A laboratory fibre-fed spectrometer uses a step-index fibre of core diameter and…

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Geometrical optics and imaging · Photodetection, photonic sources and quantum optics

What an Instrument Can See, Ch Radiance, etendue and a throughput budget that is spent

Problem

A laboratory fibre-fed spectrometer uses a step-index fibre of core diameter

100\,\mu\text{m}

and numerical aperture

\mathrm{NA} = 0.10

, coupled losslessly into free space. ((a)) Treating the fibre exit face as a circular Lambertian aperture, find its etendue

\Xi = \pi A\,\mathrm{NA}^2

, where

A

is the face area. ((b)) The detector records a photoelectron signal of

50\,\text{e}^-\,\text{s}^{-1}

over an integration time of

20\,\text{s}

, with dark current

0.02\,\text{e}^-\,\text{s}^{-1}

and read noise

3\,\text{e}^-

(rms, added once per read, independent of exposure time). Find the signal-to-noise ratio and state which noise term dominates. ((c)) A

650\,\text{nm}

laser diode of optical power

2.0\,\text{mW}

is coupled into the fibre. Using the photopic luminous efficiency

V(650\,\text{nm}) = 0.107

and the peak luminous efficacy

K_m = 683\,\text{lm}\,\text{W}^{-1}

, find the luminous flux this power corresponds to, and state why this conversion is only valid for this one wavelength.

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