Past exam of the physics A course of the University of Cambridge / 2021 / Paper 2 / Section B / B7 / c / Solution

Use SI units throughout: $d=1.0\times 10^{-3}\mathrm{m}$, $\lambda =1.0\times 10^{-6}\mathrm{m}$, and $p=q=1.0\mathrm{m}$. The effective propagation distance is $R_{\mathrm{eff}}=0.50\mathrm{m}$, givingIn the supplied Cornu spiral, the endpoint at $w=1$ is approximately $(C,S)=(0.78,0.44)$. HenceMore accurate Fresnel integrals, $C(1)=0.7798934$ and $S(1)=0.4382591$, give $I_P/I_0=2.562$. The central light intensity is increased to about $2.6$ times the unobscured value, an increase of roughly $160\%$.

There is no violation of conservation of energy: the aperture reduces the total transmitted power, while removing contributions that partly cancel the central field redistributes the transmitted power across the observation plane. In particular, the square aperture does not give the same central field as a circular aperture of a superficially similar size.