Basic case

Figure 3. Light traveling opposite directions go different distances before reaching the moving source

The shift in interference fringes in a ring interferometer can be viewed intuitively as a consequence of the different distances that light travels due to the rotation of the ring.(Fig. 3)^[32] The simplest derivation is for a circular ring of radius R, with a refractive index of one, rotating at an angular velocity of ${\displaystyle \omega }$, 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 ${\displaystyle t_{1}}$ that it takes to catch up with the light source is given by:

${\displaystyle t_{1}={\frac {2\pi R+\Delta L}{c}}}$

${\displaystyle \Delta L}$ is the distance (black bold arrow in Fig. 3) that the mirror has moved in that same time:

${\displaystyle \Delta L=R\omega t_{1}.\,}$

Eliminating ${\displaystyle \Delta L}$ from the two equations above we get:

${\displaystyle t_{1}={\frac {2\pi R}{c-R\omega }}.}$

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:

${\displaystyle t_{2}={\frac {2\pi R}{c+R\omega }}.}$

The time difference is

${\displaystyle \Delta t=t_{1}-t_{2}={\frac {4\pi R^{2}\omega }{c^{2}-R^{2}\omega ^{2}}}.}$

For ${\displaystyle R\omega =v\ll c}$, this reduces to

${\displaystyle \Delta t\approx {\frac {4\pi R^{2}\omega }{c^{2}}}={\frac {4A\omega }{c^{2}}},}$

where A is the area of the ring.

Figure 4. The Sagnac area formula applies to any shape of loop.

Although this simple derivation is for a circular ring with an index of refraction of one, the result holds true for any shape of rotating loop with area A.(Fig. 4)

For more complicated shapes, or other refractive index values, the same result can be derived by calculating the optical phase shift in each direction using Fermat's principle and taking into account the different phase velocities for the different propagation directions in an inertial laboratory frame, which can be calculated using relativistic addition of velocities.^[33]

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 ${\displaystyle \Delta \phi ={\frac {2\pi c\,\Delta t}{\lambda }}}$, which causes fringes to shift in proportion to ${\displaystyle A}$ and ${\displaystyle \omega }$ .

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.^[32]

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 is well understood in the context of special relativity where from the rotating light source's point of view the phase difference is due to the line of simultaneity along the light path not forming a closed loop in spacetime.^[34]

Generalized formula

Figure 5. Conceptually, a conventional fibre optic gyro (FOG), shown on the left, can be divided into two semicircular sections with extended fibre connecting the end sections as shown on the right, creating a fibre optic conveyor (FOC).

Modified versions of the experiment have been proposed^[35] with the light source allowed to move along a (not necessarily circular) light path. This configuration introduces another reason for the phase difference: according to the light source the two signals now follow different paths in space. Some authors refer to this effect as Sagnac effect^[31][36] although in this case the discrepancy need not be due to the lines of simultaneity not forming closed loops.

An example of the modified configuration is shown 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.

This equation is invalid, however, if the light source's path in space does not follow that of the light signals, for example in the standard rotating platform case (FOG) but with a non-circular light path. In this case the phase difference formula necessarily involves the area enclosed by the light path due to Stokes' theorem.^[34]

Fig. 4: A closed optical fiber moving arbitrarily in space without stretching.

Consider a ring interferometer where two counter-propagating light beams share a common optical path determined by a loop of an optical fiber, see Figure 4. The loop may have an arbitrary shape, and can move arbitrarily in space. The only restriction is that it is not allowed to stretch. (The case of a circular ring interferometer rotating about its center in free space is recovered by taking the index of refraction of the fiber to be 1.)

Consider a small segment of the fiber, whose length in its rest frame is ${\displaystyle d\ell '}$. The time intervals, ${\displaystyle dt'_{\pm }}$, it takes the left and right moving light rays to traverse the segment in the rest frame coincide and are given by

$${\displaystyle dt'_{\pm }={n \over c}d\ell '}$$

Let ${\textstyle d\ell =|d\mathbf {x} |}$ be the length of this small segment in the lab frame. By the relativistic length contraction formula, ${\textstyle d\ell '=\gamma d\ell \approx d\ell }$ correct to first order in the velocity ${\displaystyle \mathbf {v} }$ of the segment. The time intervals ${\displaystyle dt_{\pm }}$ for traversing the segment in the lab frame are given by Lorentz transformation as:

$${\displaystyle dt_{\pm }=\gamma \left(dt'\pm {\frac {\mathbf {v} \cdot d\mathbf {x} '}{c^{2}}}\right)\approx {\frac {n}{c}}d\ell \pm {\frac {\mathbf {v} }{c^{2}}}\cdot d\mathbf {x} }$$

correct to first order in the velocity ${\displaystyle \mathbf {v} }$. In general, the two beams will visit a given segment at slightly different times, but, in the absence of stretching, the length ${\textstyle d\ell }$ is the same for both beams. It follows that the time difference for completing a cycle for the two beams is

$${\displaystyle \Delta T=\int \left(dt_{+}-dt_{-}\right)\approx {\frac {2}{c^{2}}}\oint \mathbf {v} \cdot d\mathbf {x} }$$

