Past exam of the physics 2 paper of the University of Oxford / 2017 / 10 / b / Solution

Let the slit run over $-w/2<x<w/2$ and let its coherent illumination have constant complex amplitude $A_0$. In a direction making angle $\theta$ to the normal, the relative propagation phase from position $x$ is $-kx\sin \theta$, with $k=2\pi /\lambda$. Summing the secondary amplitudes givesThe intensity is proportional to the squared modulus, hence within the usual scalar Fraunhofer diffraction approximationThe central value is defined by the continuous limit $\sin \beta /\beta \to 1$. The nonzero minima satisfy $w\sin \theta =m\lambda$ for integer $m\ne 0$. This expression neglects the slowly varying obliquity factor and uses a uniform slit; it describes the narrow-angle diffraction pattern for $w\gg \lambda$.