s761bd6b6af2a39a9.jimcontent.com€¦ · Tome 15 (2014) – No 10 Georges Sagnac’s setup for detecting the optical whirling effect. Dispositif mis en ouvre par Georges Sagnac pour

Tome 15 (2014) – No 10

Georges Sagnac’s setup for detecting the optical whirling effect.Dispositif mis en ouvre par Georges Sagnac pour mettre en évidence l’effet tourbillonnaire optique.From / D’après Georges Sagnac, La preuve de la réalité de l’éther lumineux par l’expérience de

l’interférographe tournant, C. R. Acad. Sci. Paris 157 (1913) 1410–1413 (p. 1412).

DOSSIERThe Sagnac effect: 100 years later / L’effet Sagnac : 100 ans aprèsCoordinator / Coordinateur : Alexandre Gauguet

• Foreword / Avant-proposAlexandre Gauguet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787

• Georges Sagnac: A life for opticsOlivier Darrigol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789

• Towards a solid-state ring laser gyroscopeNoad El Badaoui, Bertrand Morbieu, Philippe Martin, Pierre Rouchon, Jean-Paul Pocholle, François Gutty, Gilles Feugnet, Sylvain Schwartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841

• The fiber-optic gyroscope, a century after Sagnac’s experiment: The ultimate rotation-sensing technology?Hervé C. Lefèvre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851

• The centennial of the Sagnac experiment in the optical regime: From a tabletop experiment to the variation of the Earth’s rotationKarl Ulrich Schreiber, André Gebauer, Heiner Igel, Joachim Wassermann, Robert B. Hurst, Jon-Paul R. Wells 859

• A ring lasers array for fundamental physicsAngela Di Virgilio, Maria Allegrini, Alessandro Beghi, Jacopo Belfi, Nicolò Beverini, Filippo Bosi, Bachir Bouhadef, Massimo Calamai, Giorgio Carelli, Davide Cuccato, Enrico Maccioni, Antonello Ortolan, Giuseppe Passeggio, Alberto Porzio, Matteo Luca Ruggiero, Rosa Santagata, Angelo Tartaglia . . . . . . . . . . . . . . . 866

• The Sagnac effect: 20 years of development in matter-wave interferometryBrynle Barrett, Rémy Geiger, Indranil Dutta, Matthieu Meunier, Benjamin Canuel, Alexandre Gauguet, Philippe Bouyer, Arnaud Landragin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875

Continued on the next page

Contents (continued)• Large-area Sagnac atom interferometer with robust phase read out

Gunnar Tackmann, Peter Berg, Sven Abend, Christian Schubert, Wolfgang Ertmer, Ernst Maria Rasel . . . . . . . 884

• Sagnac-based rotation sensing with superfluid helium quantum interference devicesYuki Sato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898

• Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909

• Index des mots clés . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911

• Keyword index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914

ErratumAmong the photographs displayed in Fig. 33 of the article by Olivier Darrigol, “Georges Sagnac: A life for optics” (this issue, pp. 789–840), contrary to what the caption indicates, only the first one is with absolute certainty a shot of Sagnac (taken at ENS Ulm, Paris). The third one (bottom left) is in reality a photograph of Pierre Curie, whereas the two others cannot be ascertained as depicting Sagnac.Parmi les photos reproduites dans la Fig. 33 de l’article d’Olivier Darrigol, « Georges Sagnac : Une vie pour l’optique », contrairement au libellé de la légende, seule la première est de façon certaine une photo de Sagnac (prise à l’ENS Ulm, à Paris). La troisième représente en réalité Pierre Curie, tandis que l’identité des personnes photographiées sur les deux autres reste incertaine.

C. R. Physique 15 (2014) 787–788

Contents lists available at ScienceDirect

Comptes Rendus Physique

www.sciencedirect.com

The Sagnac effect: 100 years later / L’effet Sagnac : 100 ans après

Foreword

This dossier on the Sagnac effect follows the colloquium held at the “Fondation Simone-et-Cino-del-Duca” (Paris) on 10 October 2013 to celebrate the centenary of the Sagnac effect, published by Georges Sagnac in 1913 in the Comptes rendus de l’Académie des sciences, vol. 157, pp. 708–710.

The Sagnac effect plays an important role in physics, stimulating much discussion between physicists from various fields in optics, quantum physics, and relativity. It is also a wonderful example of the interplay between fundamental physics (general relativity, matter waves in non-inertial frame) and applied physics (navigation, geophysics).

Olivier Darrigol presents a scientific biography of Georges Sagnac, from his student years at the École normale supérieure to his last lectures at the Sorbonne. This paper (Georges Sagnac: A life for optics) presents the Sagnac effect and the research of Georges Sagnac showing the importance of optics in his career. Originally planned to test the theory of special relativity, the Sagnac effect was applied very quickly on measurements of rotations, in particular the measurement of the Earth’s rotation rate by A.A. Michelson, F. Pearson, and H.G. Gale in 1925. However, it was not until the discovery of the laser that Sagnac interferometers were used as optical gyroscopes. In particular, the ring-laser gyros have provided an enormous increase in sensitivity. Nowadays, the ring-laser gyros are a backbone of modern navigation systems. Noad El Badaoui et al. (Towards a solid-state ring laser gyroscope) show innovative developments in ring-laser gyros for navigation systems. Hervé Lefèvre (The fiber-optic gyroscope, a century after Sagnac’s experiment: the ultimate rotation-sensing technology?) presents recent developments in fibre optical gyroscopes and their potential applications in navigation. In addition, technological advances in laser physics have matured to a point where they make ring-lasers many orders of magnitude more sensitive than early instrumentation. In particular, Ulrich Schreiber et al. (The centennial of the Sagnac experiment in the optical regime: From a tabletop experiment to the variation of the Earth’s rotation) show an impressive sensor stability over several months with new applications of ring-laser gyroscopes in the fields of geophysics, geodesy, and seismology. Angela Di Virgilio et al. (A ring lasers array for fundamental physics) report on a Sagnac interferometer project that plans to detect general relativity effects such as the Lense–Thirring frame dragging in a ground-based experiment. The Sagnac effect is a wave phenomenon, and does not depend on the nature of the wave: it can be a light wave or a matter wave. In particular, atom gyroscopes based on Sagnac atom interferometers have shown very good sensitivity and accuracy. Brynle Barrett et al. (The Sagnac effect: 20 years of development in matter-wave interferometry) review some of the key developments that have taken place over the last 20 years regarding matter-wave Sagnac interferometers. G. Tackmann et al. (Large-area Sagnac atom interferometer with robust phase read out) focus on a possible transportable cold atoms gyroscope. Finally, Yuki Sato (Sagnac-based rotation sensing with superfluid helium quantum interference devices) opens perspectives with a new type Sagnac interferometer using a macroscopic quantum system based on a superfluid helium interferometer.

Avant-proposCe dossier sur l’effet Sagnac fait suite au colloque qui s’est tenu à la Fondation Simone-et-Cino-del-Duca, à Paris, le

10 octobre 2013 pour célébrer le centenaire de l’effet Sagnac, publié par Georges Sagnac en 1913 dans les Comptes rendus de l’Académie des sciences, vol. 157, pp. 708–710.

L’effet Sagnac a joué un rôle important en physique, stimulant de nombreuses de discussions entre physiciens de diffé-rents domaines, que ce soit l’optique, la physique quantique et la relativité. Il est aussi un très bon exemple d’interaction entre la physique fondamentale (la relativité générale, les ondes de matière dans des référentiels non inertiels) et la phy-sique appliquée (navigation, géophysique).

Olivier Darrigol présente une biographie scientifique de Georges Sagnac, depuis ses années d’études à l’École normale supérieure jusqu’à ses dernières conférences à la Sorbonne. Cet article (Georges Sagnac : une vie pour l’optique) présente l’effet Sagnac et les recherches de Georges Sagnac, en insistant sur l’importance de l’optique dans sa carrière. Initialement prévu pour tester la théorie de la relativité restreinte, l’effet Sagnac a été appliqué très rapidement à des mesures de rotations, en particulier à la mesure de la vitesse de rotation de la Terre en 1925 par A.A. Michelson, F. Pearson et H.G. Gale. Cependant,

http://dx.doi.org/10.1016/j.crhy.2014.11.0021631-0705/© 2014 Published by Elsevier Masson SAS on behalf of Académie des sciences.

788 The Sagnac effect: 100 years later / L’effet Sagnac : 100 ans après

il a fallu attendre la découverte du laser pour que les interféromètres de Sagnac soient utilisés comme des gyromètres optiques. En particulier, les gyromètres laser ont permis une amélioration considérable de la sensibilité. De nos jours, les gyromètres laser sont un pilier des systèmes de navigation modernes. Noad El Badaoui et al. (Vers un gyrolaser à état solide) décrivent les innovations dans le domaine des gyromètres laser. Hervé Lefèvre (Le gyromètre à fibre optique, cent ans après l’expérience de Sagnac : la technologie ultime de mesure inertielle de rotation ?) évoque les développements récents de gyromètres à fibre optique et leurs applications potentielles pour la navigation inertielle. En outre, les progrès technologiques en phy-sique des lasers ont permis la réalisation de gyromètres laser géants, avec une sensibilité améliorée de plusieurs ordres de grandeur par rapport aux instruments initiaux. En particulier, Ulrich Schreiber et al. (Le centenaire de l’expérience de Sagnac en régime optique : d’une expérience de laboratoire à la variation de la rotation de la Terre) décrivent un capteur d’une stabilité impressionnante pendant plusieurs mois, permettant de nouvelles applications dans les domaines de la géophysique, de la géodésie et de la sismologie. Angela Di Virgilio et al. (Un réseau de lasers en anneaux pour la physique fondamentale) rapportent un projet d’interféromètre de Sagnac dont le but est de détecter des effets prévus par la relativité générale, comme l’effet Lense–Thirring, dans une expérience de laboratoire sur Terre. L’effet Sagnac est un phénomène ondulatoire, qui ne dépend pas de la nature de l’onde : il peut s’agir d’une onde lumineuse ou d’une onde de matière. En particulier, les gyromètres atomiques fondés sur l’utilisation d’interféromètres de Sagnac atomiques ont montré une excellente sensibilité et une très grande exactitude. Brynle Barrett et al. (L’effet Sagnac : 20 ans de développements des interféromètres à ondes de matière) passent en revue les principaux développements qui sont intervenus au cours des vingt dernières années dans le domaine des in-terféromètres de Sagnac à ondes de matière. G. Tackmann et al. (Interféromètre Sagnac atomique avec une acquisition de signal robuste) mettent l’accent sur le développement de gyromètres à atomes froids transportables. Enfin, Yuki Sato (Capteurs de rotation fondés sur l’effet Sagnac avec interférences quantiques dans l’hélium superfluide) ouvre des perspectives avec un nouvel interféromètre de Sagnac utilisant un système quantique macroscopique fondé sur un interféromètre à hélium superfluide.

Alexandre GauguetLaboratoire Collisions Agrégats Réactivité (LCAR)

Université Paul-Sabatier, Toulouse, FranceE-mail address: [email protected]

C. R. Physique 15 (2014) 789–840





Georges Sagnac: A life for optics

Georges Sagnac : Une vie pour l’optique

Olivier DarrigolLaboratoire SPHERE, UMR 7219, CNRS/Université Paris-Diderot, bâtiment Condorcet, case 7093, 5, rue Thomas-Mann, 75205 Paris cedex 13, France

a r t i c l e i n f o a b s t r a c t

Article history:Available online 23 October 2014

Keywords:SagnacSagnac effectX-raysOpticsFluorescenceInterferometry

Mots-clés :SagnacEffet SagnacRayons XOptiqueFluorescenceInterférométrie

Georges Sagnac is mostly known for the optical effect in rotating frames that he demon-strated in 1913. His scientific interests were quite diverse: they included photography, optical illusions, X-ray physics, radioactivity, the blue of the sky, anomalous wave prop-agation, interferometry, strioscopy, and acoustics. An optical theme nonetheless pervaded his entire œuvre. Within optics, an original theory of the propagation of light motivated most of his investigations, from an ingenious explanation of the Fresnel drag, through the discovery of the Sagnac effect, to his quixotic defense of an alternative to relativity theory. Optical analogies efficiently guided his work in other domains. Optics indeed was his true passion. He saw himself as carrying the torch of the two great masters of French optics, Augustin Fresnel and Hippolyte Fizeau. In this mission he overcame his poor health and labored against the modernist tide, with much success originally and bitter isolation in the end.

© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

r é s u m é

Georges Sagnac est principalement connu pour l’effet optique des faisceaux tournants, qu’il démontra en 1913. Ses intérêts scientifiques étaient très divers, incluant la photographie, les illusions d’optique, la physique des rayons X, le bleu du ciel, la propagation anormale des ondes, l’interférométrie, la strioscopie et l’acoustique. Le thème de l’optique habite néanmoins son œuvre toute entière. Dans le domaine de l’optique, une théorie originale de la propagation de la lumière a motivé la plupart de ses recherches, depuis une explication ingénieuse de l’entraînement de Fresnel, en passant par la découverte de l’effet Sagnac, jusqu’à son combat de Don Quichotte en faveur d’une alternative à la théorie de la relati-vité. Les analogies optiques ont efficacement guidé son travail dans d’autres domaines. En effet, l’optique était sa vraie passion. Il se voyait comme porte-flambeau de deux grands maîtres de l’optique française, Augustin Fresnel et Hippolyte Fizeau. Dans cet apostolat, il surmonta sa faible santé pour travailler à contre courant du modernisme, rencontrant d’abord beaucoup de succès, puis un isolement amer à la fin.


E-mail address: [email protected].

http://dx.doi.org/10.1016/j.crhy.2014.09.0071631-0705/© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

790 O. Darrigol / C. R. Physique 15 (2014) 789–840

1. Introduction

In 1997 the director of the Commission des plis cachetés of the Académie des sciences, Roger Balian, asked me to report on two plis cachetés (sealed letters) deposited on 28 March and 24 July 1898 by Georges Sagnac. At that time, I only knew this physicist for the effect that bears his name. I had assumed he was just one more of these French masters of experimental optics who collected the fruits of a superior interferometry. So I was surprised to see that the two plis did not belong to optics per se. They dealt with the secondary rays emitted by matter under the impact of X-rays. The pli of 24 July bears the title “Transformation des rayons X par la matière : influence de l’azimuth des rayons X et des rayons secondaires S émis” and describes an oscillation of the penetrating power of the secondary rays as a function of the angle under which they are emitted. Sagnac suggests that this oscillation might have to do with the diffraction of waves of wavelength smaller than the interatomic spacing of the target. The following year, Sagnac publicly confirmed the heterogeneity of the secondary rays, but he gave up the idea of a diffraction-related oscillation. The pli of 28 March describes an experiment demonstrating the existence of an electrically charged component of the secondary rays. I soon found out that a third pli of 18 July 1898 contained an improved version of this experiment and had been opened and published at Sagnac’s request in 1900.1

I thus became aware of Sagnac’s important role in the new field of research that Wilhelm Röntgen’s discovery of X-rays had opened in late 1895. As I learned from the historian Bruce Wheaton, Sagnac had discovered that X-rays were trans-formed by impact on matter into rays of lower penetrating power then called S rays or Sagnac rays, in a phenomenon now called X-ray fluorescence. He had established the heterogeneous and specific character of the secondary radiation emitted by heavy elements, thus anticipating later X-ray spectroscopy. And he had discovered the existence of an electrically charged component of the secondary rays, thus inaugurating studies of the X-ray photoelectric effect.

These were not the last plis cachetés deposited by Sagnac. On 23 February 1902, he wrote one in which he suggested an experimental test for the hypothesis of a gravitational origin of radioactivity. Having again to report on this pli, I found out that Sagnac had performed an improved version of this experiment a few months later and published the negative result in 1906. The last of Sagnac’s plis is the one of 18 August 1913, in which he gives the first account of the effect for which he is most famous. Sagnac’s frequent recourse to plis cachetés conveys the image of a man who knew the thrills of discovery in multiple circumstances and in different fields of physics.

Although the next generation of X-ray physicists recognized the importance of Sagnac’s pioneering work in this domain, this part of his oeuvre is now largely forgotten presumably because the techniques on which it was based became obsolete after Max Laue’s discovery of X-ray diffraction in 1912.2 In contrast, the Sagnac effect is very well known, though not in the manner hoped by his discoverer. In the relevant experiment, which dates from 1913, the interference of light in a rotating interferometer of a special kind proves to depend on the rotation (with respect to an inertial frame). Sagnac announced this result as a proof of the existence of the ether. Although there still were, in 1913, many physicists to welcome such a claim, the increasingly powerful adepts of relativity theory brushed it away. The experience remained important as an optical counterpart of Foucault’s pendulum experiment or as a rotational counterpart of the Michelson–Morley experiment, namely: the latter experiment shows the absence of fringe shift caused by the uniform translation of an interferometer, Sagnac’s shows the existence of a fringe shift caused by the uniform rotation of an interferometer. The Sagnac experiment soon became a textbook classic, and experts in relativity theory felt compelled to explain it both in special and in general relativity. Numerous variants of the experiment have been performed from the interwar period to these days. Interest in the Sagnac effect grew enormously when laser technology turned it into an efficient gyroscopic device.3

The contrast between the diversity of Sagnac’s endeavors and the modern focus on a single experiment of his raises a number of questions. Was his discovery of the Sagnac effect an isolated, felicitous hit in a fairly calm career? Is there any connection between his works on X-rays, on radioactivity, and in optics? Was he a mostly experimental physicist or was he guided by theory? The purpose of this essay is to answer these questions through a scientific biography that will take us from his student years at the École normale supérieure to his last lectures at the Sorbonne.

In Section 1, we will see how the young Sagnac developed a passion for optics and began original research on the propagation of light and on optical illusions, both theoretical and experimental. As is recounted in Section 2, in 1896 he interrupted this project to devote himself to the study of X-rays and related radiations. This change of topic did not imply a change of perspective. Optics remained Sagnac’s main source of inspiration, in three manners: he showed that some optical illusions had X-ray counterparts that jeopardized some of his colleagues results; he systematically explored the analogy between the (inelastic) scattering of X-rays and optical fluorescence; and he discussed the propagation of X-rays through matter by means of an extension of his earlier theory of the propagation of light. The fluorescence analogy was also important in bridging Sagnac’s researches on S rays with the Curies’ work on radioactivity. Marie Curie indeed regarded radioactivity as a kind of fluorescence induced by otherwise undetected radiation from the cosmos. Sagnac and Pierre Curie’s

1 The plis cachetés, introduced by the Academy in 1735, have often been used by physicists who wished to protect their priority without publication. In relatively rare cases, after some time the author of the pli judges its contents to be ripe for publication, and he or she requests its opening. In most cases the pli remains sealed. In 1976, the Academy created a commission in charge of opening the plis that had remained sealed hundred years after being deposited. Cf. Berthon [1], Carosella and Buser [2]. My report on Sagnac’s two plis is in the Sagnac folder in the archive of the Académie des sciences.

2 Cf. Quentin [3].3 Ollivier [4, vol. 3, pp. 574–582] for a first textbook account; Pauli [5, p. 565] for a review. On relativistic interpretation and on variants, cf. Martinez-

Chavanz [6]. On recent, laser-based developments, cf. MacKenzie [7].

O. Darrigol / C. R. Physique 15 (2014) 789–840 791

collaboration on the charged component of secondary rays resulted from their common interest in generalized fluorescence. Sagnac’s aforementioned speculation about a gravitational origin of radioactivity also resulted from this interest.

As soon as he could, in 1899, Sagnac returned to optics proper and to the theory of the propagation of light in which he had placed his highest hopes. Section 3 is devoted to this theory and some interesting byproducts. Through a simple kinematics of waves multiply scattered by clouds of material points, Sagnac retrieved basic laws of the propagation of light through transparent bodies, including the Fresnel drag for waves propagating in moving bodies. He thus established the ingenious principle of the effect of motion according to which the motion of a transparent body affects the propagation time through it by the same shift as if the body were empty of all matter. And he showed that this principle implied the absence of effect of a global uniform translation on optical experiments to first order in the translation velocity. Through the same picture of multiple scattering by a cloud of point-like molecules, he derived and verified laws for large-angle diffraction by a glass grating and he also argued (wrongly) that the blue of the sky was caused by the upper layers of the atmosphere only. In 1903, he studied the distribution and phase of light near a focus. This last study soon entered optics textbooks as an important extension of the Gouy phase shift. It was a byproduct of Sagnac’s early idea that the scattering of light by atoms was analogous to the behavior of light near a focus.

The intermezzo Section 4 is about Sagnac’s only venture in mathematical physics: his attempt of 1905 to mathematically develop the idea of a delayed counter-reaction of photographic plates to their impression by light. An exchange with Lorentz on this matter seems to have discouraged Sagnac from completing the theory. Section 5 opens with Sagnac’s more fertile realization, in 1908, that his principle of the effect of motion implied a phase-shift on any optical circuit for which the circulation of the ether flow with respect to the earth does not vanish. In the following years he designed contrary-beams interferometers that would detect this phase-shift. In an experiment of 1910 he thus excluded any significant drag of the ether by the earth (a reasonably extended drag would imply a shearing, therefore rotational, flow of the ether). The ether being stationary, its flow relative to the earth still is slightly rotational owing to the diurnal rotation of the earth. The effect being too small to be detectable by his interferometer, Sagnac had the idea of installing the interferometer on a turning table. This is how, in 1913, he discovered the “optical whirling effect.”

Section 6 is devoted to anticipations of Sagnac’s ether-wind considerations by Oliver Lodge, Albert Michelson, and Theodor Kaluza, and to a related experiment of 1911 by Franz Harress. Lodge [8] and Michelson [9] both had the idea of a double-beam interferometric experiment for optically detecting the rotation of the earth. Lodge even thought of replac-ing the rotating earth with a rotating table, but he judged the experiment to be too difficult to be worth trying. Sagnac’s stronger determination resulted from the theoretical motivation of his experiment, and his success from the related superi-ority of his interferometric technique. In particular, his reliance on an even number of mirrors (instead of the three mirrors in a typical square-circuit arrangement) provided the necessary stability of the interference fringes with respect to aerial perturbations and optical defects. The Harress case is more complicated. His rotating interferential setup, which was meant to measure the value of the Fresnel drag in moving glass, can retrospectively be seen as a Sagnac-effect experiment with glass instead of air as the propagation medium. Unfortunately, a blunder in the definition of the dragging coefficient led Harress to a crudely flawed interpretation of his data. Several physicists, including Albert Einstein and Max Laue, contributed to correct this interpretation. The end result is good evidence for a Sagnac effect in glass of the same magnitude as in air, with a precision similar to Sagnac’s.

Section 7 is a second intermezzo about Sagnac’s war efforts to develop acoustic means of detection and communication, again by optical analogy. Section 8 shows an aging Sagnac engaged in a quixotic quest for a conservative substitute to relativity theory along the lines of his principle of the effect of motion. The basic idea was to distinguish the behavior of wave groups from the behavior of elementary sine waves. While the wave groups traveled like in an emission theory with the velocity c with respect to their source, the elementary waves traveled very nearly as disturbances in a stationary ether. A peculiar kind of dispersion conciliated these “two mechanics.” The theory agreed with all known experiments in the optics of moving body and had two original predictions that Sagnac believed to be confirmed: silences of radio-communication, and oscillations of the spectrum of some double stars. Being presented in a sketchy, immodest, and somewhat obscure manner, it failed to attract attention beyond the province of incompetent anti-relativists.

In sum, Sagnac’s researches were either in optics or guided by optical analogy, expect for a few scattered contributions to electrodynamics.4 Two strong ideas shaped them: the propagation of light as a multiple scattering by clouds of point-like atoms, and the transformation of radiation by fluorescence. These ideas connect his seemingly disparate contributions to X-ray physics, optical illusions, photography, interferometry, ether-wind detection, and anomalous propagation. Although Sagnac developed them mostly by himself, he was no scientific marginal. This will be appreciated in Section 9, which is devoted to Sagnac’s friendships, to his career, to the recognition of his accomplishments, and to his temperament.

While preparing this work, I have benefited from a few insightful historical studies on various aspects of Sagnac’s sci-entific life. In his Paris dissertation of 1980, Regino Martinez-Chavanz [6] gave a thorough critical account of Sagnac’s ether-wind experiments, their theoretical context, their anticipations, later experiments of the same kind, and various relativistic interpretations. Most recently, in 2011, Roberto Lalli [15,16] gave a penetrating history of Sagnac’s ether-wind experiments, including Sagnac’s motivations and the French reception. Lalli discovered that the Philippe Sagnac collection of documents (fonds in French) of the Archives nationales contained a box of papers of Georges Sagnac: Fonds Sagnac, dossier

4 Sagnac [10–14].


Fig. 1. Periodic layers of silver deposit on a Lippmann plate (on the left); Sagnac’s geometric representation of the sum of the reflected vibrations in the case λ = λ′ . From Sagnac [21].

AB/XIX/3534 (Philippe, the brother of Georges, was a renowned historian). Lalli has made good use of this archive, and so have I tried to do. On Sagnac’s X-ray works and on possible anticipations of his discovery of X-ray fluorescence, there is a useful article by Michel Quentin [3]. In a lucid history of Paul Langevin’s early works, Benoît Lelong [17] compared Langev-in’s and Sagnac’s approaches to secondary rays. Lastly and most recently, Roberto de Andrade Martins [18] demonstrated the importance of the Sagnac–Curie connection in sustaining a fundamental guiding hypothesis in the Curies’ researches. Altogether, these historical studies suggest the optical theme that I am now going to develop.5

2. A passion is born

Georges Marc Marie Sagnac was born in Périgueux on 14 October 1869 into an old bourgeois family of the Périgord. After the early loss of his father, his mother raised him with the help of an important uncle who would soon become Mayor of Périgueux. He entered the École normale supérieure in 1890 with a strong predisposition for optics and some familiarity with Isaac Newton’s and Christiaan Huygens’s writings on this subject. At the École, he studied the works of the French masters Augustin Fresnel and Hippolyte Fizeau, and he began to meditate on a new theory of the propagation of light. Optics indeed was the great passion of his life, and it motivated most of his scientific activity. Plagued by poor health, he spent much time at home with his mother. His main distraction, alpinism, gave him an opportunity to verify an optical theory of his. His fondness for music and his good hearing helped him to develop acoustic analogues of optical instruments during the war.6

Optics was the obvious choice for an aspiring French physicist. Owing to Fresnel’s accomplishments and to the connection with astronomy, this field had enormous prestige and its cultivation had helped more than one to climb the French Academic ladder. At the end of his studies at the École normale in 1893, Sagnac became agrégé préparateur in charge of optical experiments in Edmond Bouty’s teaching laboratory at the Sorbonne. He then projected a dissertation on general optics. His first publication, in 1893, was an account of Gabriel Lippmann’s theory of a new process of color photography. In this process, monochromatic light induces a periodic deposit of silver particles in the sensitive layer of the photographic plate, as a consequence of the standing wave formed by reflection of the light on the silvered back of this plate; subsequently exposed to white light, this periodic reflecting deposit produces colors by interference in the manner of iridescent bodies. In order to compute this interference, Sagnac constructed a curve by vector addition of the infinitesimal vectors representing the vibrations reflected by successive infinitesimal layers for a given incoming periodic vibration. The vector between the origin and the end of this curve gave him the amplitude and phase of the resulting vibration. In this way, he could easily show that the net reflected light had much higher amplitude for the Fourier components of the incoming light whose wavelength λ′ agreed with the wavelength λ of the light that had produced the periodic deposit (see Fig. 1). A few years later, Sagnac published a theory of diffraction by parallel slits along the same lines. The implied style of reasoning is typical of Sagnac: he usually avoided algebraic calculation and tried to reason on simple theoretical ideas in geometric or kinematic guise.7

For two more years Sagnac worked on two optical topics: the propagation of light through transparent matter, and optical illusions. We will later see how he hoped to renew the first topic. On the second, in 1896 he explained an illusion produced by the rotating beam of electric lighthouses: for someone far enough from the lighthouse and looking in the opposite direction, the beam (visible by diffusion) seems to be rotating around a center situated on the horizon. This illusion, Sagnac explained, is an effect of perspective similar to the apparent rotation of parallel tree lines around their vanishing point when seen from a train. The analogy derives from the following circumstance: when the beam is close enough to the direction

5 The archive of the Wellcome Library in London holds a few Sagnac manuscripts under the reference MSS. 4332–4334. I have not been able to consult them.

6 Cf. P. Sagnac [19]; Sagnac to Lorentz, 6 January 1901, Archive for the history of quantum physics (AHQP). Sagnac’s brother Philippe, a renowned historian of the French revolution, was his best biographer. Most biographical information in this paper comes from him. Further institutional information is found in Sagnac [20] and in Archives nationales, “Dossier professionnel de Georges Sagnac,” AJ/16/6149 (see Lalli [15, p. 65, note 27]).

7 Sagnac [21,22]; Lippmann [23]. Sagnac borrowed the vector representation of vibrations from Fresnel and Alfred Cornu.


Fig. 2. (a) (left): Mutual attraction of two shadows. From Sagnac [33]. (b) (right): Deformation or doubling of the shadow of a wire when passing through the penumbra of a ring. From Sagnac [34].

of the observer, its successive positions are approximately parallel lines in relative translational motion with respect to the observer. In 1901, Sagnac gave a similar explanation of another illusion: the divergence of sunbeams through holes in a cloudy sky. In perspective, the quasi-parallel beams seem to be diverging from a point below the horizon at 180 from the sun.8

3. The optics of X-rays

In December 1895, Wilhelm Röntgen’s discovery of X-rays shook the entire world of science. As is well known, Röntgen encountered the new rays and their amazing properties in the course of experiments on the cathode rays created by electric discharge in rarefied gases. Even physicists who had never worked with discharged tubes switched to the newly open field. Following Bouty’s “insistent advice” Sagnac gave up his first dissertation project and devoted himself to the study of X-rays.9

Sagnac’s approach was decidedly optical. As a working hypothesis, he assumed that X-rays were electromagnetic waves of very high frequency, a kind of ultra-ultra-violet light. They were therefore analogous to ordinary light and, at first glance, should be expected to enjoy the same properties of reflection, refraction, diffraction, and polarization. Röntgen’s original failure to observe these properties led him to identify his new rays with longitudinal waves in the ether. Other investigators believed they could diffract X-rays, thus supporting the analogy with light. In contrast, Sagnac and a few British experts understood that electromagnetic radiation of very high frequency would not be dispersed if the frequency was much higher than the proper frequency of material oscillators and that it would not be diffracted if the wavelength was much smaller than the characteristic length of the diffracting system. From his own failure to diffract X-rays by a dense alignment of wires, Sagnac concluded that their wavelength could not exceed 0.04 µm. When he and J.J. Thomson failed to polarize them by means of crystal blades, he refused to see this negative result as an argument in favor of Röntgen’s hypothesis of longitudinal vibrations.10

3.1. Optical X-ray illusions

Most interestingly, Sagnac proceeded to show that early claims of X-ray diffraction collapsed under analogy with the following optical illusion. Let the extended luminous source S project a shadow of the opaque object A on a screen (Fig. 2a). This shadow has a penumbra delimited by the cone C of rays from S through the extremity of the body A. Let a second opaque body B penetrate this penumbra. Then “the shadow of A seems to be attracted by the shadow of B.” The reason is that the body B, while penetrating the cone C, blocks an increasing fraction F of the light rays that feed the penumbra of A, so that its shadow seems to be increasing. Through more detailed reasoning, Sagnac explained the strange appearance of the shadow of wires near or within the penumbra of a ring (Fig. 2b). Possibly, he had investigated effects of this kind before working on X-rays, as part of his interest in optical illusions. He now saw that the bending of X-ray alleged in some recent publications was the exact counterpart of the effects optically generated in Fig. 2b. Their principal cause was the large width of early X-ray sources, which were the glass walls of Crookes tubes. They disappeared when the source was properly narrowed by a diaphragm or when the source became the small anticathode of the next generation of X-ray tubes.

8 Sagnac [24,25].9 Röntgen [26–28]; Sagnac to Poincaré, 15 Sep 1900, in Walter [29, p. 328]. In conformity with French usage, I capitalize the X in X-rays (the most

common old English spelling was “x rays”).10 Sagnac [30]; [31, p. 538]. On early interpretations of X-rays, cf. Wheaton [32, pp. 16–20].


Fig. 3. The attraction of unfocused ocular images according to Sagnac. From Sagnac [35].

Sagnac drew the moral: “It is clear that the comparison of the effects of X-rays with the effects of light under the same conditions is indispensible in every research.”11

A purely optical corollary of Sagnac’s insights into the illusions of penumbra was his explanation of illusions of ill-accommodated sight, published in the following year. Let a point-like object A be placed too close to the eye for accommo-dation on the retina to be possible. Then an image A′ of A is formed beyond the retina bb′ , so that A is seen unfocused as the intersection a of the retina with the cone of rays falling on A′ from the pupil S. Introducing the opaque body B in the incoming cone of rays C blocks the shaded fraction F of the rays and thus crops the diffuse image a on one side, so that this image seems to be sharper and to have moved to the opposite side. The net effect is a seeming attraction of the image of A by the body B. Anyone can observe this phenomenon with a minimal apparatus: eye and fingers (Fig. 3).12

3.2. From dissemination to fluorescence

In 1897 Sagnac reviewed his and others’ works on the propagation of X-rays for the readers of L’Éclairage électrique. He began with a clear statement of his own method:

“I have constantly endeavored to compare the properties of X-rays with the properties that ultra-violet light of extremely small wavelength could have. This method of work will remain necessary and logical as long as no new fact will force us to formally distinguish X-rays from the transverse vibrations of the ether that constitute light waves. So far the hypothesis of X-rays as ultra-violet rays suffices for the interpretation of every known fact.”Sagnac concluded that X-rays failed to be reflected, refracted, diffracted, or polarized by current experimental means. The only property they shared with light (besides ray propagation) was their “dissemination” when traveling long distances through air. Sagnac had suspected this phenomenon from the veiling of radiographies taken at large distance from the ob-ject; Röntgen had recently confirmed it; he and Sagnac had further observed dissemination by metallic objects; and Sagnac believed that the maximum of intensity observed on photographic plates at the border between shadow and penumbra was due to a non-local action of the X-rays on the plate, implying secondary radiation. His review ended with the following observation:13

“As the extreme smallness of the wavelength of X-rays prevents us from realizing most of optical experiments with them, for further progress it is natural to study the phenomena of diffusion that truly dominates the history of these new rays.”

This is exactly what Sagnac began to do in mid-1897. In his first communications on this matter, he explained that by “dissemination” he meant either true diffusion or luminescence. By true diffusion he meant Rayleigh scattering, which re-quired a wavelength of the X-rays not too much larger than the size of the scattering molecules. This was compatible with the failure of usual diffraction experiments, and it could explain the higher penetrability of rays after traveling a long dis-tance of air as an effect similar to the dominant scattering and absorption of blue light in the optical case (the transmitted

11 Sagnac [33, p. 880]; [34, p. 409].12 Sagnac [35,36].13 Sagnac [31, pp. 531, 539]; Röntgen [26, pp. 6–7] (reflection–diffusion by metals); Röntgen [28, pp. 18–20] (diffusion by the air).


