705:. However, since the gravitational field would have to be significant, Laue (1920) concluded it is more likely that the effect is a result of changing the distance of the path by its movement through space. "The beam traveling around the loop in the direction of rotation will have farther to go than the beam traveling counter to the direction of rotation, because during the period of travel the mirrors and detector will all move (slightly) toward the counter-rotating beam and away from the co-rotating beam. Consequently the beams will reach the detector at slightly different times, and slightly out of phase, producing optical interference 'fringes' that can be observed and measured."
2773:
123:
2740:
proven performance in the telecom industry, with lifespans measured in decades. 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. Use of longer and larger coils increases sensitivity at the cost of greater sensitivity to temperature variations and vibrations.
1513:
20:
1360:
2554:
and westward around the world. In the case of a Sagnac interferometer a measure of difference in arrival time is obtained by producing interference fringes, and observing the fringe shift. In the case of a relay of pulses around the world the difference in arrival time is obtained directly from the actual arrival time of the pulses. In both cases the mechanism of the difference in arrival time is the same: the Sagnac effect.
2480:
1557:
739:
721:
MichelsonâGale 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.
676:
of motion of the emitting body". 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.
2345:
644:, 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".
774:, 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
1986:
63:, 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.
2562:
that are transported so slowly that time dilation effects arising from the transport are negligible the amount of time difference between the clocks when they arrive back at the starting point will be equal to the time difference that is found for a relay of pulses that travels around the world: 207 nanoseconds.
2752:. 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
2799:
A variety of competing optical systems are being explored for third generation enhancements beyond
Advanced LIGO. 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
2735:
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
2731:
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
2655:
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
1491:
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
2697:
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
2668:
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
1564:
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
1548:
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
720:
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
700:
which states that gravity and acceleration are equivalent. Spinning or accelerating an interferometer creates a gravitational effect. "There are, however, two different types of such motion; it may for instance be acceleration in a straight line, or circular motion with constant speed." Also, Irwin
614:
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
2561:
is also recognized as a counterpart to Sagnac effect physics. In the actual HafeleâKeating experiment the mode of transport (long-distance flights) gave rise to time dilation effects of its own, and calculations were needed to separate the various contributions. For the (theoretical) case of clocks
1381:
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
675:
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
377:
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
2553:
A relay of pulses that circumnavigates the Earth, verifying precise synchronization, is also recognized as a case requiring correction for the Sagnac effect. In 1984 a verification was set up that involved three ground stations and several GPS satellites, with relays of signals both going eastward
666:
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
150:
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
2639:
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
2660:
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
4476:
Punturo, M.; Abernathy, M.; Acernese, F.; Allen, B.; Andersson, N.; Arun, K.; Barone, F.; Barr, B.; Barsuglia, M.; Beker, M.; Beveridge, N.; Birindelli, S.; Bose, S.; Bosi, L.; Braccini, S.; Bradaschia, C.; Bulik, T.; Calloni, E.; Cella, G.; Chassande Mottin, E.; Chelkowski, S.; Chincarini, A.;
2739:
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
2780:
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
2693:
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
2706:
Ring laser gyroscopes suffer from an effect known as "lock-in" at low rotation rates (less than 100°/h). At very low rotation rates, the frequencies of the counter-propagating laser modes become almost identical. In this case, crosstalk between the counter-propagating beams can result in
1545:, 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.
683:
on the Sagnac effect stating, "General relativity would of course be capable of giving some statements about it, and we want to show at first that no noticeable influences of acceleration are expected according to it." He makes a footnote regarding discussions with German physicist,
632:, who claimed the results were at odds with special relativity. 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.
2475:{\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} }
627:
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. Harress' analysis of the results contained an error, and they were reanalyzed in 1914 by
2800:
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
2768:
fluctuations from the warm air above the match. Sagnac interferometers are almost completely insensitive to displacements of the mirrors or beam-splitter. This characteristic of the Sagnac topology has led to their use in applications requiring exceptionally high stability.
2694:
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.
1842:
2804:
at frequencies of astronomical interest. 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.
2764:, 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
2166:
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
2141:
2624:
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.
2781:(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
1484:. 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
2539:
233:
1503:
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.
2280:
1029:
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:
1532:
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
1476:
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.
1347:
663:(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).
4587:
Eberle, T.; Steinlechner, S.; Bauchrowitz, J. R.; HĂ€ndchen, V.; Vahlbruch, H.; Mehmet, M.; MĂŒller-Ebhardt, H.; Schnabel, R. (2010). "Quantum
Enhancement of the Zero-Area Sagnac Interferometer Topology for Gravitational Wave Detection".
1214:
1516:
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
635:
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
280:
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
2340:
724:
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.
2215:
1431:
2651:
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.
619:
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.
2039:
667:
viewpoint of an external inertial coordinate frame the speed of light of course remains constant â so the Sagnac effect arises no matter whether one uses inertial coordinates (see the formulas in section
2698:
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.
51:. 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
1684:
2648:
1781:
1378:
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.
2719:), lock-in will only occur during the brief instances where the rotational velocity is close to zero; the errors thereby induced approximately cancel each other between alternating dead periods.
1093:
1024:
3544:
859:
659:, which results into the so-called Born metric or Langevin metric. From these coordinates, one can derive the different arrival times of counter-propagating rays, an effect which was shown by
2617:
561:
2886:
2485:
179:
3567:
2686:
4658:
2220:
3943:
Wang, R.; Zheng, Y.; Yao, A.; Langley, D (2006). "Modified Sagnac experiment for measuring travel-time difference between counter-propagating light beams in a uniformly moving fiber".
933:
2302:
1981:{\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} }
174:
623:
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
1728:
696:
predicted that light would slow down in a gravitational field which is why it could predict the curvature of light around a massive body. Under
General Relativity, there is the
1529:
paths in space. Some authors refer to this effect as Sagnac effect although in this case the discrepancy need not be due to the lines of simultaneity not forming closed loops.
3031:
1525:
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
1627:
1252:
2711:, so that the standing wave "gets stuck" in a preferred phase, locking the frequency of each beam to each other rather than responding to gradual rotation. By rotationally
750:
that light travels due to the rotation of the ring.(Fig. 3) The simplest derivation is for a circular ring of radius R, with a refractive index of one, rotating at an
4534:
Freise, A.; Chelkowski, S.; Hild, S.; Pozzo, W. D.; Perreca, A.; Vecchio, A. (2009). "Triple
Michelson interferometer for a third-generation gravitational wave detector".
2008:
1803:
1594:
701:
Shapiro in 1964 explained
General Relativity saying, "the speed of a light wave depends on the strength of the gravitational potential along its path". This is called the
330:
255:
2854:
2796:, 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.
307:
4219:
1833:
512:
485:
959:
884:
655:). 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
275:
2578:
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
1471:
772:
458:
1260:
799:
414:
2031:
589:
142:
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
2161:
1488:
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.)
1451:
438:
1367:
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
1104:
4739:
2748:
The Sagnac topology was actually first described by Michelson in 1886, who employed an even-reflection variant of this interferometer in a repetition of the
1374:
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
3600:
A. Einstein, 'Generalized theory of relativity', 94; the anthology 'The Principle of Relativity', A. Einstein and H. Minkowski, University of Calcutta, 1920
591:) 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
3183:
3262:
146:
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
103:
4204:
4129:
385:
in 1911, two years before Sagnac conducted his experiment. By continuing the theoretical work of Michelson (1904), von Laue confined himself to an
610:
The first interferometry experiment aimed at observing the correlation of angular velocity and phase-shift was performed by the French scientist
1565:
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.)
2785:
Sagnac interferometer. The result is an interferometer that exhibits the stability of the Sagnac topology while being insensitive to rotation.
2640:
light inside the laser cavity is at first a range of frequencies, corresponding to the statistical distribution of velocities. The process of
352:
102:
can be replaced by those with no moving parts in many modern inertial navigation systems. A conventional gyroscope relies on the principle of
4906:
2628:
It is also possible to construct a ring interferometer that is self-contained, based on a completely different arrangement. This is called a
4953:
651:(1921, 1937) and others described the same effect when viewed from rotating reference frames (in both special and general relativity, see
99:
2307:
147:
2170:
716:. 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
373:
in 1897 proposed that a giant ring interferometer be constructed to measure the rotation of the Earth; a similar suggestion was made by
4870:
4772:
3253:= 1 - 1/Îœ - (λ/Îœ)(âÎœ /âλ) , by which the aberration becomes dependent on the movement of the medium, although to only a small degree.)
3239:= 1 - 1/Îœ - (λ/Îœ)(âÎœ/âλ) bestimmt, bei der die Aberration von der Bewegung der Mittel, wennschon in nur geringem Grade, abhĂ€ngig wird."
2732:
other performance characteristics that need to be considered when evaluating these distinct technologies for a particular application.
