How do they know the number of times the laser bounces back and forth, in the 4 km arms of the interferometer, before the laser light is picked out to the reading sensor?
See under “The Longer The Better” https://www.ligo.caltech.edu/page/ligos-ifo
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This number is the "effective number" of times the light bounces up and down, as judged by the sharpness of the resonance in the Fabry Perot arms of the interferometer.
The arms consist of a pair of mirrors; the end one is almost totally reflective, whereas the one closest to the laser input does have some finite transmission coefficient to allow the laser light in and out.
What happens is that on each trip up and down the arms, a small fraction of the trapped power can escape; this fraction is set by the reflectivity of the two mirrors and is parameterised by the "finesse". The larger the finesse, the larger the effective number of times the light must be bouncing up and down and the sharper the resonance (only light of a very narrow range of wavelengths will constructively interfere within the arms).
To more directly answer your question, we know because we know the reflectivity of the two mirrors involved and then we calculate the power amplification factor in a Fabry Perot resonator (effectively, the number of round trips the light makes) as: $$ n = \left(\frac{4 \sqrt{R_1 R_2}}{(1 - \sqrt{R_1 R_2})^2}\right)^{1/2} \, , $$ where $R_1 = 0.986$ and $R_2=0.999995$ are the two mirror reflectivities for LIGO, which gives $n \simeq 280$.
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$\begingroup$ In laymans terms, a Fabry Perot cavity's mirrors are slightly "leaky". They have to be, otherwise you can't get plane wave light in and out. I'm pretty sure a hand-waving $\Delta E \Delta t \ \rightarrow \ \Delta \omega \Delta t$ relationship can be found where $\Delta t$ is the round-trip time multiplied by the effective number of bounces. But don't quote me on that :-) $\endgroup$– uhohCommented Nov 30, 2023 at 14:33
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$\begingroup$ I've linked back here in How close to reality is my handwaving argument about Fabry Perot interferometers? $\endgroup$– uhohCommented Nov 30, 2023 at 22:27