Sagnac effect

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The Sagnac effect (also called Sagnac interference), named after French physicist Georges Sagnac, is a phenomenon encountered in interferometry that is elicited by rotation. The Sagnac effect manifests itself in a setup called ring interferometry. A beam of light is split and the two beams are made to follow a trajectory in opposite directions. To act as a ring the trajectory must enclose an area. On return to the point of entry the light is allowed to exit the apparatus in such a way that an interference pattern is obtained. The position of the interference fringes is dependent on the angular velocity of the setup. This arrangement is also called a Sagnac interferometer.

Schematic representation of a Sagnac interferometer.

Usually several mirrors are used, so that the light beams follow a triangular or square trajectory. Fiber optics can also be employed to guide the light. The ring interferometer is located on a platform that can rotate. When the platform is rotating the lines of the interference pattern are displaced as compared to the position of the interference pattern when the platform is not rotating. The amount of displacement is proportional to the angular velocity of the rotating platform. The axis of rotation does not have to be inside the enclosed area.

When the platform is rotating, the point of entry/exit moves during the transit time of the light. So one beam has covered less distance than the other beam. This creates the shift in the interference pattern. Therefore, the interference pattern obtained at each angular velocity of the platform features a different phase-shift particular to that angular velocity.

In the above discussion, the rotation mentioned is rotation with respect to an inertial reference frame. Since this experiment does not involve a relativistic velocity, the same wording is valid both in the context of classical electrodynamics and special relativity.

The Sagnac effect is the electromagnetic counterpart of the mechanics of rotation. A gimbal mounted gyroscope remains pointing in the same direction after spinning up, and thus can be used as the reference for an inertial guidance system. A Sagnac interferometer measures its own angular velocity with respect to the local inertial frame; hence just as a gyroscope it can provide the reference for an inertial guidance system.

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[edit] Ring lasers

The type of ring interferometer that was described in the opening section is sometimes called a 'passive ring interferometer'. 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.

Schematic representation of a ring laser setup.

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". 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.

Image: frequency shift.

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.
Animation: propagating photons.

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

For the sake of simplicity, assume that all emitted photons are emitted in a direction parallel to the ring. (That is in fact a huge simplification, but it does not affect the content of this exposition.)

The image 'frequency shift' illustrates the effect of the ring laser's rotation.

In a linear laser the laser light that is generated fits the length of the laser cavity exactly; an integer multiple of the wavelength fits the length of the laser cavity. This means that in traveling back and forth the laserlight 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 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 laserlight 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.

[edit] No calibration

In contrast with the case of passive ring interferometry, in the case of ring laser interferometry there is no need for calibration. With passive ring interferometry there is no way of establishing which position of the interference fringes corresponds to zero angular velocity of the ring interferometer setup. Ring laser interferometry, on the other hand, is self-calibrating. The beat frequency will be zero if and only if the ring laser setup is non-rotating with respect to inertial space.

The animation 'propagating photons' illustrates the physical property that makes the ring laser interferometer process a self-calibrating process. 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: the invariance of the speed of light is the reference for the self-calibrating property of the ring laser interferometer.

[edit] Lock-in

Because of the way the laser light is generated, light in laser cavities has a strong tendency to be monochromatic (and usually that is precisely what laser apparatus designers want). This tendency to not split in two frequencies is called 'lock-in'. The ring laser devices incorporated in navigational instruments (to serve as a ring laser gyroscope) are generally too small to go out of lock spontaneously. By "dithering" the gyro through a small angle at a high audio frequency rate, going out of lock is ensured.

[edit] Synchronization procedures

The procedures for synchronizing clocks all over the globe must take the rotation of the Earth into account. The signals used for the synchronizing procedure can be in the form of electric pulses conducted in electric wires, they can be light pulses conducted in fiber optic cables, or they can be radio signals.

If a number of stations situated on the equator relay pulses to one another, will the time-keeping still match after the relay has circumnavigated the globe? One condition for handling the relay correctly is that the time it takes the signal to travel from one station to the next is taken into account each time. On a non-rotating planet that ensures fidelity: two time-disseminating relays, going full circle in opposite directions around the globe, will arrive at the originating station simultaneously. However, on a rotating planet, it must also be taken into account that the receiver moves during the transit time of the signal, shortening or lengthening the transit time compared to what it would be in the situation of a non-rotating planet.

It is recognized that the synchronization of clocks and ring interferometry are related in a fundamental way. Therefore the necessity to take the rotation of the Earth into account in synchronization procedures is also called the Sagnac effect.

