Four-slit Fraunhofer diffraction

For four identical long slits of width $a$ centred at $x=\pm D/2,\pm 3D/2$, the aperture Fourier transform givesHere $\mathrm{sinc}x=\sin x/x$. Normalized to the on-axis light intensity,Principal array directions have $qD=2\pi m$; a single-slit diffraction envelope zero cancels such an order when $ma/D$ is a nonzero integer. The closest central array zeros satisfy $qD=\pm \pi /2$. Finite slit width generally shifts the actual maxima of the complete product slightly away from the nominal principal array directions.