Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 70 / 2 / c / Solution

For the given Gaussian radial spectrum, the only needed integral isIf $S$ is interpreted with the self-reciprocal Hankel pair in the supplied hint, the coherent field isIn this convention $B_n(r)=2{\mu }^{2}L{e}^{-r^2/L^2}$ and hence $B_n(0)=2{\mu }^{2}L$, directly verifying the attenuation coefficient. If instead the stated spectrum is the unnormalized Cartesian Fourier transform, use its inverse normalization and obtainIts transverse covariance coefficient is then $B_n(r)={\mu }^{2}L{e}^{-r^2/L^2}/\pi$. The apparent difference is precisely the inconsistent Fourier/Hankel normalization in the printed hint, not a difference in stochastic propagation. For a unit constant entrance field the Fresnel factor is one; coherent intensity is the square of the displayed mean amplitude.