It is generally accepted that the fringe shift produced in a fiber optic Sagnac interferometer is independent of the refractive index of the fiber. In fact, the waveguide aspect of the fiber is not of any consequence, the counter propagating beams may propagate in any dielectric medium. Does anybody know a physical explanation for this effect, other than the special relativistic (On the Experiment of F. Harress) or general relativistic (Sagnac Effect) explanations? It has been reported that(Modified Sagnac experiment for measuring traveltime difference between counter-propagating light beams in a uniformly moving fiber) the Sagnac effect is produced even when the moving path is effectively a straight line. This later configuration approximates an inertial frame, and with in an inertial frame one should not be able to measure one's state of motion!
In order for the refractive index to be of no concequence it must be the case that the refractive index has at least two effects, in such a way that they drop away against each other. As we know, the higher the refractive index, the slower the propagation of the light, and the wavelength is correspondingly smaller. That is: the frequency of the light is independent of the refractive index of the medium.
As we know, the frequency of the light comes into effect in the following way. The Sagnac device executes comparison of the amount of cycles of the counter-propagating light and the amount of cycles of the co-propagating light. if the counter-propagating light goes through half a cycle more than the co-propgating light the shift of the interference pattern will be half the distance between two lines of the interference pattern
I'm guessing that the refraction index effects drop away against each other in such a way that the difference in amount of cycles remains the same, whatever the refractive index of the medium.
For a deeper discussion than that I recommend the following article by Olaf Wucknitz:
The Sagnac effect, Twin Paradox, and space-time topology
Olaf Wucknitz also discusses the fiber optic conveyer experiments
In my opinion the article by Olaf Wucknitz is by far the most insightful article about the Sagnac effect, way ahead of anyone else.