1385:
4932:
4901:
4732:
717:
4777:
1632:
2661:
wavelength shift (hence a frequency shift) in such a way that the number of cycles is the same in both directions of propagation.
2010:. In general, the two beams will visit a given segment at slightly different times, but, in the absence of stretching, the length
4838:
3431:
Guido Rizzi; Matteo Luca Ruggiero (2003). "The relativistic Sagnac Effect: two derivations". In G. Rizzi; M.L. Ruggiero (eds.).
55:. The relative phases of the two exiting beams, and thus the position of the interference fringes, are shifted according to the
4875:
4833:
4808:
4400:
4244:
3458:
3055:
2689:
Figure 8. The red and blue dots represent counter-propagating photons, the grey dots represent molecules in the laser cavity.
389:(which he called a "valid" reference frame), and in a footnote he wrote "a system which rotates in respect to a valid system
4948:
4725:
4974:
2945:
1036:
967:
807:
596:
2881:
2583:
2136:{\displaystyle \Delta T=\int \left(dt_{+}-dt_{-}\right)\approx {\frac {2}{c^{2}}}\oint \mathbf {v} \cdot d\mathbf {x} }
746:
The shift in interference fringes in a ring interferometer can be viewed intuitively as a consequence of the different
641:
2602:, need to take the rotation of the Earth into account in the procedures of using radio signals to synchronize clocks.
517:
4843:
4823:
3673:
Michelson, Albert Abraham; Gale, Henry G. (1925). "The Effect of the Earth's Rotation on the Velocity of Light, II".
1737:
2793:
563:. He concluded that this interferometer experiment would indeed produce (when restricted to terms of first order in
4782:
4767:
2644:
makes one frequency quickly outcompete other frequencies, and after that the light is very close to monochromatic.
1629:, it takes the left and right moving light rays to traverse the segment in the rest frame coincide and are given by
4885:
1480:
When light propagates in fibre optic cable, the setup is effectively a combination of a Sagnac experiment and the
4762:
3196:
3103:
2850:"L'éther lumineux démontré par l'effet du vent relatif d'éther dans un interféromÚtre en rotation uniforme"
616:
358:
52:
86:
remains pointing in the same direction after spinning up, and thus can be used as a rotational reference for an
4828:
4792:
3533:
L.D. Landau, E.M. Lifshitz, (1962). "The Classical Theory of Fields". 2nd edition, Pergamon Press, pp. 296â297.
2908:
Anderson, R.; Bilger, H.R.; Stedman, G.E. (1994). "Sagnac effect: A century of Earth-rotated interferometers".
2599:
892:
600:
637:
5010:
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157:
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Clark, J.; Coccia, E.; Colacino, C.; Colas, J.; Cumming, A.; Cunningham, L.; Cuoco, E.; et al. (2010).
4276:
Udd, E.; Watanabe, S. F.; Cahill, R. F. (1986). "Comparison of ring laser and fiber-optic gyro technology".
5000:
4865:
4818:
4813:
693:
72:
3352:
4748:
713:
386:
111:
60:
4344:
1689:
4078:
Allan, D. W., Weiss, M. A., & Ashby, N. (1985). "Around-the-World Relativistic Sagnac Experiment".
2887:
On the proof of the reality of the luminiferous aether by the experiment with a rotating interferometer
2761:
2674:
2534:{\displaystyle \Delta \phi \approx {\frac {8\pi }{\lambda c}}{\boldsymbol {\omega }}\cdot \mathbf {A} }
1492:
frequency shifts in opposite directions. This frequency shift is not a Doppler shift, but is rather an
228:{\displaystyle \Delta \phi \approx {\frac {8\pi }{\lambda c}}{\boldsymbol {\omega }}\cdot \mathbf {A} }
87:
3762:
3385:
2882:"Sur la preuve de la réalité de l'éther lumineux par l'expérience de l'interférographe tournant"
742:
Figure 3. Light traveling opposite directions go different distances before reaching the moving source
3870:
2944:
2595:
2587:
1222:
2669:
the angular velocity of the ring laser with respect to inertial space. This is the principle of the
4979:
3730:
3276:"Bemerkungen zu P. Harzers Abhandlung 'Ăber die MitfĂŒhrung des Lichtes in Glas und die Aberration'"
2636:. The light is generated and sustained by incorporating laser excitation in the path of the light.
2571:
2275:{\displaystyle \Delta \phi \approx {\frac {4\pi }{\lambda c}}\oint \mathbf {v} \cdot d\mathbf {x} }
709:
374:
340:
3314:
3275:
1991:
1786:
312:
238:
5005:
3624:
3278:[Comments on P. Harzer's article 'On the entrainment of light in glass and aberration'].
2849:
106:
whereas the sensitivity of the ring interferometer to rotation arises from the invariance of the
1599:
284:
4969:
3725:
1836:
1808:
886:
is the distance (black bold arrow in Fig. 3) that the mirror has moved in that same time:
624:
490:
463:
366:
4144:
3249:
for light of the wavelength λ as measured in a stationary medium is determined by the formula
1342:{\displaystyle \Delta t\approx {\frac {4\pi R^{2}\omega }{c^{2}}}={\frac {4A\omega }{c^{2}}},}
941:
866:
381:
The first description of the Sagnac effect in the framework of special relativity was done by
260:
4198:
4123:
3370:
2819:
2542:
1571:
1456:
1375:
757:
697:
443:
131:
95:
3071:
2217:
where the first factor is the frequency of light. This gives the generalized Sagnac formula
4678:
4607:
4553:
4490:
4435:
4357:
4308:
4281:
4156:
4087:
4041:
3962:
3917:
3879:
3777:
3717:
3682:
3633:
3487:
3446:
3397:
3326:
3287:
3208:
3165:
3115:
2958:
2917:
2824:
2670:
2633:
2611:
2575:
777:
392:
91:
4511:
8:
4478:
2641:
1209:{\displaystyle \Delta t=t_{1}-t_{2}={\frac {4\pi R^{2}\omega }{c^{2}-R^{2}\omega ^{2}}}.}
566:
362:
68:
4682:
4611:
4565:
4557:
4502:
4494:
4439:
4361:
4312:
4285:
4278:
In Agard Guided Optical Structures in the Military Environment 14 P (See N87-13273 04-74
4160:
4091:
4045:
3966:
3921:
3883:
3781:
3721:
3686:
3637:
3491:
3450:
3401:
3330:
3291:
3212:
3169:
3119:
2962:
2921:
2013:
1521:
Modified versions of the experiment have been proposed with the light source allowed to
4857:
4701:
4666:
4631:
4597:
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4543:
4516:
4373:
4180:
4111:
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3510:
3475:
3436:
2801:
2146:
1731:
1550:
1436:
689:
680:
423:
127:
59:
of the apparatus. In other words, when the interferometer is at rest with respect to a
3997:
3974:
5015:
4922:
4787:
4706:
4623:
4451:
4396:
4377:
4324:
4172:
4103:
4061:
3797:
3747:
3739:
3515:
3454:
3413:
3104:"Die Versuche von F. HarreĂ ĂŒber die Geschwindigkeit des Lichtes in bewegten Körpern"
3051:
2855:
The demonstration of the luminiferous aether by an interferometer in uniform rotation
2749:
2708:
1481:
336:
152:
4520:
4343:
4184:
4115:
3789:
4696:
4686:
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4049:
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3641:
3619:
3505:
3495:
3405:
3334:
3295:
3216:
3173:
3123:
2999:
2966:
2946:"Experiments on the Absence of Mechanical Connexion between Ether and Matter"
2925:
2814:
2765:
2757:
2743:
2715:
the laser cavity back and forth through a small angle at a rapid rate (hundreds of
2665:
751:
656:
652:
56:
4479:"The third generation of gravitational wave observatories and their science reach"
4251:
3982:
2772:
2647:
2482:
This gives Sagnac formula for ring interferometers of arbitrary shape and geometry
2282:
In the special case that the fiber moves like a rigid body with angular frequency
4420:
4168:
3908:
Anandan, J. (1981). "Sagnac effect in relativistic and nonrelativistic physics".
3708:
Stedman, G. E. (1997). "Ring-laser tests of fundamental physics and geophysics".
1485:
592:
4447:
4099:
3106:[The experiments of F. Harress on the speed of light in moving bodies].
47:. The Sagnac effect manifests itself in a setup called a ring interferometer or
4053:
3645:
3409:
3235:
fĂŒr Licht von der im ruhenden Mittel gemessenen WellenlĂ€nge λ durch die Formel
2036:
It follows that the time difference for completing a cycle for the two beams is
1493:
611:
604:
107:
64:
40:
36:
4717:
4369:
3891:
3003:
1560:
Fig. 4: A closed optical fiber moving arbitrarily in space without stretching.
514:
of the counter-propagating ray, and consequently obtained the time difference
4994:
4853:
4653:
3659:
3417:
3338:
3299:
3220:
3177:
3127:
2579:
702:
660:
648:
135:
122:
3929:
3392:. The Sagnac effect: 100 years later / L'effet Sagnac : 100 ans aprĂšs.
