A short attenuated laser pulse, LIGO, and a fast detector

[Spinnor]

Edit, after some thought I want to remove the power recycling mirror from the interferometer below. I think that will simplify my thought experiment.

Assume we have a LIGO type interferometer which is adjusted so that all light exits the detector port. Fire a single short attenuated laser pulse at the input of our apparatus. Assume we have a high efficiency photon number detector that has a very short dead time. Assume we can use the initial laser pulse to set off a timer. Should we expect the photons to arrive at the detector spread out in time as there are many paths of different length to the detector (multiple possible trips through each arm of the interferometer)? If so, working backwards can we infer from the photon arrival times the number of photons in the interferometer as a function of time? Should we expect an approximate "exponential decay" of the number of photons in the interferometer as a function of time?

Basic operating diagram of LIGO from https://physicsopenlab.org/2020/05/16/michelson-morley-interferometer/,

A Snipet from https://www.ligo.caltech.edu/page/ligos-ifo,

"While 4-km-long arms already seems enormous, if LIGO's interferometers were simple Michelson interferometers, they would still be too short to enable the detection of gravitational waves. But there are practical limitations to building a precision instrument much larger than 4km. So how can LIGO possibly make the measurements it makes?

The paradox was solved by altering the design of the Michelson interferometer to include something called "Fabry Perot cavities". The figure at left shows this modification to the basic design illustrated above. An additional mirror is placed in each arm near the beam splitter and 4km from the mirror at the end of that arm. This 4-km-long space constitutes the Fabry Perot cavity. After entering the instrument via the beam splitter, the laser in each arm bounces between these two mirrors about 300 times before being merged with the beam from the other arm."

Thanks for any help.

 

[DaveE]

Should we expect the photons to arrive at the detector spread out in time as there are many paths of different length to the detector (multiple possible trips through each arm of the interferometer)? If so, working backwards can we infer from the photon arrival times the number of photons in the interferometer as a function of time? Should we expect an approximate "exponential decay" of the number of photons in the interferometer as a function of time?

Yes. Like ring down spectroscopy*, for example. If the pulse is short compared to the cavity length, and there isn't dispersion, then you should get a pulse train as opposed to "smeared out".

*Maybe not the best example, since the exponential decay in this case is primarily due to losses in the sample, not the output coupling losses. But it's a similar idea, the cavity energy goes away at a rate proportional to its intensity.