Fig. 4. Two ways of detecting secondary rays. The X-rays from the anticathode l hit the metal plate M to produce the secondary rays S. These rays, selected by the lead screen E, impress the photographic plate cc (producing a radiographic image of the interposed object OO); or they discharge an electroscope through the thin aluminum foil aa and the gold foil f . From Sagnac [42, p. 37].

red light being less easily absorbed). Sagnac nonetheless believed luminescence to be the more likely cause of the dissemi-nation, by analogy with the fluorescence excited by UV light. In particular, he argued that the electric conductivity of gases subjected to X-rays was comparable with the physical and chemical alterations observed in fluorescent or phosphorescent bodies.14

At this point, it is good to remember that luminescence had been generically defined by Eilhard Wiedemann as the absorption of a radiation of a given type (wavelength) followed by the emission of radiation of different type, with no delay in the case of fluorescence and a significant delay in the case of phosphorescence. In optics, luminescence was known to imply an increase of wavelength, a law first stated by George Gabriel Stokes in 1852. Sagnac expected the same thing to happen in the case of X-rays conceived as ultra-ultra-violet light.15

As explained by Quintin, Sagnac was not the first to suggest X-ray fluorescence. Röntgen had raised this possibility: “I have not yet been able to decide whether the rays emerging from the irradiated bodies are of the same kind as the in-coming rays; in other words, whether a diffuse reflection or a process analogous to fluorescence occurs.” Other investigators had found some evidence in favor of the latter option, but Sagnac was the first to provide unambiguous proofs and to launch a systematic study of the secondary rays. In the summer of 1897 he demonstrated that the surface of most metals except aluminum, when exposed to X-rays, emitted rays that could penetrate only a few millimeters of air and then impress a photographic plate. These rays were easily absorbed by thin sheets of metals or other materials. The activity of the metallic surface depended both on the hardness of the impacting X-rays and on the nature of the metal. It was stronger for the heavier elements. Sagnac further suggested that the transformed rays could be transformed again by impact on a second metal. He concluded: “Plausibly, we will be able to gradually fill up the unoccupied interval that separates the X-rays from the known ultra-violet rays and perhaps to identify them with such rays.”16

Sagnac soon proved that the “secondary rays” were in many ways similar to X-rays: they propagated in straight lines; they could not be reflected in a mirror-like manner; they could not be refracted; and they were able to discharge electrified metallic conductors. Sagnac insisted on this last property, for it provided a more precise electroscopic method to detect and measure the secondary rays (see Fig. 4). The secondary rays nonetheless differed from X-rays by being much more easily absorbed, to an extent depending on the generating metal.17

3.3. S-rays

Sagnac called the secondary rays “S-rays,” perhaps because he wanted to emphasize the transformed character of the secondary radiation (although he meant the letter S to stand for secondary, other physicists soon spoke of “Sagnac rays”). In early 1898, he introduced what he henceforth advertised as the best proof of transformation: the asymmetry of the effect of an absorbing plate placed before and after the diffusing body (see Fig. 5). In his teaching (see Fig. 6) and in systematic expositions of his researches, he liked to explain this asymmetry through an optical analogy: in the setups of Fig. 7, the

14 Sagnac [31, p. 534]; [37, pp. 169–171].15 Wiedemann [38]; Sagnac [39]. On Stokes’s “fluorescence”, cf. Darrigol [40, p. 250].16 Quentin [3]; Röntgen [28, pp. 19–20]; Sagnac [41, p. 232].17 Sagnac [42,43].


Fig. 5. Sagnac’s proof of transformation of X-rays by permutation of absorber and scatterer. (Ironically, Sagnac introduced this setup in the case of scattering by air, for which the transformation soon became controversial (see below)). Later variants of this setup (e.g., Sagnac [44, p. 11]) included a transforming metal plate in the region M.) The X-rays from the tube induce S-rays in the region M. These rays discharge the foil f of the electroscope. A metal sheet has a stronger attenuating effect when placed at the entrance of the electroscope (position A′A′) than when placed at the exit of the tube (position AA). From Sagnac [45, p. 522].

Fig. 6. Sagnac teaching (with two assistants on his sides). From the inscription on the blackboard, it appears that Sagnac was teaching luminescence, photographic action, X-rays, and transformed rays in the same lesson. The two drawings illustrate the permutation of scatterer and absorber as a proof of transformation (probably for light, as the letters L and L′ suggest). X-ray tubes and Ruhmkorff coils can be seen on the desk. The lamp on the right corner may be part of setup for showing the transformation of light by a solution of fluorescein. From Archives nationales.


Fig. 7. Sagnac’s optical analogy for the effect of permuting scatterer and absorber. The intensity of the light perceived by the eye is much smaller when the absorbing plate of blue glass AA is placed after the scattering solution of fluorescein M. From Sagnac [44, p. 11].

transformation of light when scattered by a solution of fluorescein is proved by comparing the effects of an absorbing plate placed before and after the fluorescent scatterer.18

3.4. S-ray analysis and a controversy

In 1898–1899, Sagnac showed that the penetrating power of the secondary rays from metal “most often” decreased with their atomic weight, thus anticipating Moseley’s law. He also established a rough connection between emission and absorption: the rays most absorbed by a given element are those that best excite secondary emission by this element. He recognized that the secondary radiation from the same element was heterogeneous. And he had the idea that the S-rays could be used for chemical analysis, although the means he had for separating rays of different absorbing power were rather primitive. A pervasive difficulty of his experiments, which he himself emphasized and tried to circumvent, was the fact that any absorber also emitted secondary or tertiary radiation that acted on the detector placed behind the absorber. The matters were further complicated by the ambient air, which considerably absorbed and diffused secondary rays.19

Although air, water, and aluminum were far less active than heavier metals and although their secondary rays did not clearly differ from X-rays, Sagnac assumed that all elements transformed X-rays, to a degree decreasing with their lightness. In early 1898, he even believed to have proved that air transformed X-rays by his permutation method (see Fig. 5) and he compared this transformation with the transformation of light by a solution of fluorescein. In the April 1898 issue of the proceedings of the Academia dei Lincei, the Modena-based X-ray physicists Carlo Bonacini and Riccardo Malagoli denied that Sagnac’s method was precise enough for the lighter diffusing bodies. In their opinion, there were two classes of bodies: lighter bodies for which ordinary diffusion occurred alone, and heavier bodies for which a mixture of ordinary diffusion and transformation occurred. Their own method of observation, based on photography and simultaneous comparison of the radiation from two different bodies, gave no evidence of a transformation for air or aluminum. In early 1899, Sagnac defended the superiority of his electric method and criticized many aspects of the experiments of his Italian competitors. The exchange turned sour. Bonacini and Malagoli sharpened their objections, and a visibly annoyed Sagnac dismissed the whole discussion as pointless: his adversaries had read him too superficially and too selectively; although his hypothesis of a gradually decreasing transformation was simpler than the Italian two-class hypothesis, he after all had “no preference for one hypothesis or the other.”20

The later history of the subject has partially confirmed Bonacini and Malagoli’s opinion: the action of X-rays on air and aluminum differs qualitatively from the action on metals for which the absorption edge belongs to the spectrum of the X-rays. The early methods of X-ray physics were just too imprecise to decide this question. Sagnac was not entirely wrong, however: the ionization of the air and the Compton effect do transform the incident rays; and Rayleigh (Thomson) scattering globally alters their spectrum.21

3.5. Perrin’s metal effect

As a consequence of his optical method and interest, Sagnac did not pay much attention to the electric effects of X-rays and S-rays. He used them only as a means of detection. In contrast, Jean Perrin, who had experimented to prove that cathode rays carried an electric charge, focused on the electric effects induced by X-rays. With the help of the normalienPaul Langevin, he studied the discharge of electrified conductors by X-rays, a phenomenon simultaneously discovered by Röntgen in Würzburg, by J.J. Thomson in Cambridge, by Louis Benoist and Dragomir Hurmuzescu in the research laboratory

18 Sagnac [43, p. 469]; [45, p. 522]; [46, pp. 69–70].19 Sagnac [47–50] (heterogeneity of the secondary rays), [51–53] (on p. 208 of [53], Sagnac speculates that the transforming power might be related to

the size of atoms).20 Sagnac [45,54]; [55, p. 110]; Malagoli and Bonacini [56,57].21 Cf. Quentin [3].


Fig. 8. Sagnac’s experiment reducing Perrin’s metal effect to secondary radiation. The X-rays fall on a condenser made of the (mostly inactive) aluminum foils AA and aa. The zinc plate ZZ, placed at the distance α of aa and electrically connected to aa emits S-rays that contribute to the discharge of the condenser. The discharge is maximal when ZZ touches aa. Removing aa at that stage increases the discharge by 10% only, as expected if the metal effect is caused by S-rays slightly absorbed by the foil aa. From Sagnac [59, p. 47].

of the Sorbonne, and by Augusto Righi in Bologna. Whereas Benoist and Hurmuzescu believed this effect to result from an action of the rays on the surface of the conductor (in analogy with the photo-electric effect), Perrin and J.J. Thomson traced it to an induced conductivity of the gas around the conductor. In the thesis he defended at the Sorbonne in June 1897, Perrin opted for a dual explanation: ionization of the gas, and “metal effect” implying ionization next to the metal at a rate depending on the nature of the metal.22

Sagnac soon denied the existence of the latter effect and argued that the metal-dependent part of the discharge de-pended on secondary radiation from the metal. Through careful experiments he showed that this radiation acted at a much larger distance from the metal than Perrin’s explanation would allow (see Fig. 8). In a letter to Langevin, Perrin lamented: “Sagnac has just demolished one third of my dissertation.”23

3.6. Langevin’s parallel discovery

Since the summer of 1897 Langevin was working at the Cavendish laboratory in Cambridge. The director, J.J. Thomson, had just approved Perrin’s metal effect and wanted to know more about it, if only because he did not want this effect to interfere with his studies of ionization in gases. Langevin set about studying the effect quantitatively by measuring how the discharge of a double-plate capacitor depended on the distance of the plates, on the nature of the active plate, and on the pressure of the gas (see Fig. 9). He found that for distances inferior to 1 cm the discharge depended on the distance between the two plates in a non-linear manner and that at every distance the discharge remained proportional to the pressure of the gas. These results contradicted Perrin’s localization of the metal effect on the surface of the metal and instead suggested the production of secondary rays that ionized the gas. A few years later Langevin described this unpublished work with the following preamble: “At the beginning of my stay in the Cavendish Laboratory (October 1897), I was led, at the same time as Mr. Sagnac, and in a totally independent way, to discover the existence of these secondary rays, and to carry on their study using purely electrometric methods.”24

As Lelong remarks, Sagnac’s approach was mainly optical, qualitative, and macrophysical, while Langevin’s was electric, quantitative, and microphysical. Sagnac found theoretical guidance in optical (dys)analogies, Langevin in the atomistic of ions. Whereas Sagnac vaguely measured the discharging time of an electroscope in vaguely defined geometrical conditions, Langevin used plane condensers and precise electrometers. Being less quantitative, Sagnac’s experiments were more flexible but less rigorous. Their optical motivation gave central importance to the secondary rays as a kind of fluorescence. For Langevin and J.J. Thomson, those rays were only a parasitic phenomenon in an investigation of the kinetics of ions. This might explain why Sagnac published his findings before Langevin even thought of doing so.25

3.7. Radioactivity, luminescence, and S-rays

As we just saw, Sagnac’s secondary rays mattered in X-ray studies of a different kind than his. As recently argued by Martins, they also mattered in early studies of radioactivity. On 20 January 1896, the mathematician Henri Poincaré communicated a few early radiographies to the French Academy. Henri Becquerel, who was in the attendance, asked Poincaré whether Röntgen had determined the origin of the rays. Poincaré answered that the rays came from the luminous spot of the glass wall that received the cathode rays. Becquerel “immediately thought of examining whether the new emission was not a manifestation of the vibratory motion that induced the phosphorescence and whether every phosphorescent body emitted such rays” and communicated this idea to Poincaré. The next day Becquerel started a series of experiments in which he had cathode rays fall on phosphorescent bodies placed next to a photographic plate wrapped in black paper within a Crookes tube. The plate remained unimpressed. On January 30th, the Revue générale des sciences published a text in which

22 Perrin [58]. On the earlier works, cf. Lelong [17, pp. 95–96].23 Sagnac [59]; Perrin to Langevin, undated, cited in Lelong [17, p. 106].24 Langevin [60, pp. 521–534, p. 517 (citation)]. Cf. Lelong [17, pp. 100–103].25 Lelong [17, pp. 105–106].


Fig. 9. Langevin’s apparatus for studying the metal effect. The discharges induced by the same X-ray tube T in the two condenser ABCD and A′B′C′D′ are balanced by adjusting the opening PQ of the lead screen FG, the balance being judged by the electrometer E. The X-rays enter the condenser ABCD through a thin aluminum window and impact the active metal CD, whose distance can be varied through the screw L. The cavity ABMN can be filled with various gases at various pressures. From Langevin [60, p. 522].

Poincaré wondered “whether any body whose fluorescence is sufficiently intense emits, besides light rays, the X-rays of Röntgen, whatever the cause of their fluorescence may be? Then the phenomena would no longer depend on an electric cause. This is not very probable, but this is possible and should be fairly easy to check.” The query prompted a few physicists, including Becquerel, to investigate whether phosphorescent bodies emitted X-rays even when they were not impacted by cathode rays. On February 24th, Becquerel announced to the Académie des sciences that he had obtained a positive result with a uranium salt earlier exposed to sunlight. He believed that this activity depended on the previous exposition to the sun and that it could be understood as an anomalous kind of phosphorescence in which radiation of optical frequency was turned into X-ray radiation of higher frequency (against Stokes’ law).26

Young Marie Curie soon contradicted this view by showing that the ionizing activity of the rays from uranium salts did not depend on previous illumination. She also established that radioactivity depended only on the presence of an active element (uranium or thorium) and not on the way it was chemically combined. A simple explanation of the mysterious phenomenon then came to her mind:

“Analogy with the secondary rays of the Röntgen rays. – The properties of the rays emitted by uranium and thorium are very similar to those of the secondary rays of the Röntgen rays, recently studied by Mr. Sagnac. . . . To elucidate the spontaneous radiation of uranium and thorium we could imagine that the entire space is permanently crossed by rays analogous to the Röntgen rays, but much more penetrating and absorbable only by certain elements with a large atomic weight, such as uranium and thorium.”Pierre and Marie Curie’s close friendship with Sagnac probably eased this analogy. Marie Curie’s paper (her first) was read to the Académie des sciences on 15 April 1898. In the same month, an article by Sagnac on luminescence and X-rays appeared in the Revue générale des sciences with the footnote:27

“It is fit to remind here the discovery due to H. Becquerel of new invisible radiations emitted for several months, without noticeable weakening, by uranium salts and especially by uranium constantly kept in darkness. Up to this day there seems to be no limit for the duration of these phenomena, for which S.-P. Thompson proposed the name hyperphosphorescence. We do not know whether there really is a transformation of radiations or simply a spontaneous radiation due to a new mechanism. Anyhow, these remarkable uranium rays are fairly close to the X-rays by their electrical properties.”

The analogy between uranium rays and Sagnac rays reinforced Marie Curie’s conviction that radioactivity was a property of chemical elements (contrary to ordinary fluorescence, which is a property of compounds). When the Curies discovered that pitchblende, against that view, was more active than its uranium content implied, they suspected the presence of small quantities of an unknown element of higher activity than uranium. This insight, together with an extraordinary amount

26 Becquerel [61, p. 3]; Poincaré [62, p. 56]; Becquerel [63–65]. There is no textual evidence for the traditional view that the conjecture of a relation between X-ray and phosphorescence emanated from Poincaré. And there is no reason to distrust Becquerel’s account (cited above). However, Poincaré was plausibly the first to entertain the possibility of X-ray emission without cathode-ray excitation (the emphasis is his in the citation from [62]).27 M. Curie [66, p. 1103]; Sagnac [39, p. 314n]. Cf. Martins [18]. The closeness of the friendship between Sagnac and the Curie is well illustrated in the

moving letter that Sagnac, wrote to Pierre Curie on 23 Apr 1903 to alarm him about his wife’s physical condition and to press the frenetic couple to adopt a healthier lifestyle: cf. Reid [67, pp. 127–129].


Fig. 10. Sagnac’s device for proving the electric charge of S-rays. The secondary rays from the X-ray beam are subjected to the electric field Fe toward the electroscope and to the field Fi of the charged foil f . The electroscope is covered with the metal sheet C and with a metallic grid or with a pierced lead screen pp on the left side. From Sagnac [73].

of chemical labor, led to their discovery of polonium and radium. In the fall of the same year, they learned that Johann Elster and Hans Geitel had failed to observe any decrease of radioactivity in an 850-meter-deep pit. The Curies nevertheless maintained the fluorescence hypothesis. When, in 1899, they discovered induced radioactivity, they again relied on analogy with Sagnac’s secondary radiation:28

“The phenomenon of induced radioactivity is a type of secondary radiation due to the Becquerel rays. However, this phenomenon is different from the one that is known for Röntgen rays. Indeed, the secondary rays of the Röntgen rays that have been studied up to now are born immediately when the bodies that emit them are hit by the Röntgen rays and they cease immediately with the suppression of the latter rays.”

3.8. The charged component of secondary rays

In the same year 1899, it became clear that part of the uranium rays could be magnetically deviated. This contradicted the analogy between radioactive rays and S-rays, following which they should both be electromagnetic radiation of very high frequency. The Curies nevertheless confirmed that the rays from radium carried electric charge by precise electrometry à la Pierre Curie. The analogy could still be restored if X-rays themselves carried electric charge; or if the S-rays had an electrically charged component. The Curies must have known that Sagnac had contemplated the latter possibility and that in unpublished experiments of 1898 he had obtained evidence for it. Pierre Curie and Sagnac soon collaborated to investigate whether X-rays and S-rays carried electric charge. Their experiments proved that X-rays did not and that S-rays did. From then on, Sagnac regarded the secondary rays as a mixture of electromagnetic radiation and charged particles similar to the β component of the Becquerel rays.29

In his experiment of July 1898, Sagnac used an electroscope and the secondary rays emitted by air traversed by an X-ray beam (see Fig. 10). The electroscope was covered with a metal sheet to make it a Faraday cage, and the secondary rays entered it through holes in the cage. Sagnac found that an external field applied to the secondary rays outside the cage accelerated or decelerated the discharge of the electroscope according as the field was directed toward or outward the cage. He concluded that the electroscope was bombarded by a flux of charged particles, and compared this flux to a slow version of cathode rays. The complicated electric circumstances of this experiment may explain why Sagnac deposited his results as a pli cacheté to the Académie des sciences and waited more definite results before publishing the contents in early 1900.30

The later experiment of Curie and Sagnac benefited from Pierre Curie’s refined electrometric skills. In the first of the two devices they used (Fig. 11), a foil cc of the metal M is placed in the middle of the rectangular cavity ABCD, which is made of a different metal N. The X-rays from the anticathode l enter the cavity through the foils f and induce secondary rays both on the internal foil cc and on the walls of the cavity, at a rate depending on the choice of the metals M and N. The air is evacuated from the cavity in order to avoid absorption and ionization. A quartz-controlled electrometer, connected to the foil cc, detects any electric charge associated with the secondary rays. Curie and Sagnac found the highest charge for platinum and lead, and the least for aluminum. They thus established the existence of an electrically charged component of the secondary radiation of heavy metals. They understood that this charged component shared the high ionizing power of cathode rays:

28 P. and M. Curie [68, pp. 715–716]. Cf. Martins [18].29 P. and M. Curie [69]; Sagnac, plis cachetés of 28 March and 18 July 1898, Archives de l’Académie des sciences; Curie and Sagnac [70–72]. Cf. Martins [18].30 Sagnac, plis cacheté of 18 July 1898; Sagnac [73].


Fig. 11. One of Curie and Sagnac’s setups for proving the existence of an electrically charged component of secondary rays. From Curie and Sagnac [70, p. 1014].

“The whole set of observed effects suggests that the secondary electric emission of heavy metals has properties analogous to those of cathode rays and of the deflectable rays of radium: the particles of negative electricity of the rays are able to dissociate the neutral electricity of gases in positive and negative quantities of electricity, considerably superior to the negative quantity of electricity of the rays, as long as the gas under study is not overly rarefied.”They did not fail to note that their new effect, the production of electrically charged radiation by the impact of X-rays on metals, much resembled the electric flux Philipp Lenard had recently obtained by exposing non-electrified metal plates to extreme UV light. They concluded:

“The negative electrification of secondary rays brings a new analogy between X-rays and ultra-violet rays. It thus becomes more and more probable that the secondary rays contain rays of the same species as the incoming X-rays that produce them by diffusion or transformation.”Through his collaboration with Pierre Curie, Sagnac thus gained a new argument in favor of the analogy between X-rays and ultraviolet light. In return, the Curies could protect Marie Curie’s hypothesis of a deep analogy between radioactivity and secondary emission.31

3.9. Radioactivity and gravitation

Although Ernest Rutherford and Frederick Soddy’s contemporary discovery of the link between radioactivity and trans-mutation reduced the appeal of Marie Curie’s hypothesis, Sagnac and the Curies defended it for a few more years. At the first international congress on radiology and ionization held in 1905 in Liège, Sagnac explained how his and Pierre Curie’s discovery protected the analogy between radioactive emission and secondary rays, and he offered the following wisdom:

“Some tried to ruin [Marie Curie’s] hypothesis by arguing that it is hardly philosophical to imagine an unknown, un-detected radiation to explain effects that have their evident seat in the radioactive atoms. However, pushing this objection to the extreme, we could say that luminous or electromagnetic vibrations have no existence outside their source and their material receiver since they do not produce any observable effect in vacuum and since you know them only through their action on matter. Then we would suppress the concept of luminous ether as anti-philosophical. We may describe luminous phenomena without mentioning the ether. But that is not a satisfactory description. . . . It does not give the satisfaction we feel by eliminating direct action at a distance. . . . As long as we refuse to renounce this satisfaction, the hypothesis according to which radioactive bodies borrow their energy from an ambient medium that still exists in a vacuum will be as worth conserving as the hypothesis of the luminous ether.”Sagnac went on to explain that the same ambient medium could provide the solar, gravitational, and radioactive energies.32

The following year in the Journal de physique, Sagnac published a more detailed speculation of this kind together with an experimental test he had performed in Pierre Curie’s laboratory in 1902. The starting idea was that gravitation, in analogy to Georges Lesage’s old theory and according to a more recent speculation by Lorentz, could be explained as a shadow effect for the continual bombardment of celestial bodies by a pervasive, isotropic, and highly penetrating radiation. Whereas this radiation could only rebound on ordinary atoms, Sagnac made it able to dissociate radioactive atoms. This dissociation would modify the momentum transfer from which gravity derives in Lesage’s conception. Consequently, the gravitational mass of radioactive bodies should be smaller than their inertial mass. Sagnac’s tested this effect by measuring the period of

31 Curie and Sagnac [72, pp. 20, 21].32 Sagnac [74, p. 155].


Fig. 12. Sagnac’s diagram to explain the softening of X-rays by particle diffraction. For a finite wave train of incoming X radiation, the vibration at M originating from the point p′ of the diffracting particle is still active when the vibration from p has ceased. From Sagnac [81, p. 184].

oscillation of a torsion balance loaded successively by equal weights of a radium salt and of a barium salt. The result was negative at the precision of the measurement (1%).33

3.10. The thesis, a turning point

On 21 December 1900, Sagnac defended his doctoral thesis at the Sorbonne in front of Gabriel Lippmann (president), Bouty (director and examiner), and Émile Duclaux (examiner). The title of the dissertation, De l’optique des rayons de Rönt-gen et des rayons secondaires qui en dérivent, captures Sagnac’s optical approach. The exposition is roughly chronological. The first part deals with the (dys)analogy between X-rays and light, the second with the secondary rays and their properties beginning with the absorption properties and finishing with the experiments of Curie and Sagnac. Besides the optical lead-ing thread, a general characteristic of the dissertation is its avoidance of microphysical considerations. Ions, electrons, or corpuscles (J.J. Thomson’s word for the electron) nowhere occur in the text. Although Sagnac does occasionally approve J.J. Thomson’s concept of X-rays as an electromagnetic disturbance caused by the stopping of charged corpuscles and al-though he holds a molecular conception of matter, he ignores the British and German tendency to consider cathode rays, X-rays, and their ionizing properties as an entry into a new microphysics of ions and electrons. In this regard he follows the French preference for a more phenomenological approach to the new radiations, of which the Cambridge-trained Langevin may have been the only significant exception. Even Jean Perrin, who would later offer proofs of molecular reality, avoided atomistic considerations in his early researches on X-ray ionization.34

This does not mean that Sagnac did not have any molecular theory in the back of his mind. As was already mentioned, he believed that his own theory of the propagation of light through matter implied vanishing dispersion and refraction when the wavelength became small compared to the spacing of the molecules of matter.35 In this range of wavelengths, he argued that the absorption of vibrations by matter should increase as the wavelength increased. For this reason and by analogy with Stokes’ law, he associated the higher absorption of transformed radiation with a longer wavelength. He explained the transformation of X-rays as an effect of the finite size of the particles of matter: if the incoming X-rays are damped wave trains of the form e−αt cos(ωt + ϕ), he reasoned, then the wave trains resulting from diffraction by a finite particle are longer and more regular because of the different traveling times from the various points of the diffracting particles (see Fig. 12). Their Fourier spectrum is therefore shifted toward lower frequencies.36

After completing his dissertation, Sagnac taught physics as maître de conférences at the University of Lille until 1904. He ceased to work on X-rays and secondary radiation, except for systematic reports he had to prepare for a treatise on

33 Sagnac [75]. Sagnac’s pli cacheté of 23 Feb 1902 propounds a simpler pendulum experiment, with the additional conjecture that the ratio of the (inertial) atomic mass to the atomic number was the same for barium and radium, which gives a 20% relative difference in the pendulum frequencies. On Lesage’s theory, cf. Chabot [76–78].34 Sagnac [79]; [80, pp. 431–432] (J.J. Thomson’s X-ray concept). On the French approach, cf. Lelong [17]. On contemporary debates on the nature of

X-rays, cf. Wheaton [32, Chap. 2]. The dominant hypothesis was the electromagnetic impulse hypothesis, which did not much differ from Sagnac’s wave packet hypothesis for practical purposes.35 Sagnac was also aware of Helmholtz’s theory of dispersion, based on the coupling between ether waves and material oscillators, and leading to the

same result in the limit of vanishing wavelength (Sagnac [80, pp. 431–432]).36 Sagnac [39, p. 316]; [49, p. 555]; [81, pp. 183–185].


radiology and for the international congress of 1905. Major events in this field, such as Charles Glover Barkla’s discovery of X-ray polarization in 1904 and Max Laue’s discovery of X-ray diffraction in 1912 failed to win him back. He left to Barkla, William Henry Bragg, Henry Moseley, Maurice de Broglie, and a few others the pleasure to develop X-ray spectroscopy, his idea of chemical analysis based on X-rays, and the X-ray photoelectric effect he had discovered with Pierre Curie. Judging from a letter he wrote to Lorentz in 1901, he seems to have regarded his work on X-rays as an unwanted distraction from his interest in pure optics: “Since 1896 I have unfortunately been carried away in researches [entraîné dans des recherches] on X-rays which, added to my obligatory occupations, hardly leave me any time. It is in fact for this reason that I left aside my researches in optics since 1897.”37

4. The propagation of light

Sagnac was impatient to resume his optical researches. A year before he defended his dissertation, he found time to publish some of his ideas on the propagation of light in matter. In order to understand the timeliness of these ideas, some of the earlier history of wave optics must be recalled.

4.1. The separation of ether and matter

In the oldest mechanical theories of the optical medium, those of Fresnel, Charles Augustin Cauchy, and Franz Neumann, there was only one effective medium with characteristic properties (mainly density and elasticity) depending on the amount and kind of matter mixed with the ether. Although this approach correctly represented the propagation, reflection, and refraction of light, it failed to account for the observed laws of optical dispersion. This is the main reason why the next generation of optical theories involved two separate media, ether and matter, affecting each other in various manners. In a sense the separation was not yet complete, because the matter molecules modified the structure of the interstitial ether, typically its density or its elasticity. In the 1860s Adhémar Barré de Saint-Venant and his disciple Joseph Boussinesq argued that such modification was implausible and Boussinesq devised a theory in which the ether between the molecules remained completely unchanged. The molecules only modified the ether at their location, through a coupling inspired by hydrodynamic analogy. In this framework, Boussinesq could account for all known optical phenomena, even those of the optics of moving bodies (to which I will return in a moment).38

In 1870s, the discovery of anomalous dispersion prompted a few physicists, including Wolfgang Sellmeier and Hermann Helmholtz, to represent the interaction between light and transparent matter as the mutual coupling between ether waves and material oscillators. Anomalous dispersion occurred when the frequency of the waves exceeded the characteristic fre-quency of the material oscillators. In conformity with Saint-Venant’s views, this mechanism made any modification of the interstitial ether irrelevant. In the 1878, the young Hendrik Antoon Lorentz, who was one of the few continental believers in James Clerk Maxwell’s electromagnetic theory of light, gave an electromagnetic version of the Sellmeier–Helmholtz the-ory. More broadly, Lorentz conceived an electromagnetic counterpart of Boussinesq’s theory in which the electromagnetic ether remained totally unaltered in the presence of material molecules and in which the interaction between ether and matter depended on the electromagnetic coupling between electromagnetic fields and molecular ions. In the 1890s, Joseph Larmor and Emil Wiechert developed similar views, and electrons came to replace Lorentz’s ions. By the end of the century the leading experts on optics and electromagnetism agreed that Lorentz’s theory was the only one that could account for the whole range of known optical and electromagnetic phenomena. In particular, Lorentz was able to explain seemingly incompatible results of the optics of moving bodies.39

4.2. Early inspirations

When he developed his own theory of the propagation of light through matter, Sagnac was unaware of Lorentz’s theory. According to letters he wrote to Lorentz and Poincaré in 1900–1901, he reached the main ideas of his theory while studying at the École normale in 1890–1891 and after being exposed to atomism in chemistry, mineralogy, and crystal optics. Later, in 1892–1893, he integrated ideas from Louis Georges Gouy’s memoir on the anomalous propagation of light near a focus and from Heinrich Hertz’s theory of electric oscillators, which he heard Poincaré teach at the Sorbonne. From the beginning Sagnac had been concerned with the optics of moving bodies and with Fizeau’s result regarding the drag of light waves by running water. After being long lost in complicated calculations, he arrived at a very simple explanation of this phenomenon around 1895 and at a new theory of reflection and refraction in 1896. But his X-ray work forced him to delay publication.40

Sagnac’s early concern with the discontinuity of matter contradicts the well-spread view that French physicists of the time favored a macroscopic approach to physics and liked to question atomism. It is true that most of them were ill-informed of the recent kinetic-molecular theories of Maxwell and Boltzmann and that they did not let atomistic consid-erations guide their experimental work. Nevertheless, most of them believed in the molecular structure of matter and some

37 Sagnac to Lorentz, 6 Jan 1901, AHQP. On the development of X-ray physics after Sagnac, cf. Wheaton [32], Heilbron [82].38 Boussinesq [83]. Cf. Darrigol [84, pp. 244–258]. On the first ether theories of the nineteenth century, cf. Buchwald [85].39 Lorentz [86–88]. Cf. Buchwald [89]; Darrigol [90, Chap. 8].40 Sagnac to Lorentz, 6 Jan 1901, AHQP; Sagnac to Poincaré, 15 Sep 1900, in Walter [29].


Fig. 13. Fizeau’s experiment on the dragging of light waves by running water. The light from the source O is turned into two parallel beams though the combination of the lens L, the double diaphragm and the double thick glass plate. Those beams travel a first time through two water columns AB and A′B′ that move in opposite directions. After crossing the lens L′ and being reflected on the mirror M, they exchange path and direction of motion. Their superposition is observed at S after reflection on an inclined glass plate.

of them, especially Saint-Venant and Boussinesq, pursued molecular theories in the wake of the Laplacian and Amperean tradition of the early nineteenth century.41

Although in his letter to Lorentz Sagnac claimed to have known only Fresnel’s and Franz Neumann’s ether theories at the beginning of his researches, he plausibly heard about Boussinesq’s theory (Boussinesq was then a Sorbonne professor) and from it he may have drawn the general idea of molecules interspersed in an immovable ether. If he did, he must have disliked the dryly formal way in which Boussinesq represented the coupling between ether and molecules. What Sagnac wanted was a simple, visual picture of the way in which an incoming wave interacted with a succession of molecules regarded as centers of diffusion:

“I consider the luminous vibrations inside a body as propagating by means of a medium identical to the ether of vacuum. The density and the elasticity of this vibrating medium do not differ from their values in a vacuum. I do not either con-sider mechanical reactions between ether and matter. I directly exploit the discontinuity of matter according to a principally kinematic mechanism.”Through this statement, Sagnac introduced the first sketch of his theory in the November 1899 issue of the Comptes rendus. The first two sentences seem reminiscent of Boussinesq’s theory or later French variants. The exclusion of “mechanical reaction” signals how Sagnac departs from that theory. The emphasis on kinematics became a leitmotiv of Sagnac’s approach. He purported to show that effects that had so far been explained by complex, highly formalized dynamics could in fact be explained by simple kinematic considerations.42

Sagnac went on to assume that the “particles or atoms” of matter reemitted a well-defined fraction of the energy of the incoming vibration in every direction, as a tiny Hertz resonator would do when interacting with an incoming electromag-netic wave. This “reflection–diffraction,” thus described in broad terms, sufficed to explain how the presence of matter af-fected the velocity of light. Suppose the transparent body to be delimited by a plane P and to be met by a parallel monochro-matic plane wave. This wave is multiply reflected by the particles of the body. The net transmitted wave on a plane S within the body is the resultant of waves that have been reflected an even number of times. The phase of the component waves depends on the length of their zigzagging path, and their amplitude decreases exponentially with the number of reflections. The phase of the net wave is obtained by superposing these component waves. The spacing of the particles being much smaller than the wavelength, this phase is easily seen to increase linearly with the distance from the limit P of the transpar-ent body except for very small values of this distance. The rate of increase yields the wave number. The optical index is the excess of this rate over its vacuum value. As is typical with Sagnac, there are no equations and all reasoning is intuitive.43

4.3. The Fresnel drag

In his second communication on the propagation of light, Sagnac dealt with the dragging of light waves in a moving transparent media. Early in the century, Fresnel had shown that in order that the laws of refraction be unaltered by the motion of the earth through the ether, light waves had to be dragged by a moving transparent body at a rate given by (1 − 1/n2)u, if n denotes the optical index of the body and u its velocity with respect to the ether. In an interferometric experiment of 1851, Fizeau found that the phase of waves traveling through running water was shifted to the amount predicted by Fresnel (see Fig. 13). An ingenious feature of Fizeau’s setup should be emphasized: it involves the interference between two beams of light that have traveled on the same circuit in opposite directions. This is indeed the only way to detect a phase shift occurring over a travelling length much longer than the coherence length. As we will see in a moment, Michelson, Lodge, and Sagnac later relied on the same subterfuge for the same reason.44

In the 1870s, thanks to the efforts of Éleuthère Mascart in Paris and of Wilhelm Veltmann in Holzminden, it became clear that the Fresnel drag played a central role in explaining the absence of (first-order) effects of the ether wind in a great variety of terrestrial optical experiments. At the same time, Fresnel’s old explanation of the drag by conservation of the

41 On late nineteenth-century French molecular theories, cf. Principe [91].42 Sagnac [92, p. 756]. On this theory, cf. Martinez-Chavanz [6, pp. 37–43].43 Sagnac [92].44 Fresnel [93]; Fizeau [94]. Cf. Whittaker [95, pp. 109–113]; Darrigol [40, pp. 258–261]. As Mascart and Lorentz later realized, in Fizeau’s setup the

Doppler frequency-shift in the running water slightly modifies the Fresnel drag.