2880:
2616:
1512:
4710:
4627:
4455:
4328:
4176:
4107:
4077:
3763:"Sagnac effect in a rotating frame of reference. Relativistic Zeno paradox"
3519:
2971:
685:
679:
Max von Laue in his 1920 paper gave serious consideration to the effect of
382:
370:
143:
67:
set up this experiment in 1913 in an attempt to prove the existence of the
3821:
Tartaglia, A.; Ruggiero, M. L. (2004). "Sagnac effect and pure geometry".
3024:
Laue, Max von (1911). "Ăber einen Versuch zur Optik der bewegten Körper".
1730:
be the length of this small segment in the lab frame. By the relativistic
4320:
3846:
3609:"General Relativity", Lewis Ryder, Cambridge University Press (2009). P.7
2753:
1568:
Consider a small segment of the fiber, whose length in its rest frame is
629:
4691:
3957:
3500:
3241:(According to the electromagnetic theory of light and also according to
1359:
19:
2685:
2629:
2342:
and the line integral can be computed in terms of the area of the loop:
2143:
Remarkably, the time difference is independent of the refraction index
1382:
two counter-rotating beams to traverse the circuit. The phase shift is
3827:
3441:
3317:[Answer to a reply of Paul Harzer (Nr. 4753, pp. 10 and 11)].
2722:
4586:
2907:
2712:
353:
History of special relativity § Experiments by Fizeau and Sagnac
83:
4418:
4299:
Hariharan, P. (1975). "Sagnac or MichelsonâSagnac interferometer?".
3153:
2987:
2929:
2335:{\textstyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {x} }
4036:
3694:
2570:
The Sagnac effect is employed in current technology. One use is in
2210:{\textstyle \Delta \phi \approx {\frac {2\pi c}{\lambda }}\Delta T}
708:
In 1926, an ambitious ring interferometry experiment was set up by
44:
4602:
4548:
4345:"Influence of Motion of the Medium on the Velocity of Light"
4145:"Around the world atomic clocks:predicted relativistic time gains"
378:
negative, while a stationary aether would give a positive result.
2664:
By bringing the two frequencies of laser light to interference a
2591:
1426:{\displaystyle \Delta \phi ={\frac {2\pi c\,\Delta t}{\lambda }}}
2744:
Zero-area Sagnac interferometer and gravitational wave detection
4275:
4220:"Evaluation of Ring Laser and Fiber Optic Gyroscope Technology"
3430:
3315:"Antwort auf eine Replik Paul Harzers (Nr. 4753, S. 10 und 11)"
3199:[On the entrainment of light in glass and aberration].
1556:
1363:
Figure 4. The Sagnac area formula applies to any shape of loop.
738:
79:
3545:"Sur la théorie de la relativité et l'expérience de M. Sagnac"
23:
Figure 1. Schematic representation of a Sagnac interferometer.
4475:
3227:"Nach der elektromagnetischen Lichttheorie und auch nach dem
2716:
801:
that it takes to catch up with the light source is given by:
4341:
3197:"Ăber die MitfĂŒhrung des Lichtes in Glas und die Aberration"
2656:
and forth the laser light goes through an integer number of
671:
below) or rotating coordinates (see the formulas in section
2789:
4022:
Ori, A. (2016). "Generalized Sagnac-Wang-Fizeau formula".
4533:
4419:
Sun, K-X.; Fejer, M.M.; Gustafson, E.; Byer R.L. (1996).
3371:"Volume 6: The Berlin Years: Writings, 1914â1917 page 28"
2620:
Figure 6. Schematic representation of a ring laser setup.
1835:
for traversing the segment in the lab frame are given by
4421:"Sagnac Interferometer for Gravitational-Wave Detection"
3820:
176:
and is given by a formula originally derived by Sagnac:
4659:
Ring-laser tests of fundamental physics and geophysics
4142:
3622:(December 1964). "Fourth Test of General Relativity".
2848:
2310:
2173:
2016:
1740:
1692:
647:
While Laue's explanation is based on inertial frames,
339:, that is, the platform's rotation with respect to an
4245:"Fiber-Optic Gyroscopes Key Technological Advantages"
2488:
2348:
2288:
2223:
2149:
2042:
1994:
1845:
1811:
1789:
1635:
1602:
1574:
1459:
1439:
1388:
1263:
1225:
1107:
1039:
970:
944:
895:
869:
810:
780:
760:
569:
520:
493:
466:
446:
426:
395:
315:
287:
263:
241:
182:
160:
3816:
3814:
4661:(Extensive review by G E Stedman. PDF-file, 1.5 MB)
4395:(Second ed.). Academic Press. pp. 28â29.
3942:
3091:
Die Geschwindigkeit des Lichtes in bewegten Körpern
3072:"Franz Harress - The Mathematics Genealogy Project"
2790:
Laser Interferometer Gravitational-Wave Observatory
2541:If one also allows for stretching one recovers the
1088:{\displaystyle t_{2}={\frac {2\pi R}{c+R\omega }}.}
1019:{\displaystyle t_{1}={\frac {2\pi R}{c-R\omega }}.}
3263:List of scientific publications by Albert Einstein
3231:schen RelativitÀtsprinzipe wird aber der Wert von
3097:] (in German). Erfurt, Germany: Georg Richter.
2533:
2474:
2334:
2296:
2274:
2209:
2155:
2135:
2025:
2002:
1980:
1827:
1797:
1775:
1722:
1678:
1621:
1588:
1465:
1445:
1425:
1341:
1246:
1208:
1087:
1018:
953:
927:
878:
854:{\displaystyle t_{1}={\frac {2\pi R+\Delta L}{c}}}
853:
793:
766:
583:
555:
506:
479:
452:
432:
408:
324:
301:
269:
249:
227:
168:
4217:
4211:
3989:
3811:
2736:lasers have relatively high power requirements.
1433:, which causes fringes to shift in proportion to
4992:
3618:
3353:"Otto Knopf - The Mathematics Genealogy Project"
1776:{\textstyle d\ell '=\gamma d\ell \approx d\ell }
556:{\displaystyle \Delta \tau =\tau _{+}-\tau _{-}}
4747:
3245:principle of relativity, however, the value of
3032:On an Experiment on the Optics of Moving Bodies
2985:
2878:
2846:
3840:
3838:
3565:
3542:
4733:
4667:"Relativity in the Global Positioning System"
4384:
4292:
3672:
3476:"Relativity in the Global Positioning System"
3151:
3023:
2942:
1679:{\displaystyle dt'_{\pm }={n \over c}d\ell '}
4342:Michelson, A. A. & Morley, E.W. (1886).
4203:: CS1 maint: multiple names: authors list (
4128:: CS1 maint: multiple names: authors list (
3754:
3701:
361:of 1887 had suggested that the hypothetical
117:
4469:
3907:
3835:
4740:
4726:
4580:
4527:
4414:
4412:
4236:
3903:
3901:
4902:Tests of relativistic energy and momentum
4700:
4690:
4664:
4601:
4547:
4510:
4390:
4298:
4035:
3956:
3868:E.J. Post (April 1967). "Sagnac effect".
3867:
3826:
3729:
3509:
3499:
3473:
3440:
3435:. Dordrecht: Kluwer Academic Publishers.
2970:
2874:
2872:
2842:
2840:
2776:Figure 9. Zero-area Sagnac interferometer
2680:
1410:
928:{\displaystyle \Delta L=R\omega t_{1}.\,}
924:
440:, and setting the rotational velocity as
4073:
4071:
3383:
3312:
3273:
3147:
3145:
3143:
3141:
3139:
2771:
2684:
2646:
2615:
2163:and the velocity of light in the fiber.
1555:
1511:
1496:resonance effect, as explained below in
1358:
737:
121:
18:
4839:Lorentz-violating neutrino oscillations
4409:
3898:
3760:
3707:
3590:On the Electrodynamics of Moving Bodies
3088:
3041:
3039:
3019:
3017:
3015:
3013:
2519:
2460:
2438:
2405:
2375:
2320:
2297:{\displaystyle {\boldsymbol {\omega }}}
2290:
1988:correct to first order in the velocity
1783:correct to first order in the velocity
213:
169:{\displaystyle {\boldsymbol {\omega }}}
162:
4993:
4242:
3194:
2869:
2837:
2760:of the white light). However, being a
1507:
420:valid". Assuming constant light speed
4907:KaufmannâBuchererâNeumann experiments
4876:Experimental testing of time dilation
4834:Antimatter tests of Lorentz violation
4809:Modern searches for Lorentz violation
4721:
4143:Hafele J., Keating, R. (1972-07-14).
4068:
4017:
4015:
3467:
3424:
3136:
3101:
3045:
2988:"Relative Motion of Earth and Aether"
2794:MichelsonâFabryâPĂ©rot interferometers
2582:, which can also be termed "mode 1".
