A relativistically exact Eikonal equation for optical fibers with application to adiabatically deforming ring interferometers

We derive the relativistically exact Eikonal equation for ring interferometers undergoing deformations. For ring interferometers that undergo slow deformation we describe the two leading terms in the adiabatic expansion of the phase shift. The leading term is independent of the refraction index $n$ and is given by a line integral generalizing results going back to Sagnac \cite{sagnac1913,wang2004,Ori} for non-deforming interferometers to all orders in {$\beta=|\mathbf{v}|/c$}. In the non-relativistic limit {this term} is $O(\beta)$. The next term in the adiabaticity has the form of a double integral, it is of order $\beta^0$ and depends on the refractive index $n$. It accounts for non-reciprocity due to changing circumstances in the fiber. The adiabatic correction is often comparable to the Sagnac term. In particular, this is the case in Fizeau's interferometer. Besides providing a mathematical framework that puts all ring interferometers under a single umbrella, our results generalize and strengthen results of \cite{Ori, shupe} to fibers with chromatic dispersion.