Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / Section A / A10 / Solution

Normalize the central slit amplitude to one. The two outer amplitudes are $1/2$ each, not $1/\sqrt{2}$, because the specified attenuation concerns amplitude. For centre spacing $d=5\lambda$, adjacent slits have phase difference $\delta =(2\pi d/\lambda )\sin \theta =10\pi \sin \theta$. The far-field amplitude and intensity are thereforewhere $I_0$ is the central slit's intensity alone in the same narrow-slit approximation. Principal maxima have $\sin \theta =m/5$ and height $4I_0$; between them, zeros occur at $\sin \theta =(2m+1)/10$. Only values in $[-1,1]$ are physical. The endpoints are grazing-direction limits of this ideal angular model.

The amplitude weights form a binomial pattern. They remove the secondary maxima that an equal-amplitude three-slit array would have between its principal peaks. No phase shift is needed: changing the coherent weights already changes the interference structure.