Past exam of the physics 2 paper of the University of Oxford / 2018 / 5

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.

The groove spacing is $d=(2000\times 100{)}^{-1}\mathrm{m}=5.00\mu \mathrm{m}$. The grating equation at normal incidence is $d\sin {\theta }_m=m\lambda$. For diffraction order $m=2$,ThereforeThe ideal resolving power of a diffraction grating is $|m|N$, so the Rayleigh criterion for this doublet requires about $N\ge 145$ 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.