Fig. 14. Sagnac’s drawing of a light path in a moving transparent body. From Sagnac [96].

ether flux45 fell apart because a different dragging coefficient was needed for different frequencies in dispersive media or for different directions of propagation in anisotropic media. In the aforementioned memoir of 1868, Boussinesq succeeded in deriving the Fresnel drag formula with a completely stationary ether, as an effect of the convection of the molecules of the moving transparent bodies. In 1892 Lorentz did the same in the context of his own electromagnetic theory. Assuming that Sagnac heard about these theories, they involved more algebra than he could digest. His own explanation of the Fresnel drag required no more than the multiple reflection defined in his first communication.46

4.4. The principle of the effect of motion

First consider a solid narrow transparent body moving at the velocity u through the ether. After entering the body in O, the light zigzags at the vacuum velocity c in the intermolecular space and exits the body in S0 as indicated on Fig. 14. The motion of the body from right to left increases the time of the travel from the reflection point r2 j to the reflection point r2 j+1 in the proportion c/(c − u) because the latter point recedes during this travel; and it decreases the traveling time from r2 j+1 to r2 j+2 in the proportion c/(c + u). Call t2 j−1 the point of the segment [r2 j, r2 j+1] that is at the same distance from the entrance of the body as the point r2 j−1. To first order in u/c, the motion of the body does not affect the traveling time in the portion [r2 j−1, r2 j, t2 j−1] of the path; and it affects the remainder of the path in the proportion c/(c − u). Since this remainder has the same length as the direct path from O to S0, the net first-order effect of the motion of the body is to increase the traveling time by ul/c2, wherein l is the length of the body. Sagnac thus reached the following remarkable result: the correction to the traveling time of light in a moving transparent body is the same as if the body had been emptied of all its matter. This is what Sagnac called the principle of the effect of motion.47

In the case of water moving through a pipe as it does in Fizeau’s experiment (Fig. 13), the water flows off the pipe laterally at B while the light proceeds from A to B. Consequently, the light travels a length of water that differs from the length l of the tube by the amount −uτ if τ denotes the traveling time of the light from one end of the tube to the other. For vanishing u, the velocity of light in water is c/n and τ = nl/c. The resulting first-order correction to the traveling time in the moving water is (−uτ )/(c/n) = −n2ul/c2. Adding this outflow correction to the convective correction ul/c2, Sagnac got δt = (1 − n2)ul/c2 for the total Fizeau effect. This prediction agrees with Fresnel’s partial drag hypothesis because according to this hypothesis the velocity of light in the moving water is V = c/n + (1 − 1/n2)u (in the laboratory frame) and the corresponding decrease in traveling time is l/V − l/(c/n) ∼ (1 − n2)ul/c2.

4.5. Kinematics first

When he presented his theory to the Société française de physique, Sagnac placed much weight on his principle of the effect of motion. On the one hand, this principle results from Sagnac’s concept of multiple reflection in an immovable ether. On the other, it can be inferred from Fizeau’s experimental result by subtracting from the measured phase-shift the phase-shift corresponding to the outflow of the running water. Sagnac further showed that his principle immediately explained the negative result of all of Mascart’s first-order ether-wind experiments. For instance, the independence of birefringence from the motion of the earth results from Sagnac’s principle separately applied to the proper modes of propagation in a crystal. This principle even more evidently implies the absence of fringe shift when altering the orientation of an interfer-ometer in which the interfering rays travel through different substances. In retrospect, Sagnac’s principle does exactly what a first-order Lorentz transformation does in explaining the lack of first-order effects of the motion of the earth.48

As was mentioned, Sagnac also believed that his theory, once extended to the limit of wavelengths much smaller than the interval between two molecules, explained or at least was compatible with the behavior of X-rays in matter. In this case he expected the reflection by individual atoms to be so small that much of the ethereal vibrations travelled through matter

45 If the slower velocity of the waves in a medium of higher optical index is traced to a higher density of the ether in this medium, then the conservation of the ether flux at the interface between two different media implies the Fresnel drag.46 Boussinesq [83]; Lorentz [87,88]; Sagnac [96]. Cf. Janssen annd Stachel [97]; Darrigol [40, p. 257].47 Sagnac [96].48 Sagnac [98, pp. 170–171]. As we will see in a moment, Sagnac later became aware of the connection with Lorentz’s local time.


without any reflection. Consequently, there was no phase delay and no refraction. The residual scattered light was isotropic because the wave packets reflected by different atoms were too far apart to interfere.49

Sagnac did not believe that his theory, in its present form, could explain every optical phenomenon. However, he re-garded it as methodologically important to introduce the hypotheses of the theory one by one, in the order of increasing complexity of the investigated phenomena. For the moment he knew that rectilinear propagation, refraction, the Fresnel drag, and much of the optics of moving bodies could be understood in a mostly kinematic manner, through the simple idea of scattering centers in a homogeneous ether. Polarization properties, anisotropic propagation, and anomalous dispersion surely required a more precise description of the scattering centers. But this could be done in a cumulative manner, without losing the transparency of the simpler phenomena:

“In a general manner, I believe it is useful to eliminate the dynamical or electromagnetic hypotheses as much as possible and to reduce, in each case, the hypotheses to the minimum necessary to solve the problem under study. Whenever it will be indispensable to complete or to modify the already made assumptions, by treating the problems as independently as possible from each other we will better see from what phenomenon the insufficiency of the theory depends, which are the hypotheses to be conserved and which are the hypotheses to be modified or to be added to the former ones.”To a broader audience he explained:50

“This way of simplifying the theory equally satisfies the philosopher who wishes to account for the nature of things in the most direct possible manner and to entangle the essential causes in the clearest possible manner, and the physicist who, in order to work efficiently needs the research instrument provided by a theory to be able to give direct solutions as simple as the nature of each problem allows.”

4.6. The Lorentz jubilee

Around that time Sagnac was invited to contribute to the Lorentz jubilee volume which appeared in 1900. Most likely, his reputation as an X-ray expert justified this invitation. Yet Sagnac did not write on the Sagnac rays. Instead he contributed a piece on optical theory, which was both dearer to him and closer to Lorentz’s interests. As Sagnac explained in a footnote to this memoir and as he told Lorentz in private, he studied Lorentz’s electromagnetic only in the spring of 1900, after Lorentz sent him his memoir of 1899 on a simplified optics of moving bodies (in reaction to Sagnac’s having sent his own memoir on this topic). Sagnac’s footnote went on as follows:

“My elementary optical theory and the theory both electrical and optical of Prof. H.A. Lorentz have extremely different forms. I was all the more struck by the following fact, which the reader will easily verify: the very simple law of the effect of motion which is the essential part of my elementary theory is in the end equivalent to the change of variable introduced by Lorentz.”Indeed the propagation-time correction derived by Sagnac exactly corresponds to the difference between absolute time (t) and “local time” (t −ux/c2) in Lorentz’s first-order theory, and it serves the same purpose of justifying the absence of effects of the earth’s motion through the ether.51

As Sagnac conceded, Lorentz’s theory has much wider scope than Sagnac’s and it is based on a precise, fundamental description of the interaction between electromagnetic radiation and electronic motion. Through averaging procedures it yields relations between macroscopic fields and thus determines their propagation and their behavior at the interface be-tween two different media. However, it does not provide any simple picture of how the induced vibrations of electrons contribute to the total field. In many experimentally accessible cases, no one knew or knows how to solve Lorentz’s equa-tions. A good example of this difficultly is large-angle diffraction, for which the Kirchhoff approximation no longer holds. In contrast, Sagnac’s theory leads to fairly simple calculations in that case. In 1895 Sagnac found that his idea of scattering centers, completed by the hypothesis that the radiation from every scattering center was identical with the radiation from a Hertzian dipole even at distances very small compared to the wavelength, had precise quantitative implications for the intensity and polarization of the light diffracted by a thin glass grating at large angle. He verified the resulting laws in 1896–1897 in Bouty’s laboratory, long kept them unpublished for the aforementioned reason, and finally presented them to Lorentz for his jubilee.52

The naïveté of some of Sagnac’s assumptions and the lack of a precise mathematical formulation do not inspire much trust in his results. The very few later references to his theory are vague and only retain the idea of summing over individual scattering centers.53 Lorentz’s reaction was nonetheless positive. It may be inferred from Sagnac’s reply to his lost thank-you letter: “I am very pleased to see that my article interests you and that you intend to treat [the same question] by your own methods. . . . What you kindly say about my works is very flattering and I thank you very sincerely for that.” In the same letter Sagnac admits that his approximations need better control and he alludes to unpublished attempts to treat the case of

49 Sagnac [98, p. 173].50 Sagnac [98, p. 174]; [99, p. 20]. Cf. Lalli [15, pp. 59–60].51 Sagnac [100, pp. 377n–378n]; [101]. Retrospectively, Lorentz’s local time is the first-order approximation of the Lorentz-transformed time. Lorentz used

the local time in a formal manner, as a way to retrieve the electromagnetic equations for a body at rest in the ether. The interpretation of the local time as the time given by optical synchronization was first given by Poincaré in his own contribution to the Lorentz jubilee. See Darrigol [90, Chaps. 8–9].52 Sagnac [100, p. 393n for the dates].53 Natanson [102], for example.


Fig. 15. Sagnac alpinist. Ascending La Tournette (Haute-Savoie) in August 1901 (with the boater), and resting (with the white beret) near Mont Blanc. Archives nationales.

crystals. He also gives the information I have earlier used about the origins of his theory. He concludes his letter in modest terms:54

“When I received your memoir [101], I clearly saw that my publication came too late. I was nonetheless comforted to see that I had unveiled just the principle that you too regard as fundamental; also, the methods were very different and I thought, as you kindly write to me, that one method could be better adapted than the other to the resolution of a given problem. However, my work is very modest in comparison to your oeuvre, and the benevolence with which you are willing to judge it deeply touches me.”

4.7. Out of the blue

In his theory of the propagation of light and in his discussions of X-ray scattering, Sagnac liked to refer to the blue color of the sky as evidence for the existence of discrete scattering centers in all matter. He knew about Rayleigh scattering, according to which the amount of scattered light varies as the fourth power of the frequency of the incoming light, and he was aware of David Brewster’s suggestion that the particles responsible for the blue color of the sky were the molecules of the air (and not dust particles as Rayleigh originally suggested). In 1902, Sagnac published a long discussion of this matter in the Annuaire du Club alpin français, including semi-popular accounts of the contributions of Rayleigh, Brewster, and John Tyndall.55

Sagnac’s own contribution was the assertion that at low elevation the air was too dense for significant scattering to occur. He reasoned that in a dense medium there were many scattering particles within a wavelength, so that the secondary waves from the particles of a layer parallel to the wave planes interfered destructively in every direction except the forward one (in analogy with Fresnel’s explanation of rectilinear propagation by the Huygens–Fresnel principle). If this reasoning applies to atmospheric air, the amount of Rayleigh scattering should be negligible at moderate elevations and the intensity of the blue light from the sky should be about the same in the Alps as at sea level. Enter Sagnac the alpinist (Fig. 15). Through photographic measurements he verified that this intensity did not significantly vary between 400 and 3000 meters, and he

54 Sagnac to Lorentz, 6 Jan 1901, AHQP.55 Sagnac [103]. On the history, cf. Rayleigh [104], Lilienfeld [105].


called fellow alpinists to perform similar measurements. Without waiting for the result, he confirmed the poet’s intuition: “To some extent the appearance of an azure vault corresponds to a reality.”56

Later accounts of the blue of the sky ignored Sagnac’s contribution, for good reason. It is indeed not true that the scattering of light vanishes in a gas of density higher than the inverse cube of the wavelength. As Rayleigh suspected, the intensity of the light scattered by a volume element of the gas remains the sum of the intensities of the lights scattered by the individual molecules because the irregular spatial distribution of the molecules prevents the kind of interference needed in the application of the Huygens–Fresnel principle. This interference would occur if the molecules were regularly distributed on a lattice (as in a crystal) or if they were so closely packed that the distances between nearest neighbors would vary little (as in a liquid or in a glass). The impurity of the air at lower elevation might explain why Sagnac failed to see any decrease of the intensity of the blue diffuse light at low elevation.57

4.8. The Gouy phase

Sagnac was more successful in his analysis of anomalous propagation near a focus. In the aforecited letter to Lorentz, he mentioned that in an early version of his propagation theory he assimilated the particles of matter with centers of anomalous propagation in the sense defined by the Lyon-based physicist Georges Gouy in an important memoir of 1891. Near a center of emission, Gouy argued, acoustic or optical vibrations do not have a well-defined velocity and their phase varies rapidly. The center of emission may be a small material oscillator or it may simply be the focus of converging waves. In the latter case, Gouy demonstrated, the phase of the incoming waves differs from the phase of the outgoing waves by π(far from the focus). This is what is now called the Gouy phase shift. Gouy gave two different proofs. In a simplified version of the first proof, the vibration is given as an isotropic solution to the scalar wave equation (ϕ − 1

c2∂ϕ∂t2 = 0. It therefore has

the general form ϕ = 1r f (r − ct) + 1

r g(r + ct), which is the superposition of a converging and of a diverging spherical wave. There being nothing but air or vacuum at the center, the wave (more exactly the associated energy) must remain finite at the origin. This implies g = − f . Consequently, the sign of a positive perturbation of finite length changes when passing the center; and the phase of a wave train is therefore shifted by π . This argument is not very realistic, as it implies complete spherical waves. In the laboratory, the converging waves are laterally limited by the extent of the lens that produces them. This is why Gouy favors another reasoning in which the Huygens–Fresnel principle is applied to a truncated spherical wave. Summing the partial waves from the spherical calotte, he finds that the resulting vibration undergoes the expected fringe shift at a large distance beyond the focus of the calotte. In principle this method should have enabled him to compute the exact phase variation around the focus. He did write the relevant integral, but he computed only its asymptotic value.58

Others, including Pieter Zeeman, believed they could determine the phase variation near a focus by analogy with the phase variation near an oscillator. This gave a gradual variation confined to a distance of the order of the wavelength. Gouy and Zeeman tried to measure the phase shift by observing the interference between the converging–diverging wave and a (quasi-)plane wave. Their experiments sufficed to observe the global fringe shift, but were too imprecise to determine the variation near the focus. In 1903, Sagnac applied the Huygens–Fresnel principle to Gouy’s problem, in a manner slightly different from Gouy’s: he summed the partial vibrations from the flat circular opening of a plane screen exposed to a converging wave. Calling s the distance of a point P of this opening from its center, r its distance from a point M on the axis, f the focal distance, and λ the wavelength, the phase of the partial vibration from P at M is approximately given by s2

2λ ( 1r − 1

f ). In his usual manner, Sagnac summed the partial vibrations graphically and arrived at the diagram of Fig. 16 for the variation of the total vibration on the axis.59

In order to verify this theoretical prediction, Sagnac improved on Zeeman’s astute exploitation of a birefringent lens. The lens, being made of Iceland spar,60 has two different foci corresponding to the ordinary and extraordinary rays, and it therefore produces two different images of a given point source. In the spirit of chromatic polarization, the ordinary light and the extraordinary light are brought to interfere by placing the lens between two orthogonal polarizers. In the vicinity of one of the images, one of the vibrations is approximately plane, so that its interference with the other yields the desired information on the phase. By using a lens less convergent than Zeeman’s, a smaller diaphragm, and a magnifying lens to observe the fringes, Sagnac succeeded in seeing the fringes very near the image, and he thus confirmed his theory of the propagation anomaly.61

Sagnac contributed this theory to the Boltzmann Festschrift of 1904. In 1903 he proposed a less fortunate application of the same theory to the N rays of Prosper-René Blondlot. When passed through a quartz lens, these rays allegedly produced multiple foci, which Sagnac identified with the intensity oscillation given by his lens-diffraction theory. The mutual distance of the successive foci required a wavelength of a fraction of a millimeter. Blondlot rejected this estimate even before the

56 Sagnac [103, p. 501]; [106].57 Rayleigh [104, p. 383]; Lorentz [107]. For a modern approach, cf. Jackson [108, pp. 422–423]. I thank Jean-Michel Raimond for an illuminating discussion

on this subject.58 Gouy [109].59 Sagnac [110–112].60 The optical axis of the crystal must of course coincide with the optical axis of the lens.61 Sagnac [114–116].


Fig. 16. Sagnac’s diagram for the phase variation of monochromatic light near a focus F . The curve gives the intensity on the optical axis around the focus F . The little arrows indicate the phase. From Sagnac [113, p. 535].

N rays had gone to the dustbins of history. He, who then believed to have firmly determined the wavelength of the new rays with a grating, should have applied to himself the comment he made about his colleague: “Mr. Sagnac will have an opportunity to meditate on the dangers of precipitation.”62

5. Intermezzo: an inconclusive venture

In 1904, besides the honor of contributing to the Boltzmann Festschrift, Sagnac received the Jérôme Ponti prize of the Academy of Science, and, most important, he returned to Paris to become chargé de cours at the Sorbonne. This should have given him more time for research. Yet he did not publish anything new between 1905 and 1910. One reason may have been that he was a devoted teacher, as may be inferred from a letter of Pierre Curie to Gouy: “Sagnac succeeded me at the P. C. N. [chargé de cours du certificat d’études physiques, chimiques et naturelles] and I must say he makes much stronger efforts that I did; he does many simple, clearly demonstrative experiments, and I believe his teaching his excellent.” At least four additional circumstances contributed to the pause in Sagnac’s research output: recurrent health problems, years of efforts to set up a large facility for optical experiments in the basement of his laboratory of the Rue Cuvier,63 the secrecy he wished for his new experiments on the ether wind, and the failure of a venture into a more mathematical physics.64

That venture occurred in 1905, soon after Sagnac visited Lorentz at his home in Leiden. Back in Paris, Sagnac sent to Lorentz the manuscript of two memoirs on a mathematical theory of phosphorescence and photographic action, based on the idea of ionization followed by delayed recombination. A similar idea already occurred in his dissertation, in an expla-nation of photographic illusions that some early X-ray experimenters had mistaken for diffraction fringes: the photographic layer, when subjected to light, undergoes not only a direct alteration proportional to the intensity of the light but also a retarded reaction to this alteration at a rate proportional to the amount of alteration. This reaction implies that for an in-creasing time of exposure the impression first reaches a maximum, then goes to a fairly low minimum and oscillates for a while until it reaches a stable value. The first minimum explained the well-known phenomenon of photographic inversion, according to which an increased exposure may annihilate the initial impression; and the oscillation explained the fringes sometimes observed at the limit of a shadow and sometimes mistaken for diffraction fringes.65

The 1900 version of this reasoning was mostly qualitative, although Sagnac gave the relevant delay differential equation in a footnote: f ′(t) = a −bf (t −τ ) for the impression i = f (t). When he returned to this problem in 1905, he sharpened the physical basis and developed the mathematics. The alteration of the photographic layer now became an ionization process, and the reaction became a delayed recombination of the ions. Sagnac solved the delayed differential equation of the problem through the characteristic equation. Unfortunately, he missed some roots of the latter equation. Lorentz wrote a twelve-page letter of comments to Sagnac, warning Sagnac about the preceding error and yet expressing a vivid interest in his theory. Here is an extract of Sagnac’s reply:

“I thank you with all my heart for having read with so much benevolence the two memoirs which you kindly returned to me. Your precious encouragements give me a renewed ardor. I am happy to feel like you are sheltering me from the grave errors of the leading theory. I committed a grave and inexcusable error by attributing only a real root to the characteristic equation.”Lorentz’s encouragements did not suffice, as Sagnac never published this theory.66

62 Sagnac [117–119]; Blondlot to Poincaré, 23 Dec 1903, in Walter [29], on p. 70. On N rays, cf. Nye [120].63 On this laboratory, cf. P. Sagnac [19, p. 43].64 P. Curie to Gouy, 31 Jan 1905, cited in Barbo [121, p. 267].65 Sagnac [80, pp. 422–426]. Sagnac also conceived a spatial diffusion of the negative reaction. On Sagnac’s visit to Lorentz, see Lorentz to Sagnac, 18 Apr

1905, Archives nationales.66 Sagnac [80, pp. 425n–426n]; Lorentz to Sagnac, 11 Aug 1905, Archives nationales; Sagnac to Lorentz, undated reply to the former, AHQP.


Fig. 17. Sagnac’s second interferometer. From Sagnac [124, p. 1677].

6. The optical whirling effect

6.1. From Sagnac’s principle to rotational ether flow

Sometime in 1908 Sagnac started a new series of researches based on an important new idea. As was mentioned, he knew that his “principle of the effect of motion” accounted for the failure of first-order ether-wind experiments. This is true in particular for experiments based on interference. In 1873 Veltmann had treated this case on the basis of the following theorem: the time taken by light to travel on a closed loop is unaffected by the motion of the earth no matter how many reflections or refractions it undergoes on its path. Veltmann’s proof of this theorem was based on the remark that the laws of reflection and refraction in the earth-based frame are the same as in the ether frame if the ether is dragged in the amount assumed by Fresnel. As Sagnac explained in 1905, his principle gives a very simple and direct proof the theorem: the resulting expression of the time correction, δt =

!u · dr/c2, evidently vanishes over a loop (u being the velocity of

the earth through the ether). In 1905 Sagnac used this result to give a new derivation of stellar aberration, based on the vanishing of the Veltmann integral on the triangle made by the star and two points of a wave front from the star. In this reasoning, the ether is strictly stationary and u denotes the translational velocity of the earth through it, or the opposite of the velocity of the uniform ether wind with respect to the earth. In 1908 Sagnac realized that his principle more generally implied δt = −

!v(r) · dr/c2, if v(r) denotes the velocity of the ether wind at point r on the earth.67

Being friend to Vilhelm Bjerknes, who had extended Kelvin’s circulation theorem to compressible fluids, Sagnac recog-nized Kelvin’s circulation in the integral

"v · dr. In fluid-mechanical terms, the time correction on a light circuit may be

non-zero if the ether wind within this circuit is rotational. This will for instance be the case if the ether is dragged by the earth at a rate decreasing with the elevation. In this way Sagnac knew he could test a basic hypothesis of his and Lorentz’s optics of moving bodies, the stationary ether, by observing (the lack of) a fringe shift in a vertical interferometer. The coher-ence length of ordinary light being small, the only practical way to do so is to observe the interference of two beams that have traveled on the same closed path in opposite directions. This was Sagnac’s original motivation for constructing double inverse path interferometers, now called Sagnac interferometers. The inspiration plausibly came from Fizeau.68

6.2. A new interferometer and a test of the ether drag

After some groping about, Sagnac arrived at the arrangement of Fig. 17 in which the light from the source C is partially reflected and partially transmitted by the air gap ll’ between the two glass prisms P1 and P2 (according to Sagnac, this

67 Veltmann [122]; Sagnac [123]. The date 1908 is from Sagnac [124, p. 1676].68 Sagnac [125, p. 311] for the reference to Kelvin and Bjerknes. In 1910, Sagnac [126,124] described two versions of his interferometer before giving the

motivation. On the relevance of Bjerknes’s theorem, cf. Lalli [15, p. 62].


Fig. 18. The reversal of the effect of the ether drag on Sagnac’s interferometer between noon and midnight. KM1M′2 is the vertical projection of the optical

circuit KM1M2 (a triangle). The letters T and R refer to the transmitted and reflected beams. From Sagnac [127, p. 224].

is better than a semi-reflecting plate à la Michelson because the incidence can be adjusted so that the transmitted and reflected light have exactly the same intensity for a given wavelength when the light from C is polarized perpendicularly to the plane of reflection). The two light beams are then reflected on the mirrors M1 and M2 and their superposition is observed through the telescope L.69

When the interferometer is perfectly adjusted, the phase difference is the same for all interfering rays in the field of observation and this field is uniform. Fringes are obtained by slightly rotating the separating double prism (for instance). The shift of the central fringe owing to the circulation of the ether is proportional to the surface embraced by the closed light path and to the transverse gradient of the ether wind (according to Stokes’ theorem). For a vertical, East–West oriented circuit, this shift should reach two extreme opposite values at noon and midnight since the main ether wind is the one caused by the rotation of the earth around the sun (see Fig. 18).70

With an optical circuit of 20 m2 and with painstaking attention to the possible perturbations of his sensitive apparatus, Sagnac determined that the ether shear, if any, could not exceed the fraction 0.3 × 10−7 of the velocity of the earth for an elevation of one meter (see Fig. 19). He finished this experiment in early 1910 and gave a full account of it at the International congress of radiology and electricity in Brussels in the fall of the same year. There he framed his experiment in the context of the “relativity principle” of Mascart, Poincaré, and Einstein, following which the result of optical experi-ments can only depend on the relative motion of the implied material bodies. The experiment indeed tested the stationary character of the ether, which, together with the principle of the effect of motion, Sagnac regarded as the basis of first-order relativity.71

6.3. Interferential strioscopy

While adjusting his ultra-sensitive interferometer, Sagnac noticed that the central dark field he observed through his telescope under the condition of destructive interference became luminous whenever a warm object like the hand was close to the beam. He thus got the idea of new kind of strioscopie. This word comes from the German Schierenmethode, through which August Töpler denoted the visualization of optical perturbations in a transparent medium by a method he developed in the 1860s. This technique, which turned out to be very useful to visualize flows, originated in an idea by Léon Foucault whose principle is illustrated on Fig. 20. The lens centered in O forms the image A′B′ of the bright opening AB of a diaphragm. An opaque object exactly covers A′B′ and therefore blocks all the light from AB on the right side of A′B′ . The screen E contains the image S′ of the point S on the axis before the lens. If any thermal or motional perturbation of the air occurs near S, the light from AB is deviated by this perturbation and thus fails to be blocked by A′B′ . Luminous “streaks” (Schieren, or stries) appear on the screen.72

69 Sagnac [124].70 Sagnac [125,127,128]. Sagnac operated with white light, so that his central fringe had the “teinte sensible” for which the brightest spectral component

is extinguished.71 Sagnac [127, p. 217] for the reference to Mascart, Poincaré, and Einstein. On this experiment, cf. Martinez-Chavanz [6, pp. 43–49].72 On the earlier history of strioscopy, cf. Sagnac [129, p. 241].


Fig. 19. Sagnac’s drawing of the collimator C, the telescope L, and the separating double prism between the auxiliary mirrors m0, m1, m2 (from Sagnac [127, p. 227]), and a photography of this apparatus (from Archives nationales). The beams from the separator intersect at K and are reflected by the distant mirrors M1 and M2 on the walls of the room. In the margin, Sagnac wrote: “Léon Bloch came to see the fringes adjusted by the slow rotation of the door!” (the weight of the door (porte) was indeed sufficient to bend the wall to which the mirrors M1 and M2 were attached and thus controlled the spacing of the interference fringes: see Sagnac [124, p. 232]).

Fig. 20. Foucault’s strioscopic setup. From Sagnac [130, p. 82].

Although this effect in some cases involves diffraction, it is understandable in the context of geometrical optics and it does not involve interference. In contrast, Sagnac’s interferential strioscopy implies the perturbation of a destructive interference. In the case of Sagnac’s original two-mirror interferometer (and for any even number of mirrors), the rays of the two opposite interfering beams travel exactly on the same path, so that aerial perturbations affect both of them equally. This circumstance minimizes the effect of the perturbations, which was an advantage in Sagnac’s tests of ether drag. For enhanced sensitivity to the perturbations, Sagnac switched to the three-mirror interferometer of Fig. 21. Half of the parallel beam from the collimator C is blocked by the diaphragm B. The other half it both transmitted and reflected by the double prism P1P2. The transmitted beam and the reflected beam travel on the two sides of the central ray until they finally overlap on their way to the telescope L. The interferometer is adjusted so that the field of observation is uniformly dark by destructive interference. A jet of gas introduced at g through the little pipe (bottom right of the figure) or a rotating helix h on the way of the transmitted beam illuminates the observed field in a manner depending on the index variation. Sagnac did the experiments as well as the theory of the images thus formed.73

6.4. The Sagnac effect

In the following months, Sagnac described a less sensitive but more practical variant of his interferential strioscope, and he also showed that his interferometer could be used to measure phase shifts at an interface (a silvered plate for instance)

73 Sagnac [131,129,132,130]. About odd/even number of mirrors, see Sagnac [129, p. 251].


Fig. 21. Sagnac’s strioscope. From Sagnac [129, p. 243].

with exquisite precision.74 Plausibly, these results were but an instrumental diversion and he already had a grander project: the demonstration of a rotational ether wind. Remember that he originally conceived his interferometer for this purpose. The first case of rotational ether wind he investigated, an altitude-dependent drag, was not the most obvious. For someone familiar with Bjerknes’ atmospheric vortices, the most obvious case should have been the relative ether wind implied by the earth’s diurnal rotation in a stationary-ether theory. Plausibly, that was Sagnac’s first idea75 but he immediately realized that the detection of this tiny effect would require an excessively large interferometer. In 1909 he realized that a much stronger effect of the same kind would be obtained by placing the interferometer on a uniformly rotating table.76 The difficulty of this experiment probably explains why four years elapsed before he completed it. He described the results in a pli cachetéof 18 August 1913 (Fig. 22) and then in two communications of 27 October and 22 December 1913 to the Académie des sciences.77

In the final setup, represented in Fig. 23, a circular table of 50 cm in diameter is brought to rotate at two turns per second. The entire interferometer, including source and camera, is rigidly connected to this table. Once uniform rotation has been reached, the lamp O is turned on. Its light goes through the objective C0, the Nicol polarizer N, the mirror m, and the vertical slit F; then it is partly reflected partly transmitted by the air-gap separator s; the two resulting beams travel a full cycle by reflection on the four lateral mirrors and then recombine before reaching the telescope L and the photographic plate pp′ . The four mirrors (instead of the two mirrors of Sagnac’s original interferometer) are needed to maximize the surface S embraced by the optical circuit. Inverting the rotation velocity, Sagnac found a relative fringe shift of 0.07 for indigo light, in agreement with his circulation formula (see Fig. 24). The absolute shift had the sign predicted by the theory. Therefore, this shift could not be caused by a centrifugal deformation of the apparatus, which would not depend on the sign of rotation. Nor could it be caused by relative rotation of the air above the interferometer, for a fan placed above the interferometer at rest failed to displace the fringes.78

74 Sagnac [133,134].75 Sagnac suggests so much in Sagnac [135, p. 709]. In a letter to Lucien Poincaré of 20 Jan 1919, Sagnac mentioned that in 1914 Brillouin had come to

see his experiments “on the motion of the earth,” presumably meaning that his experiments of 1910 and 1913 had to do with his early idea to detect an effect of the rotation of the earth: cf. Lalli [15, p. 65].76 Sagnac confided this idea to Lippmann in 1909. At that time, he regarded a negative outcome of the experiment as possible in a proper emission theory

(Walther Ritz had published his theory the preceding year). Cf. Sagnac [136, p. 194].77 Sagnac [135,137]. On this experiment, cf. Martinez-Chavanz [6, pp. 27–36].78 Sagnac [135,137,136]. Although Sagnac does not explicitly state the precision of his fringe-shift measurement, it must have been about 10% judging

from the photographs and the numbers he gave.


Fig. 22. The first page of Sagnac’s pli cacheté of 18 August 1913. Sagnac writes: “This is the experimental proof of the whirling relative ether wind that the rotating system creates through its motion.” From the Archives de l’Académie des sciences.


Fig. 23. Sagnac’s setup for detecting the optical whirling effect. From Sagnac [137, p. 1412].

Fig. 24. One of Sagnac’s fringe-shift photographs. The upper half of the fringes, slightly shifted to the left, corresponds to the dextrorsum (clockwise) rotation of the table; the lower half to the sinistrorsum (counter-clockwise) rotation of the table. From Archives nationales.

As we saw, the reasoning that led Sagnac to this experiment was based on his theory of the propagation of light, which required a stationary ether. In his view his experiment was analogous to the Michelson–Morley ether-drift experiment, except that it gave a positive result. In the new experiment, motion with respect to the ether had a measurable effect, and this effect was of first order in the implied velocities. Sagnac believed he had struck a fatal blow on relativity theory by proving the existence of the ether. He concluded:

“The observed interferential effect proves to be the optical whirling effect caused by the motion of the system with respect to the ether, and it directly manifests the existence of the ether, necessary carrier of the luminous waves of Huygens and Fresnel.”The title of the first note in the Comptes rendus expresses the same conviction: “The luminous ether proved by the effect of the relative ether wind in a uniformly rotating interferometer.”79

79 Sagnac [135, pp. 708, 710].


Fig. 25. Diagram for Sagnac’s direct derivation of the rotational fringe shift.

6.5. Optical gyroscopy and the angular effect

Sagnac published a fuller account of his experiment in the March 1914 issue of the Journal de physique. There he sug-gested that his optical whirling effect could be used to detect and record the rolling of ships by means of an optical circuit around the ship hull. He also improved on his earlier theoretical derivation of the effect. That derivation implicitly assumed that the propagation of light in vacuum or air remained rectilinear in the approximation of geometrical optics. Sagnac knew this was not the case. In 1911 he had derived the “angular optical whirling effect” according to which the (uniform) curl ∇ × v of the velocity v of the ether implies the angular deviation (∇ × v) × D/c of the image of the focus of a telescope by a second telescope pointed toward the first at the large distance D. In 1914 he traced this deviation to the curvature |∇ ×v|/c of the light rays between the two telescopes, with a Poincarean comment: “Beings living in a sufficiently rotational ether [of large |∇ × v|] in which they would establish their geometry on the basis of experimental optics, would be brought, if they did not discover the rotationality of their ether, to establish a non-Euclidean geometry.” At any rate, the radius of curvature of the light rays was too small in comparison with the distances between the mirrors of Sagnac’s setup to affect the calculation of the phase shift.80

Sagnac derived the angular optical effect in a manner similar to his derivation of stellar aberration, by compensating for the time shift on a closed light path made of two different rays connecting the focus of the first telescope to that of the second. Alternatively, we may combine his principle of the effect of motion with Fermat’s principle of least time. This gives, for rays in an ether moving at the velocity v(r), paths obeying δ

! BA dt = 0, with dt = nds/c − v · dr/c2. Varying under the

integral and integrating by parts, we get the differential equation of the paths:

dnTds

− ∇n + 1c(∇ × v) × T = 0, with T = dr

dsThis means that in a vacuum (n = 1), the tangent vector rotates along the path of a ray at the constant rate |∇ × v|/c: the rays have the constant curvature |∇ × v|/c.81

6.6. A simple derivation of the Sagnac effect

The negligibility of the rays’ curvature being established, there is a simple direct way to derive the fringe-shift formula for the Sagnac effect. Sagnac gave it in his memoir of 1914, as a favor to readers unfamiliar with his personal theory of the propagation of light. The ether being stationary, he reasoned, the only significant effect of the rotation of the table (as judged by an ether-bound observer) is that the element ds of the optical circuit moves ahead of the waves when they travel from the beginning to the end of the element. The resulting delay is

δt = dsc − ωr cosα

− dsc

∼ ωr2 cosαdsc2

if ω denotes the angular velocity of the turntable, r the distance of the element ds from the center of rotation, and α the angle that this element makes with the linear velocity of the corresponding point of the table (Fig. 25). If dθ denotes the angle under which ds is seen from the center, we also have ds = rdθ/ cosα. Consequently, the time shift is simply given by δt = ωr2dθ/c2 = 2ωdS/c2, wherein dS is the surface of the infinitesimal triangle defined by the center of rotation and the element ds. The total phase shift on the circuit is therefore equal to 4πωS/λc, and the shift of the phase difference between the lights traveling in the two opposite direction on the same circuit is twice this quantity.82

80 Sagnac [136], p. 191 (ship rolling), 185n (citation); Sagnac [125, pp. 312–313] (angular effect). On the angular effect, cf. Martinez-Chavanz [6, pp. 48–49].81 Sagnac [125, 312–313]; [136, pp. 183–186]. At first order in υ/c, special relativity gives the same result since it is optically equivalent to Sagnac’s

principle.82 Sagnac [136, pp. 180–182].