[edit] History of the Sagnac Effect

The first ring interferometry experiment aimed at observing the correlation of angular velocity and phase-shift was performed by the Frenchman Georges Sagnac in 1913, which is why the effect is named for him. Its purpose was to detect "the effect of the relative motion of the ether". An experiment conducted in 1911 by Francis Harress, aimed at making measurements of Fresnel drag of light propagating through moving glass, was later recognized as actually constituting a Sagnac experiment. Harress had ascribed the "unexpected bias" to something else.

In 1926 a very ambitious ring interferometry experiment was set up by Albert Michelson and Henry Gale. The aim was to find out whether the rotation of the Earth has an effect on the propagation of light in the vicinity of the Earth. The Michelson-Gale experiment was a very large ring interferometer, (a perimeter of 1.9 kilometer), large enough to detect the angular velocity of the Earth. The outcome of the experiment was that the angular velocity of the Earth as measured by astronomy was confirmed to within measuring accuracy. The ring interferometer of the Michelson-Gale experiment was not calibrated by comparison with an outside reference (which was not possible, because the setup was fixed to the Earth). From its design it could be deduced where the central interference fringe ought to be if there would be zero shift. The measured shift was 230 parts in 1000, with an accuracy of 5 parts in 1000. The predicted shift was 237 parts in 1000.

[edit] Theory

The shift in interference fringes can be viewed simply as a consequence of different distances light travels with versus against the direction of rotation. The simplest derivation is for a circular ring rotating at an angular velocity of ωrot, 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 with the rotation direction will travel more than one circumference around the ring and hit the light source from behind after a time t1

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

t_1 = \frac {2 \pi R + \Delta L_1}{c}

ΔL1 is the distance (0 to 0' in the figure) the mirror has moved in that same time:

ΔL1 = Rωrott1 .

Eliminating ΔL1 from the two equations above we get:

t_1 = \frac {2 \pi R }{c ( 1 - R \omega_{rot}/ c)} .

Likewise, the light traveling against 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:

t_2 = \frac {2 \pi R }{c ( 1 + R \omega_{rot}/ c)} .

Since Rωrot = v < < c one can use the binomial approximation: the time difference traveled by the light to the screen for an interference pattern is

\Delta t = \frac {4 \pi R^2 \omega_{rot}} {c^2} = \frac {4 A \omega_{rot}} { c^2} ,

where A is the area of the ring. This result happens to be general for any shape of loop with area A.

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). If the light were pulses shorter than Δt, there would be no interference. But applications use steady light, and shifting interference fringes are seen due to the presence of the two beams of light on the screen that left the source at different times and hence have different phases at the screen. The phase shift is \Delta \phi=\frac {2 \pi c \Delta t }{\lambda} , which causes fringes to shift in proportion to A and ωrot .

In the case of light propagating in vacuum pre-relativistic theories and relativistic physics predict the same. In other words, in the case of propagation in vacuum a Sagnac experiment does not distinguish between pre-relativistic physics and relativistic physics. Both predict the same.

When the Sagnac setup has the light propagating in fibre optics then 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 fibre optics itself is a 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. Hence only relativistic physics can account for the physics of fibre optic Sagnac interferometers.)

A clock attached to the ring would run slower due to its velocity than an inertial observer's, the light frequency of the moving source would increase to cancel that.

Also, Doppler effects cancel out, so the Sagnac effect does not involve 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 related to Doppler shift.

[edit] Reference frames

The Sagnac effect is not an artifact of the choice of reference frame. It is independent of the choice of reference frame, as is shown by a calculation that invokes the metric tensor for an observer at the axis of rotation of the ring interferometer and rotating with it yielding the same outcome. If one starts with the Minkowski metric and does the coordinate conversions x = r \cos \left ( \theta + \omega t \right ) and y = r \sin \left ( \theta + \omega t \right ) (see Born coordinates), the line element of the resultant metric is

ds2 = (c2 − r2ω2)dt2 − dr2 − r2dθ2 − dz2 − 2r2ωdtdθ

where

  • t is proper time for the central observer,
  • r is distance from the center,
  • θ is the angular distance along the ring from the direction the central observer is facing,
  • z is the direction perpendicular to the plane of the ring, and
  • ω is the rate of rotation of the ring and the observer.

Under this metric, the speed of light tangent to the ring is c \pm r\omega depending on whether the light is moving against or with the rotation of the ring. Note that only the case of ω = 0 is inertial. For \omega \ne 0 this frame of reference is non-inertial, which is why the speed of light at positions distant from the observer (at r = 0) can vary from c.

[edit] Practical uses of the Sagnac effect

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 correct results.

Global navigation systems, such as NAVSTAR, GLONASS, COMPASS or Galileo, need to take the rotation of the Earth into account in the procedures of using radio signals to synchronize clocks.

[edit] See also

[edit] References

[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/Sagnac_effect"
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