 Solution

Expand the propagation distance at small angles:The first two terms supply an aperture-independent phase. If the aperture quadratic phase is small, $k(x^2+y^2)/(2L)\ll 1$, it too may be dropped from the integrand. The remaining Fraunhofer diffraction amplitude is the aperture's Fourier transform at transverse wavevector $(k{x}_0/L,k{y}_0/L)$, as required.

For slit width $w$ and center spacing $d$, adding the two shifted top-hat transforms gives the normalized intensityThe aperture outer half-width is $17.5\mu \mathrm{m}$; its maximum quadratic phase is only about $1.5\times 10^{-3}$ radians at the given wavelength and distance, so the Fraunhofer approximation is excellent. Fringe spacing is $\lambda L/d=25.28\mathrm{mm}$ and the first single-slit envelope zero is at $\lambda L/w=63.2\mathrm{mm}$.

The side maximum of a finite-width double-slit pattern shifts slightly inward from $25.28\mathrm{mm}$. Differentiating the full intensity gives, away from zeros,Its first side maximum is at $X\simeq 23.88\mathrm{mm}$ with $I/I(0)\simeq 0.592$. The expected rate is therefore $59.2{\mathrm{s}}^{-1}$. Poisson shot noise gives signal-to-noise ratio $N/\sqrt{N}=\sqrt{N}$, so one needs $N=100$ counts:Using the common fringe-center approximation instead gives relative intensity ${\mathrm{sinc}}^2(\pi w/d)\simeq 0.573$ and $t\simeq 1.75\mathrm{s}$. Both express the same order of exposure; the first uses the true maximum. Dark counts, background subtraction and uncertainty in the reference rate are excluded by the ideal stated noise model.