One derivation of the etalon's amplitude transfer function runs this way: “The…

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Wave propagation and light in media · Interference and diffraction

What an Instrument Can See, Ch Multiple beams: the stack as a matrix and the etalon as a filter

Problem

One derivation of the etalon's amplitude transfer function runs this way: “The impulse response is h(t)=t1t2∑m≥0(r1r2)mδ(t−mT)h(t)=t_1t_2\sum_{m\ge0}(r_1r_2)^m\delta(t-mT). To transform it I use e−i2πνte^{-i2\pi\nu t}, the same sign this book uses for every transform, so H(ν)=∫h(t) e−i2πνt dt=t1t2∑m≥0(r1r2)me−i2πνmT=t1t21−r1r2e−i2πνT.H(\nu)=\int h(t)\,e^{-i2\pi\nu t}\,dt=t_1t_2\sum_{m\ge0}(r_1r_2)^m e^{-i2\pi\nu mT}=\frac{t_1t_2}{1-r_1r_2e^{-i2\pi\nu T}}. Squaring the modulus gives exactly the same Airy function T(ν)=∣H(ν)∣2\mathcal T(\nu)=|H(\nu)|^2 as the book's H(ν)=t1t2/(1−r1r2e+i2πνT)H(\nu)=t_1t_2/(1-r_1r_2e^{+i2\pi\nu T}), so the sign of the exponent cannot matter physically.” ((a)) Find the specific error in this argument. ((b)) Apply the chapter's own shift theorem to this exponent and show what time it assigns to the mmth reflected pulse. Say what is physically wrong with the result, even though ∣H(ν)∣2|H(\nu)|^2 came out unchanged.

Related theory

Browse the problem library