Past exam of the physics A course of the University of Cambridge / 2022 / Paper 2 / Section A / 1 / Solution

Use Fresnel diffraction, since the detector moves along the axis through successive near-field interference extrema. Write $z$ for its distance from the opening and $\rho$ for the integration coordinate within the opening. For a uniform incident plane wave, the on-axis complex amplitude, in the paraxial approximation, is proportional toThus the light intensity relative to the incident value isThe Fresnel number $F=r^2/(\lambda z)$ is an odd integer at a maximum and an even integer at a minimum. It decreases as the detector moves away. Adjacent extrema therefore differ by one:ConsequentlyAs a check, $F=5$ at $z=1\mathrm{m}$ and $F=4$ at $z=1.25\mathrm{m}$, so these are indeed a maximum followed by the next minimum. Also $r/z\simeq 1.5\times 10^{-3}$, consistent with the paraxial approximation.