Past exam of the physics A course of the University of Cambridge / 2012 / Paper 2 / Section B / B8 / Solution

The Fraunhofer approximation drops the aperture-dependent quadratic propagation phase. For illuminated extent $b$ and observation distance $R$, a sufficient paraxial condition is $b^2/(\lambda R)\ll 1$, with coherent nearly plane incident illumination. Collimation followed by observation in a lens's back focal plane produces the same angular Fourier pattern without an astronomical propagation distance.

Let $(x,y)$ label the aperture field ${\psi }_{\Sigma }$ and let $(x_0,y_0)$ be transverse coordinates of the distant detector, with $R$ the axial separation. Expanding the propagation distance retains a common phase and the cross term $-(x_{0}x+y_{0}y)/R$, while dropping the aperture's quadratic term. Summing the scalar Huygens contributions gives a field proportional to the Fourier transform at $(k{x}_0/R,k{y}_0/R)$. In one dimension, writing $q=k{x}_0/R$, $\tilde{\psi }(q)=\int \psi (x)e^{-iqx}dx$ givesRemove the known propagation prefactor and any detector-plane quadratic phase before applying this inverse transform. Recovery needs the complex diffracted field, including phase; its intensity alone is insufficient for this inversion.

The whole-aperture Fraunhofer distance of a coherent array uses the full ten-dish extent, which is about $b=9D+25\mathrm{m}=7.225\mathrm{km}$. Thus $b^2/\lambda \simeq 8.7\times 10^8\mathrm{m}$, and a conventional boundary $2b^2/\lambda$ is about $1.7\times 10^9\mathrm{m}$. Far-field propagation must be of order a million kilometres or more, with a stronger inequality for accurate Fraunhofer behaviour. Applying the condition to only one dish or one spacing would miss the array extent.

For equal synchronized dish fields, the array sum is ${\sum }_{j=0}^9{e}^{-ij{k}_{0}D\sin \theta }$. Its intensity isNear the centre, $\sin \theta \simeq \theta$ and the single-dish envelope $I_1(\theta )$ is nearly constant, giving the requested approximation. A uniformly illuminated circular dish instead supplies the broader envelope $[2J_1(\pi d\sin \theta /\lambda )/(\pi d\sin \theta /\lambda ){]}^2$, with first zero near $1.22\lambda /d=2.93\times 10^{-3}\mathrm{rad}$. It cannot be ignored over the entire pattern.

Principal array maxima have $\sin {\theta }_m=m\lambda /D$, height $100I_1({\theta }_m)$, with spacing approximately $7.5\times 10^{-5}\mathrm{rad}=15.5$ arcseconds. There are narrow sidelobes and many grating orders within the dish envelope because $D\gg d$. The central maximum's first zeros obey $10\beta =\pm \pi$. By reception/transmission reciprocity, the source must therefore move throughfrom zero-order maximum to first minimum. The equatorial geometry fixes the sweep direction; no elapsed time is requested.