Past exam of the physics A course of the University of Cambridge / 2010 / Paper 1 / Section D / D12 / a / Solution

One instrument is a diffraction grating spectrometer. Its principal maxima satisfy $d\sin \theta =m\lambda$ at normal incidence. N coherently illuminated grooves give a narrow principal peak; applying the Rayleigh criterion to the next wavelength yields the resolving power of a diffraction gratingHigher order and more illuminated grooves improve resolution, but finite entrance-slit width, aberrations, detector sampling and overlap between spectral orders limit useful resolution. Angular dispersion $d\theta /d\lambda =m/(d\cos \theta )$ is distinct from the ability to resolve two finite-width peaks.

A second instrument is a Fabry–Pérot interferometer. For plate spacing L, refractive index n and internal angle theta, transmission resonances satisfy $2nL\cos \theta =m\lambda$. Multiple coherent reflections produce an Airy transmission profile. For identical lossless mirrors of intensity reflectivity $\mathcal{R}$, the high-finesse limit givesThe frequency free spectral range is approximately $c/(2nL\cos \theta )$ for nondispersive n. A larger optical spacing narrows a line but also shrinks the unambiguous frequency interval; nearby orders need sorting. Mirror flatness, absorption, alignment and aperture broadening lower the effective finesse. This resolving-power formula assumes separated narrow lines and a stated line-separation convention, rather than equating transmission peak spacing with peak width.