961:from the two equations above we get:
672:
640:(which follows from the relativistic
257:is the oriented area of the loop and
4335:
3936:
3036:
3010:
1805:of the segment. The time intervals
90:. With the development of so-called
4975:Test theories of special relativity
4021:
3386:"Georges Sagnac: A life for optics"
3095:The speed of light in moving bodies
2584:Global navigation satellite systems
1723:{\textstyle d\ell =|d\mathbf {x} |}
460:, he computed the propagation time
367:was completely dragged by the Earth
16:Relativistic effect due to rotation
13:
4012:
2489:
2224:
2201:
2174:
2043:
1411:
1389:
1264:
1108:
945:
896:
870:
839:
521:
316:
309:, determines the phase difference
183:
98:based on the Sagnac effect, bulky
14:
5027:
4933:MichelsonâGaleâPearson experiment
4844:Lorentz-violating electrodynamics
4824:Experiments of Rayleigh and Brace
4646:
3995:
3844:
3357:www.genealogy.math.ndsu.nodak.edu
3076:www.genealogy.math.ndsu.nodak.edu
2565:
718:MichelsonâGaleâPearson experiment
642:velocity addition in moving media
335:The rotation thus measured is an
39:, is a phenomenon encountered in
4886:Length contraction confirmations
4783:de Sitter double star experiment
3384:Darrigol, Olivier (2014-12-01).
2527:
2468:
2449:
2424:
2413:
2394:
2383:
2364:
2353:
2328:
2312:
2268:
2257:
2129:
2118:
1996:
1974:
1951:
1900:
1888:
1791:
1711:
1497:
607:) would give a negative result.
243:
221:
104:conservation of angular momentum
4269:
4218:Juang, J.-N.; Radharamanan, R.
4136:
3861:
3790:10.1070/pu2002v045n08abeh001225
3666:
3652:
3612:
3603:
3594:
3582:
3568:"Sur l'expérience de M. Sagnac"
3559:
3536:
3527:
3377:
3363:
3345:
3306:
3267:
3256:
3188:
3184:On the Experiment of F. Harress
2548:
1247:{\displaystyle R\omega =v\ll c}
595:). He also concluded that only
35:, named after French physicist
4954:Refutations of emission theory
4793:Measurements of neutrino speed
4620:10.1103/PhysRevLett.104.251102
4512:11858/00-001M-0000-0011-2EAE-2
3082:
3064:
2979:
2936:
2901:
2605:
1716:
1703:
138:in a single or multiple loops.
1:
4566:10.1088/0264-9381/26/8/085012
4536:Classical and Quantum Gravity
4503:10.1088/0264-9381/27/8/084007
4483:Classical and Quantum Gravity
3975:10.1016/S0375-9601(03)00575-9
3433:Relativity in Rotating Frames
2830:
2792:(LIGO) consisted of two 4-km
733:
668:
4949:Refutations of aether theory
4871:Moessbauer rotor experiments
4773:Moessbauer rotor experiments
4768:KennedyâThorndike experiment
4169:10.1126/science.177.4044.166
3154:"Zum Versuch von F. Harress"
2033:is the same for both beams.
2003:{\displaystyle \mathbf {v} }
1798:{\displaystyle \mathbf {v} }
694:Theory of General Relativity
688:. The reason for looking at
599:models (such as the ones of
325:{\displaystyle \Delta \phi }
250:{\displaystyle \mathbf {A} }
112:inertial frames of reference
73:theory of special relativity
7:
4763:MichelsonâMorley experiment
4749:Tests of special relativity
4448:10.1103/PhysRevLett.76.3053
4100:10.1126/science.228.4695.69
2808:
2675:inertial navigation systems
617:MichelsonâMorley experiment
387:inertial frame of reference
369:. To test this hypothesis,
359:MichelsonâMorley experiment
134:, can be realized using an
10:
5032:
4829:TroutonâRankine experiment
4054:10.1103/physreva.94.063837
3740:10.1088/0034-4885/60/6/001
3646:10.1103/PhysRevLett.13.789
3588:Albert Einstein. (1905).
3410:10.1016/j.crhy.2014.09.007
3110:. 4th series (in German).
3026:MĂŒnchener Sitzungsberichte
2762:common-path interferometer
2701:
2609:
1622:{\displaystyle dt'_{\pm }}
350:
346:
302:{\displaystyle c/\lambda }
130:Sagnac interferometer, or
88:inertial navigation system
4962:
4941:
4915:
4894:
4881:HafeleâKeating experiment
4852:
4801:
4755:
4370:10.2475/ajs.s3-31.185.377
3892:10.1103/RevModPhys.39.475
3871:Reviews of Modern Physics
3675:The Astrophysical Journal
3319:Astronomische Nachrichten
3280:Astronomische Nachrichten
3201:Astronomische Nachrichten
3004:10.1080/14786440409463244
2572:inertial guidance systems
2559:HafeleâKeating experiment
1828:{\displaystyle dt_{\pm }}
1356:is the area of the ring.
728:
507:{\displaystyle \tau _{-}}
480:{\displaystyle \tau _{+}}
277:the wavelength of light.
118:Description and operation
4980:Standard-Model Extension
4866:IvesâStilwell experiment
4819:TroutonâNoble experiment
4814:HughesâDrever experiment
3339:10.1002/asna.19141990304
3300:10.1002/asna.19141990104
3221:10.1002/asna.19141982001
3178:10.1002/andp.19203671303
3128:10.1002/andp.19203671302
3046:Pauli, Wolfgang (1981).
2986:Michelson, A.A. (1904).
2879:Sagnac, Georges (1913),
2847:Sagnac, Georges (1913),
2673:, widely used in modern
954:{\displaystyle \Delta L}
879:{\displaystyle \Delta L}
375:Albert Abraham Michelson
341:inertial reference frame
270:{\displaystyle \lambda }
4590:Physical Review Letters
4428:Physical Review Letters
4250:. iXSea. Archived from
3930:10.1103/physrevd.24.338
3761:Malykin, G. B. (2002).
3625:Physical Review Letters
3566:Langevin, Paul (1937).
3543:Langevin, Paul (1921).
3390:Comptes Rendus Physique
3089:Harress, Franz (1912).
2723:Fibre optic gyroscopes
1596:. The time intervals,
1589:{\displaystyle d\ell '}
1466:{\displaystyle \omega }
1098:The time difference is
767:{\displaystyle \omega }
673:§ Reference frames
615:the carefully prepared
453:{\displaystyle \omega }
4970:One-way speed of light
4393:Optical Interferometry
4391:Hariharan, P. (2003).
3152:Laue, Max von (1920).
2992:Philosophical Magazine
2972:10.1098/rsta.1897.0006
2943:Lodge, Oliver (1897).
2777:
2690:
2681:Zero point calibration
2652:
2621:
2545:interference formula.
2535:
2476:
2336:
2298:
2276:
2211:
2157:
2137:
2027:
2004:
1982:
1837:Lorentz transformation
1829:
1799:
1777:
1724:
1680:
1623:
1590:
1561:
1518:
1467:
1447:
1427:
1364:
1343:
1248:
1210:
1089:
1020:
955:
929:
880:
855:
795:
768:
743:
692:is because Einstein's
585:
557:
508:
481:
454:
434:
410:
326:
303:
271:
251:
229:
170:
139:
96:fiber optic gyroscopes
24:
4778:Resonator experiments
4280:. McDonnell-Douglas.
3313:Einstein, A. (1914).
3274:Einstein, A. (1914).
3195:Harzer, Paul (1914).
3182:English translation:
3030:English translation:
2951:Philos. Trans. R. Soc
2820:Fiber optic gyroscope
2775:
2727:ring laser gyroscopes
2688:
2650:
2619:
2610:Further information:
2576:Ring laser gyroscopes
2536:
2477:
2337:
2299:
2277:
2212:
2158:
2138:
2028:
2005:
1983:
1830:
1800:
1778:
1725:
1681:
1624:
1591:
1559:
1515:
1468:
1448:
1428:
1362:
1344:
1249:
1211:
1090:
1021:
956:
930:
881:
856:
796:
794:{\displaystyle t_{1}}
769:
741:
698:equivalence principle
586:
558:
509:
482:
455:
435:
411:
409:{\displaystyle K^{0}}
327:
304:
272:
252:
230:
171:
132:fibre optic gyroscope
125:
100:mechanical gyroscopes
49:Sagnac interferometer
22:
5011:Theory of relativity
4321:10.1364/AO.14.2319_1
3048:Theory of Relativity
2825:Ring laser gyroscope
2671:ring laser gyroscope
2634:ring laser gyroscope
2612:Ring laser gyroscope
2486:
2346:
2308:
2286:
2221:
2171:
2147:
2040:
2014:
1992:
1843:
1809:
1787:
1738:
1690:
1633:
1600:
1572:
1457:
1437:
1386:
1261:
1223:
1105:
1037:
968:
942:
893:
867:
808:
778:
758:
597:complete-aether-drag
567:
518:
491:
464:
444:
424:
393:
313:
285:
261:
239:
180:
158:
148:interference fringes
43:that is elicited by
5001:Physics experiments
4895:Relativistic energy
4692:10.12942/lrr-2003-1
4683:2003LRR.....6....1A
4671:Living Rev. Relativ
4612:2010PhRvL.104y1102E
4558:2009CQGra..26h5012F
4495:2010CQGra..27h4007P
4440:1996PhRvL..76.3053S
4362:1886AmJS...31..377M
4313:1975ApOpt..14.2319H
4307:(10): 2319_1â2321.
