How close to reality is my handwaving argument about Fabry Perot interferometers?

Question

Under this answer to https://astronomy.stackexchange.com/q/55437/7982 I wrote a comment where propose the uncertainty inequality roughly written¹ as $\Delta E \Delta t \ge h$ or for photons $\Delta \nu \Delta t \ge 1$ can be applied to Fabry Perot interferometers where $\Delta t$ is the round-trip time multiplied by the mean number of traversals of a photon before leaking out. I guess that would be when the probability that the photon is still inside drops to 1/e.

Factors of order $2\pi$ aside, is it basically right?

I just ran across this related item in HSM SE: Uncertainty principle / Fourier results

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¹the proper quantum mechanical uncertainty principle for energy and time is written $\sigma_E \sigma_t \ge \frac{h}{2\pi}$

Answer 1

The Finesse of the LIGO Fabry Perot resonators is (or was when I last updated my lecture notes) about $Q=450$. I call it $Q$ because this is effectively the Q-factor of the resonator.

The relationship between this and the power amplification gain - which is often hand-wavingly used to describe how many time the light travels up and down the arms - is $2Q/\pi = n = 286$ for LIGO.

The Q-factor is the ratio of the frequency of the resonance to the FWHM of the resonance. The relevant frequency is the free spectral range of the interferometer arms, which is $c/2L$. Thus $$\Delta \nu = \frac{c}{2LQ} = \frac{c}{\pi nL}\ .$$

The "round trip time" for the photons would be $\Delta t = 2nL/c$, and so $$\Delta \nu \Delta t = \frac{2}{\pi}$$