special relativity - Interferometers and Earth's motion relative to the Aether

Why was the Michelson and not the Fabry-Perot interferometer used to detect the motion of Earth relative to the Aether?

Maybe the Fabry-Perot was used but we all know that the most famous experiment was made with the Michelson interferometer, so:

Are there features of the Michelson interferometer that makes it better than the Fabry-Perot one in this kind of experiment?

I know that the fringe shift $n$ in both cases is

$$n\propto \frac{Lv^2}{\lambda c^2}$$

where $L$ is the distance between the mirrors and $v$ is Earth's speed relative to the Aether. In others words, the experiment would lead to the "same" result in both cases (apart by a constant factor). So this makes me think that Michelson interferometer has something special in comparison with the Fabry-Perot one.

Answer

Modern replications of the Michelson-Morley experiment do indeed use Fabry-Pérot cavities in each arm, in exactly the way you describe. See the heading "Recent Optical Resonator Experiments" on the Michelson-Morley Interferometer wikipedia page. Exactly as you reason, the sensitivity is increased, through optical resonance, by a factor of $F$, the cavity's finesse. These modern experiments achieve astounding bounds on the velocity of the putative Aether wind. As stated on the Wiki page:

Herrmann, S.; Senger, A.; Möhle, K.; Nagel, M.; Kovalchuk, E. V.; Peters, A., "Rotating optical cavity experiment testing Lorentz invariance at the $10^{−17}$ level", Phys. Rev. D 80 #100, 2009

shows that the difference between the speed of light in the two arms fulfils $\frac{\Delta\,c}{c}\leq 10^{-17}$. These interferometers are small versions of the kind of device used in gravitational wave sensing.

The reason Michelson and Morley didn't do it this way is simply practicality, as Dmckee points out. If you want to increase your interferometer's sensitivity to phase through resonance, as in a Fabry-Pérot interferometer, then you must ensure that the cavity stays at the exact length where resonance happens. The sensitivity swiftly drops off if the cavity's strain (from temperature drift, vibration and so forth) as you move away from resonance. The cavity's high sensitivity amplifies phase noise exactly as it does the signal, so for many limitations on the experiment, the method will not help you. In many cases, resonance is therefore simply like using an amplifier to amplify a noisy signal.

To make resonance work properly for you in an MM experiment, this implies extremely high technology to suppress phase noise A major source of noise is the jitter of the source laser's cavity, if you're going to use a laser, so cryogenic sources are used in applications like these. The fact that the interferometer needs to rotate is a big potential problem: you need to find bearings that will not vibrate. Michelson and Morley used a pool of mercury, but once you introduce resonance the requirements go through the roof!

The Michelson interferometer is optimal for many applications similar to the MM experiment for the reason that I explore in my paper:

R. W. C. Vance and R. Barrow, "General linear differential interferometers", J. Opt. Soc. Amer. A, 12 #2, pp 346-352, 1995

That is, the Michelson interferometer is the highest sensitivity interferometer whose output intensity is a pure differential function of the two arm lengths. This means that any effect that bears on the arms equally - large scale vibration, temperature drift and so forth - is cancelled out in such a device. I prove in this paper that it is impossible to have a globally differential function of phase and exceed the sensitivity of the Michelson interferometer.