I wrote a code to calculate thermo-refractive noise in a finite-sized cylindrical substrate as given in Heinert etal 2011. The noise is very small ~10^-7 [^Hz/_rtHz] compared to other noise in the cavity ( no surprise here). The code can be used to estimate the TR noise in fiber optic. The calculation should be correct as I double checked with the calculation by Koji and Deep.

I followed the calculation for TR noise in cylindrical substrate [Heinert etal 2011] for our setup (1" diameter , 0.25" thick, fused silica). The result is in [m/rtHz].

To convert it to frequency noise of the laser:

1. Convert the displacement noise to phase noise in the beam first,  S_phi = S_x * 2*pi*n/lambda_0  (n is index of refraction).
2. S_f = S_phi * f   (f is fourier frequency), multiply by 4 to get the contribution from 4 mirrors.

above: TR noise in substrate. It just so small compared to other noise sources in the noise budget(~ 10^-3 - 10^-1 Hz/rtHz level that I don't see the need to add it in the complete noise budget.

Since we will use the same substrate, the noise level will be the same for short and long cavities. The different in beamsize will vary the noise level a bit.

Note: this calculation is for a Gaussian beam profile in a cylindrical substrate, to use this calculation for  fiber optic TR noise, some assumption about the mode of the beam is required.