4286:1986gosm.agar.....U
4161:1972Sci...177..166H
4092:1985Sci...228...69A
4046:2016PhRvA..94f3837O
3998:"Sagnac and Fizeau"
3967:2003PhLA..312....7W
3922:1981PhRvD..24..338A
3884:1967RvMP...39..475P
3847:"The Sagnac Effect"
3782:2002PhyU...45..907M
3722:1997RPPh...60..615S
3687:1925ApJ....61..140M
3660:"The Sagnac Effect"
3638:1964PhRvL..13..789S
3501:10.12942/lrr-2003-1
3492:2003LRR.....6....1A
3480:Living Rev. Relativ
3451:2003gr.qc.....5084R
3402:2014CRPhy..15..789D
3331:1914AN....199...47E
3292:1914AN....199....8E
3213:1914AN....198..377H
3170:1920AnP...367..448L
3120:1920AnP...367..389K
3050:. New York: Dover.
2963:1897RSPTA.189..149L
2922:1994AmJPh..62..975A
2802:gravitational waves
2642:stimulated emission
2026:{\textstyle d\ell }
1651:
1618:
1508:Generalized formula
584:{\displaystyle v/c}
363:luminiferous aether
82:mounted mechanical
75:makes superfluous.
33:Sagnac interference
4858:Length contraction
4802:Lorentz invariance
4665:Ashby, N. (2003).
3474:Ashby, N. (2003).
3158:Annalen der Physik
3108:Annalen der Physik
3102:Knopf, O. (1920).
2778:
2691:
2653:
2622:
2531:
2472:
2332:
2304:, the velocity is
2294:
2272:
2207:
2153:
2133:
2023:
2000:
1978:
1825:
1795:
1773:
1732:length contraction
1720:
1676:
1639:
1619:
1606:
1586:
1562:
1519:
1463:
1443:
1423:
1376:Fermat's principle
1365:
1339:
1254:, this reduces to
1244:
1206:
1085:
1016:
951:
925:
876:
851:
791:
764:
744:
690:General Relativity
681:General Relativity
581:
553:
504:
477:
450:
430:
406:
322:
299:
267:
247:
225:
166:
140:
25:
4988:
4987:
4928:Sagnac experiment
4923:Fizeau experiment
4788:Hammar experiment
4654:The Sagnac Effect
4434:(17): 3053â3056.
4402:978-0-12-311630-7
4155:(4044): 166â168.
4024:Physical Review A
3945:Physics Letters A
3460:978-0-486-64152-2
3057:978-0-486-64152-2
2750:Fizeau experiment
2709:injection locking
2586:(GNSSs), such as
2516:
2251:
2199:
2156:{\displaystyle n}
2112:
1965:
1937:
1919:
1663:
1482:Fizeau experiment
1446:{\displaystyle A}
1421:
1334:
1306:
1201:
1080:
1011:
849:
638:dragging of light
433:{\displaystyle c}
365:, if it existed,
337:absolute rotation
210:
153:angular frequency
61:nonrotating frame
5023:
4742:
4735:
4728:
4719:
4718:
4714:
4704:
4694:
4640:
4639:
4605:
4584:
4578:
4577:
4551:
4531:
4525:
4524:
4514:
4473:
4467:
4466:
4464:
4462:
4425:
4416:
4407:
4406:
4388:
4382:
4381:
4356:(185): 377â386.
4347:
4339:
4333:
4332:
4296:
4290:
4289:
4273:
4267:
4266:
4264:
4262:
4256:
4249:
4240:
4234:
4233:
4231:
4229:
4224:
4215:
4209:
4208:
4202:
4194:
4192:
4191:
4140:
4134:
4133:
4127:
4119:
4075:
4066:
4065:
4039:
4019:
4010:
4009:
4007:
4005:
3993:
3987:
3986:
3960:
3940:
3934:
3933:
3905:
3896:
3895:
3865:
3859:
3858:
3856:
3854:
3842:
3833:
3832:
3830:
3818:
3809:
3808:
3806:
3804:
3767:
3758:
3752:
3751:
3733:
3705:
3699:
3698:
3670:
3664:
3663:
3656:
3650:
3649:
3620:Irwin I. Shapiro
3616:
3610:
3607:
3601:
3598:
3592:
3586:
3580:
3579:
3563:
3557:
3556:
3540:
3534:
3531:
3525:
3523:
3513:
3503:
3471:
3465:
3464:
3444:
3428:
3422:
3421:
3381:
3375:
3374:
3367:
3361:
3360:
3349:
3343:
3342:
3310:
3304:
3303:
3271:
3265:
3260:
3254:
3224:
3192:
3186:
3181:
3149:
3134:
3131:
3098:
3086:
3080:
3079:
3068:
3062:
3061:
3043:
3034:
3029:
3021:
3008:
3007:
2983:
2977:
2976:
2974:
2948:
2940:
2934:
2933:
2905:
2899:
2898:
2884:
2876:
2867:
2866:
2852:
2844:
2815:Born coordinates
2766:refractive index
2758:coherence length
2540:
2538:
2537:
2532:
2530:
2522:
2517:
2515:
2507:
2499:
2481:
2479:
2478:
2473:
2471:
2463:
2452:
2441:
2427:
2416:
2408:
2397:
2386:
2378:
2367:
2356:
2341:
2339:
2338:
2333:
2331:
2323:
2315:
2303:
2301:
2300:
2295:
2293:
2281:
2279:
2278:
2273:
2271:
2260:
2252:
2250:
2242:
2234:
2216:
2214:
2213:
2208:
2200:
2195:
2184:
2162:
2160:
2159:
2154:
2142:
2140:
2139:
2134:
2132:
2121:
2113:
2111:
2110:
2098:
2093:
2089:
2088:
2087:
2072:
2071:
2032:
2030:
2029:
2024:
2009:
2007:
2006:
2001:
1999:
1987:
1985:
1984:
1979:
1977:
1966:
1964:
1963:
1954:
1949:
1938:
1930:
1925:
1921:
1920:
1918:
1917:
1908:
1907:
1903:
1891:
1885:
1880:
1858:
1857:
1834:
1832:
1831:
1826:
1824:
1823:
1804:
1802:
1801:
1796:
1794:
1782:
1780:
1779:
1774:
1751:
1729:
1727:
1726:
1721:
1719:
1714:
1706:
1685:
1683:
1682:
1677:
1675:
1664:
1656:
1647:
1628:
1626:
1625:
1620:
1614:
1595:
1593:
1592:
1587:
1585:
1472:
1470:
1469:
1464:
1452:
1450:
1449:
1444:
1432:
1430:
1429:
1424:
1422:
1417:
1399:
1348:
1346:
1345:
1340:
1335:
1333:
1332:
1323:
1312:
1307:
1305:
1304:
1295:
1291:
1290:
1274:
1253:
1251:
1250:
1245:
1215:
1213:
1212:
1207:
1202:
1200:
1199:
1198:
1189:
1188:
1176:
1175:
1165:
1161:
1160:
1144:
1139:
1138:
1126:
1125:
1094:
1092:
1091:
1086:
1081:
1079:
1065:
1054:
1049:
1048:
1025:
1023:
1022:
1017:
1012:
1010:
996:
985:
980:
979:
960:
958:
957:
952:
934:
932:
931:
926:
920:
919:
885:
883:
882:
877:
860:
858:
857:
852:
850:
845:
825:
820:
819:
800:
798:
797:
792:
790:
789:
773:
771:
770:
765:
752:angular velocity
710:Albert Michelson
657:Minkowski metric
653:Born coordinates
590:
588:
587:
582:
577:
562:
560:
559:
554:
552:
551:
539:
538:
513:
511:
510:
505:
503:
502:
486:
484:
483:
478:
476:
475:
459:
457:
456:
451:
439:
437:
436:
431:
415:
413:
412:
407:
405:
404:
331:
329:
328:
323:
308:
306:
305:
300:
295:
276:
274:
273:
268:
256:
254:
253:
248:
246:
234:
232:
231:
226:
224:
216:
211:
209:
201:
193:
175:
173:
172:
167:
165:
92:laser gyroscopes
71:that Einstein's
57:angular velocity
5031:
5030:
5026:
5025:
5024:
5022:
5021:
5020:
4991:
4990:
4989:
4984:
4958:
4937:
4911:
4890:
4856:
4848:
4797:
4751:
4746:
4649:
4644:
4643:
4585:
4581:
4532:
4528:
4474:
4470:
4460:
4458:
4423:
4417:
4410:
4403:
4389:
4385:
4340:
4336:
4297:
4293:
4274:
4270:
4260:
4258:
4257:on 5 March 2012
4254:
4247:
4243:Napolitano, F.