This simple derivation of the Sagnac effect does not require the concept of ether wind and it can easily be transposed in the context of Lorentz’s theory or in the context of special relativity. Yet Sagnac never ceased to see his experiment as a proof of a relative ether whirling. Remember that he arrived at this experiment by exploring the consequences of his “principle of the effect of motion,” which implied a relation between the phase shift on a light circuit and the circulation of the relative ether flow. Sagnac meant his experiments of 1910 and 1913 to test the rotational character of that flow. In France, he was not alone in regarding the motion of the ether as a basic open question. Reviewing recent advances in optics for the Revue générale des sciences in 1914, the Marseilles professor Louis Houllevigue adorned his account of Sagnac’s experiment with the comment:

“M. Sagnac seems to have taken all the cautionary measures we might imagine in order to shelter himself from errors. If he has not omitted anything, the result of his experiment is one of the most important that optics has registered since Fresnel, since it would unquestionably prove the existence of a wave-carrying medium independent from material media. Thus would be closed, in favor of the ether, a still much undecided debate.”Foreign experts in the optics and electrodynamics of moving bodies would not have so easily embraced this conclusion. A good proportion of them had already been seduced by Einstein’s ether-less theory. Most of them accepted Lorentz’s electromagnetic theory, in which the question of the motion of the ether became futile. In earlier times, however, this question had been the center of attention of several important physicists, so much so that one may wonder whether the kind of experiments Sagnac conceived in the 1910s had not been discussed before him. Indeed they had been.83

7. Anticipations of the Sagnac effect

7.1. Dilemmas of the optics of moving bodies

Before discussing anticipations of Sagnac’s ether-wind experiments, it will be useful to remember some nineteenth-century optics of moving bodies. Until the 1880s there were roughly two competing theories in this field. In Fresnel’s theory, the ether was stationary in a vacuum and it was partially dragged by moving transparent bodies. The stationary character explained stellar aberration. The partial drag was adjusted so that the motion of the earth did not affect the laws of refraction. In Stokes’ theory, the ether was completely dragged by the earth (its relative motion with respect to the earth vanished near the earth). This assumption immediately implied the absence of effect of the motion of the earth on terrestrial optical experiments; and Stokes believed he could conciliate it with stellar aberration if the ether motion was irrotational. Most French physicists of course favored Fresnel’s theory, the more so because they believed that Fizeau had confirmed the Fresnel drag in 1851. British physicists tended to favor Stokes’ theory in part because it was in harmony with Maxwell’s electromagnetic theory of light, which assumed a single ether-matter medium moving at a well-defined velocity; and also because, Fizeau’s result, which no one had yet cared to confirm, weighed little against the accumulated evidence that the motion of the earth had no effect on terrestrial optical experiments.84

In 1881 the American physicist Albert Michelson performed an experiment in which he had two light beams from the same source interfere after making roundtrips in the two perpendicular arms of an interferometer of his own (Fig. 26). The idea was to compare the relative velocities of light in two perpendicular directions. If the ether was fully dragged as Stokes required, these two velocities should of course be equal. If it was stationary as Fresnel required, a computable and measurable second-order fringe-shift had to occur. Michelson found none and concluded that Fresnel’s theory had to be abandoned. Although Michelson’s announcement soon turned out to be premature, he was able to confirm the absence of fringe-shift in 1887 with a much more sophisticated apparatus in collaboration with Edward Morley.85

One year earlier, in 1886, the two Americans had performed an accurate variant of Fizeau’s running-water experiment of 1851. They thereby relied on Fizeau’s idea of splitting a light beam into two beams traveling on opposite circuits, except that the splitting was now obtained with Michelson’s signature semi-reflecting plate (Fig. 27). They thus confirmed the Fresnel–Fizeau drag.86 In combination with the double-arm experiment of 1887, this result perplexed Michelson: whereas the former experiment seemed to confirm Stokes’ theory, the latter seemed to confirm Lorentz’s. One could perhaps imagine a variant of Stokes’ theory that would integrate the Fresnel–Fizeau drag. Alas this subterfuge was no longer available, because Lorentz had recently detected a serious internal contradiction in Stokes’ theory: the complete drag of the ether and the irrotational character of its motion are mathematically incompatible.87 As is well known, George Francis FitzGerald (in 1889) and Lorentz (in 1892) rescued Fresnel’s theory by assuming a proper contraction of the longitudinal arm of Michelson’s interferometer. To many, this sounded like an ad hoc move and the question of ether motion remained open.88

83 Houllevigue [138, p. 442]. On the anticipations of Sagnac’s ether-wind experiments, cf. Post [139]; Martinez-Chavanz [6, Chap. 2]; Anderson, Bidger, and Stedman [140].84 On the history of the optics of moving bodies, cf. Whittaker [95]; Janssen and Stachel [97]; Darrigol [90, pp. 314–319]; [84, pp. 258–261].85 Michelson [141]; Michelson and Morley [142]. On Michelson’s ether-drift experiments, cf. Swenson [143].86 Michelson and Morley [144]. On Michelson’s interferometry and silvered plates, cf. Staley [145, pp. 47–49].87 The irrotational motion of an incompressible fluid is completely determined by the Neumann boundary condition of zero normal velocity on the surface

of the earth, and the resulting motion has a non-zero tangential velocity.88 Cf. Hunt [146, pp. 189–197]; Darrigol [90, pp. 317–319].


Fig. 26. Michelson’s setup for comparing the relative velocities of light in two orthogonal directions. The light from the source s is split into two orthogonal beams by the silvered glass plate a, then reflected by the mirrors b and c. The interference of the recombined lights along d is observed through the telescope f . If the setup moves with respect to the ether (case 2), the silvered plate moves from the position a to the position a/ in the ether during the roundtrip aba/ of the reflected light. From Michelson [142, p. 335].

Fig. 27. Michelson and Morley’s set up for measuring the Fresnel–Fizeau drag. The parallel light from the source a and the lens is divided into two beams by the silvered glass plate b. By reflection on the mirrors c and f and on the rear faces d and e of a glass prism, they travel on two opposite circuits, along or against the flow in two water pipes on the cd and ef axes. Their interference is observed through the telescope g . From Michelson and Morley [144, p. 381].

7.2. Lodge on ether whirling

In order to test the stationary character of the ether, the leading Maxwellian physicist Oliver Lodge built an “ether-whirling machine” in which the light beams of an interferometric device traveled in the thin gap between two heavy, fast rotating steel disks. From Michelson, Lodge borrowed the idea of making the light travel both ways on a (multiple) circuit after splitting by a semi-reflecting plate (see Fig. 28). If the ether followed the rotation of the disks, then light would take a different time to travel in opposite directions on the optical circuit, and the interference fringes would be shifted. In 1893 Lodge informed the Royal Society that the expected shift did not occur. The stationary ether seemed confirmed and the Michelson–Morley result looked more mysterious than ever.89

89 Lodge [147]. Cf. Martinez-Chavanz [6, pp. 14–16]; Hunt [148].


Fig. 28. Lodge’s interferometric technique for his ether-whirling experiment. A narrow parallel beam from the source is both reflected and transmitted by the horizontal silvered plate. The two resulting beams make three turns in opposite direction by reflection on the four mirrors before recombining. The big square represents the rigid wooden box that holds the mirrors. The inserted circle represents the projection of the limit of the rotating steel disks. From Lodge [147, p. 757].

Lodge’s and Sagnac’s ether-whirling experiments differ in several respects. In the former experiment, the interferometric apparatus (source, mirrors, and telescope) are at rest (with respect to the earth), whereas they are on a rotating table in the latter. Whereas Lodge is testing the drag of the ether by nearby moving matter, Sagnac is testing the absence of drag of the ether by nearby moving matter. For Lodge a fringe shift would imply ether drag; for Sagnac, the observed fringe-shift results from the absence of ether drag. Sagnac’s setup is a detector of absolute rotation and can work as a kind of gyroscope, Lodge’s cannot.

Lodge nonetheless derived the Sagnac formula 4πωa2/λc for the phase shift in the case of a square optical circuit of radius a immersed in ether rotating at the angular velocity ω, with the parenthetical comment:

“(Evidently the larger the square the better, and a large enough square might show even the earth’s rotation effect, only it is difficult to see how to imitate the effect of stopping and reversing the rotation, at least with the unwieldy size of frame necessary.)”Lodge seems to have meant an experiment in which the optical frame (source, lamps, and telescope) would be at absolute rest while the ether would rotate together with the earth. Again, that is not the Sagnac effect. The humorous remark about reversing the rotation of the earth points to a genuine difficulty in any interferential detection of the rotation of the earth: the fringe shift remains undetectable as long as no way to determine the absolute position of the central fringe is known.90

Four years later at the Royal Society, Lodge made his suggestion a little clearer:“Now, by staking out mirrors at the corners of a field, it is arithmetically quite possible to arrange for a perceptible shift

of the bands due to the rotation of the earth, if it carries ether round with it; but it does not seem possible to experimentally observe that shift, unless some method could be devised of making the observer and his apparatus independent of the rotation.”Most interestingly, he went on to consider the symmetric case in which the apparatus and observer move together with the earth and the ether does not:

“It is to be observed, that since a motion of the disks relatively to the observer and the light causes no effect, the ether being stationary, it follows that a motion of the light and observer would produce an effect, since they would be moving relatively to the ether. Hence if, instead of spinning only the disks, the whole apparatus, lantern, optical frame, telescope, observer and all were mounted on a turn-table and caused to rotate, a reversible shift of the bands should be seen.”This is exactly the Sagnac effect. Lodge went on:

“In an actual experiment of this kind, centrifugal force would give some trouble by introducing strains, and rapid rotation would be uncomfortable for the observer; but really rapid rotation should be unnecessary to show the effect. My present

90 Lodge [147, pp. 773–774].


Fig. 29. Michelson’s setup for detecting a vertical velocity gradient of the ether. From Michelson [149, p. 475].

optical apparatus mounted on a turn-table revolving 4 times a minute should show something, viz.: 1100 th band shift each

way.”Be it by lack of time or by fear of discomfort, Lodge never performed this experiment (unlike Sagnac, he did not think of using photographic recording with remote control). He also considered the effect of the motion of earth on a large terrestrial interferometer:

“If the ether is stationary near the earth, that is, if it be neither carried round nor along by that body, then a single interference square, 1 kilometer in the side, would show a shift of rather more than one band width, due to the earth’s rotation in these latitudes.”The difficulties with the size of the interferometer and with the irreversibility of the rotation of the earth deterred him from attempting this second experiment.91

7.3. Michelson’s ether-drag test of 1897

In the same year 1897, Michelson had a more practicable idea for testing the drag of the ether by the earth. He presum-ably reasoned that his setup for measuring the drag of the ether by running water could as well be used to detect different velocities of the ether on two parallel light paths. His new setup (Fig. 29) was indeed very similar. The light from the source s is split by the silvered glass plate o into two beams that travel around the squares ocba and oabc and then recombine in the telescope e. The rectangle being set vertically, this interferometer detects the vertical difference of horizontal ether velocity that a drag of the ether by the earth would imply. Calling υ the velocity difference for the paths ab and oc of length L, the expected phase shift is 4πυL/λc. With his 60 m × 15 m rectangle and with L/λ = 108, Michelson found that the fringe-shift difference between noon and midnight did not exceed 1/20. This gives 4υL/λc ≤ 1/20, so that the velocity gradient of the ether must be less than 4 cm/s over 15 m (or less than 10−7 the velocity of the earth around the sun (about 30 km/s), to be compared with 0.3 × 10−7 in Sagnac’s similar experiment). Assuming an exponential diminution of the relative velocity, Michelson found that the earth’s influence on the ether would have to extend over distances compa-rable to the diameter of the earth. This being highly implausible, he judged that Fresnel’s stationary ether may after all be preferable: “One is inclined to return to the hypothesis of Fresnel and to try to reconcile in some other way the negative results obtained in the [Michelson–Morley experiment].”92

We thus see that thirteen years before Sagnac, Michelson conceived and performed a similar test of ether drag with the same negative result and a comparable sensitivity. The main differences are the geometry of the experiment (square instead of triangular circuit) and the way in which the effect of fluctuations in the air is avoided. Sagnac minimized these perturbations by using two separate rooms for the optical circuit and for the observation of the interference fringes, and he reduced their perturbing effect by using an even number of mirrors. In contrast, Michelson, whose circuit was significantly longer than Sagnac (150 m instead of 30 m), could not obtain stable fringes in the air and had to make his light beams travel through pipes evacuated to a hundredth of an atmosphere (one for each side of the rectangle, with glass windows at both ends).93

91 Lodge [8, p. 151]. Lodge discussed these issues in contemporary correspondence with Joseph Larmor. At the end of murky reasoning and discussion, they privately arrived at Sagnac’s phase-shift formula for a circuit of arbitrary shape. Cf. Anderson, Bilger, and Stedman [140, pp. 981–983].92 Michelson [149, p. 478] (after this remark, Michelson deplores that any theory that takes all experimental results into account, including Lorentz’s, has

to make fantastic assumptions about the relation between ether and matter). Cf. Martinez-Chavanz [6, pp. 17–20]; Lalli [15, pp. 60–61].93 Cf. Sagnac [127, p. 227]. Despite the rectangular geometry of Michelson’s circuit, the number of mirrors is even, because the reflection at the corners of

the rectangle is produced by pairs of mirrors. It is not clear why Michelson used pairs instead of single mirrors. He did not relate the parity of the number of mirrors with lower sensitivity to thermal fluctuations. As Jacques Vigué explained to me, Sagnac’s success in air may have to do with better mechanical stability of his setup and perhaps also with the smaller number and better quality of his mirrors.


7.4. Michelson’s ether-wind experiment of 1904

After his experiment of 1897, Michelson was more willing to assume the stationary character of the ether. For a stationary ether, the motion of the earth through the ether should affect the traveling time of light between two points of the earth. The failure of the Michelson–Morley experiment of 1887 did not mark the end of attempts to detect such consequences. For instance, in 1904 Wilhelm Wien proposed an experiment in which two toothed wheels rotate at the same speed and with the same phase around a common axis that is parallel to the velocity of the earth. Light is sent with the same initial intensity in the parallel and anti-parallel directions between the teeth of the two wheels and the final intensities of the beams are compared. If the ether is stationary, Wien reasoned, the traveling time of light between the two wheels depends on the direction of propagation, so that the final intensities of the two beams should be different. As Emil Cohn soon pointed out, the result depends on the method for synchronizing the two wheels. Optical synchronization should give a null effect, since it assumes isotropic propagation. Mechanical synchronization could give a positive effect if mechanical processes essentially differ from electromagnetic processes.94

In an article of 1904 for the Philosophical magazine, Michelson made the same comment, adding in a footnote: “Perhaps, however, even mechanical impulses would be affected by the earth’s motion in such a way as to neutralize the expected effect.” He went on to offer a better way to detect the effect of the earth’s motion in the ether, by having beams of light travel on the same parallel around the earth in opposite directions and then interfere. Owing to the rotation of the earth, one of the beams would be retarded while the other would be advanced and the interference fringes should be shifted. More realistically, any optical circuit of large dimensions on the surface of the earth should produce the differential time delay 2

c2

"u · dr, wherein u denotes the velocity of the point r of the earth in the ether. From this formula Michelson

estimated that the interference fringes would be shifted by about one fringe for a circuit of one square kilometer at a latitude of 45 (he thereby overlooked a factor 2). Like Lodge he noted the difficulty of defining a reference for the position of the fringes, and he suggested using the image of a slit for one of the beams. Some twenty years elapsed before he could measure this effect, in collaboration with Henry Gale and Fred Pearson, astutely centering the fringes by means of a smaller interferometer inserted in the large one.95

7.5. Kaluza

A last theoretical anticipation of the Sagnac effect is found in a short communication by the young Theodor Kaluza intended for the Naturforscherversammlung of September 1910 in Königsberg. For a Born-rigid96 rotating disk, Kaluza defined the Eigengeometrie of the disk through the orthogonal section of the congruence of world-lines of the material points of the disk and found it to be Lobachevskian, thus anticipating a famous consideration by Einstein. In addition, Kaluza considered the synchronization of clocks on the disk and found that the local synchronicity condition was not integrable. For a loop containing the center of rotation and defined by r(θ) in polar coordinates, he gave the synchronization error “measured in proper time”:

δ = 2#

(ωr2/c2)dθ$

1 − r2ω2/c2

where ω is the angular velocity of the disk. Kaluza concluded:“From the existence of this synchronization error there follows the possibility of a proof of the rotation of the earth by

pure optical or electromagnetic means. (No contradiction with relativity theory.) The idea may not be practically feasible at present; in the best case, we are talking about 2 · 10−7 s.”Kaluza was unaware of Michelson’s similar suggestion of 1904. His own considerations long remained unnoticed, in part because an illness prevented him from attending the Königsberg meeting. Another reason is the laconic, formal way in which he published his results.97

A plausible reconstruction of this non-integrable synchronization argument runs as follows. Assuming that for co-moving observers in the neighborhood of a point of the disk the velocity of light still is the constant c (more exactly, synchronization is defined so that this condition holds), then the Lorentz transformations with translation velocity u = rω and with the axis in the orthoradial direction can be used to derive the time and space given by optical synchronization in this neighborhood. Hence we have dt = γ (dt′ + udx′/c2) for the time difference between two neighboring events with respect to observers at rest, if dt′ and dx′ denote the increments of the time and orthoradial coordinates with respect to the disk’s observers and with γ = (1 − u2/c2)−1/2. Consequently, the t-time lag of two neighboring synchronized clocks of the disk is dt = γ udx′/c2. The coordinate x′ being measured with rods contracted in the ratio γ −1, and radial rods being unchanged, we also have

94 Wien [150]; Cohn [151, pp. 1408–1409]. Cf. Darrigol [90, pp. 368–369].95 Michelson [9, p. 716n]; Michelson, Gale, and Pearson [152]. Cf. Martinez-Chavanz [6, pp. 20–21, 63–66]; Anderson, Bilger, and Stedman [140,

pp. 975–976].96 Born-rigidity, introduced by Max Born in 1909, means constant distance between two neighboring material points in the tangent inertial frame.97 Kaluza [153]. Cf. Walter [154, pp. 68–70]. The formulas in Kaluza’s article seem to be given for ω = c = 1.


dx′ = γ rdθ . The synchronization error for a clock situated at the distance r0 and for a synchronization loop embracing the origin is, when measured in proper time,

δ = γ (r0)

#γ 2%ωr2/c2&dθ

Kaluza’s formula gives twice this value, only when the synchronization look is the circle r = r0. Kaluza may have reasoned correctly and forgotten to mention the restriction to a circular loop in his condensed, published statement. The factor two may result from a different convention in the definition of the synchronization error.

In any case, Kaluza’s imprecisions do not affect the following deductions. For events occurring at two distant points, the local time difference depends on the line of clocks on which the synchronization is done. For instance, two identical events will be judged non-synchronous if their synchronicity is judged by a dense series of clocks arranged on a circle centered on r = 0. As another consequence, light traveling on a loop around the center of rotation will take the local time t′± ≈ t± ∓ 2Sω/c2 (the + index being for travel in the direction of rotation, and the index − for travel in the opposite

direction), wherein S is the area enclosed by the loop. Since the velocity of light with respect to the disc has been assumed to be the constant c, we must have t′

+ = t′− and t+ − t− = 4Sω/c2, in conformity with the Sagnac time-shift (in the

laboratory frame).Langevin gave the latter reasoning in 1935, without knowledge of Kaluza’s earlier considerations, and with a different

derivation of the synchronization error (based on a direct exploitation of the isotropy of light propagation with respect to the disk’s observers).98 The Kaluza–Langevin reasoning shows that the Sagnac effect is compatible with the validity of special relativity in the instantaneous tangent frames at every point of the rotating disk. It also shows that the effect derives from the non-integrabiliy of proximate clock synchronization on the disk. This deep insight of the young Kaluza is preserved in general relativity. Of course, it does not invalidate the more elementary derivations of the effect that Lodge, Michelson, and Sagnac obtained in the ether-frame.99

7.6. Lodge, Michelson, and Sagnac

To summarize, Lodge [8], Michelson [9], and Kaluza [153] all conceived the Sagnac effect by the rotation of the earth well before Sagnac publicly discussed it. Lodge also anticipated the Sagnac effect on a turntable in 1897. In the same year, Michelson performed a test of ether drag similar to Sagnac’s test of 1910. One might wonder how Sagnac could remain unaware of the latter experiment, since it had been published in the same journal (The American journal of science) as Michelson’s other famous experiments and since two well-read experts on the optics of moving bodies, Lorentz and Wien, attended the Brussels congress in which Sagnac announced his result. Then one should also wonder why Michelson and Sagnac were both unaware of Lodge’s suggestions, which appeared in the widely read Philosophical transactions. The explanation is simple: the international circulation of knowledge and the refereeing system of the time were not what they are now, despite the relatively small amount of publications.

More interestingly, the motivations and the heuristics of Lodge, Michelson, and Sagnac were quite different, despite the similarity of the experiments they conceived. Whereas Sagnac’s experiments of 1910 and 1913 were a consequence of his discussion of interference by means of his principle of the effect of motion, Lodge’s and Michelson’s experiments derived from the instrumental opportunities offered by Fizeau’s inverse-path trick and by Michelson’s separating plate. Whereas Sagnac’s interest in the effects of rotation (of the ether) derived from the theoretical occurrence of the curl of the ether flow in his phase-shift formula, Lodge’s own concern with rotation (of the ether or of the optical apparatus) derived from his appeal to an optical circuit à la Fizeau. Whereas Sagnac’s test of the ether drag derived from his focus on rotational ether flow, Michelson’s similar test derived from direct analogy with the setup used in his verification of Fizeau’s result of 1951. These differences in motivation had consequences on the experimental setups. In particular, Sagnac used optical circuits of variable shape with a variable number of mirrors because he knew in advance that the phase-shift only depended on the (oriented) surface of the circuit (for a constant curl of the ether flow). In contrast, Lodge and Michelson favored a circuit involving two parallel beams, by geometrical simplicity and by continuity with Fizeau’s experiment. They were both unaware of the superiority of circuits involving an even number of mirrors (for the broad, weakly coherent beams of the time).100 So too are modern commentators of Sagnac’s experiments. What is now called a Sagnac interferometer truly is a three-mirror rectangular-circuit interferometer, in which Sagnac rather saw a strioscope.

Granted that Lodge was first to conceive the Sagnac effect on a turntable, it remains true that Sagnac was first to experimentally demonstrate the effect. This priority is neither a consequence of Lodge’s fear of centrifugal force nor an

98 Langevin starts with the Minkowskian metric formula ds2 = c2dt2 − dr2 − r2dθ2 in polar coordinates and for observers at rest. The coordinate change θ ′ = θ − ωt gives ds2 = (c2 − r2ω2)dt2 − 2r2ωdθ ′dt − dr2 − r2dθ ′2. With respect to these coordinates the propagation of light is anisotropic because of the rectangular term in dθ ′dt . In order to restore the isotropy of light propagation with respect to the disk, Langevin introduces the proper time t′ =

!(γ −1dt − γωr2c−2dθ ′) for which ds2 = c2dt′2 − dr2 − (1 − ω2r2/c2)−1r2dθ ′2. The synchronization of neighboring clock obtains for dt′ = 0, which is

equivalent to the condition dt = γ 2ωr2c−2dθ ′ derived from the local Minkowskian structure.99 Langevin [155,156]. Cf. Dieks [157]. For a modern argument, cf. Gourgoulhon [158, Chap. 13]. For the argument in general relativity, cf. Landau and

Lifshitz [159, §89].100 See footnote 73 above.


Fig. 30. Harress’ rotating glass-ring interferometer (the average radius of the ring is about 20 cm). From Harress [160, p. 30].

automatic consequence of Sagnac’s higher commitment. Rather, Sagnac benefited from two advantages of his interferometry: higher fringe contrast thanks to the double-prism separator, and avoidance of aerial fluctuations thanks to the even number of mirrors. Had Lodge or Michelson tried the similar experiment with a square optical circuit they would probably have failed because this arrangement would have been too susceptible to the perturbations caused by the rotation of the table.

7.7. Harress

In the years 1909–1911, the doctoral candidate Franz Harress experimented on the propagation of light in moving glass in the cellars of the observatory of Jena University. Thanks to Rudolf Straubel’s support, he benefited from custom-made glass pieces from the Carl Zeiss company. In order to get a high constant velocity of the moving glass, he had the idea of using the rotating polygonal arrangements of glass prisms shown on Fig. 30. The light L from a collimator is divided by the split glass cube W (analogous to Sagnac’s double prism) and directed toward the center of the turntable by the glass reflector R. The reflectors R1 and R2 then direct the two light beams to the entrance of the polygonal arrangement of ten prisms. By total internal reflection on the walls of the prisms, these beams travel on two contrary polygonal paths within the glass, then return to the glass cube, recombine, and emerge in a single beam L′ . All the glass pieces rotate together at uniform speed (about 700 turns/minute) around a vertical axis through the center of the polygon. As the collimator and telescope do no move, the observation must be done for flashes of light sent at a given constant phase of the rotation. Harress found that the rotation caused a shift of the interference fringes (of the order of one tenth of the fringe spacing), and measured this shift with a precision of about 10%.101

On the theoretical side, Harress reproduced Fresnel’s derivation of the value 1 − 1/n2 for the dragging coefficient α; he recalled the slightly different formula that Lorentz had obtained in his electron theory: α = 1 − 1/n2 − (λ/n)(dn/dλ), and he proved that the latter formula derived (in a first approximation) from Einstein’s relativistic composition of velocities if the Doppler shift of the light entering the moving transparent body was taken into account. In the theoretical analysis of his own experiment, he simply added the time shifts caused by the partial ether drag on each portion of the polygonal light path. His result for the total time shift may be rewritten as δt = −2n2αωS/c2, wherein n is the optical index of the glass, ω the angular velocity, and S the surface of the light circuit. Drawing the dragging coefficient from the measured phase-shift, he found it to be 30% below the theoretical values given by Fresnel and Lorentz (which differ only by 2% from each other).102

Unfortunately, a grave error jeopardized Harress’ conclusions. In his derivation of the expression δt = −2n2αωS/c2 of the time shift, he used the expression l/(c/n + αu) for the time taken by light to travel the element l of glass moving at velocity u in the direction of light propagation. In reality, the velocity of light with respect to the dragged ether is c/n and the velocity of the dragged ether with respect to the laboratory is αu, so that c/n + αu represents the velocity of light with respect to the laboratory. Its velocity with respect to the glass is c/n + αu − u, and the time taken by light to travel the length l of glass is l/(c/n + αu − u). Hence α should be replaced by α − 1 in Harress’ formulas, and the correct value of the

101 Harress [160]. On the precision, cf. Knopf [161, p. 433].102 Harress [160], pp. 3–4 (Fresnel), 6 (Lorentz), 7–13 (relativistic), 59 (total time shift), 70 (results).


total first-order time shift is δt = 2n2(1 − α)ωS/c2. With this correction, Harress’ measurements agree with the theoretical dragging formulas within the margin of experimental error.

7.8. Harzer, Einstein, and Laue on Harress

Harress published his highly skilled but flawed dissertation work in 1912. The astronomer Paul Harzer corrected the calculation error in 1914 in the Astronomische Nachrichten. For Fresnel’s value of the dragging coefficient, the corrected time shift becomes independent of the index n and agrees with the value 2ωS/c2 given by Sagnac’s principle of the effect of mo-tion. For Harzer, this was only a coincidence and not a very significant one because he and Harress favored Lorentz’s value of the dragging coefficient. As Einstein soon explained in the same journal, in the case of Harress’ experiment (unlike Fizeau’s) the relevant value of the dragging coefficient is Fresnel’s because the light enters the rotating glass polygon perpendicularly and there is no modifying Doppler shift. Harress did not have a say, as he unfortunately died on the battlefield.103

After the war, Max Laue convinced Harress’ doctoral adviser, Otto Knopf, to publish the details of Harress’ experiment in Annalen der Physik, because his dissertation was not commonly available and because the analysis of the experimental results needed revision. Laue himself contributed a detailed proof that (special) relativity theory led to the value 2ωS/c2 of the time delay on an optical circuit, whatever be the optical index on this circuit. In 1911 he had already shown that the then competing theories of the optics of moving bodies, those of Lorentz, Cohn, and Einstein all yielded the time-shift formula given by Michelson in his memoir of 1904 on an optical effect of the rotation of the earth. In his article on Harress, Laue explained that the time shift could be obtained either by applying the relativistic law of composition of velocities in the inertial frame tangent to an element of the optical circuit (to put it in modern terms) or in applying the Lorentz coordinate transformation to the times at which the light reaches the two extremities of the element. To first order the latter procedure is strictly equivalent to Sagnac’s principle, which Laue ignored. In both cases, it is assumed that the rotation of the system does not significantly alter the light path. As Sagnac and Harzer had earlier argued, this rotation in fact curves the light rays. Laue confirmed that this curvature did not significantly alter the time shift. Following a hint by Wien, he also stated that any general-relativistic effect of centrifugal forces could probably be neglected, for these forces were of the same order as the usual gravitational forces.104

Laue mentioned Sagnac’s experiment at the beginning of his article, as a variant of Harress’ experiment in which air is used instead of glass as the medium of propagation. In passing he noted that “Sagnac’s interpretation of his experiment as a proof of the existence of an ‘ether’ was not compelling at all.” The remark had already been made by several physicists, including Luigi Puccianti and Hans Witte in 1914. It is indeed obvious that the Sagnac effect does not contradict special relativity since the latter theory forbids only effects of uniform translation on optical experiments. Even without Laue’s or others’ learned treatments, it is easy to understand that the Sagnac effect comes out the same in special relativity and in any stationary-ether theory, since it can be derived by tracing the progression of light from one moving mirror to the next in the laboratory frame, as known to Lodge, Michelson, and Sagnac. Since the early 1920s, many physicists have discussed relativistic explanations of the Sagnac effect in a deeper manner, also in a general-relativistic context. Their efforts have not prevented the anti-relativist sect to brandish the Sagnac effect as a proof of the ether. Those developments go beyond the intended scope of the present study.105

7.9. Harress and Sagnac

Laue’s and Knopf’s memoirs may be seen as an implicit attempt to give Harress some credit for the discovery of the Sagnac effect, although they do not contain any priority claim. When in 1926 the Hungarian physicist Béla Pogány performed an improved version of Harress’ experiment, he called it the “Harress-Sagnac” experiment. This may be seen either as a way to indicate the similarity of the two experiments, or, less likely, as a way to downplay the importance of Sagnac’s contribution. Nowadays, some authors use the expression “Harress-Sagnac effect” instead of “Sagnac effect.” It is not the historian’s business to decide which is fairer.106

It is instructive, however, to compare the motivations and achievements of the two physicists. Sagnac was expecting a certain effect for theoretical reasons, and he designed an experiment that was explicitly meant to establish this effect. In contrast, Harress wanted to measure the Fresnel dragging coefficient in glass and his glass-ring interferometer was the instrument of that measurement. In his investigations, the effect of rotation on interference intervened only as a means to relate the measured fringe-shift to the dragging coefficient. He did not regard this effect as something essentially new and rather saw it as a straightforward consequence of the Fresnel drag applied to the elements of a transparent rotating ring. His theoretical expression of this effect was so wrong that it gave a zero effect in the n = 1 case, which is precisely the case of Sagnac’s experiment.

103 Harress [160]; Harzer [162]; Einstein [163]. Einstein’s paper triggered a small polemic with Harzer: see Harzer [164], Einstein [165].104 Knopf [161]; Laue [166,167].105 Laue [166, p. 449]; Puccianti [168]; Witte [169]. On relativistic derivations of the Sagnac effect, cf. Post [139]; Martinez-Chavanz [6, Chaps. 7–8]; Malykin [170,171]; Rizzi and Ruggiero [172]; Gourgoulhon [158, pp. 218–228].106 Pogány [173].


It is sometimes said that Harress’ measurements provided much more precise evidence for the Sagnac effect than Sag-nac’s own experiment. In reality the precision was comparable, because what Harress won on higher rotation speed and smaller index fluctuation he lost on smaller surface of the optical circuit and unwanted beam deviations. This is why Knopf and Laue called for a repetition of his experiment and Wien supported Pogány’s answer to that call.107 To sum up, Harress had the general idea that rotation may induce fringe-shift; he verified the existence of such an effect in glass; and he per-formed measurements that could later be used to confirm the Sagnac effect; however, the theoretical interpretation he gave to his experiment is utterly incompatible with the true Sagnac effect.

8. Intermezzo: war acoustics

In the years following Sagnac’s publication of his ether-whirling effect, the war directed Sagnac and other physicists to more urgent problems. Harress lost his life, and Sagnac switched to acoustic researches which he thought could benefit the French army. His activity in this domain was very intensive and a large quantity of relevant manuscript materials can be found in his personal archive. His principal achievement was an acoustic interferential telescope through which the fearful Zeppelins could be detected at night time. Seven hundred fifty such instruments of large size were built and successfully used by the French army on various fronts. Toward the end of the war Sagnac also invented a high-pitch trumpet that permitted communication between the battlefront and the rearguard. The three hundred copies or so he built with his assistant at the Sorbonne served the Champagne army. Even in those times, optics was still inspiring Sagnac’s innovations.108

9. The double mechanics

After the war, Sagnac resumed his optical researches with a new aim: produce a credible alternative to the relativity theory whose counterintuitive assumptions he and many of his French colleagues strongly disliked. While Arthur Eddington was bringing Einstein’s theories to the world’s attention through his solar-eclipse expedition, Sagnac began publishing on “an absolute mechanics of undulations” that was doomed to remain a very personal enterprise.

