In principle, the noise in optical fibers should be well predicted by the Wanser formula for thermo-refractive noise. Somewhere out there in the world, people are working on making a fiber that has opposite signs for the dn/dT and the d(dL/L)/dT, but until then TR is the limit.
However, non-fundamental temperature fluctuations are typically bigger than thermodynamic temperature fluctuations in the 0-1 Hz band. Above ~50 Hz, acoustic noise also becomes important.
Here are a collection of links with sort of simple ways to reduce these effects:
Wondering if we can do something easy for the relative laser noise measurements. Like putting the fiber in a metal tube with foam inside.