= Four-slit Fraunhofer diffraction
For four identical long slits of width $a$ centred at $x=\pm D/2,\pm3D/2$, the aperture <Fourier transform> gives
$$
\psi(q)\propto2a\operatorname{sinc}(qa/2)[\cos(qD/2)+\cos(3qD/2)],\qquad
q=\frac{2\pi\sin\theta}{\lambda}.
$$
Here $\operatorname{sinc}x=\sin x/x$. Normalized to the on-axis <light intensity>,
$$
I(q)/I(0)=\operatorname{sinc}^2(qa/2)\cos^2(qD/2)\cos^2(qD).
$$
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.
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