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 a ring interferometer or Sagnac interferometer. A beam of light is split and the two beams are made to follow the same path but in opposite directions. On return to the point of entry the two light beams are allowed to exit the ring and undergo interference. The relative phases of the two exiting beams, and thus the position of the interference fringes, are shifted according to the angular velocity of the apparatus. In other words, when the interferometer is at rest with respect to a nonrotating frame, the light takes the same amount of time to traverse the ring in either direction. However, when the interferometer system is spun, one beam of light has a longer path to travel than the other in order to complete one circuit of the mechanical frame, and so takes longer, resulting in a phase difference between the two beams. Georges Sagnac set up this experiment in 1913 in an attempt to prove the existence of the aether that Einstein's theory of special relativity makes superfluous.
A gimbal mounted mechanical gyroscope remains pointing in the same direction after spinning up, and thus can be used as a rotational reference for an inertial navigation system. With the development of so-called laser gyroscopes and fiber optic gyroscopes based on the Sagnac effect, bulky mechanical gyroscopes can be replaced by those with no moving parts in many modern inertial navigation systems. A conventional gyroscope relies on the principle of conservation of angular momentum whereas the sensitivity of the ring interferometer to rotation arises from the invariance of the speed of light for all inertial frames of reference.
Description and operation
Typically three or more mirrors are used, so that counter-propagating light beams follow a closed path such as a triangle or square (Fig. 1). Alternatively fiber optics can be employed to guide the light through a closed path (Fig. 2). If the platform on which the ring interferometer is mounted is rotating, the interference fringes are displaced compared to their position 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. The phase shift of the interference fringes is proportional to the platform's angular frequency and is given by a formula originally derived by Sagnac:where is the oriented area of the loop and the wavelength of light.
The effect is a consequence of the different times it takes right and left moving light beams to complete a full round trip in the interferometer ring. The difference in travel times, when multiplied by the optical frequency determines the phase difference
The rotation thus measured is an absolute rotation, that is, the platform's rotation with respect to an inertial reference frame.
History
The Michelson–Morley experiment of 1887 had suggested that the hypothetical luminiferous aether, if it existed, was completely dragged by the Earth. To test this hypothesis, Oliver Lodge in 1897 proposed that a giant ring interferometer be constructed to measure the rotation of the Earth; a similar suggestion was made by Albert Abraham Michelson in 1904. They hoped that with such an interferometer, it would be possible to decide between a stationary aether, versus aethers which are partially or completely dragged by the Earth. That is, if the hypothetical aether were carried along by the Earth (or by the interferometer) the result would be negative, while a stationary aether would give a positive result.
The first description of the Sagnac effect in the framework of special relativity was done by Max von Laue in 1911, English translation: On an Experiment on the Optics of Moving Bodies two years before Sagnac conducted his experiment. By continuing the theoretical work of Michelson (1904), von Laue confined himself to an inertial frame of reference (which he called a "valid" reference frame), and in a footnote he wrote "a system which rotates in respect to a valid system is not valid". Assuming constant light speed , and setting the rotational velocity as , he computed the propagation time of one ray and of the counter-propagating ray, and consequently obtained the time difference . He concluded that this interferometer experiment would indeed produce (when restricted to terms of first order in ) the same positive result for both special relativity and the stationary aether (the latter he called "absolute theory" in reference to the 1895-theory of Lorentz). He also concluded that only complete-aether-drag models (such as the ones of Stokes or Hertz) would give a negative result.
The first interferometry experiment aimed at observing the correlation of angular velocity and phase-shift was performed by the French scientist Georges Sagnac in 1913. Its purpose was to detect "the effect of the relative motion of the ether". Sagnac believed that his results constituted proof of the existence of a stationary aether. However, as explained above, von Laue already showed in 1911 that this effect is consistent with special relativity. Unlike the carefully prepared Michelson–Morley experiment which was set up to prove an aether wind caused by earth drag, the Sagnac experiment could not prove this type of aether wind because a universal aether would affect all parts of the rotating light equally.
