A multimode optical fibre feeds a bench-mounted spectrograph

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Geometrical optics and imaging · Geometrical optics and imaging

What an Instrument Can See, Ch Pupils: the aperture of a linear system

Problem

A multimode optical fibre feeds a bench-mounted spectrograph. The fibre's output cone has numerical aperture NAfib=0.22\mathrm{NA}_{\mathrm{fib}} = 0.22 in air, and the light is emitted uniformly over the full solid angle of this cone (a fair approximation for a step-index fibre once the light has left the core). The fibre tip sits at the focus of the spectrograph's collimating mirror, a parabola of focal length fcol=300 mmf_{\mathrm{col}} = 300\ \mathrm{mm} and clear diameter Dcol=40 mmD_{\mathrm{col}} = 40\ \mathrm{mm}; the collimator's clear aperture is the entrance pupil that the rest of the spectrograph optics is matched to. Treat all angles as paraxial, so sin⁡θ≈tan⁡θ\sin\theta \approx \tan\theta. ((a)) State the pupil-matching condition this instrument chain must satisfy, and find the numerical aperture the collimator itself accepts. ((b)) Given that the fibre delivers a larger cone than this, find the fraction of the light leaving the fibre that the collimator actually collects (the captured fraction scales as the square of the ratio of the two numerical apertures, since solid angle goes as θ2\theta^2 paraxially). ((c)) Find the clear diameter the collimator mirror would need, at the same focal length, to accept the fibre's full output cone with no loss.

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