# How is the phase gain of a Fabry-Perot resonator for gravitational wave detection derived?

I am trying to understand the use of the Fabry-Perot (FP) resonator in the arms of a gravitational wave detector. A typical explanation is that the gain in power in the arms is equal to $$\sqrt{F}$$ (where $$F$$ is the coefficient of finesse, and which I know how to derive) and so the light effectively travels up and down the arms this many times, increasing the effective length of the Michelson interferometer and hence improving the phase sensitivity by the same factor. This derivation assumes the gravitational wave frequency, $$f$$, is small, so that the metric perturbation is constant whilst the light is in the arm.

However, this doesn't help me understand the higher frequency response of the FP resonator, which apparently goes something like $$(1 + f^2/f_p^2)^{-1}$$, where $$f_p$$ is a frequency pole that depends on the arm length and the resonator finesse.

Is there a straightforward way to derive this important relationship, which limits the performance at high frequencies? I have been unable to find anything in the gravitational wave literature or any of the optics resources that are readily available.

• I can try to write a more detailed answer later when I have some more time, but there is a nice derivation in this public technical note from LIGO: dcc.ligo.org/LIGO-P080036/public. The result you want follows from Eq 32. – Andrew Apr 26 at 0:04