Generalized Sagnac-Wang-Fizeau formula

We present a special-relativistic analysis of deformable interferometers where counterpropagating beams share a common optical path. The optical path is allowed to change rather arbitrarily and need not be stationary. We show that the phase shift has two contributions: One contribution is independent of the refractive index $n$ and has the form of a line integral. This term gives the Wang empirical formula for deformable Sagnac interferometers. The second term is quadratic in $n$ and has the form of a double integral. The analysis provides a unifying framework incorporating the Sagnac, Wang, and Fizeau effects in a single scheme, gives a rigorous proof of Wang empirical formula, and can be used to study various perturbations of Sagnac interferometers.

Figure 1

The Sagnac interferometer rotates as a rigid body in the laboratory with angular velocity $\mathrm{\Omega }$. The beam splitter is at the bottom left corner. The two counterpropagating beams are marked by red arrows: one beam propagates with and the other against the rotation. The optical path may contain comoving dielectric medium shown as a blue rectangle. The incoming source and outgoing signal are marked by dashed arrows.

Figure 2

Schematic Fizeau interferometer. The three mirrors and beams splitter are at rest in the laboratory. The two counterpropagating beams are denoted by the red arrows. Fluid, say water, is flowing in one arm of the interferometer so one beam is moving against the flow and one with the flow.

Figure 3

An example of a Wang interferometer: The optical fiber shown as the red curve moves in the direction marked by the cyan arrows. The black dot represents the comoving beam splitter and the two dashed red arrows the light source and detector. The counterpropagating light beams share a common path and are marked by red arrows.

Figure 4

The parametrized curve $\mathbf{r}(\theta ,t)$ is, for each fixed time $t$, a closed loop in space representing a fiber loop. $\theta$ is a comoving coordinate, designating fixed material points on the fiber. Shown are the tangent $\mathbf{e}$ at a point $\theta$ of the loop. The two counterpropagating beams are marked with the red arrows and the $\pm$ signs. The black dot represents the comoving beam splitter located at $\theta =0$ and the two dashed red arrows the comoving source and detector.

Figure 5

Space-time diagram describing the travel of a nodal point across an infinitesimal fiber's segment $d\theta$. The diagram shows the projections of all 3D vectors in one direction. The two parallel black lines are the world lines of the two segment's edges. The red line is the world line of the node. The lines intersect at the events $A$ and $B$ when the node enters and leaves the segment. Also shown are the vectors $\mathbf{e}d\theta =d\mathbf{\ell },d{\mathbf{r}}_{AB}$, and $\mathbf{v}dt$ which participate in Eq. (17).