV]$. The phase covariance in the first Rytov approximation therefore starts from the fluctuating wave phaseUnder the integrability assumptions needed to interchange the expectation and integral,For a complex absorbing potential, the corresponding phase fluctuation is $-\int [b\mathrm{Re}W+a\mathrm{Im}W]$; the displayed scalar formula is the real-index case. The zero mean comes from centering $V$, not from setting the mean of $V$ equal to zero. In particular the printed $⟨n⟩=0$ does not imply $⟨V⟩=0$. A physical positive refractive index usually has a nonzero background mean; a zero-mean assumption normally refers to its fluctuation. The algebra above remains meaningful for a signed real random field and explicitly retains its mean scattering potential.