9.1. Outer laws, inner laws, and their liaison

In a series of notes published in 1919–1920 in the Comptes rendus, Sagnac distinguished between “outer laws” regarding the (time-)average of the total energy of radiation and “inner laws” regarding the details of wave propagation in the ether. Whereas the former laws strictly obeyed Galilean relativity, the latter depended on the global motion of the system through the ether. At the outer level, the propagation of luminous energy had to be isotropic in a frame attached to the source; a flash of light sent by a source moving at the velocity u in the direction of motion had to travel at the velocity c + u, as it would in any emission theory; interference patterns also had to be invariant under a uniform translation of the system, as they would if the ether were completely dragged by the system.109

These assumptions seem completely at odd with the assumption of a stationary ether. Sagnac needed to conciliate the Galilean invariance of his outer laws with the inner laws of wave propagation in this ether. To this effect he relied on his old principle of the effect of motion and on analogy with Gouy’s distinction between phase velocity and group velocity. Call k0 and ω0 the wave number and angular frequency of the wave emitted by a sinusoidal point source at rest in the ether. According to Sagnac’s principle, the amplitude at point R for the radiation from a source S is obtained by summing the partial amplitudes of various paths between S and R, each path being a series of free propagations between successive point-like secondary sources (the molecules of matter in the case of propagation in transparent matter, or the points of a hole on a screen in the case of diffraction); and the only effect of the uniform translation of the whole system is a phase shift on each element of the path of propagation. This phase shift corresponds to an alteration of the wave number k depending on the original value k0 of this number, on the translational velocity u of the system, and on the angle αthat the direction of propagation makes with this velocity. In addition, Sagnac assumes that the frequency ω of the wave is altered in a manner depending on the velocity u only. Euclidean symmetry and homogeneity imply the forms110

kk0

= 1 + F%u2/c2&u

ccosα, and

ω

ω0= G

%u2/c2& (1)

No matter how the functions F and G are chosen, the relation between ω and k remains linear and there is no dispersion in the usual sense. Undeterred by this difficulty, Sagnac considered the superposition of two waves of the same kind but emit-ted by sources with the different velocities u and u + du, the increment du being parallel to the direction of propagation. In the frame moving at the velocity u, the phase difference of the two waves in this direction is

dϕ ='(ω + dω)t − (k + dk) · (r − tdu)

(− [ωt − k · r]

107 Pogány [173, p. 217].108 Cf. P. Sagnac [19, p. 144]; Berthelot [174, p. 232]; Sagnac [20, pp. 3–4].109 Sagnac [175–181]. On this theory, cf. Martinez-Chavanz [6, pp. 51–56].110 Sagnac [175, p. 531].


Fig. 31. Sagnac’s table explaining how the “absolute wave fields” F lead, by superposition according to the “new absolute mechanics of waves” (NRF) to “external relativity” and “dynamical drag” (energy projected by the source) for the average energies of radiation. The letter N stands for Newtonian relativity. The equations in the right column correspond to the Eqs. (3) below. Those in the central column correspond to Eqs. (1) and (2). From Sagnac [176, p. 644].

The resulting beats travel at the velocity

U = dω

dk+ k · du

dk

In analogy with the ordinary concept of group velocity, Sagnac identifies this velocity with the velocity of propagation of the energy from the source. According to the external principle of relativity, this velocity must be the constant c. This condition and the relations k · du = kcdυ , du2 = 2c2υdυ with υ = (u/c) cosα lead to the equation

2υG ′ + 1 + υ FF + 2υ2 F ′ = 1, or (F − 1) − υ

%F + 2G ′& − 2υ2 F ′ = 0 (2)

The left side of this last equation is a polynomial in cosα with coefficients that are functions of u only. Its vanishing requires F − 1 = 0, F + 2G ′ = 0, and F ′ = 0. The unique solution for which G(0) = 1 is F = 1 and G = 1 − u2/2c2. Sagnac thus got the “rigorously true” equations (see Fig. 31)111:

kk0

= 1 + uc

cosα, andω

ω0= 1 − u2

2c2 (3)

9.2. Empirical consequences

The first of Eqs. (3) leads to the phase shift k0(u/c)dl cosα = ω0u · dl/c2 on the elementary light path dl that makes the angle α with the velocity u. Sagnac’s old principle of the effect of motion, and the resulting invariance of interference and diffraction phenomena thus hold at any order in u/c. Consequently, the theory explains the null-result of the Michelson–Morley experiment of 1887. It also accounts for the Sagnac effect by giving the value c ± |ω|r to the orthoradial velocity of light with respect to the rotating table (r being the distance from the center, and ω the angular velocity). Yet some of its predictions differ from those of the theories of Lorentz, Poincaré, and Einstein. For Sagnac, the clocks in the moving frame still give the absolute time and the measuring rods still give the absolute length, even if they are defined by optical means. The velocity of light flashes (not the phase velocity) depends on the velocity of the source as in an emission theory. Sagnac proposed a crucial experiment in which the velocity of the light from the fast ions of canal rays would be measured by Foucault’s method of the rotating mirror. In his theory the Doppler shift of this light should imply a velocity of the emitter in the direction of the emission, which in turn should imply an alteration of the velocity of light in the laboratory frame.112

Sagnac also expected the difference between the phase velocity ω/k of the “inner mechanics” of waves and the velocity c of the energy in the “outer mechanics” to be accessible to experience and thus to yield a measurable effect of the motion of the earth through the ether. Somewhat obscurely, he imagined that wave trains sent by an emitter in the direction of the ether wind would undergo a slow periodic inversion of phase with the spatial period λc/u owing to the difference between the velocity c of the front of the train and the phase velocity (about c ± u). For a detector responding to sufficiently large positive disturbances only, the response would depend on the sign of the first (largest) wave of a damped wave train and therefore would vanish at periodic distances from the emitter. The signals of wireless telegraphy could provide the desired damped wave trains, the Branly coherer the sign-dependent detector.113

111 Sagnac [175], pp. 471 (group velocity), 531 (Eq. (3)). The derivations of Eqs. (2) and (3) are mine. Even in his most detailed accounts [176,182], Sagnac remained allusive.112 Sagnac [177], pp. 783 (Michelson), 784 (canal rays), 785 (Sagnac effect); [180].113 Sagnac [178].


Fig. 32. Sagnac’s drawing for periodic silences in telegraphic transmission. The original signal (départ) is a bump (+) followed by weaker undulations. At 40 km from the source, the bump becomes a hollow (−) and it is undetected by a Branly tube; and 40 km further the head bump reappears and the reception is reestablished. From Archives nationales.

9.3. Radio-silences and double stars

In 1920, Sagnac proudly announced that the predicted periodic silences of telegraphic communication had already been observed in the late 1890s by the British admiralty during a systematic study of communication between two ships at increasing mutual distance:

“Wireless telegraphy, unwittingly preceding the theory, discovered in 1895–1902, of the relative motion of the waves and the energy, which unprepared experimenters judged to be a simple effect of interference – an effect truly impossible under the usual conditions at sea.”The inferred value of the velocity of the earth through the ether would be of the order of 100 km/s, in agreement with the known value of the relative velocity of the stars and the solar system. The double mechanics of waves and energy was thus confirmed. Sagnac called the periods of silence “fits” (accès) in a transparent allusion to Newton’s fits of easy reflection. He indeed regarded his double mechanics of waves and projected energy as similar to Newton’s combination of projected light corpuscles and induced ether waves.114

In the same year 1920, Sagnac’s position at the Sorbonne rose to maître de conférences on theoretical physics and celestial physics. A letter to his brother Philippe informs us of the contents of his inaugural lecture. He first reminded his audience how Newton and Fresnel had introduced the ether as a medium of propagation, and how the lack of observed effects of the ether wind had recently led the relativists to reject this intuition. He commented:

“This gigantic error is comparable to the error of a visitor to an alternative-current factory who would mistake the alternative current for a continuous current and would want to use only average-energy receptors that cannot oscillate with the current.”The simple connection between matter and light expressed in the relativity principle, Sagnac continued, only applies to the total energy of light. This energy is normally recorded in a manner independent of the inner undulation, “as the energy of an alternative current is recorded by a thermal apparatus too sluggish to oscillate with the current.” This total energy occurs in “flashes projected by the source in the manner of the bullets of a gun.” There follows an aquatic metaphor:

“In contrast, the waves are bound to the non-dragged ether in which the flash propagates and with which it is bound through its undulation, as an eel swims and follows the undulation it creates on the water; this undulation run along the body of the eel, thus forming sometimes a bump, sometimes a hollow near its head.”Sagnac thus meant to explain the evolution of a telegraphic signal when the distance from the emitter increases: the detector (coherer) fails to detect this signal at periodic intervals of distance because it detects the bumps only (see Fig. 32).Sagnac had some fifteen students and a couple of colleagues in his audience. They were quite attentive and they applauded him.115

From his new chair, Sagnac was also required to teach “celestial physics.” He used this duty as an opportunity to find astronomical proofs of his theory. In 1922, he became aware of an anomaly in the Doppler shift of the light emitted by some double stars (Cepheids): the extrema of the shift did not coincide with the phases of the motion of the emitting star in which the velocity in the direction of observation was an extremum. Sagnac explained this anomaly by the different retardation of the emitted light when the source is moving toward and against the terrestrial observer. In his theory, a Doppler shift δλ/λ indeed implies the velocity c − cδλ/λ for the propagation of energy from the source. Sagnac found quantitative agreement between this differential retardation and the observed phase anomaly. His remaining publications, until the last in August 1924, all bore on double stars and their observed phase anomalies. They were meant to verify his theory and to disprove general relativity.116

114 Sagnac [179]; [181, p. 102]; Jackson [183, pp. 268–271] (observed anomaly). Captain Jackson’s explanation of the anomaly, based on wavelength differences in the successive wave trains from the emitter, seems reasonable to me.115 G. Sagnac to P. Sagnac, undated, Archives nationales.116 Sagnac [184–188]. Cf. Martinez-Chavanz [6, pp. 61–62]. Sagnac did not mention that in 1913 Willem de Sitter had failed to observe a prediction of emission theories: the distortion of the orbits of double stars.


9.4. Reception

Not much can be said on the reception of Sagnac’s theory. As we saw, the Sorbonne students were quite receptive. So too were a probably large number of Sagnac’s colleagues, as relativity theory was still unpopular among French academics.117

In 1919, Sagnac got a prestigious prize of the Académie des sciences, the Pierson–Perrin prize, for his oeuvre in physics. In his report, the freshly elected academician Daniel Berthelot summarized Sagnac’s new theory and judged: “This penetrating synthesis, based on a subtle analysis of phenomena and on a series of ingenious new experiments, reveals a great effort of thought which, beyond mathematical formulas, endeavors to reach the reality of physical phenomena.” Interestingly, in his report on Sagnac’s candidature for a vacant seat at the Académie des sciences in 1923, Berthelot dropped most of his description of Sagnac’s theory as well as the cited praise. Perhaps he then judged that the winds had turned in favor of relativity theory.118

Most enthusiastic was the French popularizer of Einstein’s theories, Lucien Fabre. In the second edition of his widely ready read The new figure of the world: Einstein’s theories, published in 1921, Fabre appended an account of Sagnac’s theory by Sagnac himself. Fabre’s pompous introduction reads:

“M. Sagnac, of whom it may have been written, in an allusion to the sentence that closes this book, that he perhaps was the new Poincaré, the only person able to give a definitive answer about the value of Einsteinian theories, agreed to entrust to this humble volume an original note whose extraordinary importance shall not escape anyone’s attention. . . . I am extremely happy to give to my readers fresh news about a work that seems to contain in germs the most beautiful discoveries.”The final paragraph of the book reads:

“The human mind remains perplexed. On the one hand, the mind’s taste for simplification, the incontrovertible fact of universal relativity, the clarity of the doctrine, the mystical drive toward unity, the marvels of Einstein’s discoveries, incite the mind to adhere to relativity theory. On the other hand, the quest of the absolute, the instinct of perenniality, the urge for intellectual images, the confusion between the necessary and the given, the sense of immediate causality, bend the mind toward less audacious theories such as Lorentz’s. At this moment, what we miss is the lights of a Poincaré.”Fabre’s introduction of Sagnac as the Poincaré of the situation is not as farfetched as it might seem. Poincaré was not entirely pleased with relativity theory, although his reasons differed from Sagnac. With Sagnac he agreed that the relativity principle and the associated symmetry were not a sufficient basis for the unification of physics, that they should not compel us to give up our ancestral concepts of space and time, and that the ether should still play an essential role in the new physics. Unlike Sagnac, however, he believed in the strict validity of the relativity principle and he agreed with Einstein that the times and lengths measured by optical means depended on the velocity of the inertial frame (although in his view the measurement performed in the ether frame were the only ones yielding the “true” values).119

In the spring of 1922 Einstein visited Paris and gave a detailed report on the theories of relativity at the Collège de France, followed by two rounds of discussions with physicists and mathematicians. Sagnac spoke in the first round. Here is a contemporary account of his intervention by one of the auditors, the physicist and poet Jean-Baptiste Pomey:120

“M. Sagnac insisted in throwing a summary of his theories to the attendance.– What was the attitude there? Did one acclaim the Frenchman who was opposing a clear doctrine to the revolutionary

doctrine of the German?– No. At first, it was a scandal because of the violence of the intervention. Mr. Einstein was quite disconcerted. Perhaps

he expected a fight on political grounds, but this brutal attack on scientific grounds flustered him. Fortunately, he was made to understand that the best was to reply nothing. At any rate Mr. Sagnac spoke very fast, with strong, clear, and imperative voice, in incisive sentences, as if he was bringing the absolute and incontrovertible truth and as if he had the sacred duty to make this truth heard in spite of everything. At any rate his address sounded like a manifesto. For he did not do what was needed to make himself understood, and I myself, having read several of his communications without truly grasping their content, would have been happy to get the much desired explanation in this occasion. Unfortunately, I suppose he saw himself as facing irreducible adversaries, and he made no concession to the attendance. He went on and on despite the interjections that came from certain rows, as if he was proclaiming or protesting, without bothering to bring light or to convince. The attendance let the storm pass, and, when he at last took his seat, the discussion started again as if his communication had never happened. As he received a rather important prize at the Institut [de France], I imagine his works must have some interesting content and might have deserved proper scrutiny. But that outburst was totally unfit to induce any exchange of useful observations.”

Although Pomey plausibly inflated the incident to amuse his readers, his account confirms a sadly verifiable fact: even in publications Sagnac only sketched his theory and he made little effort to be understood. As a consequence, other experts in the optics of moving bodies or in relativity theory ignored his theory, as Einstein did at the Collège de France. The praise

117 Cf. Paty [189]; Biézunsky [190]; Borella [191]; Moatti [192].118 Berthelot [174, p. 1231]; Berthelot’s report of 1923 in the Sagnac folder of the archive of the Académie des sciences.119 Fabre [193, pp. 18, 241]. On Poincaré’s views, see Darrigol [194].120 Pomey [195, pp. 204–205].


that Sagnac got in his own country came from students or physicists like Berthelot who had no expert knowledge of this domain of physics and a natural antipathy for relativity theory. Berthelot’s anti-relativist pamphlet of 1922 was full of crude misunderstandings of both Lorentz’s and Einstein’s theories.121 Fabre’s New figure, though more knowledgeable and more sympathetic to relativity, was but the work of a clever amateur.

The only competent report I have found on Sagnac’s theory is the one given by the Belgian astronomer and mathe-matician Maurice Alliaume in a review of astronomy and relativity for the year 1923. Alliaume correctly observes that an in-depth study of this theory should start with Sagnac’s old explanation of the Fresnel drag, and he puts his finger on an es-sential difference between Sagnac’s theory and relativity: whereas Sagnac admits the inner mechanics of the “hidden reality” of phase waves, relativistic mechanics is “essentially based on the principle of identity of indiscernibles.” Although Alliaume refrains from explicit judgment, the words through which he introduces Sagnac’s theory (and Lenard’s double-ether theory of 1923) are not very engaging:

“Einstein has reconciled [the Michelson–Morley result with stellar aberration] by a new theory of light that upsets our most entrenched conceptions and removes the difficulty by negating the ether. His adversaries reconcile [the antagonist facts] and will go on doing so in the framework of the more traditional physics through new theories of light, and each one will have his own theory. We have had the theory of G. Sagnac. . . ”Besides Alliaume, the only experts who publicly reacted to Sagnac’s attempt were the reviewers for the American and German abstracts of physics literature. The general feeling was that the theory could be safely ignored until Sagnac would do enough to make himself understood.122

This lack of active, competent feedback may partly be seen as a consequence of the general agreement, among experts on the electrodynamics of moving bodies, that Lorentz’s and Einstein’s theories were the best theories. Earlier in the century, say until the 1910s, it was still permitted to hesitate between several options: Einstein’s theory, its more conservative but empirically equivalent Poincaré–Lorentz version, Emil Cohn’s macroscopic theory of 1904, and Walther Ritz’s emission theory. In the course of time, however, the superiority of Einstein’s approach grew more and more evident to the specialists. New alternatives were not likely to be taken seriously.123

9.5. Critical assessment

There are, in addition, intrinsic features of Sagnac’s theory that make it less credible than earlier alternatives to Einstein’s theory. Three issues are at stake here: the compatibility of the theory with experimental facts, its inner consistency, and its scope. Regarding the first issue, around 1920 there was no clear-cut proof that the velocity of light did not depend on the velocity of this source. The most publicized proof, Willem de Sitter’s failure to observe any distortion of the apparent orbit of double stars in 1913, was still under debate.124 Perhaps this is the reason why Sagnac, despite his interest in double stars, ignored this refutation of emission theories. More puzzling is his willingness to regard single, vague observations done in complicated circumstances as strong evidence in favor of his theory. It sounds as if the aging Sagnac, who had been a very careful experimenter, now contented himself with fragile, largely subjective confirmation.

Regarding the issues of consistency and scope, Sagnac’s theory fares even worse. At first glance it is tempting to compare this theory with Ritz’s and Cohn’s theories, which also explain the absence of effects of the earth’s translational motion on (most) optical experiments (including the Michelson–Morley experiment) without leaving the framework of Galilean kinematics. Ritz’s theory does so by referring wave propagation to the moving source, thus eliminating the ether and even the possibility of simple wave equations. Cohn’s theory does so by modifying the Lorentz–Maxwell equations for the electrodynamics of moving bodies in such a manner that their form in a frame moving at the velocity u (with respect to absolute space) differs from their form in the absolute frame only by the shift t → t + u · r/c2 of the time variable at any order in u/c. This shift being unobservable in optical experiments in which stationary light patterns are observed, the theory is compatible with the observed absence of effects of the motion of the earth.125

These two theories were fully worked out in all mathematical details and they embraced electromagnetic phenomena, whereas Sagnac’s theory remained rudimentary in its mathematical development and was mostly confined to optics (save for its application to electromagnetic radio waves). This is a first important difference. Still one might hope to develop Sagnac’s theory into a theory of comparable power. This task would be easy if one could show that Sagnac’s theory is a variant either of Ritz’s theory or of Cohn’s. The first option is not open, because wave propagation in Ritz’s theory is entirely relative to the source, whereas in Sagnac’s theory it is only the propagation of the energy of the waves that is relative to the source. At first glance, the Cohn option looks more promising because Sagnac’s modification of the wave number in a

121 Berthelot [196]. For instance Berthelot confused the dilation of time with the Doppler effect (p. 28). He portrayed relativity theory as the work of a wandering Jew who did not need the home of Newtonian space and time: cf. Lalli [15, p. 74].122 Alliaume [197, pp. 177–179]. I have consulted the Science abstracts, section A: Physics, the Physikalische Berichte, and the Jahrbuch über die Fortschritte in der Mathematik. Most reviews summarize Sagnac’s results without judging them. In his review of Sagnac [181] for the Jahrbuch, Herman Müntz writes “Die Ausführung wird nur ganz flüchtig angedeutet. . . . Es wird nicht gesagt, welches Verhalten der von einer bewegten Quelle ausgehenden Wellen jenen Hypothesen entsprechen muss, wodurch die Grundlagen der Theorie erst geklärt werden würden.”123 On the early alternatives to Einstein’s theory, cf. Darrigol [90, pp. 385–392].124 Cf. Martínez [198, p. 16]. For contestations of Sitter’s conclusion, see Alliaume [197, pp. 167–168].125 On Cohn’s theory, cf. Darrigol [199]; On Ritz’s, cf. Martínez [198], Darrigol [40].


moving frame is equivalent to the shift t → t + u · r/c2 of the time variable. The similarity stops there, however, because the dispersion relations of the two theories differ:

ω2 = c2%k − ωu/c2&2for Cohn; ω

k= c

1 − u2/2c2

1 + k · u/kcfor Sagnac.

Moreover, the predictions of Cohn’s theory differ from Lorentz’s only at second order in u/c, whereas in Sagnac’s theory the projection of electromagnetic energy implies a first-order departure from Lorentz’s theory.

The way in which Sagnac conciliates the isotropy of energy propagation from a moving source with his laws (3) of wave propagation from a moving source is quite strange. As we saw, the implied wave groups involve the superposition of waves from sources moving at slightly different velocities. There is no dispersion in the ordinary sense (different phase velocities for different wavelengths), there is only a dependency of the phase velocity on the velocity of the source and the direction of propagation. This would be fine if every wave train in nature was made of the superposition of monochromatic waves from sources moving at different velocities. But I do not see why this should be the case: a typical wave train either comes from a damped source with a well-defined velocity, or it is obtained by chopping the undulation from a permanent source.

A last defect of Sagnac’s theory is the lack of any simple wave equation leading to the expressions (3) of the frequency and wave number. In particular, his laws of wave propagation do not agree with the value c − k · u/k of the relative phase velocity ω/k that one expects in a stationary-ether theory (it does so only to first order in u/c). Sagnac did not worry about these difficulties, presumably because he reasoned by simple geometric arguments that did not require any differential equation or any precise ether mechanism. All along his career he avoided the extensive mathematical apparatus that had become the norm in theoretical physics.

9.6. Delusion

To sum up, Sagnac’s last theory of wave propagation probably deserved being ignored by other theoreticians of optics and electrodynamics. It did not meet common standards of clarity and rigor, and its most specific assumption, a sort of dis-persion bridging the propagation of waves with the large-scale propagation of luminous energy, seems highly problematic. While reading Sagnac’s series of notes in the Comptes rendus one cannot help feeling that he was a victim of some sort of delusion. The occasional bursts of enthusiasm, the obscurity of many sentences, and the pompous announcements of an entirely new physics all suggest that Sagnac no longer was in possession of the qualities that permitted his earlier successes both in theory and in experiments. Intellectual solitude surely aggravated the problem. His only scientific collaborator, his beloved friend Pierre Curie, was no longer there to discuss physics with him; and he seems to have made no effort to discuss with their common friend Paul Langevin, who would no doubt have seen the weaknesses of Sagnac’s theory.

“How could a scientific life that started so happily and went on so brilliantly finish so early? For a number of years, the savant was no longer himself; he was a vaincu de la vie.” These are the words of Philippe Sagnac, in the concluding section of the obituary he wrote for his brother. From the same source we learn that Sagnac’s health deteriorated in 1924, so much that in 1926 he applied for retirement. Plausibly this decline began earlier: when, after the war, Sagnac indulged in a kind of speculation that he would earlier have been the first to criticize. Although the resulting publications can be safely ignored by today’s physicists, they are instructive to anyone interested in Sagnac’s scientific life for they extend the style of optical thinking that Sagnac had been defending since the early 1890s: a mostly kinematic approach with few equations and in a persistently Newtonian framework.

10. Friends, honors, and persona

Although Sagnac preferred to work alone or with his laboratory assistant, he had a few high-profile friends and support-ers, at home and abroad. The most powerful of them was Daniel Berthelot, son of Marcellin, member of the two academies, and senator. Berthelot wrote the reports that helped Sagnac win important prizes, and he supported his candidatures to the Académie des sciences. Being a physicochemist, he had little competence to judge Sagnac’s more theoretical work. His spontaneous dislike of relativity theory made up for that. Sagnac’s dearest and closest friend was Pierre Curie, his only collaborator in one occasion only. Curie’s accidental death in 1906 affected Sagnac for the rest of his life. The great Henri Poincaré shared Sagnac’s passion for optics and he appreciated his accomplishments: he supported one of his candidatures at the Sorbonne; he may have been responsible for Sagnac’s invitation to participate to the Lorentz jubilee volume (at least Sagnac thought so); and he exchanged a few letters with Sagnac. To be true, Sagnac did not show the best of himself in this correspondence: in the summer 1899 he believed to have found a violation of the principle of reaction in a unipolar induction device imagined by Wladimir de Nicolaiève. Poincaré immediately corrected him.126

In foreign countries, Sagnac’s most prominent acquaintance was Hendrik Lorentz. Sagnac’s contribution to the Lorentz jubilee prompted a friendly exchange of letters. Sagnac thus came to study the contents of Lorentz’s theory and, as we

126 On Berthelot, cf. Moatti [192, pp. 154–156]. On Curie’s friendship, cf. P. Sagnac [19, pp. 44–45]. On the relation with Poincaré, cf. Walter [29, Chap. 51]. The letters on unipolar induction and the reaction principle are ibid., pp. 324–327.


saw, he related his principle of the effect of motion to Lorentz’s local time. After visiting Lorentz and his wife in Leyden in the spring of 1905, he obtained valuable criticism of his mathematical theory of photographic action. Sagnac got to know Philipp Lenard in international congresses of radiology, and corresponded with him on various matters including Hertz’s mechanics, the photo-electric effect, and of course the Sagnac effect. In May 1914, Lenard wrote:

“I would like to express my special thanks for the interesting news about the rotating interferometer. We will have a special report on this apparatus and on your work with it at the Heidelberg physics seminar, and I am glad to have your explanations and the good illustration.”Perhaps Lenard, being a visceral anti-relativistic and anti-Semite, appreciated Sagnac’s alleged proof of the ether. He nonetheless devoted most of his long letter to shunning Sagnac’s request to arrange refereeing in a German publication, arguing that it would serve no purpose. The war put an end to the relations between the two physicists. A far more honest and faithful friend of Sagnac’s was the Norwegian meteorologist Vilhelm Bjerknes, whom he may have met while they were both students in Paris and whom he later visited in Oslo. Bjerknes’ only extant letter to Sagnac, sent in February 1914, concerns Sagnac experiment on the whirling optical effect. Then in Leipzig, Bjerknes discussed Sagnac’s experiment with the eminent optician Otto Wiener and came to the conclusion:

“It is an undeniable fact that your apparatus records its global rotational motion without recourse to any external ref-erence. In this regard your experiment will always remain a fundamental experiment, even one of the most fundamental experiments of physics. Yet it will not suffice to convince the staunch relativists of the existence of the ether. For the relativists are men who purport to imagine waves that propagate in a medium whose existence they deny.”Bjerknes may have enjoyed the discovery of whirling ether winds that Sagnac himself compared to Bjerknes’ atmospheric eddies. The rest of his letter shows some familiarity with Sagnac’s principle of the effect of motion:

“We would be very curious to know, Mr. Wiener and I, if in your experiment you still measure the velocity of light in vacuum even if the light propagates in water. A positive answer to this question would provide a brilliant confirmation of your theory, and would certainly be much unexpected for most physicists.”Indeed, this variant of Sagnac’s experiment is comparable to Harress’ experiment for which we saw that Sagnac’s principle and relativity theory both give the same phase shift as in a vacuum.127

Sagnac’s contemporaries recognized and rewarded Sagnac’s merits in various manners. They appointed him to increas-ingly high positions at the Sorbonne: agrégé préparateur in 1893–1900, chargé de cours in 1904–1912, professeur adjoint in 1912–1920, and maître de conférences in 1920–1926. He won several academic prizes: the Jérôme Ponti prize in 1904 to finance his optical researches, the Henry Wilde prize in 1917 for his war work, the coveted Pierson–Perrin prize in 1919, and the La Caze grand prize of physics in 1920. In the same year, he celebrated his election to the Conseil des observatoires du Mont-Blanc. For all that, he never obtained a full professorship at the Sorbonne, and he failed in his repeated candida-tures to the Académie des sciences (on Violle’s seat and on Bouty’s seat in 1923, and on Berthelot’s seat in 1927). Not too much can be inferred from this failure as his brilliant contemporary Langevin, for instance, did not enter the Académie until 1934.128

Sagnac’s few extant letters to friends and family, and the testimonies of his brother and of his colleagues convey the picture of a man modest and yet sure of himself, also gentle, attentive, and generous: un excellent cœur, summarized Henri Bénard. His occasional displays of aggressiveness, during his polemic with Italians X-ray physicists or during the Einstein event at the Collège de France, probably did not reflect his true temperament. They are easily explained by integrity in defending a lifelong scientific passion. Since his student years, Sagnac believed to have discovered a fundamentally new approach to optics that captured the essence of the propagation of light and offered better guidance in the laboratory than the more complicated theories of Lorentz and others. In spite of his fragile health (so fragile that Lenard believed him to be dead at some point),129 he spent an enormous amount of time and energy experimenting on light and on radiations he hoped to be similar to light. He thus opened a new subfield of X-rays physics; he explained fine features of the propagation of light such as the Gouy phase inversion; he developed a superior technique of interferometry; and he discovered the effect that now bears his name. Naturally he attributed these successes to his theoretical vision and to his experimental method, and he strongly defended both of them against perceived threats. This great coherence of Sagnac’s endeavors explains both his failure to catch the train of more modern theories and his success in discovering new entities, processes, and effects.130

Sagnac (Fig. 33) died on 26 February 1928 in Bellevue, aged fifty-nine, “suddenly” according to his brother, “after a long illness” according to Bénard. He lived long enough to see relativity theory and quantum theory shatter the classical-optical worldview that sustained his entire scientific life. A certain effect survived him. A lot more of his life is worth remembering.

Acknowledgments

My text has benefited from comments and suggestions by Roberto Lalli, an expert on the history of Sagnac’s and others’ ether-drift experiments. I am also grateful to Alexandre Gauguet, who induced me to write on Sagnac, to Christian Bordé,

127 Bjerknes to Sagnac, 1 Feb 1914, Archives nationales.128 For Sagnac’s Sorbonne career, cf. Maurain and Pacaud [200]. On the prizes, cf. the reports in the Comptes rendus and the Sagnac folder at the archive of the Académie des sciences. More biographical information is in Sagnac [20] and in Dostrovsky [201].129 Cf. Lenard to Sagnac, 19 Dec 1905, Archives nationales.130 Bénard [202, p. 46S]; Lenard to Sagnac, 19 Dec 1905, Archives nationales.


Fig. 33. Georges Sagnac at various ages. First picture from ENS school picture [3]; others from the Archives nationales (there being no name written on these pictures, their identity is not quite certain).

who kindly provided me with documents and information he had on Sagnac, to Jean-Michel Raimond, who helped me understand a few theoretical and experimental points, to Jacques Vigué, who shared his expert opinion on some peculiarities of Sagnac’s interferometry, to Regino Martinez-Chavanz, who lent me a copy of his valuable dissertation on the Sagnac effect, and to Sabine Clabecq for her help at the archive of the Académie des sciences. All pictures in this article are reproduced with kind permission of the Archives nationales, Paris, France.

Bibliography of Sagnac’s writings

This bibliography was compiled from Johann Christian Poggendorff’s Biographisch literarisches-Handwörterbuch and from the Catalogue of scientific papers of the Royal Society of London, with a few additions.

1893Georges Sagnac, Essai sur la photographie des couleurs par la méthode de Gabriel Lippmann, La photographie, 1893, also in Gaston-Henri Niewen-Glowkski and Armand Ernault, Les couleurs et la photographie, Paris, 1895, pp. 285–294.

1895Georges Sagnac, Sur la résistance au courant variable, Éclair. électr. 4 (1895) 66–73.

1896Georges Sagnac, Les phares tournants et l’illusion du point de fuite, Éclair. électr. 9 (1896) 373–375.Georges Sagnac, Sur la diffraction et la polarisation des rayons de M. Röntgen, C. R. Acad. Sci. Paris 122 (1896) 783–785.


Georges Sagnac, Illusions qui accompagnent la formation des pénombres. Applications aux rayons X, J. Phys. Theor. Appl. 6 (1896) 169–173.Georges Sagnac, Illusions qui accompagnent la formation des pénombres. Applications aux rayons X, C. R. Acad. Sci. Paris 123 (1896) 880–887.Georges Sagnac, Les rayons X et les illusions de pénombres, Éclair. électr. 9 (1896) 408–409.Georges Sagnac, Les expériences de M. H. Becquerel sur les radiations invisibles émises par les corps phosphorescents et par les sels d’uranium, J.Phys. Theor. Appl. 5 (1896) 193–202.

1897Georges Sagnac, Illusions qui accompagnent la formation des pénombres. Applications aux rayons X, Bull. Soc. Fr. Phys. (1897) 9–14.Georges Sagnac, Illusions de la vue qui accompagnent les défauts d’accommodation, Bull. Soc. Fr. Phys. (1897) 14–21.Georges Sagnac, Sur les propriétés des gaz traversés par les rayons X et sur les propriétés des corps luminescents ou photographiques, C. R. Acad. Sci. Paris 125 (19 July 1897) 168–171.Georges Sagnac, Sur la transformation des rayons X par les métaux, C. R. Acad. Sci. Paris 125 (26 July 1897) 230–232, 942–944.Georges Sagnac, Recherches sur la propagation des rayons X, Éclair. électr. 13 (1897) 531–539.Georges Sagnac, Illusions qui accompagnent les défauts de mise au point, J. Phys. Theor. Appl. 6 (1897) 169–173.Georges Sagnac, Sur la théorie des diélectriques et la formule de Clausius, J. Phys. Theor. Appl. 6 (1897) 273–276.Georges Sagnac, Sur une interprétation diélectrique de la formule de Fresnel, J. Phys. Theor. Appl. 6 (1897) 277–279.

1898Georges Sagnac, Sur le mécanisme de la décharge des conducteurs frappés par les rayons X, C. R. Acad. Sci. Paris 126 (1898) 36–40.Georges Sagnac, Transformation des rayons X par transmission, C. R. Acad. Sci. Paris 126 (1898) 467–470.Georges Sagnac, Emission de rayons secondaires par l’air sous l’influence de rayons X, C. R. Acad. Sci. Paris 126 (1898) 521–523.Georges Sagnac, Caractères de la transformation des rayons X par la matière, C. R. Acad. Sci. Paris 126 (1898) 887–890.Georges Sagnac, Mécanisme de la décharge par les rayons X, C. R. Acad. Sci. Paris 127 (1898) 46–49.Georges Sagnac, Transformations des rayons X par la matière, Bull. Soc. Fr. Phys. (1898) 115–140.Georges Sagnac, Luminescence et rayons X, Rev. Gén. Sci. Phys. Pures Appl. 9 (1898) 314–320.Georges Sagnac, Recherches sur la transformations des rayons X par la matière, Eclair. électr. 14 (1898) 466–474, 509–514, 547–555.Georges Sagnac, Théorie géométrique de la diffraction à l’infini des ondes planes par un écran percé par des fentes parallèles, J. Phys. Theor. Appl. 7 (1898) 28–36.Georges Sagnac, Sur un nouvel électro-aimant de laboratoire pour la production des champs très intenses, construit d’après les indications de M. Pierre Weiss, Bull. Soc. Fr. Phys. (1898) 37*–38*.

