A Fabry-Pérot etalon used as a narrowband filter is built from two identical

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

Optics and Photonics, Ch Multiple-beam interference: thin films, multilayers and the Fabry-Perot etalon

Problem

A Fabry-Pérot etalon used as a narrowband filter is built from two identical, lossless, flat mirrors of power reflectance R=0.90\mathcal R = 0.90, facing each other with a round-trip phase δ\delta between them (as defined in the text, δ=2k0ndcos⁡θ\delta = 2k_0nd\cos\theta). Light of unit amplitude E~0\tilde E_0 is incident from outside; the amplitude reflection coefficient of each mirror seen from outside is r=Rr=\sqrt{\mathcal R}, and the amplitude transmission coefficients satisfy tt′=T=1−Rtt'=\mathcal T=1-\mathcal R. ((a)) By summing every beam that returns to the incident side, the beam reflected directly off the front mirror plus every beam that has made one or more extra round trips inside the cavity before coming back out, show that the reflected amplitude is E~r=E~0 r 1−eiδ1−Reiδ\tilde E_r = \tilde E_0\, r\,\frac{1-e^{i\delta}}{1-\mathcal R e^{i\delta}} and hence that the reflected fraction of the power is RFP=Fsin⁡2(δ/2)1+Fsin⁡2(δ/2),F=4R(1−R)2\mathcal R_{FP} = \frac{F\sin^2(\delta/2)}{1+F\sin^2(\delta/2)}, \qquad F = \frac{4\mathcal R}{(1-\mathcal R)^2} ((b)) Using the Airy transmission function TFP=1/(1+Fsin⁡2(δ/2))\mathcal T_{FP}=1/(1+F\sin^2(\delta/2)) already derived in the text, show algebraically that RFP=1−TFP\mathcal R_{FP}=1-\mathcal T_{FP} for every δ\delta. ((c)) Evaluate RFP\mathcal R_{FP} and TFP\mathcal T_{FP} for this etalon at a round-trip phase δ=π/3\delta = \pi/3 rad and confirm numerically that they sum to 1.

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