Einstein was aware of the phenomenon of the Sagnac effect through the earlier experiments of Franz Harress in 1911. Harress' experiment had been aimed at making measurements of the Fresnel drag of light propagating through moving glass. Not aware of the Sagnac effect, Harress had realized the presence of an "unexpected bias" in his measurements, but was unable to explain its cause. English translation: On the Experiment of F. Harress Harress' analysis of the results contained an error, and they were reanalyzed in 1914 by Paul Harzer, who claimed the results were at odds with special relativity. From p. 377: "Nach der elektromagnetischen Lichttheorie und auch nach dem Einsteinschen Relativitätsprinzipe wird aber der Wert von k für Licht von der im ruhenden Mittel gemessenen Wellenlänge λ durch die Formel k = 1 - 1/ν2 - (λ/ν)(∂ν/∂λ) bestimmt, bei der die Aberration von der Bewegung der Mittel, wennschon in nur geringem Grade, abhängig wird." (According to the electromagnetic theory of light and also according to Einstein's principle of relativity, however, the value of k [Fresnel's coefficient of the entrainment of aether] for light of the wavelength λ as measured in a stationary medium is determined by the formula k = 1 - 1/ν2 - (λ/ν)(∂ν /∂λ) [where ν is Fresnel's refraction exponent], by which the [stellar] aberration becomes dependent on the movement of the medium, although to only a small degree.)List of scientific publications by Albert Einstein This was rebutted by Einstein. Harress himself died during the First World War, and his results were not publicly available until von Laue persuaded Otto Knopf, whose assistant Harress had been, to publish them in 1920.
Harress' results were published together with an analysis by von Laue, who showed the role of the Sagnac effect in the experiment. Laue said that in the Harress experiment there was a calculable difference in time due to both the dragging of light (which follows from the relativistic velocity addition in moving media, i.e. in moving glass) and "the fact that every part of the rotating apparatus runs away from one ray, while it approaches the other one", i.e. the Sagnac effect. He acknowledged that this latter effect alone could cause the time variance and, therefore, "the accelerations connected with the rotation in no way influence the speed of light".
While Laue's explanation is based on inertial frames, Paul Langevin (1921, 1937) and others described the same effect when viewed from rotating reference frames (in both special and general relativity, see Born coordinates). So when the Sagnac effect should be described from the viewpoint of a corotating frame, one can use ordinary rotating cylindrical coordinates and apply them to the Minkowski metric, which results into the so-called Born metric or Langevin metric. (Open access)L.D. Landau, E.M. Lifshitz, (1962). "The Classical Theory of Fields". 2nd edition, Pergamon Press, pp. 296–297. From these coordinates, one can derive the different arrival times of counter-propagating rays, an effect which was shown by Paul Langevin (1921). Or when these coordinates are used to compute the global speed of light in rotating frames, different apparent light speeds are derived depending on the orientation, an effect which was shown by Langevin in another paper (1937).
This does not contradict special relativity and the above explanation by von Laue that the speed of light is not affected by accelerations. Because this apparent variable light speed in rotating frames only arises if rotating coordinates are used, whereas if the Sagnac effect is described from the viewpoint of an external inertial coordinate frame the speed of light of course remains constant – so the Sagnac effect arises no matter whether one uses inertial coordinates (see the formulas in section below) or rotating coordinates (see the formulas in section below). That is, special relativity in its original formulation was adapted to inertial coordinate frames, not rotating frames. Albert Einstein in his paper introducing special relativity stated, "light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body".Albert Einstein. (1905). On the Electrodynamics of Moving Bodies Einstein specifically stated that light speed is only constant in the vacuum of empty space, using equations that only held in linear and parallel inertial frames. However, when Einstein started to investigate accelerated reference frames, he noticed that "the principle of the constancy of light must be modified" for accelerating frames of reference.A. Einstein, 'Generalized theory of relativity', 94; the anthology 'The Principle of Relativity', A. Einstein and H. Minkowski, University of Calcutta, 1920