Remarkably, the time difference is independent of the refraction index ${\displaystyle n}$ and the velocity of light in the fiber. Imagine a screen for viewing fringes placed at the light source (alternatively, 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 given by ${\textstyle \Delta \phi \approx {\frac {2\pi c}{\lambda }}\Delta T}$ where the first factor is the frequency of light. This gives the generalized Sagnac formula^[37]

$${\displaystyle \Delta \phi \approx {\frac {4\pi }{\lambda c}}\oint \mathbf {v} \cdot d\mathbf {x} }$$

In the special case that the fiber moves like a rigid body with angular frequency ${\displaystyle {\boldsymbol {\omega }}}$, the velocity is ${\textstyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {x} }$ and the line integral can be computed in terms of the area of the loop:

$${\displaystyle \oint \mathbf {v} \cdot d\mathbf {x} =\oint {\boldsymbol {\omega }}\times \mathbf {x} \cdot d\mathbf {x} =\oint {\boldsymbol {\omega }}\cdot \mathbf {x} \times d\mathbf {x} =2\oint {\boldsymbol {\omega }}\cdot d\mathbf {A} =2{\boldsymbol {\omega }}\cdot \mathbf {A} }$$

This gives Sagnac formula for ring interferometers of arbitrary shape and geometry

$${\displaystyle \Delta \phi \approx {\frac {8\pi }{\lambda c}}{\boldsymbol {\omega }}\cdot \mathbf {A} }$$

If one also allows for stretching one recovers the Fizeau interference formula.^[37]

Practical uses

The Sagnac effect is employed in current technology. One use is in inertial guidance systems. Ring laser gyroscopes are extremely sensitive to rotations, which need to be accounted for if an inertial guidance system is to return accurate results. The ring laser also can detect the sidereal day, which can also be termed "mode 1". Global navigation satellite systems (GNSSs), such as GPS, GLONASS, COMPASS or Galileo, need to take the rotation of the Earth into account in the procedures of using radio signals to synchronize clocks.

Ring lasers

Figure 6. Schematic representation of a ring laser setup.

Fibre optic gyroscopes are sometimes referred to as 'passive ring interferometers'. A passive ring interferometer uses light entering the setup from outside. The interference pattern that is obtained is a fringe pattern, and what is measured is a phase shift.

It is also possible to construct a ring interferometer that is self-contained, based on a completely different arrangement. This is called a ring laser or ring laser gyroscope. The light is generated and sustained by incorporating laser excitation in the path of the light.

To understand what happens in a ring laser cavity, it is helpful to discuss the physics of the laser process in a laser setup with continuous generation of light. As the laser excitation is started, the molecules inside the cavity emit photons, but since the molecules have a thermal velocity, the light inside the laser cavity is at first a range of frequencies, corresponding to the statistical distribution of velocities. The process of stimulated emission makes one frequency quickly outcompete other frequencies, and after that the light is very close to monochromatic.

Figure 7. Schematic representation of the frequency shift when a ring laser interferometer is rotating. Both the counterpropagating light and the co-propagating light go through 12 cycles of their frequency.

For the sake of simplicity, assume that all emitted photons are emitted in a direction parallel to the ring. Fig. 7 illustrates the effect of the ring laser's rotation. In a linear laser, an integer multiple of the wavelength fits the length of the laser cavity. This means that in traveling back and forth the laser light goes through an integer number of cycles of its frequency. In the case of a ring laser the same applies: the number of cycles of the laser light's frequency is the same in both directions. This quality of the same number of cycles in both directions is preserved when the ring laser setup is rotating. The image illustrates that there is wavelength shift (hence a frequency shift) in such a way that the number of cycles is the same in both directions of propagation.

By bringing the two frequencies of laser light to interference a beat frequency can be obtained; the beat frequency is the difference between the two frequencies. This beat frequency can be thought of as an interference pattern in time. (The more familiar interference fringes of interferometry are a spatial pattern). The period of this beat frequency is linearly proportional to the angular velocity of the ring laser with respect to inertial space. This is the principle of the ring laser gyroscope, widely used in modern inertial navigation systems.

Zero point calibration

Figure 8. The red and blue dots represent counter-propagating photons, the grey dots represent molecules in the laser cavity.

