Sagnac effect in ring lasers and ring resonators. How does the refractive index of the optical medium influence the sensitivity to rotation?

The Sagnac effect in a ring laser (RL) results in a frequency difference of counterpropagating waves that is proportional to the RL angular rotation rate. We address the question of how an optical medium filling the whole RL or a part of it influences the frequency difference of counterpropagating waves. While the formulas for this difference in a rotating RL abound in the literature, there is no agreement among them as to whether the medium increases or decreases this difference or indeed leaves it unchanged. Nor do the available (and often contradictory) experimental data fully clarify the situation. Because the Sagnac effect is a special relativity effect, the relativistic velocity addition law is used here to calculate the frequency difference of counterpropagating waves in an RL. When a homogeneous optical medium fills the entire perimeter of the resonator of a rotating RL, we show that the frequency difference of counterpropagating waves is inversely proportional to the refractive index of the medium. The results obtained can also be used to calculate the difference between the resonant frequencies of counterpropagating waves in rotating ring resonators in the presence of an optical medium.