1899Georges Sagnac, Émission de différents rayons très inégalement absorbables dans la transformation des rayons X par un même corps, C. R. Acad. Sci. Paris 128 (1899) 300–303, 380.Georges Sagnac, Sur la transformation des rayons X par les différents corps, C. R. Acad. Sci. Paris 128 (1899) 546–549.Georges Sagnac, Nouvelle manière de considérer la propagation des vibrations lumineuses à travers la matière, C. R. Acad. Sci. Paris 129 (1899) 756–758.Georges Sagnac, Théorie nouvelle des phénomènes optiques d’entraînement de l’éther par la matière, C. R. Acad. Sci. Paris 129 (1899) 818–821.Georges Sagnac, Sur la transformation des rayons X par la matière, Éclair. électr. 18 (1899) 41–48.Georges Sagnac, Sur la transformation des rayons X par la matière, Éclair. électr. 21 (1899) 109–110.Georges Sagnac, Sur la transformation des rayons X par les différents corps simples, Éclair. électr. 18 (1899) 64–66.Georges Sagnac, Sur la transformation des rayons X par les différents corps, Éclair. électr. 19 (1899) 201–208.Georges Sagnac, Transformation des rayons X par la matière, J. Phys. Theor. Appl. 8 (1899) 65–88.Georges Sagnac, Remarques sur l’interprétation des expériences de MM. H. Haga et C.-H. Wind, J. Phys. Theor. Appl. 8 (1899) 333–335.Georges Sagnac, Théorie nouvelle de la transmission de la lumière dans les milieux en repos ou en mouvement, Bull. Soc. Fr. Phys. (1899) 162–174.Georges Sagnac, Théorie nouvelle de la propagation de la lumière à travers les corps en repos et en mouvement, Bull. Soc. Gens Sci. (Nov 1899) 17–20.

1900Georges Sagnac, Rayons X et décharge: généralisation de la notion de rayons cathodiques, C. R. Acad. Sci. Paris 130 (1900) 320–323.Georges Sagnac, Pierre Curie, Électrisation négative des rayons secondaires produits au moyen des rayons Röntgen, C. R. Acad. Sci. Paris 130 (1900) 1013–1016.


Georges Sagnac, Relations nouvelles entre la réflexion et la réfraction vitreuses de la lumière, Arch. Neerl. 5 (1900) 377–394.Georges Sagnac, Explication nouvelle de la propagation de la lumière à travers les milieux doués d’une absorption élective, Bull. Soc. Fr. Phys. (1900) 3*–4*.Georges Sagnac, Théorie nouvelle de la transmission de la lumière dans les milieux en repos ou en mouvement, J. Phys. Theor. Appl. 9 (1900) 177–189.Georges Sagnac, La propagation de la lumière à travers les corps en repos ou en mouvement, Rev. Gén. Sci. Phys. Pures Appl. 11 (1900) 243–249Georges Sagnac, De l’Optique des Rayons X et des Rayons Secondaires qui en Dérivent, Gauthier-Villars, Paris, 1900. Also in [80,203,81].

1901Georges Sagnac, Mode de production de rayons lumineux divergents à 180 du Soleil, C. R. Acad. Sci. Paris 133 (1901) 703–704.Georges Sagnac, Pierre Curie, Électrisation négative des rayons secondaires issus de la transformation des rayons X, Bull. Soc. Fr. Phys. (1901) 179–187.Georges Sagnac, Nouvelles recherches sur les rayons de Röntgen, J. Phys. Theor. Appl. 10 (1901) 668–685.Georges Sagnac, Propagation des rayons X de Röntgen, Ann. Chim. Phys. 22 (1901) 394–432.Georges Sagnac, Rayons secondaires dérivés des rayons de Röntgen, Ann. Chim. Phys. 22 (1901) 493–563.Georges Sagnac, Relation des rayons X et de leurs rayons secondaires avec la matière et l’électricité, Ann. Chim. Phys. 23 (1901) 145–198.Georges Sagnac, Nouvelle expérience d’interférence avec le biprisme de Fresnel et avec les glaces argentées de Jamin, Rev. Gén. Sci. Phys. Pures Appl. 12 (1901) 639.

1902Georges Sagnac, Pierre Curie, Électrisation négative des rayons secondaires issus de la transformation des rayons X, J. Phys. Theor. Appl. 1 (1902) 13–21.Georges Sagnac, Principes d’un nouveau réfractomètre interférentiel, C. R. Acad. Sci. Paris 134 (1902) 820–821.Georges Sagnac, Sur la résistance électrique d’un conducteur magnétique ou diamagnétique parcouru par un courant variable et placé dans un champ magnétique, J. Phys. Theor. Appl. 1 (1902) 237–238.Georges Sagnac, L’origine du bleu du ciel, Annu. Club Alp. Fr. 29 (1902) 462–501.

1903Georges Sagnac, La longueur d’onde des rayons N déterminée par diffraction, C. R. Acad. Sci. Paris 136 (1903) 1435–1437.Georges Sagnac, La longueur d’onde des rayons n déterminée par la diffraction, J. Phys. Theor. Appl. 2 (1903) 503–508.Georges Sagnac, La longueur d’onde des rayons n déterminée par la diffraction, Bull. Soc. Fr. Phys. (1903) 184–186.Georges Sagnac, Les propriétés nouvelles du radium, J. Phys. Theor. Appl. 2 (1903) 545–548.Georges Sagnac, De la propagation anomale des ondes, J. Phys. Theor. Appl. 2 (1903) 721–727.Georges Sagnac, De la propagation anomale des ondes, Bull. Soc. Fr. Phys. (1903) 177–219.Georges Sagnac, Étude de l’absorption et des actions électriques des rayons X, in: Charles-Jacques Bouchard (Ed.), Traité de Radiologie, Steinheil, Paris, 1903, pp. 396–420.

1904Georges Sagnac, Lois de la propagation anomale de la lumière dans les instruments d’optique, C. R. Acad. Sci. Paris 138 (1904) 479–481.Georges Sagnac, Vérifications des lois expérimentales de la lumière le long de l’axe d’un instrument d’optique, C. R. Acad. Sci. Paris 138 (1904) 619–621.Georges Sagnac, Nouvelles lois relatives à la propagation de la lumière dans les instruments d’optique, C. R. Acad. Sci. Paris 138 (1904) 678–680.Georges Sagnac, Sur la propagation de la lumière au voisinage d’une ligne focale et sur les interférences des vibrations dont les amplitudes sont des fonctions différentes de la distance, C. R. Acad. Sci. Paris 139 (1904) 186–188.Georges Sagnac, Lois de la propagation anomale des ondes au voisinage d’un foyer, in: Festschrift Ludwig Boltzmann gewid-met zum sechszigsten Geburtstage, Barth, Leipzig, 1904, pp. 528–536.Georges Sagnac, Remarques au sujet de l’article de M. Kimball intitulé: ‘Note on the application of Cornu’s spiral to the diffraction grating’, J. Phys. Theor. Appl. 3 (1904) 211–212.Georges Sagnac, Opinion de M. Sagnac sur les rayons N, Rev. Sci. 2 (1904) 624.

1905Georges Sagnac, Sur la propagation de la lumière dans un système en translation et sur l’aberration des étoiles, C. R. Acad. Sci. Paris 141 (1905) 1220–1223.

1906Georges Sagnac, Les méthodes d’études expérimentales de la transformation des rayons X et des rayons secondaires qui en résultent, Radium 3 (1906) 9–18.Georges Sagnac, Une relation possible entre la radioactivité et la gravitation, J. Phys. Theor. Appl. 5 (1906) 455–462.


Georges Sagnac, Le bleu du ciel et les couleurs du disque solaire, Sci. XXe siècle 4 (1906) 69–72.Georges Sagnac, Les méthodes expérimentales de la transformation des rayons X et des rayons secondaires qui en résultent, in: Congrès International pour l’Étude de la Radiologie et de l’Ionisation, Liège, 1905, Dunod, Paris, 1906, pp. 146–163.Georges Sagnac, Classification et mécanisme des diverses actions électriques dues aux rayons X, in: Congrès International pour l’Étude de la Radiologie et de l’Ionisation, Liège, 1905, Dunod, Paris, 1906, pp. 164–176.Georges Sagnac, Die Methoden der Experimentaluntersuchung über die Umwandlung der X-Strahlen und der daraus resul-tierenden Sekundärstrahlen, Phys. Z. 7 (1906) 41–50 [transl. by Max Iklé of [204]].Georges Sagnac, Klassifikation und Mechanismus verschiedener elektrischer Wirkungen welche von X-Strahlen herrühren, Phys. Z. 7 (1906) 50–56 [transl. by Max Iklé of [204]].

1908Georges Sagnac, Remarques sur une communication récente de M. Righi, Bull. Soc. Fr. Phys. (1908) 75–76.

1910Georges Sagnac, Sur les interférences de deux faisceaux superposés en sens inverse le long d’un circuit optique de grandes dimensions, C. R. Acad. Sci. Paris 150 (1910) 1302–1305.Georges Sagnac, Interféromètre à faisceaux lumineux superposés inverses donnant en lumière blanche polarisée une frange centrale étroite à teinte sensible et des franges colorées étroites à intervalles blancs, C. R. Acad. Sci. Paris 150 (1910) 1676–1679.

1911Georges Sagnac, Les systèmes optiques en mouvement et la translation de la terre, C. R. Acad. Sci. Paris 152 (1911) 310–313, 480.Georges Sagnac, Limite supérieure d’un effet tourbillonnaire optique dû à un entrainement de l’éther lumineux au voisinage de la terre, in: Congrès International de Radiologie et d’Électricité, 3–15 Sep 1910, in: Comptes Rendus, vol. 1, Severeyns, Brussels, 1911, pp. 217–235, 2 vols.Georges Sagnac, Limite supérieure d’un effet tourbillonnaire optique dû à un entrainement de l’éther lumineux au voisinage de la terre, Radium 8 (1911) 1–8.Georges Sagnac, La translation de la terre et les phénomènes optiques purement terrestres, C. R. Acad. Sci. Paris 152 (1911) 1835–1838.Georges Sagnac, Strioscopie et striographie interférentielles analogues à la méthode optique des stries de Foucault et de Töpler, C. R. Acad. Sci. Paris 153 (1911) 90–93.Georges Sagnac, Paradoxes au sujet des actions optiques du premier ordre de la translation de la terre, C. R. Acad. Sci. Paris 153 (1911) 243–245.Georges Sagnac, Strioscope et striographe interférentiels. Forme interférentielle de la méthode optique des stries, Radium 8 (1911) 241–253.

1912Georges Sagnac, Strioscope et striographe interférentiels, Bull. Soc. Fr. Phys. (1912) 72–74.Georges Sagnac, Mesure directe des différences de phase dans un interféromètre à faisceaux inverses. Application à l’étude optique des argentures transparentes, C. R. Acad. Sci. Paris 154 (1912) 1346–1349.

1913Georges Sagnac, Strioscopes interférentiels et interféromètres simplifiés à circuits inverses. Vibrations stationnaires sur une argenture transparente, C.R. Acad. Sci. Paris 156 (1913) 1838–1840.Georges Sagnac, L’éther lumineux démontré par l’effet du vent relatif d’éther dans un interféromètre en rotation uniforme, C. R. Acad. Sci. Paris 157 (1913) 708–710.Georges Sagnac, La preuve de la réalité de l’éther lumineux par l’expérience de l’interférographe tournant, C. R. Acad. Sci. Paris 157 (1913) 1410–1413.Georges Sagnac, Strioscope et striographe interférentiels. Forme interférentielle de la méthode optique des stries, J. Phys. Theor. Appl. 3 (1913) 81–88, 292–304.

1914Georges Sagnac, Effet tourbillonnaire optique. La circulation de l’éther lumineux dans un interférographe tournant, J. Phys. Theor. Appl. 4 (1914) 177–195.

1919Georges Sagnac, Éther et mécanique absolue des ondulations, C. R. Acad. Sci. Paris 169 (1919) 469–471, 529–531.Georges Sagnac, Mécanique absolue des ondulations et relativité newtonienne de l’énergie, C. R. Acad. Sci. Paris 169 (1919) 643–646.Georges Sagnac, Comparaison de l’expérience et de la théorie mécanique de l’éther ondulatoire, C. R. Acad. Sci. Paris 169 (1919) 783–785.Georges Sagnac, Le problème de la comparaison directe des deux vitesses de propagation et de la révélation de la vitesse de la terre, C. R. Acad. Sci. Paris 169 (1919) 1027–1029, 1128.


1920Georges Sagnac, Les longueurs d’accès de la radiation lumineuse newtonienne et les zones de silence des signaux amortis de la T.S.F., C. R. Acad. Sci. Paris 170 (1920) 800–803.Georges Sagnac, La relativité réelle de l’énergie des éléments de radiation et le mouvement dans l’Éther des ondes, C. R. Acad. Sci. Paris 170 (1920) 1239–1242.Georges Sagnac, Les deux mécaniques simultanées et leurs liaisons réelles, C. R. Acad. Sci. Paris 171 (1920) 99–102.Georges Sagnac, Notice sur les Titres et Travaux Scientifiques [including biographical information and Berthelot’s report for the Pierson–Perrin prize], Gauthier-Villars, Paris, 1920.

1921Georges Sagnac, Note sur une double mécanique de la lumière liée au temps et à l’espace newtoniens, in: Lucien Fabre (Ed.), Une Nouvelle Figure du Monde: Les Théories d’Einstein, 2nd ed., Payot, Paris, 1921, pp. 252–255.

1922Georges Sagnac, Les invariants newtoniens de la matière et de l’énergie radiante, et de l’éther mécanique des ondes variables, C. R. Acad. Sci. Paris 174 (1922) 29–32.Georges Sagnac, La projection de la lumière des étoiles doubles périodiques et les oscillations des raies spectrales, C. R. Acad. Sci. Paris 174 (1922) 376–378.Georges Sagnac, Les oscillations des raies spectrales des étoiles double expliquées par la loi nouvelle de projection de l’énergie de la lumière, C. R. Acad. Sci. Paris 175 (1922) 89–91.

1923Georges Sagnac, Sur le spectre variable périodique des étoiles doubles: incompatibilité des phénomènes observés avec la théorie de la relativité générale, C. R. Acad. Sci. Paris 176 (1923) 161–163.

1924Georges Sagnac, La classification véritable des étoiles doubles définies par la loi précise de la projection de leur lumière rapportée à l’arrivée au Soleil de leurs signaux dans le spectre, C. R. Acad. Sci. Paris 179 (1924) 437–440.Georges Sagnac, Le mécanisme de la projection de la lumière dans les étoiles doubles, C. R. Acad. Sci. Paris 179 (1924) 621–623.

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[192] Alexandre Moatti, Einstein, un Siècle Contre Lui, Odile Jacob, Paris, 2006.[193] Lucien Fabre, Une Nouvelle Figure du Monde : Les Théories d’Einstein, 2nd ed., Payot, Paris, 1921.[194] Olivier Darrigol, Henri Poincaré’s criticism of fin de siècle electrodynamics, Stud. Hist. Philos. Mod. Phys. 26 (1995) 1–44.[195] Jean-Baptiste Pomey, Les conférences d’Einstein au Collège de France, Le Producteur 8 (1922) 201–206.[196] Daniel Berthelot, La Physique et la métaphysique des Théories d’Einstein, Payot, Paris, 1922.[197] Maurice Alliaume, Atronomie et relativité : bilan de 1923, Publ. Lab. Astron. Géod. Louvain 1 (1924) 163–200.[198] Alberto Martínez, Ritz, Einstein, and the emission hypothesis, Phys. Perspect. 6 (2004) 4–28.[199] Olivier Darrigol, Emil Cohn’s electrodynamics of moving bodies, Am. J. Phys. 63 (1995) 908–915.[200] Charles Maurain, Anatole Pacaud, La faculté des sciences de l’université de Paris de 1906 à 1940, Presses universitaires de France, Paris.[201] Sigalia Dostrovsky, Sagnac, Georges, Marc, Marie, in: Dictionary of Scientific Biography, vol. 12, 1975, pp. 69–70.[202] Henri Bénard, Georges Sagnac, J. Phys. Radium 9 (1928) 45S–46S [Bull. Soc. Fr. Phys. No 259].[203] Georges Sagnac, Rayons secondaires dérivés des rayons de Röntgen, Ann. Chim. Phys. 22 (1901) 493–563.[204] Georges Sagnac, Classification et mécanisme des diverses actions électriques dues aux rayons X, in: Congrès international pour l’étude de la radiologie

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C. R. Physique 15 (2014) 841–850





Towards a solid-state ring laser gyroscope

Vers un gyrolaser à état solide

Noad El Badaoui a,b, Bertrand Morbieu a, Philippe Martin b, Pierre Rouchon b, Jean-Paul Pocholle c, François Gutty c, Gilles Feugnet c, Sylvain Schwartz c,∗

a Thales Avionics, 40, rue de la Brelandière, BP 128, 86101 Châtellerault, Franceb Centre Automatique et Systèmes, Mines ParisTech, PSL Research University, 60, boulevard Saint-Michel, 75272 Paris cedex 06, Francec Thales Research and Technology France, Campus Polytechnique, 1, avenue Augustin Fresnel, 91767 Palaiseau, France



Keywords:Ring laser gyroscopeSolid-state laserSagnac effect

In this paper, we report our recent progress towards a solid-state ring laser gyroscope (RLG), where mode competition is circumvented by active control of differential losses, and nonlinear effects are mitigated by longitudinal vibration of the gain medium. The resulting dynamics is significantly different from that of a classical helium–neon RLG, owing in particular to parametric resonances that occur when the Sagnac frequency is an integer multiple of the crystal vibration frequency. We describe the main experimental and theoretical results obtained so far, and the prospects of practical applications in the near future.

Published by Elsevier Masson SAS on behalf of Académie des sciences.

r é s u m é

Nous décrivons dans cet article nos récents progrès vers la réalisation d’un gyrolaser à état solide. Dans ce dispositif, le problème de la compétition entre modes est résolu par un contrôle actif des pertes différentielles, et les effets non linéaires sont fortement atténués par la mise en vibration du milieu à gain. La dynamique d’un tel système est significativement différente de celle d’un gyrolaser à hélium–néon classique, en particulier à cause des résonances paramétriques qui surviennent lorsque la fréquence Sagnac est un multiple entier de la fréquence de vibration du cristal. Nous décrivons les principaux résultats expérimentaux et théoriques obtenus jusqu’ici et discutons les perspectives d’applications pratiques à court et moyen termes.

Published by Elsevier Masson SAS on behalf of Académie des sciences.

1. Introduction

A century after Sagnac pioneering experiments [1,2] and more than fifty years after the first demonstration of a ring laser gyroscope by Macek and Davis [3], optical rotation sensing is still a remarkably active field of research. It includes integrated

* Corresponding author.E-mail address: [email protected] (S. Schwartz).

http://dx.doi.org/10.1016/j.crhy.2014.10.0081631-0705/Published by Elsevier Masson SAS on behalf of Académie des sciences.

842 N. El Badaoui et al. / C. R. Physique 15 (2014) 841–850

Fig. 1. Basic principle of the solid-state ring laser gyroscope. The double arrow relates to the vibration of the Nd–YAG crystal, which will be discussed in Sections 3 and 4. (Color online.)

optics [4–6], slow and fast light [7–14], hollow core fibers [15–17] and large instruments for fundamental science [18–21]. From an industry perspective, two solutions have emerged and are routinely used for guidance, navigation and control: the ring laser gyroscope (RLG) [22] and the interferometric fiber-optic gyroscope (IFOG) [23]. Although the two devices can be shown to be equivalent in theory (in the sense that they have the same shot-noise limit for equal size and optical power, under the additional hypothesis that the number of fiber turns in the IFOG is equal to the finesse of the cavity in the RLG [24,14]), they differ by their practical implementation: the IFOG is shot-noise limited, but more sensitive to external perturbations (especially time-dependent temperature gradients [25]), while the RLG is more robust to its environment, but has an additional source of noise resulting from mechanical dither (which is the best known solution so far to circumvent the lock-in phenomenon [26,27]).

A key requirement common to all kinds of optical gyroscopes is reciprocity, which means that the two counter-propagating beams must share the same optical path, in order to make the variations of the latter common-mode. For the RLG, which is an active device, this implies that the two counter-propagating modes must also share the same gain medium, and are thus subject to mode competition, which tends to hinder bidirectional emission. This problem is classi-cally solved by using a gaseous gain medium for the RLG, typically a helium–neon mixture. The trick is to tune the cavity out of resonance with the atoms at rest, such that the two counter-propagating modes are resonant, owing to the Doppler effect, with two different classes of atoms (corresponding to opposite classes of velocity), ensuring stable bidirectional emis-sion. From a practical point of view, it would be a strong asset to be able to replace the gaseous mixture with a solid-state component, taking advantage of the recent progress in cost reduction, lifetime and reliability driven by markets much big-ger than inertial sensing. In this case, however, the Doppler trick cannot be used anymore, and one has to implement new techniques to circumvent mode competition and nonlinear couplings.

In this manuscript, we report our recent progress towards the achievement of a diode-pumped neodymium-doped yt-trium aluminium garnet (Nd–YAG) RLG. We will first describe the technique of active control of the differential losses that we have implemented on this device, enabling bidirectional emission and rotation sensing. We will then discuss the non-linearity of the resulting frequency response curve, which is mostly due to the existence of a population inversion grating in the amplifying medium. Based on theoretical predictions from a semiclassical model and on experimental results, we will show how the grating can be washed out by vibrating the gain crystal along the laser axis, significantly improving the linearity of the frequency response. We will also describe the residual nonlinearities, due for the most part to a para-metric resonance between the Sagnac frequency and the crystal vibration frequency. Finally, we will discuss the expected performance of this novel rotation sensor, and prospects for future applications.

2. Circumventing mode competition in the solid-state ring laser

Our basic setup is sketched in Fig. 1. It is made of a four-mirror ring cavity, containing a diode-pumped Nd–YAG crystal as the gain medium. The readout system combines the beams emitted from the two counter-propagating modes to form a beat signal on a photodiode.

The issue of mode competition is addressed by an active control of the differential losses between the counter-propagating modes [28–30]. The basic idea is to measure independently the intensity of the two beams, and to make the differential losses proportional to the intensity difference using a feedback loop, with the appropriate sign such that the more intense mode gets the higher losses at any time. In practice, the differential losses are created by polarization effects, based on the combination of a nonreciprocal rotation (obtained by Faraday effect in the YAG crystal placed inside a solenoid), a reciprocal rotation (obtained by a slight non-planarity of the cavity) and a polarizing effect (obtained by an appropriate coating on one of the mirrors). The amount of differential losses is proportional (in the limit of small rotations) to the current flowing in the solenoid, which is controlled by the feedback loop as described above.

The experimental frequency response curve of the solid-state ring laser with active stabilization of the differential losses is shown in Fig. 2. Below a critical rotation rate on the order of 10 deg/s, nonlinear couplings in the gain medium dominate and no stable signal is observed. Above this critical rotation rate, the feedback loop becomes efficient and a stable beat signal is obtained. As can be seen in Fig. 2, the frequency response curve of the solid-state RLG is nonlinear, with an upwards deviation from the ideal Sagnac line. The main reason for this nonlinearity is the presence of a population inversion grating

N. El Badaoui et al. / C. R. Physique 15 (2014) 841–850 843

Fig. 2. Typical (experimental) frequency response of the solid-state ring laser gyroscope without crystal vibration. (Color online.)

in the gain medium [30], which couples the two counter-propagating modes. This grating results from the inhomogeneous saturation of the gain by the interfering counter-propagating modes. Its effect on the frequency response decreases for increasing rotation rates, because it becomes more and more difficult for the population inversion density, which has a finite response time (about 230 µs in our case), to follow the variations of the light pattern.

Based on the latter observation, we have introduced in the laser cavity a mechanical device to vibrate the gain crystal along the light propagation axis [31], with a view to washing out the population inversion grating. As will be discussed in the following, this can significantly improve the linearity of the frequency response curve at low rotation rates provided the appropriate choice of experimental parameters is made. This requires a careful description of the laser dynamics including crystal vibration [32], which will be presented in the next sections.

3. Semiclassical description of the solid-state RLG with crystal vibration

In order to describe the electric field inside the ring cavity, we assume that there is only one laser mode in each direction of propagation, something which is experimentally obtained owing to crystal vibration, which counteracts spatial hole burning effects [33,34]. We furthermore assume that the two modes have the same polarization state. This eventually leads, in the plane wave approximation, to the following expression for the electric field:

E(x, t) = ℜ[ 2∑

p=1

E p(t)ei(ωct+µpkx)

]

where µp = (−1)p and where ωc and k are respectively the angular and spatial average frequencies of the laser, whose longitudinal axis is associated with the x coordinate. The dynamics of the solid-state RLG is then ruled, in the rotating wave approximation, by the following semiclassical equations [35]:

dE1,2

dt= −γ1,2

2E1,2 + i

m1,2

2E2,1 − iµ1,2

Ω

2E1,2 + σ

2T

(

E1,2

l∫

0

Nc dx + E2,1e2ikxc

l∫

0

Nce−2iµ1,2kx dx

)

(1)

where γ1,2 are the cavity losses associated with the counter-propagating modes, m1,2 are the backscattering coefficients, Ω is the (angular) frequency difference between the counter-propagating modes, σ is the laser cross section, T is the cavity round-trip time, l is the size of the gain medium, Nc is the population inversion density in the frame of the vibrating crystal and xc is the coordinate (in the laser frame) of a fixed point attached to the crystal, given by:

xc = xm

2cos(2π fmt) (2)

where xm is the amplitude of the movement and fm = ωm/(2π) its frequency. The backscattering coefficients, which depend on the spatial inhomogeneities of the propagation medium, have the following expression [35]:

m1,2(t) = m c1,2e−2iµ1,2kxc(t) + m m

1,2 (3)

where m c1,2 is the backscattering coefficient associated with the diffusion inside the YAG crystal, and m m

1,2 is the backscat-tering coefficient associated with any other source of diffusion inside the laser cavity, essentially the mirrors. The stabilizing coupling described in the previous section is modelled (neglecting finite-bandwidth effects) by taking losses of the form:


Fig. 3. Beat frequency as a function of the amplitude of the crystal movement for θ = 200 deg/s with the crystal vibrating at fm ≃ 40 kHz. The ex-perimental data (red circles) are in good agreement with numerical simulations (black crosses) obtained using the following measured parameters [38]: γ = 15.34 106 s−1, η = 0.21, |mc

1,2| = 1.5 104 s−1, |mm1,2| = 8.5 104 s−1, arg(mc

1/mc2) = arg(mm

1 /mm2 ) = π/17, K = 107 s−1. Integration step is 0.1 µs,

average values are computed between 8 and 10 ms. (Color online.)

γ1,2 = γ − µ1,2 Ka(|E1|2 − |E2|2

)(4)

where γ is the average loss coefficient, K > 0 represents the strength of the stabilizing coupling and a is the saturation parameter. The population inversion density function in the frame of the vibrating crystal Nc(x, t) is ruled by the following equation:

∂Nc

∂t= W th(1 + η) − Nc

T1− aNc|E1e−ik(x+xc) + E2eik(x+xc)|2

T1(5)

where η is the relative excess of pumping power above the threshold value W th and T1 is the lifetime of the population inversion. The difference Ω between the eigenfrequencies of the counter-propagating modes is induced by the combined effects of rotation (Sagnac effect [36]) and crystal vibration (Fresnel–Fizeau drag effect [37]), resulting in the following expression:

Ω = Ωs − 4π xc(t)lc(n2 − 1)

λL(6)

where Ωs = 8π Aθ/(λL) is the Sagnac angular frequency, A is the area enclosed by the ring cavity, L is the (optical) length of the cavity, θ is the angular velocity of the cavity around its axis, λ = 2πc/ωc is the emission wavelength, and lc and nare respectively the length and the refractive index of the crystal. In Eq. (6), we have neglected the effect of dispersion in the YAG crystal, which is much smaller than the Fresnel–Fizeau drag effect in our case.

To summarize, the dynamics of the solid-state RLG with crystal vibration is described by Eqs. (1), in conjunction with Eqs. (2), (3), (4), (5) and (6). Of course, this model is too complex to be solved analytically. However, it can be integrated numerically with realistic laser parameters, providing a good agreement with the experimental results. This is illustrated by the data shown in Fig. 3, where the beat frequency is plotted as a function of the amplitude of the crystal vibration xm for fm ≃ 40 kHz. The experimental frequency response curve has in this case a non-trivial shape, which is well reproduced by the numerical simulation.

4. Dynamics of the solid-state RLG with high-frequency crystal vibration

The linearity of the solid-state RLG can be significantly improved by making the crystal vibration frequency ωm much bigger than all other frequencies involved in the laser dynamics, namely |Ωs|, |mm

1,2|, |mc1,2| and the relaxation frequency

ωr = √γ η/T1 [35], as will be discussed in this section.

4.1. Analytical derivation of the beat frequency in the limit of high rotation rates

In this subsection, we furthermore assume that the rotation rate is high enough for the following conditions to be fulfilled:

ωm > |Ωs| ≫∣∣mm

1,2

∣∣,∣∣mc

1,2

∣∣,ωr (7)


In this case, the nonlinearity of the frequency response of the solid-state RLG can be analytically estimated. To do so, we first introduce N0 and N1, respectively the mean and first spatial harmonics of the population inversion density function Nc(x, t), defined by:

N0 = 1l

l∫

0

Nc(x, t) dx and N1 = 1l

l∫

0

Nc(x, t)e2ikx dx

If we furthermore assume that the pumping rate is close to the threshold value (i.e. η ≪ 1), then we can rewrite Eqs. (1)and (5) as follows:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

dE1,2

dt=

(Z2

N0 − γ1,2

2

)E1,2 + im1,2

2E2,1 − iµ1,2Ω

2E1,2 + Z

2N1,2e−2iµ1,2kxc E2,1

dN0

dt= W th(1 + η) − N0

T1− a

2T1Nth

(|E1|2 + |E2|2

)

dN1

dt= − N1

T1− a

2T1Nthe−2ikxc E1 E∗

2

(8)

where Z = σ l/T , N2 = N∗1 and Nth is the value of N0 at the laser threshold. It obeys the following relation:

W th = Nth

T1≃ γ

Z T1(9)

where the contribution of backscattering has been neglected in the expression of Nth. We will now derive an approximation of this system using the mathematical theory of averaging over nonlinear dynamical systems [39,40]. For simplicity, we neglect the Fresnel–Fizeau term introduced in Eq. (6), such that Ω = Ωs. Going into the rotating frame by setting:

E1 = eiΩst/2 F1 and E2 = e−iΩst/2 F2 (10)

the system (8) becomes:⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

dF1,2dt

=(

Z2

N0 − γ1,22

)F1,2 + Z

2N1,2e−iµ1,2kxm cos(ωmt)eiµ1,2Ωst F2,1 + i

2

[m m

1,2 + m c1,2e−iµ1,2kxm cos(ωmt)]eiµ1,2Ωst F2,1

dN0dt

= W th(1 + η) − N0T1

− a2T1

Nth(|F1|2 + |F2|2

)

dN1dt

= − N1T1

− a2T1

Nthe−ikxm cos(ωmt)eiΩst F1 F ∗2

Using the classical series involving the Bessel functions Jn:

eikxm cos(ωmt) = J0(kxm) ++∞∑

n=1

in Jn(kxm)(einωmt + e−inωmt)

we can perform a first order averaging of the previous system of equations considering the fact that the frequencies nωm ±Ωs are much larger than the other frequencies involved in the laser dynamics (see (7)). This provides the following model:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

d F1,2

dt=

(Z2

N0 − γ1,2

2

)F1,2

dN0

dt= W − N0

T1− a

2T1Nth

(| F1|2 + | F2|2

)

dN1

dt= − N1

T1

(11)

leading to the following equilibrium values for the averaged parameters:

γ1 = γ2 = γ , a|F1|2 = a|F2|2 = η, N0 = γ /Z and N1 = 0 (12)

Let us compute the second order correction for ( F1, F2, N0, N1) around the equilibrium of (11). The small fluctuations denoted with a δ around the average values (denoted with a bar) admit the following form:


Fig. 4. Sketch of the optical standing wave inside and outside the YAG crystal. When the crystal is physically translated by λ/2, the relative position between the optical standing wave and a fixed point in the crystal is changed by λ/(2n). (Color online.)

δF1 = 12

[−m m

1 + m c1 J0(kxm)

Ωs+ im c

1 J1(kxm)

(eiωmt

ωm − Ωs− e−iωmt

ωm + Ωs

)]e−iΩst F2

δF2 = 12

[m m

2 + m c2 J0(kxm)

Ωs− im c

2 J1(kxm)

(eiωmt

ωm + Ωs− e−iωmt

ωm − Ωs

)]eiΩst F1

δN0 = 0

δN1 = iaγ

2Z T1

[J0(kxm)

Ωs− i J1(kxm)

(eiωmt

ωm + Ωs− e−iωmt

ωm − Ωs

)]eiΩst F1 F ∗

2

where we have neglected the Jn terms with n ≥ 2 for simplicity. In order to compute the second-order approximation, we replace (F1, F2, N0, N1) by their first-order value ( F1 + δF1, F2 + δF2, N0 + δN0, N1 + δN1) in the second members of differential equations (11), and re-average the fast-oscillating terms. The second order approximation eventually leads to:

d F1,2

dt=

(Z N0 − γ1,2

2+ i+1,2

)F1,2 (13)

where:

+1 − +2 = γ η

2T1

[J 2

0(kxm)

Ωs+ 2Ωs J 2

1(kxm)

Ω2s − ω2

m

]+ m c

1 m c2

2ΩsJ 2

0(kxm) + m m1 m m

2

2Ωs

+ m c1 m m

2 + m m1 m c

2

2ΩsJ0(kxm) + m c

1 m c2 Ωs J 2

1(kxm)

Ω2s − ω2

m(14)

The measured (angular) beat frequency Ωbeat is defined in this formalism by the time average of arg(E1/E2). Based on Eqs. (10) and (13), it is equal to Ωbeat = Ωs + ℜ(+1 − +2), where +1 − +2 is given by Eq. (14).

4.2. Physical interpretation and comparison with experiment

In a first approximation, we neglect the J1 terms and keep only the dominant backscattering term m c1 m c

2 in Eq. (14), leading to:

Ωbeat ≃ Ωs + γ η

2ΩsT1J 2

0(kxm) + ℜ(m c1 m c

2 )

2ΩsJ 2

0(kxm) (15)

Physically, the first correcting term is due to the residual effect of the population inversion grating, which goes to zero either for very high rotation rates or for a set of discrete values of the vibration amplitude corresponding to J0(kxm) = 0. The second correcting term, resulting from backscattering in the crystal, is also reduced by crystal vibration, owing to the Doppler effect, which makes the backscattered light from one mode non-resonant anymore with the counter-propagating mode. The two effects go simultaneously to zero when J0(kxm) = 0, corresponding to the situation where all the ions of the gain medium see on average the same intensity. The smallest value of xm for which this happens is kxm ≃ 2.405, where k = 2π/λ is the wavevector outside the YAG crystal, and shall not be confused in this context with its counterpart inside the YAG crystal kc = n2π/λ, as illustrated in Fig. 4. With λ = 1.064 µm, this leads to the following requirement on the vibration amplitude: xm ≃ 0.41 µm, which we typically achieve in practice using a resonant mechanical device excited by a piezoelectric ceramic.