4241:
4237:
4227:
4225:
4222:
4216:
4212:
4196:
4195:
4189:
4187:
4141:
4137:
4121:
4120:
4086:(4695): 69â71.
4076:
4069:
4020:
4013:
4003:
4001:
3994:
3990:
3958:physics/0609222
3941:
3937:
3906:
3899:
3866:
3862:
3852:
3850:
3843:
3836:
3819:
3812:
3802:
3800:
3770:Physics-Uspekhi
3765:
3759:
3755:
3710:Rep. Prog. Phys
3706:
3702:
3671:
3667:
3658:
3657:
3653:
3632:(26): 789â791.
3617:
3613:
3608:
3604:
3599:
3595:
3587:
3583:
3564:
3560:
3541:
3537:
3532:
3528:
3472:
3468:
3461:
3429:
3425:
3382:
3378:
3369:
3368:
3364:
3351:
3350:
3346:
3311:
3307:
3272:
3268:
3261:
3257:
3207:(20): 377â392.
3193:
3189:
3164:(13): 448â463.
3150:
3137:
3114:(13): 389â447.
3087:
3083:
3070:
3069:
3065:
3058:
3044:
3037:
3022:
3011:
2998:(48): 716â719.
2984:
2980:
2941:
2937:
2930:10.1119/1.17656
2916:(11): 975â985.
2906:
2902:
2877:
2870:
2845:
2838:
2833:
2811:
2746:
2729:
2704:
2683:
2614:
2608:
2568:
2551:
2526:
2518:
2508:
2500:
2498:
2487:
2484:
2483:
2467:
2459:
2448:
2437:
2423:
2412:
2404:
2393:
2382:
2374:
2363:
2352:
2347:
2344:
2343:
2327:
2319:
2311:
2309:
2306:
2305:
2289:
2287:
2284:
2283:
2267:
2256:
2243:
2235:
2233:
2222:
2219:
2218:
2185:
2183:
2172:
2169:
2168:
2148:
2145:
2144:
2128:
2117:
2106:
2102:
2097:
2083:
2079:
2067:
2063:
2059:
2055:
2041:
2038:
2037:
2015:
2012:
2011:
1995:
1993:
1990:
1989:
1973:
1959:
1955:
1950:
1948:
1929:
1913:
1909:
1899:
1898:
1887:
1886:
1884:
1873:
1869:
1865:
1853:
1849:
1844:
1841:
1840:
1819:
1815:
1810:
1807:
1806:
1790:
1788:
1785:
1784:
1744:
1739:
1736:
1735:
1715:
1710:
1702:
1691:
1688:
1687:
1668:
1655:
1643:
1634:
1631:
1630:
1610:
1601:
1598:
1597:
1578:
1573:
1570:
1569:
1551:Stokes' theorem
1510:
1458:
1455:
1454:
1438:
1435:
1434:
1400:
1398:
1387:
1384:
1383:
1371:.(Fig. 4)
1328:
1324:
1313:
1311:
1300:
1296:
1286:
1282:
1275:
1273:
1262:
1259:
1258:
1224:
1221:
1220:
1194:
1190:
1184:
1180:
1171:
1167:
1166:
1156:
1152:
1145:
1143:
1134:
1130:
1121:
1117:
1106:
1103:
1102:
1066:
1055:
1053:
1044:
1040:
1038:
1035:
1034:
997:
986:
984:
975:
971:
969:
966:
965:
943:
940:
939:
915:
911:
894:
891:
890:
868:
865:
864:
826:
824:
815:
811:
809:
806:
805:
785:
781:
779:
776:
775:
759:
756:
755:
736:
731:
669:§ Theories
573:
568:
565:
564:
547:
543:
534:
530:
519:
516:
515:
498:
494:
492:
489:
488:
487:of one ray and
471:
467:
465:
462:
461:
445:
442:
441:
425:
422:
421:
400:
396:
394:
391:
390:
355:
349:
314:
311:
310:
291:
286:
283:
282:
262:
259:
258:
242:
240:
237:
236:
220:
212:
202:
194:
192:
181:
178:
177:
161:
159:
156:
155:
151:the platform's
120:
17:
12:
11:
5:
5029:
5019:
5018:
5013:
5008:
5006:Interferometry
5003:
4986:
4985:
4983:
4982:
4977:
4972:
4966:
4964:
4960:
4959:
4957:
4956:
4951:
4945:
4943:
4939:
4938:
4936:
4935:
4930:
4925:
4919:
4917:
4913:
4912:
4910:
4909:
4904:
4898:
4896:
4892:
4891:
4889:
4888:
4883:
4878:
4873:
4868:
4862:
4860:
4850:
4849:
4847:
4846:
4841:
4836:
4831:
4826:
4821:
4816:
4811:
4805:
4803:
4799:
4798:
4796:
4795:
4790:
4785:
4780:
4775:
4770:
4765:
4759:
4757:
4756:Speed/isotropy
4753:
4752:
4745:
4744:
4737:
4730:
4722:
4716:
4715:
4662:
4656:
4648:
4647:External links
4645:
4642:
4641:
4596:(25): 251102.
4579:
4526:
4468:
4408:
4401:
4383:
4334:
4301:Applied Optics
4291:
4268:
4235:
4210:
4135:
4067:
4011:
3996:Brown, Kevin.
3988:
3935:
3916:(2): 338â346.
3897:
3878:(2): 475â493.
3860:
3845:Brown, Kevin.
3834:
3810:
3776:(8): 907â909.
3753:
3731:10.1.1.128.191
3716:(6): 615â688.
3700:
3695:10.1086/142879
3665:
3651:
3611:
3602:
3593:
3581:
3572:Comptes Rendus
3558:
3549:Comptes Rendus
3535:
3526:
3466:
3459:
3423:
3376:
3362:
3344:
3305:
3266:
3255:
3225:From p. 377:
3187:
3135:
3133:
3132:
3081:
3063:
3056:
3035:
3009:
2978:
2935:
2900:
2891:Comptes Rendus
2868:
2859:Comptes Rendus
2835:
2834:
2832:
2829:
2828:
2827:
2822:
2817:
2810:
2807:
2745:
2742:
2728:
2721:
2703:
2700:
2682:
2679:
2666:beat frequency
2607:
2604:
2567:
2566:Practical uses
2564:
2550:
2547:
2529:
2525:
2521:
2514:
2511:
2506:
2503:
2497:
2494:
2491:
2470:
2466:
2462:
2458:
2455:
2451:
2447:
2444:
2440:
2436:
2433:
2430:
2426:
2422:
2419:
2415:
2411:
2407:
2403:
2400:
2396:
2392:
2389:
2385:
2381:
2377:
2373:
2370:
2366:
2362:
2359:
2355:
2351:
2330:
2326:
2322:
2318:
2314:
2292:
2270:
2266:
2263:
2259:
2255:
2249:
2246:
2241:
2238:
2232:
2229:
2226:
2206:
2203:
2198:
2194:
2191:
2188:
2182:
2179:
2176:
2152:
2131:
2127:
2124:
2120:
2116:
2109:
2105:
2101:
2096:
2092:
2086:
2082:
2078:
2075:
2070:
2066:
2062:
2058:
2054:
2051:
2048:
2045:
2022:
2019:
1998:
1976:
1972:
1969:
1962:
1958:
1953:
1947:
1944:
1941:
1936:
1933:
1928:
1924:
1916:
1912:
1906:
1902:
1897:
1894:
1890:
1883:
1879:
1876:
1872:
1868:
1864:
1861:
1856:
1852:
1848:
1822:
1818:
1814:
1793:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1750:
1747:
1743:
1718:
1713:
1709:
1705:
1701:
1698:
1695:
1674:
1671:
1667:
1662:
1659:
1654:
1650:
1646:
1642:
1638:
1617:
1613:
1609:
1605:
1584:
1581:
1577:
1537: = 2
1509:
1506:
1494:optical cavity
1462:
1442:
1420:
1416:
1413:
1409:
1406:
1403:
1397:
1394:
1391:
1350:
1349:
1338:
1331:
1327:
1322:
1319:
1316:
1310:
1303:
1299:
1294:
1289:
1285:
1281:
1278:
1272:
1269:
1266:
1243:
1240:
1237:
1234:
1231:
1228:
1217:
1216:
1205:
1197:
1193:
1187:
1183:
1179:
1174:
1170:
1164:
1159:
1155:
1151:
1148:
1142:
1137:
1133:
1129:
1124:
1120:
1116:
1113:
1110:
1096:
1095:
1084:
1078:
1075:
1072:
1069:
1064:
1061:
1058:
1052:
1047:
1043:
1027:
1026:
1015:
1009:
1006:
1003:
1000:
995:
992:
989:
983:
978:
974:
950:
947:
936:
935:
923:
918:
914:
910:
907:
904:
901:
898:
875:
872:
862:
861:
848:
844:
841:
838:
835:
832:
829:
823:
818:
814:
788:
784:
763:
735:
732:
730:
727:
612:Georges Sagnac
580:
576:
572:
550:
546:
542:
537:
533:
529:
526:
523:
501:
497:
474:
470:
449:
429:
403:
399:
348:
345:
321:
318:
298:
294:
290:
266:
245:
223:
219:
215:
208:
205:
200:
197:
191:
188:
185:
164:
119:
116:
108:speed of light
65:Georges Sagnac
41:interferometry
37:Georges Sagnac
31:, also called
15:
9:
6:
4:
3:
2:
5028:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4998:
4996:
4981:
4978:
4976:
4973:
4971:
4968:
4967:
4965:
4961:
4955:
4952:
4950:
4947:
4946:
4944:
4940:
4934:
4931:
4929:
4926:
4924:
4921:
4920:
4918:
4916:Fizeau/Sagnac
4914:
4908:
4905:
4903:
4900:
4899:
4897:
4893:
4887:
4884:
4882:
4879:
4877:
4874:
4872:
4869:
4867:
4864:
4863:
4861:
4859:
4855:
4854:Time dilation
4851:
4845:
4842:
4840:
4837:
4835:
4832:
4830:
4827:
4825:
4822:
4820:
4817:
4815:
4812:
4810:
4807:
4806:
4804:
4800:
4794:
4791:
4789:
4786:
4784:
4781:
4779:
4776:
4774:
4771:
4769:
4766:
4764:
4761:
4760:
4758:
4754:
4750:
4743:
4738:
4736:
4731:
4729:
4724:
4723:
4720:
4712:
4708:
4703:
4698:
4693:
4688:
4684:
4680:
4676:
4672:
4668:
4663:
4660:
4657:
4655:
4651:
4650:
4637:
4633:
4629:
4625:
4621:
4617:
4613:
4609:
4604:
4599:
4595:
4591:
4583:
4575:
4571:
4567:
4563:
4559:
4555:
4550:
4545:
4542:(8): 085012.
