= Solution
A <grating spectrometer> places an illuminated narrow slit at the front focal plane of a <collimator>. The resulting parallel beam reaches the <diffraction grating> at known incidence, here taken normal. A rotatable telescope focuses each outgoing diffracted parallel beam onto its crosshair or detector. Align the slit with the grating rulings, focus the <collimator> at infinity, and illuminate many grooves across a suitable aperture. A narrow slit supplies sufficient <spatial coherence> and prevents slit width from blurring the angular separation; the telescope must also resolve the two images.
\Image[/past-exam-of-the-physics-2-paper-of-the-university-of-oxford/2018-grating-spectrometer.png]
{title=A <grating spectrometer> with a narrow source slit, <collimator>, normally illuminated <diffraction grating> and rotatable observing telescope}
The groove spacing is $d=(2000\times100)^{-1}\ \mathrm m=5.00\ \mu\mathrm m$. The <grating equation> at normal incidence is $d\sin\theta_m=m\lambda$. For <diffraction order> $m=2$,
$$
\theta_{579}=\arcsin(0.2316)\simeq13.3913^\circ,\qquad
\theta_{577}=\arcsin(0.2308)\simeq13.3442^\circ.
$$
Therefore
$$
\boxed{\Delta\theta\simeq8.22\times10^{-4}\ \mathrm{rad}=0.0471^\circ=2.83\ \text{arcmin}.}
$$
The ideal <resolving power of a diffraction grating> is $|m|N$, so the <Rayleigh criterion> for this doublet requires about $N\geq145$ illuminated grooves, spanning roughly $0.73\ \mathrm{mm}$. This is an ideal lower bound; finite slit width, aberrations and detector resolution can require a larger usable aperture.
Back to article page