The J 20 shape of the frequency response curve predicted in Eq. (15) is experimentally observed in Fig. 5, where the beat

frequency is measured as a function of the amplitude of the crystal vibration for a fixed rotation rate of 55 deg/s and a crystal vibration frequency of fm ≃ 168 kHz. This curve provides an experimental signal to tune the vibration amplitude such that J0(kxm) = 0.

When the latter condition is fulfilled, the frequency response curve of the solid-state RLG gets much closer to the ideal Sagnac line for |Ωs| ≪ ωm, as illustrated in Fig. 6. Conversely, when the Sagnac frequency approaches the crystal


Fig. 5. Measured beat frequency as a function of the driving voltage of the vibrating device for θ = 55 deg/s, with the crystal vibrating at fm ≃ 168 kHz. The driving voltage is proportional to the amplitude of the vibration up to ≃ 250 mV peak-to-peak (beyond this point, saturation occurs). The point where the curve is minimum corresponds to J0(kxm) = 0, hence xm ≃ 0.41 µm.

Fig. 6. Experimental frequency response of the solid-state ring laser gyroscope with J0(kxm) = 0 and fm ≃ 168 kHz. A phenomenon of parametric resonance is observed when the Sagnac frequency approaches the crystal vibration frequency. (Color online.)

vibration frequency, the nonlinearity of the frequency response significantly increases. To give a quantitative estimate of this parametric resonance phenomenon, we include the J1 terms in Eq. (14) while setting the J0 terms to zero, leading to:

Ωbeat ≃ Ωs + γ ηΩs J 21(kxm)

T1(Ω2s − ω2

m)+ ℜ(m m

1 m m2 )

2Ωs+ ℜ(m c

1 m c2 )Ωs J 2

1(kxm)

Ω2s − ω2

m(16)

Since the value of kxm is deliberately chosen to be equal to the first zero of the J0 function, the J1 term becomes the dominant nonlinearity, with J 2

1(kxm) ≃ 0.27. Eq. (16) is in good agreement with experimental data, as reported in Fig. 7, where the beat frequency is measured as a function of the pumping rate. The experimentally measured slope is −3.7 kHz, while the theoretical estimate from Eq. (16) leads to −γ J 2

1 (kxm)Ωs/[2π T1(ω2m − Ω2

s )] ≃ −3.6 kHz (where we have used the following parameters: γ = 15.34 106 s−1, Ωs/(2π) ≃ 117.3 kHz, T1 = 230 µs and ωm/(2π) ≃ 168 kHz).

Keeping only the leading order in the small quantity Ωs/ωm, the frequency shift induced by the parametric resonance described above can be rewritten as − J 2

1(kxm)[ωr/ωm]2Ωs. In terms of gyroscope performance, this corresponds to a scale-factor nonlinearity, which is typically (with ωr/(2π) ≃ 20 kHz) on the order of 3 × 10−3. With this parameters, this means in particular that the pumping rate η has to be controlled at the 10−3 level in order to make the scale factor stable at the ppm level. A similar control on the amplitude of the crystal vibration xm is also required for the same reason. One way to relax the latter constraint would be to increase the vibration frequency fm beyond its actual value of 168 kHz.


Fig. 7. Experimental frequency response of the solid-state ring laser gyroscope as a function of the pumping rate, with J0(kxm) = 0, fm ≃ 168 kHz and θ = 160 deg/s. The measured slope is in good agreement with analytical predictions (Eq. (16)). (Color online.)

Fig. 8. Experimental frequency response of the solid-state ring laser gyroscope at low rotation rates, with J0(kxm) = 0 and fm ≃ 168 kHz. The dashed line is a fit of the experimental data by the typical frequency response curve of a helium–neon RLG [27]. (Color online.)

4.3. Case of low rotation rates

When the condition J0(kxm) = 0 is fulfilled, the nonlinearity of the frequency response curve is significantly reduced, and the deviation from the ideal Sagnac line at low rotation rates is downwards, as illustrated in Fig. 8. It is a remarkable fact that the shape of this frequency response curve is now similar to what would be expected from a helium–neon RLG, where the dominant coupling source is (linear) backscattering on the cavity mirrors. This similarity is confirmed by the numerical simulations shown in Fig. 9, where the frequency response curve of a high-performance helium–neon RLG is shown to be identical to that of a solid-state RLG with the same mirror backscattering coefficients m m

1,2. Further numerical simulations [32] including mechanical dither also show that the two devices are expected to have the same level of angular random walk if identical mirrors are used. In other words, everything happens at low rotation rates as if all sources of nonlinear coupling from the Nd–YAG crystal were effectively suppressed by the crystal vibration.

5. Conclusion

The ultimate level of performance that can be expected from the solid-state RLG (in addition to the scale factor effect outlined in the previous section) is discussed in Ref. [41]. As already mentioned, the angular random walk resulting from mechanical dither is expected to be the same as for a helium–neon RLG, provided mirrors of equivalent quality are used. The Schawlow–Townes limit [42] should be slightly higher in the case of the solid-state RLG because the finesse of the cavity is a little bit lower, although this will be partly compensated by the possibility to use more power thanks to the high gain in the Nd–YAG crystal. In total, the solid-state RLG should be able to reach the 10−3 deg/

√h range. As regards the


Fig. 9. Up: simulated frequency response curve of a solid-state RLG with crystal vibration at fm ≃ 168 kHz and J0(kxm) = 0. All parameter values are taken from experimental measurements [38], except the mirror backscattering coefficients mm

1,2, which have the typical value of a high-performance helium–neon RLG. Low: simulated frequency response curve of a high-performance helium–neon RLG (obtained by numerical integration of the Adler equation [27] with the same value of mm

1,2 as for the upper curve). (Color online.)

ultimate bias stability, it is predicted to lay in the 10−2 deg/h range if the current flowing out of the feedback loop can be measured at the 10−3 level [41]. As discussed in this paper, the same precision will be required on the pumping rate and crystal vibration amplitude to reach the ppm level on the stability of the scale factor.

Experimental work is still ongoing at Thales to raise the Technology Readiness Level of the solid-state RLG (in particular by using high-quality mirrors at 1.064 µm and by improving the stability of the control electronics). Beyond industrial applications, this technology could also be an interesting platform to test more advanced schemes of rotation sensing, involving for example anomalous dispersion [14].

References

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2006, p. ME6.[17] M.A. Terrel, M.J. Digonnet, S. Fan, Resonant fiber optic gyroscope using an air-core fiber, J. Lightwave Technol. 30 (7) (2012) 931.[18] K. Schreiber, T. Klügel, J.-P. Wells, R. Hurst, A. Gebauer, How to detect the Chandler and the annual wobble of the Earth with a large ring laser

gyroscope, Phys. Rev. Lett. 107 (17) (2011) 173904.[19] F. Bosi, G. Cella, A. Di Virgilio, A. Ortolan, A. Porzio, S. Solimeno, M. Cerdonio, J. Zendri, M. Allegrini, J. Belfi, et al., Measuring gravitomagnetic effects

by a multi-ring-laser gyroscope, Phys. Rev. D 84 (12) (2011) 122002.[20] K. Schreiber, A. Gebauer, J.-P. Wells, Long-term frequency stabilization of a 16 m2 ring laser gyroscope, Opt. Lett. 37 (11) (2012) 1925.[21] C. Clivati, D. Calonico, G.A. Costanzo, A. Mura, M. Pizzocaro, F. Levi, Large-area fiber-optic gyroscope on a multiplexed fiber network, Opt. Lett. 38 (7)

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[24] J. Gea-Banacloche, Passive versus active interferometers: why cavity losses make them equivalent, Phys. Rev. A 35 (6) (1987) 2518.[25] D.M. Shupe, Thermally induced nonreciprocity in the fiber-optic interferometer, Appl. Opt. 19 (5) (1980) 654.[26] J. Killpatrick, The laser gyro, IEEE Spectr. 4 (10) (1967) 44.[27] F. Aronowitz, Fundamentals of the ring laser gyro, in: Optical Gyros and Their Application, NATO Research and Technology Organization, 1999 (Ch. 3).[28] A. Dotsenko, E. Lariontsev, Use of a feedback circuit for the improvement of the characteristics of a solid-state ring laser, Sov. J. Quantum Electron.

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lasers, Phys. Rev. Lett. 97 (2006) 093902.[31] S. Schwartz, F. Gutty, J.-P. Pocholle, G. Feugnet, Solid-state ring laser gyro with a mechanically activated gain medium, US Patent 7,589,841.[32] S. Schwartz, F. Gutty, G. Feugnet, E. Loil, J.-P. Pocholle, Solid-state ring laser gyro behaving like its helium–neon counterpart at low rotation rates, Opt.

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C. R. Physique 15 (2014) 851–858





The fiber-optic gyroscope, a century after Sagnac’s

experiment: The ultimate rotation-sensing technology?

Le gyromètre à fibre optique, cent ans après l’expérience de Sagnac : la technologie ultime de mesure inertielle de rotation ?

Hervé C. Lefèvre ∗

iXBlue SAS, 52, avenue de l’Europe, 78160 Marly-le-Roi, France


Article history:Available online 4 November 2014

Keywords:Fiber-optic gyroscopeRing-laser gyroscopeSagnac effectInertial navigation

Mots-clés :Gyromètre à fibre optiqueGyromètre laserEffet SagnacNavigation inertielle

Taking advantage of the development of optical-fiber communication technologies, the fiber-optic gyroscope (often abbreviated FOG) started to be investigated in the mid-1970s, opening the way for a fully solid-state rotation sensor. It was firstly seen as dedicated to medium-grade applications (1/h range), but today, it reaches strategic-grade performance (10− 4/h range) and surpasses its well-established competitor, the ring-laser gyroscope, in terms of bias noise and long-term stability. Further progresses remain possible, the challenge being the ultimate inertial navigation performance of one nautical mile per month corresponding to a long-term bias stability of 10− 5/h. This paper is also the opportunity to recall the historical context of Sagnac’s experiment, the origin of all optical gyros.


r é s u m é

Profitant du développement des technologies de télécommunication à fibre optique, le gyromètre à fibre optique a commencé à être étudié au milieu des années 1970, ouvrant la voie à un gyromètre entièrement à état solide. Il a d’abord été considéré comme seulement adapté aux applications de moyenne performance (de l’ordre de 1/h), mais atteint aujourd’hui des performances de classe stratégique (de l’ordre de 10− 4/h) et surpasse, en termes de bruit et de dérive à long terme du biais, son concurrent établi, le gyromètre laser. Des progrès supplémentaires restent possibles, le défi étant la performance ultime de navigation inertielle au nautique par mois, qui correspond à une stabilité de dérive de 10− 5/h. Cet article offre aussi l’opportunité de rappeler le contexte historique de l’expérience de Sagnac, qui est à l’origine de tous les gyros optiques.


* Tel.: +33 1 30 08 88 88; fax: +33 1 30 08 88 00.E-mail address: [email protected].


852 H.C. Lefèvre / C. R. Physique 15 (2014) 851–858

Fig. 1. Original Sagnac’s setup [1] of a ring interferometer to demonstrate sensitivity to rotation rate (S stands for surface, which means “area” in French).

1. Introduction

Both optical gyroscopes, the ring-laser gyro (RLG) and the fiber-optic gyro (FOG), are based on the same Sagnac effect [1], which shows that light travelling along a closed-ring path in opposite directions allows one to detect rotation with respect to inertial space. Over one turn as in the original Sagnac’s experiment, a century ago [2], the effect is extremely weak but it can be increased with recirculation in the resonant cavity of a ring laser or using the numerous loops of a fiber coil. The RLG was demonstrated only a few years after the invention of the laser in 1960, and it is based on helium–neon (He–Ne) technology. It became very successful in the 1980s and has since overcome classical spinning-wheel mechanical gyroscopes because of its improved lifetime and reliability. It also provided an excellent scale factor performance, making strap-down navigation systems possible. Earlier mechanical systems used a stabilized gimbaled platform where the gyros work only around zero to stabilize the attitude of the platform; a strap-down system avoids the delicate mechanics of the gimbaled approach but the gyros are attached directly to the vehicle, and then they have to follow precisely the whole dynamical range of the vehicle rotation, which requires a very good stability of the scale factor. It was clear progress over mechanical gyroscopes, but gas lasers still have several drawbacks such as high-voltage discharge electrodes that tend to wear out over the long term or the need for perfect sealing of the gas enclosure. The advent of low-attenuation optical fiber and efficient semiconductor light source developed for optical communications in the 1970s opened the way for a fully solid-state device. Then, however, the FOG was seen as an approach dedicated to medium performance, and unable to compete with the RLG for top-grade applications. As we shall see, this is not the case anymore. This paper is also the opportunity to recall the historical context of Sagnac’s experiment and to outline the early theoretical contribution of Max von Laue [3].

2. Historical context of Sagnac’s experiment

If Huygens proposed in the 17th century a wave theory of light, Newton imposed his views of a corpuscular theory in the early 18th century. It is only in the early 19th century that Young’s double-slit experiment reopened the wave theory, knowing that it was not easily admitted: you did not contradict Newton! It required the exceptional quality of the theoretical and experimental work of Fresnel to convince the physicist community.

However, for the minds of that time, a wave needed some kind of propagation medium, as for acoustic waves. It was called “luminiferous Aether”, and light was seen as propagating at a constant velocity c with respect to this fixed Aether.

Even when Maxwell showed in 1864 the electromagnetic nature of light wave, Aether was not questioned. It required the famous Michelson and Morley experiment in 1887 to have a clear demonstration that the concept of Aether should be revised, and this yielded, in 1905, the special theory of Relativity, when, based on earlier theoretical works of Lorentz, Poincaré, Planck and Minkowski, Einstein abandoned the concept of Aether and stated that light is propagating at the same velocity c in any inertial frame of reference in linear translation, despite its own velocity.

This revolutionary conceptual leap was very difficult to admit, and Sagnac’s experiment [Fig. 1], the base of present optical gyroscopes, was actually performed to demonstrate that Aether did exist as clearly stated in the title of his publica-tion [2]: “The luminiferous Aether demonstrated by the effect of the relative wind of Aether in an interferometer in uniform rotation”.

Sagnac’s experiment, which takes place in a vacuum (actually in air, but it can be considered as in a vacuum), can be either explained by special Relativity or classical Aether theory, and does not allow one to demonstrate which theory is right or wrong. It was explained clearly by von Laue in 1911 [3] [Fig. 2], two years before Sagnac’s publication, and maybe the Sagnac effect should be renamed the Sagnac–Laue effect!

H.C. Lefèvre / C. R. Physique 15 (2014) 851–858 853

Fig. 2. Explanation of “Sagnac–Laue” effect in von Laue publication of 1911 [3].

Another important point of Aether theory was the hypothesis made by Fresnel in 1818 of the drag of Aether by matter, which was demonstrated experimentally by Fizeau in 1851 [4]. The velocity v of a wave that propagates in a medium of index n that moves at a speed vm is not c/n anymore, but:

v = cn

+!

1 − 1n2

"vm (2.1)

Note that in Fizeau’s experiment, one has to take into account the dispersion of the index, since the wave frequency seen by the medium is not the same in both directions because of the Doppler effect, but in a fiber gyro both opposite waves have the same frequency in the frame of the medium and there is no dispersion effect [5].

This Fresnel–Fizeau drag effect was explained by von Laue in 1907 [6] as resulting from the law of combination of speeds of Special Relativity:

v = v1 + v2

1 + v1 v2c2

(2.2)

where v1 is the speed of a mobile in a frame moving at v2 with respect to the “rest” frame, and v is the speed of this mobile in this “rest” frame. One sees that Eq. (2.2) yields Eq. (2.1) to first order in vm, considering v1 = c/n, v2 = vm and vm ≪ c. The Fresnel–Fizeau drag effect is actually a relativistic effect!

The important point is that, because of the Fresnel–Fizeau drag effect, the Sagnac effect does not depend on the index of refraction of the corotating propagation medium as it was stated by von Laue [7] as early as in 1920, and as it is clearly experienced in a fiber-gyroscope [5]. If the Sagnac effect in a vacuum can be explained by Aether theory, the Sagnac effect in a medium is related to the Fresnel–Fizeau drag effect and is then also a relativistic effect.

3. What are we looking for? Single-mode reciprocity is key

Going back to present optical gyros, despite their difference of principle, RLGs and FOGs have similar theoretical noises for the same single-turn enclosed area and the same number of recirculations [8]. The typical RLG perimeter is 20 to 30 cm with on the order of 104 recirculations in the high-Q mirror cavity [Fig. 2]. An FOG coil of 104 loops of 10 cm in diameter (i.e. 3 km long and typically 3 dB of attenuation) has the same potential. Today, RLGs are in the so-called navigation-grade performance range needed for airliners, i.e. below 10− 2/h in term of long-term bias stability, while highest-performance FOGs are in the so-called strategic-grade performance range needed for very long-term marine and submarine navigation, i.e. at least ten times better, below 10− 3/h. Translated in path length difference induced by the Sagnac effect, it means a relative change on the order of 10− 18 for the RLG, and 10− 19 to 10− 20 for the FOG! These incredible numbers may look unrealistic, but there is the fundamental principle of reciprocity of light propagation which acts as a perfect common-mode rejection between both counter-rotating waves, when there is single-mode propagation. Because of single-mode reciprocity, the transit time of both counterpropagating waves can be perfectly balanced, leaving out only the Sagnac effect. The quality of the residual bias instability (zero instability) depends on the residual lack of reciprocity.

A detailed analysis of the principle of RLG can be found in a work by F. Aronowitz [9], one of the pioneers of this technology. The RLG has naturally “quasi-reciprocity” because it operates in a single transverse laser mode as well as a


Fig. 3. Symmetrical discharge to balance. Fresnel–Fizeau drag effect due to the ionic flow in an RLG.

single longitudinal mode and the reflection birefringence of the mirrors due to the large angle of incidence ensures a single polarization in the cavity, but its reciprocity is not perfect. The electrical discharge creates an ionic flow, and because of the Fresnel–Fizeau drag effect, this matter flow yields a velocity difference between counterpropagating waves [9]. It is only on the order of 10− 15 in terms of relative velocity, but it creates a spurious non-reciprocal effect equivalent to about 1/h. It is counterbalanced by using a common cathode and two symmetrical anodes [Fig. 3], but this balancing cannot be perfect and there is a residual bias instability on the order of a few thousandths of degree per hour. One could think: why does not one use a solid-state laser to avoid this drag effect? After all, since the early 1960s, when the He–Ne laser gyro was invented, numerous kinds of lasers have been developed, but there is a key problem in laser behavior: mode competition! In principle, a CW ring laser should not work because both directions have the same lasing conditions and they “compete”, i.e. it is unstable. He–Ne ring lasers work because of a very subtle effect: with the flow, the moving amplifying ions see different frequencies for both opposite directions because of the Doppler effect, and the use of 20Ne and 22Ne isotopes with gain curves shifted in frequency allows one to get two “superimposed” lasers: one isotope amplifying one direction and the other one the opposite one, which avoids mode competition. “Magic”. . . but within the limit of the Fresnel–Fizeau drag-induced non-reciprocity!

In the case of FOG, reciprocity was much more difficult to get, mainly because of the residual birefringence of the fiber. As it is well known, a single-mode fiber has actually two orthogonal polarization modes that propagate with slightly different velocities because of fiber birefringence. One understands that if one direction uses one mode and the opposite one uses the crossed mode, there is a non-reciprocal phase difference. It was shown very early [10] that reciprocity does not require true single-mode propagation along the entire interferometer and that a single-spatial mode/single-polarization mode filter at the common input–output of the ring interferometer is sufficient. However, the requirement on polarizer rejection to fully suppress the problem can be very stringent. Because of coherence effects, the residual phase non-reciprocity in radian may be equal to the amplitude rejection of the polarizer [11], i.e. a very good rejection of − 80 dB may yield a phase non-reciprocity as high as 10− 4 rad, but today the problem is solved with the progress of the components and the use of decoherence [12,13]. Proton-exchanged lithium niobate (LiNbO3) integrated-optics yields a single-polarization waveguide that provides excellent polarization rejection (as good as − 80/–90 dB), and a polarization-maintaining (PM) fiber limits the amount of light in the crossed polarization mode, but it would not be sufficient by far. One has also to take advantage of decoherence/depolarization effects with the use of a broadband source which has a short coherence time. Because of the birefringence of PM fiber and LiNbO3 crystal, the spurious crossed polarization propagates at a different speed from the main signal and loses its coherence with respect to this main signal, which drastically reduces the parasitic effect. To further reduce defects, one can also take advantage of the natural unpolarization of ASE (Amplified Spontaneous Emission) sources based on telecom diode-pumped EDFA (erbium-doped fiber amplifier) technology. The crossed component of the input unpolarized light (the component orthogonal to the polarizer axis) compensates for the residual nonreciprocity of the main component [12,13]. Because of the residual polarization dependent loss of the components, the actual input unpolarization is not perfect, but in practice the degree of polarization of the input ASE light is only few percent and this brings an additional 30-fold reduction of polarization non-reciprocities.

Now, light travelling in a dense medium and with high-power density because of the guidance, one could have faced nonlinear effect destroying reciprocity [14] which is based on the linearity of propagation equation, but the power fluctu-ation statistics of broadband source happens to balance this effect perfectly [12,13]. Today, the FOG appears as a unique sensor that could be just limited by its theoretical white photon shot noise without any source of long-term drift.

Note that the use of a low-temporal-coherence source brings excess relative intensity noise (excess RIN) because of the random intensity beating of all its spectral components. With an erbium fiber source, it can be as high as 10− 6/

√Hz, but


Fig. 4. Principle of digital phase ramp feedback, with !τg being the transit time through the coil (5 µs/km), which depends on the group’s velocity.

RIN can be reduced by compensation techniques [12], and it is possible to get very close to the theoretical photon shot noise, which is typically 10− 7/

√Hz for a returning power of few tens of µW.

Finally, gyroscopes have to operate over a large dynamical range, and this requires signal processing techniques that will not degrade this intrinsic stability.

4. The other key issue: signal processing techniques

Among the advantages of RLG is its very simple read-out mechanism. As in any laser, there are an integral number of wavelengths along the cavity path. Path length difference created by the Sagnac effect induces a wavelength difference between both counterpropagating resonant beams, and therefore a frequency difference. Both output beams are recombined to interfere [Fig. 2] and yield a frequency beating that is proportional to the rotation rate. A simple counting electronics provides a linear read-out of the rate over a very large dynamical range.

Note however that at low rate, there is the so-called “lock-in” effect. Both laser beams have very close frequencies: about 1 Hz difference for 1/h, when light frequency is 500 THz at a He–Ne operating wavelength of 633 nm. Despite impressive technological progress, there is still some residual mirror backscattering yielding coupling that locks them on the same frequency. This is eliminated by a mechanical dithering, but it increases the theoretical RLG measurement noise by an order of magnitude [9].

The FOG is not an active resonator anymore but a passive interferometer with an external light source. It is possible to get no lock-in, so no need for dithering that avoids its related noise degradation. In particular, the backscattering of a fiber is higher than the one of RLG mirrors, but the low temporal coherence of the broad-spectrum source of an FOG limit spurious interferences between this backscattered light and the primary waves. However, the raw response is the nonlinear raised cosine response of an interferometer. This has been overcome by a very efficient phase modulation technique associated with a drift-free digital demodulation and a phase feedback. This so-called all-digital phase ramp combines square-wave biasing modulation and synchronized phase steps generated and demodulated digitally [Fig. 4] [12,13,15].

This sophisticated processing approach is conceptually much more complicated than the simple frequency readout of an RLG, but it can be easily implemented with present digital electronics. It yields an excellent scale factor linearity of 1 ppm without any degradation of the basic noise or the reciprocity of the interferometer, and it works without quantization error despite a limited number of converter bits, because of averaging effects [12,15].

Bias noise and drift are calculated with Allan variance (or deviation, its square root) [12,16] and, in a temperature stabilized environment, a high-performance FOG does yield the theoretical − 1/2 power reduction slope of a white noise over days of measurement without any visible bias stability limitation (no flicker, no rate random walk), down to the 10− 5/h range, corresponding to an interferometer phase difference close to 10− 10 rad, and with a white noise be-low 10− 2(/h)/

√Hz, which corresponds to an angular random walk in the 10− 4/

√h range, and a phase noise of few

10− 7 rad/√

Hz. This is absolutely unique, compared to any other inertial sensors, as accelerometers or mechanical and laser gyros, which all face long term drift, even in a temperature-controlled environment.

There is a residual temperature dependence in FOG that is related to the so-called Shupe effect [12,17] due to temper-ature transient. However, symmetrical winding techniques [18] as well as careful modeling reduce very significantly the effect. Typical long-term bias stability specification for a high-performance FOG is in the range of a few 10− 4/h in an unstabilized temperature environment.

The scale factor varies also in temperature. It depends on the geometrical area of the gyro coil and on the wavelength. Temperature modeling allows one to get a reproducible stability of the coil size to an accuracy of 10 ppm, and wavelength stability may be improved to this same 10 ppm value with internal spectral filtering of an erbium-doped fiber source [12].

5. Configuration of an FOG

Based on solid-state technologies of optical-fiber communications, the FOG yields a high reliability and a very long life time in addition to its performance. It is composed of (Fig. 5; see for example [12,13]):


Fig. 5. Configuration of an FOG with a Y junction as the splitter and recombiner of the interferometer, a polarizer and a pair of electro-optic phase modulators on the multi-function LiNbO3 integrated-optic circuit.

Fig. 6. (Color online.) iXBlue MARINS inertial navigation system (400 × 300 × 280 mm3) using gyro coils of 3 km over a mean diameter of 170 mm. The bias stability specification of the gyros over environment is 5 × 10− 4/h, and the scale factor specification is 10 ppm.

• a broadband source based, for high grade, on EDFA technology at a wavelength of 1550 nm; wavelength stability may be obtained with internal spectral filtering with a fiber Bragg grating,

• a polarization-maintaining (PM) fiber coil (a few hundred meters for medium grade to several kilometers for very high grade),

• a LiNbO3 integrated-optic circuit with electrodes to generate phase modulation with the electro-optic Pockels effect and that provides excellent polarization selectivity with proton-exchanged waveguide,

• a fiber coupler (or a circulator for higher returning power) to send to a detector light returning from the common input–output port of the interferometer;

• an analog–digital (A/D) converter to sample the detector signal;• a digital logic electronics that generates the phase modulation and the phase feedback through a digital-to-analog (D/A)

converter.

It is important to note that with the adequate design and components, the performance of an FOG is very reproducible in production, even for the high-performance end. It does not require trimming or selection.

Testament to the quality, accuracy and reliability of the fiber-optic gyro is its growing use for positioning and navigation instruments in navy [Fig. 6] and space [Fig. 7] applications.

As already discussed, the residual limit of FOG bias stability is the temperature transient, and Fig. 8 shows a promising laboratory result of longitude error of an iXBlue prototype of inertial navigation system that was performed “at rest” in a temperature-stabilized environment over more than one month (38 days). The longitude accuracy is in the range of one


Fig. 7. (Color online.) Four-axis (for redundancy) FOG developed by iXSpace, an iXBlue company, in cooperation with Airbus Defence & Space (formerly EADS-Astrium) for space applications; the gyro coils are 5 km long over a mean diameter of 170 mm. Bias and scale factor specifications are similar to the ones of MARINS gyros.

Fig. 8. Longitude error of iXBlue’s inertial system prototype, in a temperature controlled environment, over 38 days: 1 nautical mile (Nm) per month.

nautical mile (Nm) over one month! It corresponds to a gyro bias stability of about 10− 5/h and a gyro scale factor stability of about 1 ppm!

Understanding this result requires some explanations about inertial navigation. An inertial navigation system (INS) cal-culates the trajectory with respect to inertial space with the mathematical integration of the measurements of the rate of rotation by the gyros and of the acceleration by the accelerometers. Note that mathematical integration is similar to averaging and then it filters out short-term noise to keep only the effect of the long-term drift.

“At rest” with respect to the Earth means actually to follow the movement of rotation of the Earth, which is quite fast. At the latitude of 48 where the experiment was performed, the tangential velocity due to the Earth’s rotation is 1100 km/h and it is measured by the gyros. Over the 38 days of the experiment, the system “at rest” travelled in fact over one million of kilometers, and its position in longitude is known by the integration of the tangential speed measured with the rotation rate. The linear component of the mean position drift being less than half a nautical mile (i.e. about one kilometer) over one million of kilometers, it means the measurement of the 15/h of Earth rotation rate is performed with a relative stability of one millionth, i.e. a 1.5 × 10− 5/h error combining bias and scale factor. The gyro fiber coils of this experiment being 3 km long over a diameter of 200 mm and the wavelength being 1550 nm, this 1.5 × 10− 5/h error correspond to a phase difference error of only 5 × 10− 10 rad. Compared to the absolute phase of 2 × 10+10 rad accumulated over 3 km of propagation in the fiber coil, it yields a relative stability of 2.5 × 10− 20!


There is also a 24 h oscillation in this experimental result: the modulus of the measurement of the Earth rotation rate vector by the gyro triad yields, as we just saw, the linear part of the drift, but there is also a residual error in the measured direction of the axis of this Earth rotation rate vector. At rest with respect to the Earth, the measured inertial movement should follow a circle with always the same latitude, but with an orientation defect, this measured circle will become slightly tilted yielding a 24-h oscillation of the position error in latitude and longitude. This error is bounded and then it is not as harmful as the linear component of the drift in longitude that grows continuously over time.

6. Conclusion

Entering production in the 1980s, the RLG has revolutionized inertial techniques, and it is clearly the technology of reference today. However, its limited lifetime and its need for dithering have motivated the development of FOG technology based on a fully solid-state approach. Theoretical performance is similar for both technologies, but it has been more difficult for FOG to obtain it. It started as a product for medium grade (1/h range) applications in the 1990s [19]. However, with the development of fiber-optic communications components and digital signal processing techniques, it was shown that the FOG not only brings the expected improvement of lifetime, but does not face the performance limitation of the RLG in terms of noise and bias stability.

Today there is a clear change of mind, and the FOG is not seen any more as limited to medium grade, as presented during the last OFS (Optical Fiber Sensor) Conference [20] by Northrop Grumman, Honeywell and iXBlue. As shown in this paper, it even has the potential to become the “ultimate-performance” gyro that can surpass by at least one if not two orders of magnitude RLG technology. Results in a temperature-controlled environment are already impressive and the final challenge is to obtain this performance without this thermal control!

References

[1] E.J. Post, Sagnac effect, Rev. Mod. Phys. 39 (1967) 475–494.[2] G. Sagnac, L’éther lumineux démontré par l’effet du vent relatif d’éther dans un interféromètre en rotation uniforme, C. R. Acad. Sci. Paris 95 (1913)

708–710.[3] M. von Laue, Über einen Versuch zur Optik der bewegten Körper, Münch. Sitz. ber. (1911) 405–411.[4] H. Fizeau, Sur les hypothèses relatives à l’éther lumineux, et sur une expérience qui parait démontrer que le mouvement des corps change la vitesse

avec laquelle la lumière se propage dans leur intérieur, C. R. Acad. Sci. Paris 33 (1851) 349–355.[5] H.J. Arditty, H.C. Lefèvre, Sagnac effect in fiber gyroscopes, Opt. Lett. 6 (1981) 401–403.[6] M. von Laue, Die Mitführung des Lichtes durch bewegte Körper nach des Relativitätsprinzip, Ann. Phys.-Berlin 328 (10) (1907) 989–990.[7] M. von Laue, Zum Verzuch von F. Harress, Ann. Phys.-Berlin 62 (1920) 448–463.[8] C. Fabre, La limite quantique dans les gyromètres optiques, Rev. Sci. Tech. Déf. 7 (1990) 109–115.[9] F. Aronowitz, Fundamentals of the Ring Laser Gyro, in: Optical Gyros and Their Application, RTO AGARDograph 339, 1999, pp. 23–30.

[10] R. Ulrich, Fiber-optic rotation sensing with low drift, Opt. Lett. 5 (1980) 173–175.[11] E.C. Kintner, Polarization control in optical-fiber gyroscope, Opt. Lett. 6 (1981) 154–156.[12] H.C. Lefèvre, The Fiber-Optic Gyroscope, second edition, Artech House, Boston–London, 2014.[13] G.A. Pavlath, Fiber optic gyros: past, present, and future, in: 22nd International Conference on Optical Fiber Sensors, OFS 2012, in: Proc. SPIE, vol. 8421,

2012, Paper 842102-1.[14] S. Ezekiel, J.L. Davis, R.W. Hellwarth, Intensity Dependent Nonreciprocal Phase Shift in a Fiber Optic Gyroscope, Springer Series on Optical Sciences,

vol. 32, 1982, pp. 332–336.[15] H.J. Arditty, P. Graindorge, H.C. Lefèvre, P. Martin, J. Morisse, P. Simonpiétri, Fiber-optic gyroscope with all-digital processing, in: OFS 6, in: Proceedings

in Physics, Springer-Verlag, 1989, pp. 131–136.[16] IEEE standard specification format guide and test procedure for single-axis interferometric fiber optic gyros, IEEE Std 952, 1997.[17] D.M. Shupe, Thermally induced nonreciprocity in the fiber-optic interferometer, Appl. Opt. 9 (1980) 654–655.[18] N.J. Frigo, Compensation of linear sources of non-reciprocity in Sagnac interferometers, Proc. SPIE 412 (1983) 268–271.[19] S. Ezekiel, E. Udd (Eds.), Fiber Optic Gyros: 15th Anniversary Conference, Proc. SPIE, vol. 1585, 1991.[20] G. Sanders, chair, in: Fiber Optic Gyros: 35th Anniversary Workshop, OFS 22 Conference, Beijing, China, in: Proc. SPIE, vol. 8421, 2012.

C. R. Physique 15 (2014) 859–865





The centennial of the Sagnac experiment in the optical regime: From a tabletop experiment to the variation of the

Earth’s rotation

Le centenaire de l’expérience de Sagnac en régime optique :d’une expérience de laboratoire à la variation de la rotationde la Terre

Karl Ulrich Schreiber a,∗, André Gebauer a, Heiner Igel b, Joachim Wassermann b, Robert B. Hurst c, Jon-Paul R. Wells c

a Forschungseinrichtung Satellitengeodaesie, Technische Universität München, Germanyb Department fuer Geo- und Umweltwissenschaften Geophysik, Ludwig-Maximilians-Universität München, Germanyc Department of Physics and Astronomy, University of Canterbury, Christchurch, New Zealand



Keywords:Sagnac effectRing laser gyroscopeEarth rotationSeismology

Mots-clés :Effet SagnacGyroscope laser à anneauRotation de la TerreSismologie

The investigation of non-reciprocal behavior of optical beams in a rotating reference frame was the main motivation of the historic tabletop experiment of George Sagnac. His ground-breaking experiment was extended to a very large installation more than a decade later, which was sensitive enough to allow Michelson, Pearson and Gale to resolve the rotation rate of the Earth by an optical interferometer. With the advent of lasers in the early sixties of the last century, rotating laser cavities with a ring structure demonstrated superior performance and very soon matured to a point where mechanical gyroscopes were quickly superseded by laser gyroscopes in aircraft navigation. When vastly upscaled ring lasers were taken back to the laboratory at the end of the 20th century, the goal of applying the Sagnac effect to geodesy for the monitoring of tiny variations of Earth’s rotation was the main motivation. The large-ring laser G, which is the most stable instrument out of a series of instruments built by the New Zealand–German collaboration, routinely resolves the rotation rate of the Earth to better than eight orders of magnitude. Since G is directly referenced to the Earth rotation axis, the effect of diurnal polar motion, the Chandler and the Annual wobbles as well as tilts from the solid Earth tides can be found in the interferogram obtained from the ring laser. G has also demonstrated high sensitivity to rotations associated with seismic events. The toroidal eigenmodes of the Earth when they are excited by large earthquakes have been resolved. A surprisingly large amplitude has been measured for Love wave signals contained in the microseismic background signal. This paper summarizes the recent development of highly sensitive large Sagnac gyroscopes, and presents unique results from the measurements of rotations of the earth.