4541:
4537:
4530:
4522:
4518:
4513:
4508:
4504:
4500:
4496:
4492:
4489:(8): 084007.
4488:
4484:
4480:
4472:
4457:
4453:
4449:
4445:
4441:
4437:
4433:
4429:
4422:
4415:
4413:
4404:
4398:
4394:
4387:
4379:
4375:
4371:
4367:
4363:
4359:
4355:
4351:
4346:
4338:
4330:
4326:
4322:
4318:
4314:
4310:
4306:
4302:
4295:
4287:
4283:
4279:
4272:
4253:
4246:
4239:
4221:
4214:
4206:
4200:
4186:
4182:
4178:
4174:
4170:
4166:
4162:
4158:
4154:
4150:
4146:
4139:
4131:
4125:
4117:
4113:
4109:
4105:
4101:
4097:
4093:
4089:
4085:
4081:
4074:
4072:
4063:
4059:
4055:
4051:
4047:
4043:
4038:
4033:
4030:(6): 063837.
4029:
4025:
4018:
4016:
3999:
3992:
3984:
3980:
3976:
3972:
3968:
3964:
3959:
3954:
3951:(1â2): 7â10.
3950:
3946:
3939:
3931:
3927:
3923:
3919:
3915:
3911:
3904:
3902:
3893:
3889:
3885:
3881:
3877:
3873:
3872:
3864:
3848:
3841:
3839:
3829:
3828:gr-qc/0401005
3824:
3817:
3815:
3799:
3795:
3791:
3787:
3783:
3779:
3775:
3771:
3764:
3757:
3749:
3745:
3741:
3737:
3732:
3727:
3723:
3719:
3715:
3711:
3704:
3696:
3692:
3688:
3684:
3680:
3676:
3669:
3661:
3655:
3647:
3643:
3639:
3635:
3631:
3627:
3626:
3621:
3615:
3606:
3597:
3591:
3585:
3577:
3573:
3569:
3562:
3554:
3550:
3546:
3539:
3530:
3524:(Open access)
3521:
3517:
3512:
3507:
3502:
3497:
3493:
3489:
3485:
3481:
3477:
3470:
3462:
3456:
3452:
3448:
3443:
3442:gr-qc/0305084
3438:
3434:
3427:
3419:
3415:
3411:
3407:
3403:
3399:
3395:
3391:
3387:
3380:
3372:
3366:
3358:
3354:
3348:
3340:
3336:
3332:
3328:
3324:
3321:(in German).
3320:
3316:
3309:
3301:
3297:
3293:
3289:
3285:
3282:(in German).
3281:
3277:
3270:
3264:
3259:
3252:
3248:
3244:
3240:
3236:
3232:
3228:
3222:
3218:
3214:
3210:
3206:
3203:(in German).
3202:
3198:
3191:
3185:
3179:
3175:
3171:
3167:
3163:
3159:
3155:
3148:
3146:
3144:
3142:
3140:
3129:
3125:
3121:
3117:
3113:
3109:
3105:
3100:
3099:
3096:
3092:
3085:
3077:
3073:
3067:
3059:
3053:
3049:
3042:
3040:
3033:
3027:
3020:
3018:
3016:
3014:
3005:
3001:
2997:
2993:
2989:
2982:
2973:
2968:
2964:
2960:
2956:
2952:
2947:
2939:
2931:
2927:
2923:
2919:
2915:
2911:
2904:
2896:
2892:
2888:
2883:
2875:
2873:
2864:
2860:
2856:
2851:
2843:
2841:
2836:
2826:
2823:
2821:
2818:
2816:
2813:
2812:
2806:
2803:
2797:
2795:
2791:
2786:
2784:
2774:
2770:
2767:
2763:
2759:
2755:
2751:
2741:
2737:
2733:
2726:
2720:
2718:
2714:
2710:
2699:
2695:
2687:
2678:
2676:
2672:
2667:
2662:
2659:
2649:
2645:
2643:
2637:
2635:
2631:
2626:
2618:
2613:
2603:
2601:
2597:
2593:
2589:
2585:
2581:
2577:
2573:
2563:
2560:
2555:
2546:
2544:
2523:
2512:
2509:
2504:
2501:
2495:
2492:
2464:
2456:
2453:
2445:
2442:
2434:
2431:
2428:
2420:
2417:
2409:
2401:
2398:
2390:
2387:
2379:
2371:
2368:
2360:
2357:
2349:
2324:
2316:
2264:
2261:
2253:
2247:
2244:
2239:
2236:
2230:
2227:
2204:
2196:
2192:
2189:
2186:
2180:
2177:
2164:
2150:
2125:
2122:
2114:
2107:
2103:
2099:
2094:
2090:
2084:
2080:
2076:
2073:
2068:
2064:
2060:
2056:
2052:
2049:
2046:
2034:
2020:
2017:
1970:
1967:
1960:
1956:
1945:
1942:
1939:
1934:
1931:
1926:
1922:
1914:
1910:
1904:
1895:
1892:
1881:
1877:
1874:
1870:
1866:
1862:
1859:
1854:
1850:
1846:
1838:
1820:
1816:
1812:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1748:
1745:
1741:
1733:
1707:
1699:
1696:
1693:
1672:
1669:
1665:
1660:
1657:
1652:
1648:
1644:
1640:
1636:
1615:
1611:
1607:
1603:
1582:
1579:
1575:
1566:
1558:
1554:
1552:
1546:
1544:
1540:
1536:
1530:
1528:
1524:
1514:
1505:
1501:
1499:
1495:
1489:
1487:
1483:
1478:
1474:
1460:
1440:
1418:
1414:
1407:
1404:
1401:
1395:
1392:
1379:
1377:
1372:
1370:
1361:
1357:
1355:
1336:
1329:
1325:
1320:
1317:
1314:
1308:
1301:
1297:
1292:
1287:
1283:
1279:
1276:
1270:
1267:
1257:
1256:
1255:
1241:
1238:
1235:
1232:
1229:
1226:
1203:
1195:
1191:
1185:
1181:
1177:
1172:
1168:
1162:
1157:
1153:
1149:
1146:
1140:
1135:
1131:
1127:
1122:
1118:
1114:
1111:
1101:
1100:
1099:
1082:
1076:
1073:
1070:
1067:
1062:
1059:
1056:
1050:
1045:
1041:
1033:
1032:
1031:
1013:
1007:
1004:
1001:
998:
993:
990:
987:
981:
976:
972:
964:
963:
962:
948:
921:
916:
912:
908:
905:
902:
899:
889:
888:
887:
873:
846:
842:
836:
833:
830:
827:
821:
816:
812:
804:
803:
802:
786:
782:
761:
753:
749:
740:
726:
722:
719:
715:
711:
706:
704:
703:Shapiro delay
699:
695:
691:
687:
682:
677:
674:
670:
664:
662:
661:Paul Langevin
658:
654:
650:
649:Paul Langevin
645:
643:
639:
633:
631:
626:
621:
618:
613:
608:
606:
602:
598:
594:
578:
574:
570:
548:
544:
540:
535:
531:
527:
524:
499:
495:
472:
468:
447:
427:
419:
401:
397:
388:
384:
379:
376:
372:
368:
364:
360:
354:
344:
342:
338:
333:
319:
296:
292:
288:
278:
264:
217:
206:
203:
198:
195:
189:
186:
154:
149:
145:
137:
136:optical fiber
133:
129:
124:
115:
113:
109:
105:
101:
97:
93:
89:
85:
81:
76:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
29:Sagnac effect
21:
4942:Alternatives
4927:
4674:
4670:
4593:
4589:
4582:
4539:
4535:
4529:
4486:
4482:
4471:
4459:. Retrieved
4431:
4427:
4392:
4386:
4353:
4349:
4337:
4304:
4300:
4294:
4277:
4271:
4259:. Retrieved
4252:the original
4238:
4226:. Retrieved
4213:
4199:cite journal
4188:. Retrieved
4152:
4148:
4138:
4124:cite journal
4083:
4079:
4027:
4023:
4002:. Retrieved
3991:
3948:
3944:
3938:
3913:
3910:Phys. Rev. D
3909:
3875:
3869:
3863:
3851:. Retrieved
3801:. Retrieved
3773:
3769:
3756:
3713:
3709:
3703:
3678:
3674:
3668:
3654:
3629:
3623:
3614:
3605:
3596:
3584:
3575:
3571:
3561:
3552:
3548:
3538:
3529:
3483:
3479:
3469:
3432:
3426:
3393:
3389:
3379:
3365:
3356:
3347:
3325:(3): 47â48.
