= Solution
For uniform illumination, the aperture is $h(x)=1$ for $|x|<a/2$ and zero otherwise. With $q=2\pi\sin\theta/\lambda$,
$$
\widetilde h(q)=\int_{-a/2}^{a/2}e^{-iqx}\,dx
=a\frac{\sin(qa/2)}{qa/2},\qquad
\boxed{I(\theta)=I(0)\left(\frac{\sin\alpha}{\alpha}\right)^2,\quad
\alpha=\frac{\pi a\sin\theta}{\lambda}}.
$$
Zeros occur at $\sin\theta=j\lambda/a$, $j\ne0$; the central maximum extends between $\pm\lambda/a$. A monochromatic <laser> expanded into a collimated beam can illuminate the slit. A <lens> with the screen in its back <focal plane> produces the Fraunhofer pattern; alternatively use a sufficiently distant screen, $z\gg a^2/\lambda$. Align the slit and beam and avoid detector saturation.
For $N$ equally spaced slits, write the aperture as a <convolution>
$$
h_N=\operatorname{rect}_a*\sum_{j=0}^{N-1}\delta(x-jb).
$$
The <convolution theorem> converts this into a single-slit transform times the finite-array transform. Therefore
$$
\boxed{I_N(q)\propto a^2\operatorname{sinc}^2\left(\frac{qa}{2}\right)
\left[\frac{\sin(Nqb/2)}{\sin(qb/2)}\right]^2},
\qquad \operatorname{sinc}u=\frac{\sin u}{u}.
$$
Principal maxima are at $b\sin\theta=m\lambda$ for <normal incidence>, inside the single-slit envelope. Their height scales as $N^2$ times the envelope, and their first zeros are separated from a peak by $\delta(\sin\theta)=\lambda/(Nb)$.
The envelope-to-order-spacing ratio is $b/a$: increasing it admits more orders within the central envelope and changes the relative order strengths. In particular, an order is missing when $ma/b$ is a nonzero integer. Statements about absolute angular spacing need a fixed dimension: at fixed $a$, increasing $b$ reduces order spacing; at fixed $b$, decreasing $a$ broadens the envelope.
For fixed incident angle, differentiating the <grating equation> gives an order displacement $\delta(\sin\theta)=|m|\delta\lambda/b$. Equating this to the peak-to-first-zero width gives
$$
\boxed{\delta\lambda=\frac{\lambda}{|m|N}\simeq\frac{\lambda b}{|m|W}}.
$$
Here $W\simeq Nb$ is the illuminated grating width; this is the ideal Rayleigh limit, before slit width or aberrations broaden the spectral image.
With $b=(1600\ \mathrm{mm^{-1}})^{-1}=0.000625\ \mathrm{mm}$, the required count is $N\ge372.7/(2\times0.2)=931.75$. Thus \b[932 illuminated slits], or \b[$W_{\min}\simeq0.5825\ \mathrm{mm}$], suffice ideally. Checking <diffraction-order feasibility>, <normal incidence> cannot produce this second order because $2\lambda/b=1.193>1$; an oblique <spectrograph> geometry is required. A feasible reflection-Littrow example has $\sin\theta=m\lambda/(2b)=0.5963$, or $\theta\simeq36.6^\circ$.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2014/paper-2-grating-fourier.png]
{title=Original slit, finite-array convolution, diffraction pattern and Fourier-plane experimental layout}
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