A spectrograph places a diffraction grating with lines per millimetre

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Interference and diffraction · Interference and diffraction

Optics and Photonics, Ch Fraunhofer diffraction from apertures and gratings

Problem

A spectrograph places a diffraction grating with 600600 lines per millimetre, illuminated over a width of 30 mm30\ \text{mm}, in front of a Fabry-Perot etalon (Chapter 11) of plate spacing d=5.0 mmd = 5.0\ \text{mm} and mirror reflectance R=0.90\mathcal{R} = 0.90. Both are used in light near λ=589 nm\lambda = 589\ \text{nm}, and the grating is used in first order (m=1m=1). ((a)) Find the resolving power λ/Δλ\lambda/\Delta\lambda of the grating alone. ((b)) Using the etalon's finesse F=πR/(1−R)\mathcal{F} = \pi\sqrt{\mathcal{R}}/(1-\mathcal{R}) and its interference order me=2d/λm_e = 2d/\lambda (Chapter 11), find the resolving power of the etalon alone. ((c)) Find the etalon's free spectral range ΔλFSR=λ2/(2d)\Delta\lambda_{\rm FSR} = \lambda^2/(2d), and explain in one or two sentences why placing the grating in front of the etalon (rather than using either alone) is the sensible choice for resolving fine spectral detail near 589 nm589\ \text{nm}.

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