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

TL;DR

This paper derives a relativistically exact Eikonal equation for ring interferometers undergoing deformation, providing a comprehensive framework that generalizes previous results and accounts for non-reciprocal phase shifts due to fiber changes.

Contribution

It introduces a relativistically exact Eikonal equation for deforming ring interferometers, extending previous models to include all orders in velocity and fiber dispersion effects.

Findings

The leading phase shift term is independent of refractive index and generalizes Sagnac results.

The second, adiabatic correction depends on refractive index and fiber deformation.

Adiabatic correction can be comparable to the Sagnac term, especially in Fizeau's interferometer.

Abstract

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.