An intracavity etalon inside a dye laser 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

An intracavity etalon inside a dye laser is built from two identical, lossless, flat mirrors of power reflectance R=0.50\mathcal R = 0.50, facing each other with a round-trip phase δ\delta between them (δ=2k0ndcos⁡θ\delta=2k_0nd\cos\theta). Light of unit amplitude E~0\tilde E_0 is incident from outside; each mirror has amplitude reflection coefficient r=Rr=\sqrt{\mathcal R} seen from outside, with tt′=T=1−Rtt'=\mathcal T=1-\mathcal R. ((a)) By summing every beam returning 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, 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 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 function TFP=1/(1+Fsin⁡2(δ/2))\mathcal T_{FP}=1/(1+F\sin^2(\delta/2)) from the text, show 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} at δ=2π/3\delta = 2\pi/3 rad and confirm numerically that they sum to 1.

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