An absorber of area sits in a medium of refractive index behind a passive

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Earth, Climate & Energy Physics · Geometrical optics and imaging

Energy from Physics, Ch The solar resource resolved: geometry, spectrum and the limits of concentration

Problem

An absorber of area AabsA_{\rm abs} sits in a medium of refractive index nn behind a passive, lossless concentrator, but, to cut its losses through the sides of the cavity, it is built to accept light only from within a half-angle θr\theta_r of its normal. Sunlight reaches the aperture AinA_{\rm in}, in air, from within the Sun's half-angle θ⊙\theta_\odot. ((a)) Write dΩ=sin⁡θ dθ dφ\mathrm d\Omega=\sin\theta\,\mathrm d\theta\,\mathrm d\varphi and integrate n2cos⁡θ dA dΩn^2\cos\theta\,\mathrm dA\,\mathrm d\Omega to find the largest etendue the absorber can accept. ((b)) Equate it with the etendue of the incoming sunlight and show that C=Ain/Aabs≤n2sin⁡2θr/sin⁡2θ⊙C=A_{\rm in}/A_{\rm abs}\le n^2\sin^2\theta_r/\sin^2\theta_\odot. Check that θr=90∘\theta_r=90^\circ recovers the chapter's limit. ((c)) Evaluate the bound for n=1n=1 and θr=60∘\theta_r=60^\circ, with θ⊙=4.65 mrad\theta_\odot=4.65\,\mathrm{mrad}.

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