Sagnac Effect - Theory

Theory

The shift in interference fringes in a ring interferometer can be viewed simply as a consequence of the different distances that light travels due to the rotation of the ring.(Fig. 3) The simplest derivation is for a circular ring rotating at an angular velocity of, but the result is general for loop geometries with other shapes. If a light source emits in both directions from one point on the rotating ring, light traveling in the same direction as the rotation direction needs to travel more than one circumference around the ring before it catches up with the light source from behind. The time that it takes to catch up with the light source is given by:

is the distance (black bold arrow in Fig. 3) that the mirror has moved in that same time:

Eliminating from the two equations above we get:

Likewise, the light traveling in the opposite direction of the rotation will travel less than one circumference before hitting the light source on the front side. So the time for this direction of light to reach the moving source again is:

The time difference is

For, this reduces to

where A is the area of the ring.

Although this simple derivation is for a circular ring, the result is in fact general for any shape of rotating loop with area A.(Fig. 4)

We imagine a screen for viewing fringes placed at the light source (or we use a beamsplitter to send light from the source point to the screen). Given a steady light source, interference fringes will form on the screen with a fringe displacement proportional to the time differences required for the two counter-rotating beams to traverse the circuit. The phase shift is, which causes fringes to shift in proportion to and .

At non-relativistic speeds, the Sagnac effect is a simple consequence of the source independence of the speed of light. In other words, the Sagnac experiment does not distinguish between pre-relativistic physics and relativistic physics.

When light propagates in fibre optic cable, the setup is effectively a combination of a Sagnac experiment and the Fizeau experiment. In glass the speed of light is slower than in vacuum, and the optical cable is the moving medium. In that case the relativistic velocity addition rule applies. Pre-relativistic theories of light propagation cannot account for the Fizeau effect. (By 1900 Lorentz could account for the Fizeau effect, but by that time his theory had evolved to a form where in effect it was mathematically equivalent to special relativity.)

Since emitter and detector are traveling at the same speeds, Doppler effects cancel out, so the Sagnac effect does not involve the Doppler effect. In the case of ring laser interferometry, it is important to be aware of this. When the ring laser setup is rotating, the counterpropagating beams undergo frequency shifts in opposite directions. This frequency shift is not a Doppler shift, but is rather an optical cavity resonance effect, as explained below in Ring lasers.

The Sagnac effect has stimulated a century long debate on its meaning and interpretation, much of this debate being surprising since the effect is perfectly well understood in the context of special relativity. An essential point that has not been well-understood until recent years, is that rotation is not required for the Sagnac effect to be manifest. What matters is that light moves along a closed circuit, and that an observer is in motion with respect to that circuit. In Fig. 5, the measured phase difference in both a standard fibre optic gyroscope, shown on the left, and a modified fibre optic conveyor, shown on the right, conform to the equation Δt = 2vL/c2, whose derivation is based on the constant speed of light. It is evident from this formula that the total time delay is equal to the cumulative time delays along the entire length of fibre, regardless whether the fibre is in a rotating section of the conveyor, or a straight section. In addition, it is evident that that there is no connection between the total delay and the area enclosed by the light path. The equation commonly seen in the analysis of a rotating, circular Sagnac interferometer, Δt = 4Aω/c2, can be derived from the more general formula by a simple substitution of terms: Let v = rω, L = 2πr. Then Δt = 2vL/c2 = 4πr2ω/c2 = 4Aω/c2.