Past exam of the physics 2 paper of the University of Oxford / 2013 resit / 8 / i / Solution

In the Fraunhofer approximation, contributions from positions $\xi$ across the slit have relative phase $k\xi \sin \theta$. With uniform illumination, the field relative to its axial value isThus the single-slit diffraction intensity isOn the detector $\sin \theta =y/\sqrt{d^2+y^2}$; near the axis, $\beta \simeq \pi wy/(\lambda d)$. The central maximum takes the continuous limit $\sin \beta /\beta \to 1$, and the minima satisfy $w\sin \theta =m\lambda$ for nonzero integers $m$. The stated far-field condition justifies adding the aperture contributions with a linear phase.