Chapter 3 showed that a circular pupil of area and numerical aperture carries an étendue

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

What an Instrument Can See, Ch The transfer function of an imaging system

Problem

Chapter 3 showed that a circular pupil of area AA and numerical aperture NA\mathrm{NA} carries an étendue Ξ=πA NA2\Xi = \pi A\,\mathrm{NA}^2, and that the number of independent modes, per polarisation, a beam of this étendue carries is about Ξ/λ2\Xi/\lambda^2. This chapter has shown separately that the coherent (amplitude) space-bandwidth product of an imaging system with field area AA and numerical aperture NA\mathrm{NA} is SBPcoh=Aπ(NA/λ)2\mathrm{SBP}_{\mathrm{coh}} = A\pi(\mathrm{NA}/\lambda)^2. ((a)) Show algebraically that SBPcoh\mathrm{SBP}_{\mathrm{coh}} and Ξ/λ2\Xi/\lambda^2 are the same expression. ((b)) A fibre-fed spectrograph's collimating optic accepts light over a circular field of area A=0.50 mm2A=0.50\ \mathrm{mm^2} at NA=0.22\mathrm{NA}=0.22, used at λ=900 nm\lambda=900\ \mathrm{nm}. Evaluate this common count of independent modes.

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