In passive ring interferometers, the fringe displacement is proportional to the first derivative of angular position; careful calibration is required to determine the fringe displacement that corresponds to zero angular velocity of the ring interferometer setup. On the other hand, ring laser interferometers do not require calibration to determine the output that corresponds to zero angular velocity. Ring laser interferometers are self-calibrating. The beat frequency will be zero if and only if the ring laser setup is non-rotating with respect to inertial space.

Fig. 8 illustrates the physical property that makes the ring laser interferometer self-calibrating. The grey dots represent molecules in the laser cavity that act as resonators. Along every section of the ring cavity, the speed of light is the same in both directions. When the ring laser device is rotating, then it rotates with respect to that background. In other words: invariance of the speed of light provides the reference for the self-calibrating property of the ring laser interferometer.

Fibre optic gyroscopes versus ring laser gyroscopes

Fibre optic gyros (FOGs) and ring laser gyros (RLGs) both operate by monitoring the difference in propagation time between beams of light traveling in clockwise and counterclockwise directions about a closed optical path. They differ considerably in various cost, reliability, size, weight, power, and other performance characteristics that need to be considered when evaluating these distinct technologies for a particular application.

RLGs require accurate machining, use of precision mirrors, and assembly under clean room conditions. Their mechanical dithering assemblies add somewhat to their weight but not appreciably. RLGs are capable of logging in excess of 100,000 hours of operation in near-room temperature conditions. Their lasers have relatively high power requirements.^[40]

Interferometric FOGs are purely solid-state, require no mechanical dithering components, do not require precision machining, have a flexible geometry, and can be made very small. They use many standard components from the telecom industry. In addition, the major optical components of FOGs have proven performance in the telecom industry, with lifespans measured in decades.^[41] However, the assembly of multiple optical components into a precision gyro instrument is costly. Analog FOGs offer the lowest possible cost but are limited in performance; digital FOGs offer the wide dynamic ranges and accurate scale factor corrections required for stringent applications.^[42] Use of longer and larger coils increases sensitivity at the cost of greater sensitivity to temperature variations and vibrations.

Zero-area Sagnac interferometer and gravitational wave detection

The Sagnac topology was actually first described by Michelson in 1886,^[43] who employed an even-reflection variant of this interferometer in a repetition of the Fizeau experiment.^[44] Michelson noted the extreme stability of the fringes produced by this form of interferometer: White-light fringes were observed immediately upon alignment of the mirrors. In dual-path interferometers, white-light fringes are difficult to obtain since the two path lengths must be matched to within a couple of micrometers (the coherence length of the white light). However, being a common-path interferometer, the Sagnac configuration inherently matches the two path lengths. Likewise Michelson observed that the fringe pattern would remain stable even while holding a lighted match below the optical path; in most interferometers the fringes would shift wildly due to the refractive index fluctuations from the warm air above the match. Sagnac interferometers are almost completely insensitive to displacements of the mirrors or beam-splitter.^[45] This characteristic of the Sagnac topology has led to their use in applications requiring exceptionally high stability.

Figure 9. Zero-area Sagnac interferometer

The fringe shift in a Sagnac interferometer due to rotation has a magnitude proportional to the enclosed area of the light path, and this area must be specified in relation to the axis of rotation. Thus the sign of the area of a loop is reversed when the loop is wound in the opposite direction (clockwise or anti-clockwise). A light path that includes loops in both directions, therefore, has a net area given by the difference between the areas of the clockwise and anti-clockwise loops. The special case of two equal but opposite loops is called a zero-area Sagnac interferometer. The result is an interferometer that exhibits the stability of the Sagnac topology while being insensitive to rotation.^[46]

The Laser Interferometer Gravitational-Wave Observatory (LIGO) consisted of two 4-km Michelson–Fabry–Pérot interferometers, and operated at a power level of about 100 watts of laser power at the beam splitter. After an upgrade to Advanced LIGO several kilowatts of laser power are required.

A variety of competing optical systems are being explored for third generation enhancements beyond Advanced LIGO.^[47] One of these competing proposals is based on the zero-area Sagnac design. With a light path consisting of two loops of the same area, but in opposite directions, an effective area of zero is obtained thus canceling the Sagnac effect in its usual sense. Although insensitive to low frequency mirror drift, laser frequency variation, reflectivity imbalance between the arms, and thermally induced birefringence, this configuration is nevertheless sensitive to passing gravitational waves at frequencies of astronomical interest.^[46] However, many considerations are involved in the choice of an optical system, and despite the zero-area Sagnac's superiority in certain areas, there is as yet no consensus choice of optical system for third generation LIGO.^[48][49]