Past exam of the physics A course of the University of Cambridge / 2017 / Paper 2 / Section B / B8 / Convolution and a finite slit array

Use Fourier transform convention $\hat{f}(q)=\int f(x)e^{-iqx}dx$. The convolution theorem givesA finite array of very narrow slits at $x=jD$, $j=0,\dots ,N-1$, has transmission proportional to ${\sum }_j\delta (x-jD)$. It can be constructed by multiplying an infinite Dirac comb by a finite window. The infinite comb transforms to a reciprocal comb; the finite window convolves each reciprocal peak with a narrow sinc-shaped amplitude. Thus increasing the illuminated length narrows each Fraunhofer diffraction peak.

The exact finite sum makes the argument quantitative:Its normalized intensity is $[\sin (NqD/2)/(N\sin (qD/2)){]}^2$. Principal maxima occur at $qD=2\pi m$; their first zeros are displaced by $|\Delta (qD)|=2\pi /N$. Replacing the ideal narrow slit by a finite slit function convolves the array in real space, multiplying this field by the single-slit Fourier transform. Field amplitudes are transformed and combined before their squared modulus is taken.