3322:
3318:
3308:
3283:
3279:
3269:
3258:
3250:
3246:
3242:
3238:
3234:
3230:
3226:
3204:
3200:
3190:
3161:
3157:
3111:
3107:
3094:
3090:
3084:
3075:
3066:
3047:
3025:
2995:
2991:
2981:
2954:
2950:
2938:
2913:
2909:
2903:
2894:
2890:
2862:
2858:
2798:
2787:
2782:
2779:
2747:
2738:
2734:
2730:
2724:
2705:
2696:
2692:
2663:
2657:
2654:
2638:
2627:
2623:
2580:sidereal day
2569:
2556:
2552:
2549:Applications
2165:
2035:
1567:
1563:
1549:path due to
1547:
1542:
1538:
1534:
1531:
1526:
1522:
1520:
1502:
1490:
1479:
1475:
1380:
1373:
1368:
1366:
1353:
1351:
1218:
1097:
1028:
938:Eliminating
937:
863:
747:
745:
723:
707:
686:Wilhelm Wien
678:
665:
646:
634:
625:Fresnel drag
622:
609:
417:
383:Max von Laue
380:
371:Oliver Lodge
356:
334:
279:
144:fiber optics
141:
126:Figure 2. A
77:
53:interference
48:
32:
28:
26:
4652:Mathpages:
4261:15 February
4228:15 February
4004:15 February
4000:. MathPages
3853:15 February
3849:. MathPages
3803:15 February
3681:: 140â145.
3396:(10): 824.
2957:: 149â166.
2910:Am. J. Phys
2897:: 1410â1413
2754:micrometers
2606:Ring lasers
1498:Ring lasers
630:Paul Harzer
128:guided wave
4995:Categories
4350:Am. J. Sci
4190:2006-09-18
4037:1601.01448
3578:: 304â306.
3555:: 831â834.
3243:Einstein's
3028:: 405â412.
2831:References
2630:ring laser
734:Basic case
714:Henry Gale
351:See also:
4603:1007.0574
4549:0804.1036
4378:131116577
4062:119242639
3798:250738420
3748:250854429
3726:CiteSeerX
3418:1631-0705
2865:: 708â710
2783:zero-area
2713:dithering
2524:⋅
2520:ω
2510:λ
2505:π
2496:≈
2493:ϕ
2490:Δ
2465:⋅
2461:ω
2443:⋅
2439:ω
2435:∮
2418:×
2410:⋅
2406:ω
2402:∮
2388:⋅
2380:×
2376:ω
2372:∮
2358:⋅
2350:∮
2325:×
2321:ω
2291:ω
2262:⋅
2254:∮
2245:λ
2240:π
2231:≈
2228:ϕ
2225:Δ
2202:Δ
2197:λ
2190:π
2181:≈
2178:ϕ
2175:Δ
2123:⋅
2115:∮
2095:≈
2085:−
2074:−
2053:∫
2044:Δ
2021:ℓ
1968:⋅
1946:±
1943:ℓ
1927:≈
1893:⋅
1882:±
1863:γ
1855:±
1821:±
1771:ℓ
1765:≈
1762:ℓ
1756:γ
1746:ℓ
1734:formula,
1697:ℓ
1670:ℓ
1645:±
1612:±
1580:ℓ
1527:different
1461:ω
1419:λ
1412:Δ
1405:π
1393:ϕ
1390:Δ
1321:ω
1293:ω
1280:π
1271:≈
1265:Δ
1239:≪
1230:ω
1192:ω
1178:−
1163:ω
1150:π
1128:−
1109:Δ
1077:ω
1060:π
1008:ω
1002:−
991:π
946:Δ
909:ω
897:Δ
871:Δ
840:Δ
831:π
762:ω
748:distances
549:−
545:τ
541:−
532:τ
525:τ
522:Δ
500:−
496:τ
469:τ
448:ω
320:ϕ
317:Δ
297:λ
265:λ
218:⋅
214:ω
204:λ
199:π
190:≈
187:ϕ
184:Δ
163:ω
84:gyroscope
5016:Rotation
4711:28163638
4677:(1): 1.
4628:20867358
4521:18080099
4461:31 March
4456:10060864
4329:20155007
4185:10067969
4177:17779917
4116:22556404
4108:17811569
3520:28163638
3486:(1): 1.
3286:: 8â10.
3229:Einstein
2809:See also
1905:′
1878:′
1749:′
1673:′
1649:′
1616:′
1583:′
110:for all
45:rotation
4963:General
4702:5253894
4679:Bibcode
4636:9929939
4608:Bibcode
4574:7535227
4554:Bibcode
4491:Bibcode
4436:Bibcode
4358:Bibcode
4309:Bibcode
4282:Bibcode
4157:Bibcode
4149:Science
4088:Bibcode
4080:Science
4042:Bibcode
3963:Bibcode
3918:Bibcode
3880:Bibcode
3778:Bibcode
3718:Bibcode
3683:Bibcode
3634:Bibcode
3511:5253894
3488:Bibcode
3447:Bibcode
3398:Bibcode
3327:Bibcode
3288:Bibcode
3209:Bibcode
3166:Bibcode
3116:Bibcode
2959:Bibcode
2918:Bibcode
2889:],
2857:],
2702:Lock-in
2600:Galileo
2596:COMPASS
2592:GLONASS
1486:Lorentz
593:Lorentz
347:History
4709:
4699:
4634:
4626:
4572:
4519:
4454:
4399:
4376:
4327:
4183:
4175:
4114:
4106:
4060:
3983:699912
3981:
3796:
3746:
3728:
3518:
3508:
3457:
3416:
3054:
2725:versus
2658:cycles
2543:Fizeau
1517:(FOC).
1352:where
729:Theory
601:Stokes
235:where
80:gimbal
69:aether
4632:S2CID
4598:arXiv
4570:S2CID
4544:arXiv
4517:S2CID
4424:(PDF)
4374:S2CID
4255:(PDF)
4248:(PDF)
4223:(PDF)
4181:S2CID
4112:S2CID
4058:S2CID
4032:arXiv
3979:S2CID
3953:arXiv
3823:arXiv
3794:S2CID
3766:(PDF)
3744:S2CID
3437:arXiv
3093:[
2885:[
2853:[
2756:(the
2717:hertz
605:Hertz
4707:PMID
4624:PMID
4463:2012
4452:PMID
4397:ISBN
4325:PMID
4263:2013
4230:2013
4205:link
4173:PMID
4130:link
4104:PMID
4006:2013
3855:2013
3805:2013
3516:PMID
3455:ISBN
3414:ISSN
3052:ISBN
2788:The
2557:The
1686:Let
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