Sagnac effect

The Sagnac effect manifests itself in an experimental 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 angular velocity of the setup. This arrangement is also called a Sagnac interferometer.

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Sagnac_interferometer.png

Schematic representation of a Sagnac interferometer.

Usually several mirrors are used, so that the lightbeams 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 sideways 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 kinematics of rotation. A gyroscope that can move around freely in a mounting can be used to measure the rotation of the mounting, and likewise, a Sagnac interferometer measures its angular velocity with respect to the local inertial frame.

Contents

1 Ring laser interferometers

2 Synchronisation procedures

3 History of the Sagnac Effect

4 Relativistic Physics

4.1 Calculations

5 Practical uses of the Sagnac Effect

6 External links

Ring laser interferometers

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.

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Schematic representation of a ring laser interferometer.

It is also possible to construct a ring interferometer that is self-contained. The light is generated and sustained by incorporating laser excitation in the path of the light. In normal laser operation the light inside the laser cavity is several frequencies at first, one frequency quickly outcompetes other frequencies, and after that the light is monochromatic. The frequency that outcompetes the others fits well in the cavity, a multiple of its wavelength is the length of the cavity. When a ring laser interferometer is rotating, the effective pathlengths of the two counterpropagating beams of laser light are different, and then the laser process generates two frequencies of laser light. The two resulting frequencies are such that at all times there is the same amount of cycles in both directions of propagation. Interpreted in terms of classical wave mechanics there is a standing wave in the ring laser interferometer, and this standing wave has the property that it is stationary with respect to the local inertial frame of reference when the ring laser interferometer is not rotating and is also stationary with respect to the local inertial frame of reference when the ring laser interferometer is rotating. This is why a ring laser interferometer is the interferometric counterpart of a mechanical gyroscope.

The interference pattern that can be obtained from the two frequencies is the difference between them, a beat frequency. The period of this beat frequency is linearly proportional to the angular velocity of the ring laser interferometer with respect to the local inertial frame of reference.

Ring laser interferometers are much more sensitive than passive ring interferometers. However, because of the way the laser light is generated, light in laser cavities has a tendency to be monochromatic. This tendency to not split in two frequencies is called phase-lock, and ring laser interferometer devices incorporated in navigational instruments (to serve as a ring laser gyroscope), must make sure this phase-lock is avoided.

Synchronisation procedures

The procedures for synchronizing clocks all over the globe using radio signals must take the rotation of Earth into account. For example, if a number of stations, situated on the equator, relay timing signals one to another,(or via satellite relay), 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 still match when they are compared at the end. 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 synchronisation of clocks and ring interferometry are related in a fundamental way. Therefore the necessity to take the rotation of Earth into account in sychronisation procedures is also called the Sagnac effect.

History of the Sagnac Effect

The first to perform a ring interferometry experiment aimed at observing the correlation of angular velocity and phase-shift was performed by the Frenchman G. Sagnac in 1913, which is why the effect is named for him. Later an experiment conducted in 1911 by F. Harress, aimed at making measurements of Fresnel drag of light propagating through moving glass, was recognized as actually constituting a Sagnac experiment. Harress had ascribed the "unexpected bias" to something else.

In 1925 a very ambitious ring interferometry experiment was set up by Michelson and Gale. The aim was to find out whether the rotation of Earth has an effect on the propagation of light in the vicinity of 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 Earth. The outcome of the experiment was that the angular velocity of 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.

Relativistic Physics

As seen from the perspective of a frame of reference that is co-rotating with the platform the light beams do travel the same distance. The result of the Sagnac experiment has been cited by many as a disproof of the theory of relativity, starting with Ives in 1938. The reasoning is that if the speed of light is a constant for the observer, then for the observer on the rotating ring light should take the same time to travel each way and no effect should occur. This argument does not hold because the rotating ring is an accelerated frame of reference, while the constancy of the speed of light ( $c$ ) applies only in inertial frames of reference.

Calculations

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 $y = r \sin \left ( \theta + \omega t \right )$ , the line element of the resultant metric is

$ds^2 = (c^2 - r^2 \omega^2)\, dt^2 - dr^2 - r^2 d\theta^2 - dz^2 - 2r^2 \omega \, dt \, d\theta$

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 $c$ .

Practical uses of the Sagnac Effect

The Sagnac Effect is employed in current technology. One use is in inertial guidance systems. Ring interferometers are extremely sensitive to rotations, which need to be accounted for if an inertial guidance system is to return correct results.

The Global Positioning System needs to take the rotation of Earth into account in the procedures of using radio signals to synchronize clocks.

External links

Mathpages.com article on the Sagnac Effect

Article on the Sagnac Effect and Reference Frames

Stedman Review of the Sagnac Effect