* Corresponding author.E-mail address: [email protected] (K.U. Schreiber).


860 K.U. Schreiber et al. / C. R. Physique 15 (2014) 859–865

r é s u m é

La recherche sur le comportement non réciproque des faisceaux optiques dans un référentiel tournant était la motivation principale de l’expérience historique de George Sagnac. Son expérience révolutionnaire à été étendue à une installation beaucoup plus grande plus d’une décade plus tard, avec une sensibilité suffisante pour permettre à Michelson, Pearson et Gale de déterminer la vitesse de rotation de la Terre à l’aide d’un interféromètre optique. Avec l’arrivée des lasers au début des années 1960, des cavités laser en anneau tournantes ont montré des performances supérieures et ont très vite atteint un niveau de maturité permettant à des gyro-lasers de surpasser les gyroscopes mécaniques dans le domaine de la navigation aérienne. Les lasers en anneau de très grandes dimensions ont repris le chemin des laboratoires à la fin du vingtième siècle, avec comme motivation principale l’utilisation de l’effet Sagnac pour la géodésie, afin d’accèder à de minuscules variations de la rotation de la Terre. Le laser à large anneau G, qui est l’instrument le plus stable parmi toute une série d’instruments construits dans le contexte d’une coopération germano-néo-zélandaise, résout régulièrement la vitesse de rotation de la Terre jusqu’à mieux que huit ordres de grandeur. Puisque G se réfère directement à l’axe de rotation de la Terre, l’effet du mouvement polaire diurne, le mouvement de Chandler et l’oscillation annuelle comme les inclinaisons des marées solides à la surface terrestre apparaissent dans l’interferogramme obtenu à partir du laser en anneau. G s’est aussi révélé avoir une grande sensibilité aux rotations associées aux événements sismiques. Les modes propres toroïdaux de la Terre, lorsqu’ils sont excités par des tremblements de terre de grande ampleur, ont été résolus. Une amplitude étonnamment grande a été mesurée pour des signaux d’ondes de Love contenus dans le bruit de fond micro-sismique. Cet article résume le développement récent de grands gyroscopes de grande sensibilité, et présente des résultats uniques pour ce qui est de la mesure des rotations de la Terre.


1. Introduction

Highly sensitive rotation sensing has a wide range of applications. These reach from very basic controls in robotics up to complex aircraft maneuvers based on high performance inertial navigation in real time. The available sensors for this purpose also cover a wide range. In the simplest case, they comprise two mechanical parts that move at some angular velocity with respect to each other, such as a lever in a gearbox. In robotics and telescope control, where this concept is applied, one usually finds encoders that count increments of angles. The great advantage of such encoder systems is their long-term stability for the measurement of angles. This stability results from the rigid link for the relative rotation of two mechanical parts with respect to each other. On the other hand, the movement of a pendulum in an inertial space that senses Coriolis forces, such as the trajectory of a Foucault pendulum on a rotating Earth, does not have such a rigid link and the measurement is subject to perturbations caused by friction and other small experimental imperfections. The very large optical Sagnac interferometer, set up by Michelson, Pearson and Gale [1], pointed into a new direction. It demonstrated the capability of a strapped-down optical system to observe rotation rates as low as 15 per hour to an accuracy of about 2%. The high-end regime of modern rotation sensing finally is marked by highly sensitive optical Sagnac interferometers, such as the fiber optic gyroscopes (FOG) or the ring lasers (RLG). These devices outperform a mechanical pendulum by many orders of magnitude. Although superior to a pendulum, a FOG or RLG only measures rotation rates and their most significant limitation is their long-term stability. While a navigation gyro in an airplane integrates rates of angular motion over short periods of time only, much more demanding applications, such as the precise estimation of the orientation of the instantaneous Earth rotation axis in space, are required in space geodesy.

A network of radio telescopes routinely measures the relative orientation of these telescopes with respect to distant quasars interferometrically two times per week. This generates a precise estimate of the rate of rotation of the Earth and the orientation of the rotational axis. It would be very desirable to support this measurement effort with a number of au-tonomously operated strap-down gyroscopes, rigidly attached to the Earth’s crust. While the telescopes of the International VLBI Service (IVS) [2] can resolve about 100 µs out of a total of 86400 s in length of day and the long-term stability of the measurement is given by the rigidity of the quasar constellation, large ring lasers have the sensitivity to perform to the same level of resolution, but not yet to the necessary long-term stability. However, they are much easier to operate and they do not require elaborate handling of large data volumes, including physical transport of storage media. In addition, unlike the other geodetic space techniques, such as Satellite and Lunar Laser Ranging (SLR/LLR), VLBI and the Global Navigational Satellite Systems (GNSS), ring lasers are referenced to the instantaneous axis of rotation of the Earth. Combining the inertial rotation measurement approach with the state-of-the-art measurements of the VLBI technique may eventually provide a ground-based assessment of the Lense–Thirring frame dragging process [3,4].

K.U. Schreiber et al. / C. R. Physique 15 (2014) 859–865 861

2. Active Sagnac interferometers

With the development of the laser in 1960 came the opportunity to move Sagnac interferometry from phase to frequency measurement, with a vast concomitant improvement in sensitivity. The first ring laser that operated as an inertial rotation sensor (albeit on a laboratory turntable) was reported by Macek and Davis [5] in 1963, exploiting the 1.15 µm line of the helium–neon (He–Ne) gain medium. The ring laser equation is given by [6,7]:

δ f = 4AλP

n · Ω (1)

with δ f the beat note obtained from the two counter-propagating laser beams in the cavity after superposition in a beam-splitter behind one of the turning mirrors. A is the area enclosed by the optical signal, P the corresponding perimeter, λ the wavelength of the optical beam, n the normal vector to the plane of the laser and Ω the rate at which the entire apparatus is rotating. This beat frequency is usually referred to as the ‘Sagnac frequency’. Laser gyroscopes are highly at-tractive navigational tools, as unlike mechanical gyros they have no moving parts. Modern aircraft navigational gyroscopes are commonly He–Ne lasers operated at a wavelength of 632.8 nm and usually have an area < 0.02 m2 corresponding to a perimeter of 30 cm or less, ensuring single longitudinal mode operation. The typical sensitivity of such devices is around 5 × 10−7 rad/s/

√Hz.

Small laser gyros have a tendency at very low rotation rates for the two beams to frequency-lock (known as lock-in), and this is usually overcome by mechanical angular dithering. By contrast, the first large ring laser gyroscope to unlock solely on Earth rotation was the 0.748-m2 Canterbury-I ring laser (C-I) situated in Christchurch, New Zealand [8]. It was a planar, essentially square geometry, filled with cold He–Ne gas and defined entirely by dielectric ‘super’ mirrors having a nominal reflectance of 99.9985%. A square ring configuration was chosen to optimize the signal-to-noise ratio, but also because of an expectation of reduced backscattering for mirrors used at 45 angle of incidence. Excitation occurred within a small glass capillary having no Brewster windows, thereby minimizing optical losses. In this early design, the required thermo-mechanical stability was achieved by placing the mirrors on super-invar holders, themselves attached to a 1-m2 Zerodur plate. Yielding a Sagnac frequency of 76 Hz, this device only rotation sensed for short periods due to its initial location in a high-rise building and the early-generation super-mirrors employed. Ultimately, it was shifted to an old war-time bunker in the Christchurch suburb of Cashmere, the Cashmere Caverns facility. Its operation was a tremendous step forward and the technical advances employed in this early prototype underpin the state-of-the-art ring lasers in use today. In particular, the gain starvation approach within an over-pressured (2–8 Torr) cavity ensured operation on a single longitudinal laser mode. This technique utilizes the pressure induced homogeneous line broadening to reduce the curvature of the gain profile at its most strongly saturated point, which in turn allows weak saturation to persist to higher powers than it otherwise would [9].

The ring lasers developed after C-I used progressively larger perimeters. The first to have a monolithic construction was the C-II device [10], which was constructed entirely from Zerodur (by Carl-Zeiss) and rigidly attached to the local bedrock in the Cashmere Caverns in 1997. The optical beam path is drilled into the neutral plane of the Zerodur slab, so that an area of 1 m2 is circumscribed by the counter-propagating laser beams. All four corners are beveled and polished so that ULE disks with optically contacted super mirrors can be wrung onto the laser body in order to generate a pre-aligned closed light path. Utilizing improved supermirror technology yielded a cavity Q of 8 × 1010 with a vast improvement in stability afforded by its monolithic design. Once enclosed in an ambient pressure stabilizing vessel, this ring laser structure detected already signals from solid Earth tides, exacerbated by ocean loading on the Banks Peninsula [11]. In the same year, when C-II was commissioned, the 12.25-m2 G-0 ring laser was set up as a proof of concept for the already designed, but not at that stage constructed, German ‘Gross-ring’. G-0 was only intended to demonstrate operation on a single longitudinal laser mode for an upscaled device with a significantly smaller FSR. However, it contained many design features that were be reused in later heterolithic rings, including a folded lever system which allowed for optical alignment of each cavity mirror to within ±10 seconds of arc [12]. Ultimately, the success of G-0 as an operational laser gyroscope removed the last hurdle for a more ambitious project.

The Gross-Ring (or G) is a 16-m2 monolithic ring laser constructed from the largest available piece of Zerodur. G is housed in a purpose-built underground laboratory at the Geodetic Observatory Wettzell near the Bavarian–Czech border in southern Germany [13]. As with the C-II laser the supermirrors are optically contacted, in this case to four Zerodur bars that form an X-shape, which defines the four-by-four meter perimeter of the ring. Having temperature and ambient pressure control, coupled with state-of-the-art supermirrors (the Q is 3.5 × 1012) and unparalleled stability, G yields the best overall rotation sensing performance of any ring laser developed in our group, as demonstrated by the recent observation of the Chandler and annual wobbles of the rotating Earth [14].

The success of the G-ring led to the development of ever larger lasers including the 834-m2 Ultra-Gross-Ring-2 (denoted by UG-2) located at Cashmere, which was necessarily heterolithic [15]. Built under a premise that bigger would be better, these lasers ultimately failed to deliver on their potential due to unexpectedly high losses on the intra-cavity mirrors and a susceptibility to geometric variation [16]. Perhaps more importantly, the lesson was learned that there is an optimum size for a ring laser gyroscope rigidly attached to the Earth, at least when there is no active interferometric scale-factor stabilization involved. Ironically, that value falls remarkably close to the dimensions of the G-ring.


Fig. 1. (Color online.) Time series of rotations derived from the large-ring laser G at the Geodetic Observatory Wettzell. Solid Earth tides, diurnal polar motion and the long-period Chandler and the Annual wobble are clearly visible in the data. VLBI measurements carried out twice per week are shown for comparison.

3. Ring lasers in geodesy

Applications in space geodesy typically require a superior relative sensor resolution of 1 part in 109 with respect to the measurement quantity and a similar stability over several months [2]. This is in particular true for the precise measurement of a global measurement quantity, such as Earth’s rotation based on a single local sensor [17]. The rotation rate in the Sagnac equation for an active ring cavity (Eq. (1)) then changes from Ω to ΩE , to denote the Earth’s rotation vector with a typical value of 72.7 µrad/s. Since errors in the estimation of day length accumulate with every turn of the Earth, the desired sensor resolution is as high as 0.07 prad/s at integration times of about two hours in order to also resolve sub-daily effects on the rotation rate of the Earth. In order to achieve this, it is important to have even better control over the scale factor 4A/λP , as well as all the variations of the inner product between ΩE and n, which finally relates the normal vector of the gyroscope to the instantaneous axis of rotation of the Earth. This property of an inertial sensor is unique among all the measurement techniques of space geodesy. It means that small variations in the orientation of the Earth rotation vector relative to the orientation of the ring laser structure are also contained in the observation of the beat note of the interferometer. Changes in orientation can have several causes. It is either possible to observe a variation of the normal vector with respect to the local vertical or to experience a change of attitude of the Earth rotation vector with respect to the body of the Earth. While the corresponding tilt of the solid Earth tides and ocean loading are important examples of the former, diurnal polar motion, nutation and Eulerian wobble of the rotating Earth, known as the Chandler wobble, are typical candidates for the latter. Fig. 1 illustrates the combined observation of these signals in a long-term measurement of Earth’s rotation. The G ring laser in Wettzell is a single component gyroscope, orientated nearly horizontally in the laboratory at the latitude ϕ = 49.1444 north. It is only sensitive to the projection sin(ϕ) of the normal vector to the Earth rotation vector ΩE . North–south tilts induced by geophysical signals together with a small offset in the horizontal alignment are changing the projection by a small amount denoted as δφ. Eq. (1) then changes to:

δ f = 4AλP

n · ΩE sin(ϕ + δφ) (2)

In addition to the above-mentioned rather periodic geophysical signals, the initial sensor orientation and local variations of the ring laser monument are causing an offset. Variations in the atmospheric pressure loading as well as changes in the local ground water table can alter the instantaneous gyroscope orientation significantly. In order to keep track of these irregular non-periodic changes, we are operating several high-resolution tiltmeters with a sensitivity of about 0.5 nrad on top of the ring laser body. While a tiltmeter is referenced to the local plumb line, it is sensitive to both the displacement caused by the solid Earth tides and the variation of the plumb line. A ring laser gyroscope, on the other hand, is only sensitive to the change of projection as a result of local displacement and corresponding latitude variation [18]. It is

δφtilt = − (1 + k − h)

g R∂v∂ϕ

(3)

and

δφRLG = (h − l)g R

∂v∂ϕ

(4)

where h, l, k are the respective Love numbers and g the mean equatorial gravity and R the Earth’s equatorial radius. Here v is the tidal potential and ϕ the latitude. The differences between the ring-laser-derived orientation changes and the


tiltmeter measurements are small, but can not be neglected. Up to now, it was not possible to establish the exact optical scale factor of the active ring laser cavity along with the true orientation of the gyroscope. The hot cavity path-length of the laser beams inside the cavity can be established from the optical frequency of the lasing mode, but this is insufficient to establish the corresponding gyroscope area. Therefore we are only obtaining relative changes of the ring laser beat note.

Fig. 1 compares two measurements, namely the VLBI-derived CO4 series of the north–south component of polar motion with respect to the longitude of 12.878 east of the ring laser on the Geodetic Observatory Wettzell. Comparing the change of the rate of rotation as a result of the motion of the instantaneous rotational pole with respect to the ring laser location, one can see small deviations between the VLBI inferred and the ring laser derived measurement. This is caused by the variation of the backscatter coupling between the two counter-propagating laser modes in the ring laser cavity, limiting the ring laser measurements to about 'Ω/ΩE = 8 × 10−9, within half an order of magnitude of the VLBI measurements from a global network.

A data set, recorded in 2007 during the storm event “Kyrill”, indicated a significant amount of correlation between the measured local rotational signal, regional barometric pressure loading and the corresponding wind activity. The observed phase shift between the signatures of the storm in the meteorological parameters and the ring laser response suggested an elastic local response to the wind load. With the improved data quality and sensor stability based on the newly introduced sensor stabilization methods [19], most of the earlier observations were identified as instrumentally induced effects. How-ever, locally acting wind loads are causing rotational dither within the sensitivity regime of the gyroscope. A finite element analysis approach was used to identify the transfer mechanisms and to reconstruct the amplitude and frequency regime of these perturbations. It was found that local wind, although creating small oscillations of the gyroscope with frequencies around 1 Hz, is not a limiting factor for the measurements of variations in length of day, because they average out over the longer integration times used in the geodetic application [20]. The finite-element model was chosen to be 10 km by 10 km wide and about 3 km high. It was shaped according to a digital terrain model (DTM) around the observatory with a spatial resolution of 25 m for the model surface. Since the rheology is assumed to be elastic, the results can be scaled linearly for the Young’s modulus of the soil. At the side and the bottom boundaries, the nodes were allowed to move along the plane, but not out of the plane. The wind load was applied to the nodes of the model surface from different directions. The force field is a sum of the pressure loading derived from the topography of the DTM and friction at the surface estimated from the corresponding land-use model (DLM). Forests have a much larger contribution than grass land. Every scenario generated a deformation and a rotation matrix for the ring laser site, which could be compared to the gyroscope measurements. The effect of friction is about two orders of magnitude higher than the loading effects and the modeled amplitudes are com-parable to the observations. Since the wind typically has a rather turbulent spatial structure, the resultant rotational field quickly drops off, since the individual contributions become incoherent about 300 m away from the ring laser.

4. Ring lasers in seismology

Currently there are two types of measurements that are routinely used to monitor global and regional seismic wave fields. Standard inertial seismometers measure three components of translational ground displacement (velocity, acceler-ation) and form the basis for monitoring seismic activity and ground motion. The second type aims at measuring the deformation of the Earth (strains). It has been noted for decades ([21] and previous 1980 edition) that there is a third type of measurement that is needed in seismology and geodesy in order to fully describe the motion at a given point, namely the measurement of ground rotation. The three components of seismically induced rotation have been extremely difficult to measure, primarily because previous devices did not provide the required sensitivity to observe rotations in a wide fre-quency band and distance range (the two horizontal components, equal to tilt at the free surface, are generally recorded at low frequencies, but are contaminated by horizontal accelerations). Single-component ring laser measurements of rotational ground motions—in particular the G-ring located in Wettzell (station WET)—have basically provided so far the first and only high-resolution broadband observations for seismology. The observations had a strong impact in the community and contributed substantially to the emergence of a new field, now termed “rotational seismology” (already two special issues published on the topic). In the following, we will briefly review the current state of the art and illustrate current research activities.

The collocated observations using the G-ring laser sensor at WET measuring the local vertical component of the rotation vector and a classical seismic broadband station have allowed seismologists to collect a data base with several hundred earthquakes covering magnitudes from M3 to M9 [22,23]. In seismology, a fundamental property of elastic wave fields can be exploited over a wide range of applications: assuming plane wave propagation (well justified in global seismology) the amplitude ratio of transverse acceleration (seismometer) and rotation rate (ring laser) is proportional to phase velocity (e.g., [22,24]). This has tremendous implications: the collocated point measurement contains direct information on the subsurface seismic velocity structure and propagation direction that both single observations do not have. These facts inspired the development of an entirely new approach to solving the seismic inverse problem using amplitude information: seismic tomography without using travel times [25,26]. The new method allows the tomographic reconstruction of seismic velocity models around the receivers. The locality of this approach is powerful for applications where sensors cannot be deployed in small-scale arrays (e.g., ocean floor, remote regions, boreholes, planetary missions).

A substantial quality improvement of the G-ring laser in Wettzell in summer 2009 led to the first ever made observation of the Earth’s free oscillations on a vertical rotation component [27]. In the long-period range, seismometer records are


Fig. 2. Amplitude spectra of ground motion observations following the M8.8 Maule earthquake in Chile 27 February 2010. The spectra are based on 36 hr seismograms. The vertical gray lines are theoretical predictions of toroidal nTm free oscillations based upon a spherically symmetric Earth model. The spectral lines indicate the excitation of individual modes (overtones of the fundamental modes). Top: For comparison the transverse component of ground motion at the Black Forest Observatory, Germany (BFO). Middle: Transverse component of ground motion at Wettzell, collocated with the ring laser. Bottom: Vertical component of rotational motions (ring laser observations).

severely contaminated by tilt motions [28]. Following several of the recent giant earthquakes time series with Love waves traveling around the Earth, more than four times could be observed on the ring laser. The spectrum of the observations revealed the theoretically expected spectral lines for toroidal modes (Fig. 2). The relative amplitudes of the individual modes are different from those observed in the translational components as a consequence of the differential relation between both motion types illustrating their complementary nature. It is important to note that, in particular, toroidal modes are extremely difficult to measure and ring lasers have started to fill this gap. Our pilot study opens many questions on further constraints from such observations on the Earth”s deep structure and seismic sources. In particular, the coupling between toroidal and spheroidal modes observed in some of the past earthquakes is currently being investigated [29,30].

A further quite sensational observation in the ring laser record could recently be made. Our planet is constantly vibrating—even in the absence of earthquakes—as a consequence of atmospheric pressure variations as well as interfer-ing ocean waves. The use of this ocean-generated ambient seismic noise band (approx. 5–20-s period) in connection with cross-correlation techniques is currently revolutionizing seismology, as it allows earthquake-free tomography. Taking the noise source distribution into account would substantially improve the results. So far the studies of noise almost exclusively focused on the use of vertical component seismograms (Rayleigh-type surface waves, polarized in the plane through source and receiver). It is important to note that the ambient noise also contains a substantial amount of energy in Love waves (horizontal polarization), as recently discovered by Japanese scientists. The analysis of this motion type with a vertical-component ring laser is extremely attractive as the instrument records only this type of motion. After improvements of the ring laser signal-to-noise ratio the ocean generated ambient noise signal recently appeared in the observations [31]. These observations are remarkable and offer new ways in analyzing the origin of the ocean-generated noise wave field with a dif-ferent perspective: the ring laser delivers pure Love-waves (something seismometers cannot do), and their origin and nature are not understood. The classic theory of Longuet–Higgins for the generation of ocean-generated ground motions predicts Rayleigh waves but not Love waves. Analysis of the ring laser recorded ambient noise field revealed clear evidence that Love waves are generated in the same area as the Rayleigh waves (mostly in the Northern Atlantic for European observatories). This implies that we are now ready to systematically investigate the Love wave part in the Earth’s noise field without the use of seismic arrays and the difficulties that Love and Rayleigh waves are mixed in translational records.

5. Conclusion

When Georges Sagnac reported his famous experiment in 1913, he has managed to resolve rotation rates of approxi-mately 1.8 rad/s with his optical interferometer. By means of significant upscaling Michelson, Pearson and Gale eventually achieved a resolution limit of their Sagnac interferometer of Ω ≈ 1.6 µrad/s. Their instrument covered an area of about 0.2 km2. With the availability of ring lasers and low-loss mirrors, inertial rotation sensing for navigation and attitude con-trol, based on rather compact devices, became feasible and widespread. Starting at the end of the last century, the concept of ring laser gyroscopes was revisited again with the goal of obtaining sufficient sensitivity and stability to apply Sagnac interferometry to geodesy and geophysics. Today, one hundred years after the original experiment by Georges Sagnac, we have reached a point where we can resolve 'Ω/ΩE < 10−8 routinely. Under conditions where the instrumentation can be stabilized to a constant value for the backscatter coupling over several months, long-term variations of polar motion can be observed. Improving the sensor stability rather than improving the sensitivity has become the main objective of our


research collaboration. If successful, a Sagnac interferometer has the potential to allow the detection of the Lense–Thirring frame-dragging in a ground-based experiment.

For the short term, the ring laser technology has provided seismology with a new quality of ground motion observations that is far from being fully explored. In particular, most observations so far are based on the vertical component G-ring observations. It is fair to say that the horizontal components of ground motion have never accurately been observed in the broadband seismic frequency band (tiltmeters are also sensitive to horizontal accelerations, which is why they are not able to provide pure tilt observations). This is one of the remaining research frontiers in ring laser applications to seismology and efforts are on the way to develop multi-component ring laser systems that are capable of measuring the complete vector of ground motion.

Acknowledgements

U. Schreiber and A. Gebauer acknowledge funding through the German Science foundation (DFG) under the contract SCHR 645/6-1. R. Hurst, J.-P. Wells and U. Schreiber acknowledge support from the Marsden fund of the Royal Society of New Zealand 10-UOC-80. H. Igel gratefully acknowledges contributions from the European Research Council (ROMY Project ERC-Adv. 2013 No. 339991), the Emmy Noether Program of the German Science Foundation and the Marie Curie Program (QUEST-ITN).

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C. R. Physique 15 (2014) 866–874





A ring lasers array for fundamental physics

Un réseau de lasers en anneaux pour la physique fondamentale

Angela Di Virgilio a,∗, Maria Allegrini b, Alessandro Beghi c, Jacopo Belfi a, Nicolò Beverini b, Filippo Bosi a, Bachir Bouhadef a, Massimo Calamai b, Giorgio Carelli b, Davide Cuccato c,d, Enrico Maccioni b, Antonello Ortolan d, Giuseppe Passeggio e, Alberto Porzio e,f, Matteo Luca Ruggiero g, Rosa Santagata h, Angelo Tartaglia g

a INFN Sezione di Pisa, Pisa, Italyb Department of Physics, University of Pisa, Pisa, Italyc DEI, University of Padua, Padua, Italyd INFN Legnaro National Laboratory, Legnaro (Padua), Italye INFN Sezione di Napoli, Naples, Italyf CNR-SPIN, Naples, Italyg Polytechnic of Turin, Turin, Italyh Department of Physics, University of Siena, Siena, Italy



Keywords:Sagnac effectRing-laserInertial sensorGravito-magnetismLense–Thirring effectLength of the day

Mots-clés :Effet SagnacLaser à anneauxCapteur inertielMagnétisme gravitationnelEffet Lense–ThirringLongueur du jour

After reviewing the importance of light as a probe for testing the structure of space-time, we describe the GINGER project. GINGER will be a three-dimensional array of large-size ring-lasers able to measure the de Sitter and Lense–Thirring effects. The instrument will be located at the underground laboratory of Gran Sasso, in Italy. We describe the preliminary actions and measurements already under way and present the full road map to GINGER. The intermediate apparatuses GP2 and GINGERino are described. GINGER is expected to be fully operating in few years.


r é s u m é

Après avoir passé en revue l’importance de la lumière comme sonde pour évaluer la structure de l’espace-temps, nous décrivons le projet GINGER. GINGER sera un réseau tridimensionnel de lasers en anneaux de grandes tailles capable de mesurer les effets de Sitter et Lense–Thirring. Cet instrument sera localisé dans le laboratoire souterrain du Gran Sasso, en Italie. Nous décrivons les actions préliminaires et les mesures déjà en cours, puis nous présentons la feuille de route complète de GINGER. Les équipements intermédiaires GP2 et GINGERino sont décrits. GINGER devrait être complètement opérationnel d’ici quelques années.


* Corresponding author.


A. Di Virgilio et al. / C. R. Physique 15 (2014) 866–874 867

1. Introduction

One of the pillars of the contemporary understanding of matter, energy and space-time is general relativity (GR). Its successes in explaining the behavior of the world around us and of the whole universe are well known as well as its so far unresolved conflict with quantum mechanics in the high-energy domain. It is however true that also in the very low-energy sector of the gravitational interaction, there are predictions of GR that have not been fully explored up to these days.

A typical example is the so-called gravito-magnetic component of the gravitational field, whose direct verification relies for the moment on only three experiments in space: Gravity Probe B (GP-B) that took data from 2004 to 2005 concluded in 2011 and the results were published in the same year [1]; the two LAGEOS satellites orbital nodes analysis, published in 2004 [2] and, with an improved modeling of the gravitational field of the Earth, in 2011 [3]; the LARES mission, under way and gathering data, launched in February 2012 [4].

GP-B verified the geodetic effect in the gravitational field of the Earth with an accuracy of 0.28% and the Lense–Thirring (LT) drag with an accuracy of 19%; the analysis of the precession of the nodes of the LAGEOS satellites verified the LT effect with the accuracy of 10%; finally LARES is working to determine the LT drag with an accuracy of a few percents (possibly 1%). Other evidence of gravito-magnetic effects may be found in the laser ranging of the orbit of the moon and in the study of the dynamics of binary systems composed of at least one compact massive object (neutron star).

Another example of a weak effect predicted by GR are gravitational waves. No direct measurement has been performed so far, but very strong indirect evidence for their existence is obtained from the observation of double star systems including a pulsar [5].

Besides pure GR effects, the observation of the universe on the widest scale provides also facts that can be consistent with GR, assuming that otherwise unseen entities exist, such as dark matter and dark energy. The former would produce the additional gravity required to explain the rotation curves of galaxies and the speeds of the components of star or galaxy clusters. The latter would be necessary to generate the push required by the accelerated expansion of the universe. These facts, partly going back to the thirties of the last century (dark matter) [6], partly quite recent (dark energy) [7], have stimulated ideas implying that GR might need some extension if not a complete change of paradigm. What matters here is that the phenomenology to look for and to analyze in search for differences from GR is in the domain of low and ultralow energies. The above remarks present reasonable motivations for working experimentally on the gravitational interaction in the weak domain looking for post-Newtonian effects and Parameterized Post Newtonian (PPN) descriptions that could evidence deviations from classical GR. Can such an investigation be conducted in a laboratory, besides relying on large-scale observations of the sky? The answer is yes and, among various possible experimental approaches, a perfect tool is represented by light. Light is indeed intrinsically relativistic and is affected in various ways by the gravitational field. In the classical domain and, as far as a theory is considered treating space-time as a continuous four-dimensional Riemannian manifold, light completely covers the manifold with a network of null geodesics. If we find the way of reading the local and global configuration of the null geodesics tissue, we can reconstruct the “shape” of space-time, i.e. the gravitational field, and see whether it fully corresponds to the GR description or maybe there is something missing.

While considering how to exploit light in order to explore the gravitational field, we should add that the advancement of the laser technologies has pushed the possibilities of such devices to unprecedented values of accuracy and precision. All in all, a laser, and in particular a ring laser, appears today as a most interesting apparatus to probe the structure of space-time at the laboratory scale.

These are the main motivations for the design and implementation of an experiment based on the use of ring lasers for fundamental physics. The main purpose is to explore the asymmetric propagation of light along a closed space path in the gravitational field of a rotating body. In a sense, the prototype of this type of experiments is the old Sagnac interfer-ential measurement of what we can now call the kinematic asymmetry of the propagation of light along a closed space contour, as seen by a rotating observer in a flat space-time (no gravitational field) [8]. So far ring lasers have been built as Sagnac sensors of absolute rotations (which means with respect to the “fixed stars”) for practical purposes, as compact and sensible devices replacing mechanical gyroscopes (this is the reason why ring lasers are also called gyrolasers) for navigation or, in the case of the most refined instruments, for geodesy or even for determining the length of the day in competition with VLBI (Very Long Base Interferometry). The latter application, which is fundamental, is already in the reach of the G Ring in Wettzell [9]. The latter facility is by now on the verge of being able to detect not only the kinematical rotations of the laboratory, but also the physical effects of the gravitational field due to the rotation of the source and of the laboratory.

Our experiment, named GINGER (Gyroscopes IN GEneral Relativity), is intended to further improve the technology be-yond G. The rest of the paper will describe both the theoretical framework and the final configuration of GINGER, and the steps that are under way in order to test the innovative technologies we are going to use and in order to build the final laboratory which will be located in the LNGS (Gran Sasso National Laboratories) of the Italian INFN. The ring laser appears today as a most interesting apparatus to probe the structure of space-time at the laboratory scale. At his early stage, the ex-pected sensitivity of GINGER will not be competitive with space measurements to test PPN theories, but being the apparatus on Earth, improvements will be feasible with time.

868 A. Di Virgilio et al. / C. R. Physique 15 (2014) 866–874

2. Light in the gravitational field of a rotating body

Assuming that space-time can be described by a metric theory on a four-dimensional Riemannian manifold with Lorentzian signature, the central geometric object containing the essence of the gravitational field will be the line element. If the source of curvature (i.e. gravity) is a steadily and freely rotating object, the line element is given by:

ds2 = g00c2 dt2 + grr dr2 + gθθ r2 dθ2 + gφφr2 sin2 θ dφ2 + 2g0φcr sin θ dφ dt (1)

The coordinates used in (1) are spherical in space with the radial coordinate measured from the barycenter of the central mass, assumed to be in free fall, the θ angle (colatitude) measured from the rotation axis of the source and φ (longitude) measured from a fixed direction (with respect to the “fixed stars”) in space; time t is measured by clocks located in a remote region not influenced by the gravitational field. The not fully standard notation used in (1) insures the dimensionlessness of the gµν functions; the speed of light c is here essentially a conversion factor transforming time into a length. The gµν ’s (the components of the metric tensor) depend on the variables r and θ only, because of the symmetry. If the central mass is indeed rotating, no global coordinate transformation exists converting the metric in (1) into the Minkowski metric.

The underlying assumptions so far are:

• the source of gravity is isolated and in steady rotation with respect to the ‘fixed stars’;• the central object is rigid or at least it keeps its shape and mass distribution fixed in time, and it is axially symmetric

with respect to the rotation axis;• space-time is asymptotically flat and Minkowskian.

If we consider a real system, such as the terrestrial gravitational field, none of the above conditions, strictly speaking, is satisfied. The Earth is influenced by the other bodies in the Solar System so that its axis does not keep a fixed orientation with respect to the quasars (“fixed stars”). The gravitational perturbations induced by the surrounding bodies and the differential heating of the surface cause changes in the shape and mass distribution because of the non-rigidity of the planet. Space-time is not flat anywhere in the universe because no empty asymptotic region exists.

If we are interested in tiny relativistic effects we shall be very careful while using the simple symmetries implied in (1), because they are all imperfect.

In any case, working with light and assuming that in free space its speed is the same c for all inertial freely falling observers (which is the essence of relativity), the corresponding line element will be equal to 0, and we will be able to write the coordinated time span along the world line of a light ray as:

dt =− g0φr sin θ dφ ±

!g2

0φr2 sin2 θ dφ2 − g00(grr dr2 + gθθ r2 dθ2 + gφφr2 sin2 θ dφ2)

cg00(2)

To ensure in any case an evolution towards the future (dt > 0) the + sign must be chosen.Eq. (2) permits to evaluate the coordinated time of flight of an electromagnetic signal between two successive events

in vacuo. Let us consider a closed path (in space); of course, excluding the horizon of a black hole, a closed path may be followed by light only in presence of some technical expedient (mirrors, optical fiber).

Integrating over the path, both on the right and on the left, two different results are obtained because of the off diagonal g0φ component of the metric tensor. Let us use the angular velocity of the central body as a reference for the direction of rotation: the so-defined anticlockwise sense will correspond to dφ > 0, the clockwise will correspond to dφ < 0. Finally we see that the difference between the corotating time of flight, t+ , and the counter-rotating one, t− , will be:

δt = t+ − t− = − 2c

"g0φ

g00r sin θ dφ (3)

If at the start and arrival point, imagined as being fixed in the chosen reference frame, there is a device sensible to (3)its proper time τ difference will be:

δτ = − 2c√

g00

"g0φ

g00r sin θ dφ (4)

The δτ difference is the basis of the way a ring laser works. δτ may be measured letting the two counter-rotating beams interfere and this is the typical Sagnac technique; the way of a ring laser is however different. Since the emission of light is continuous and steady, two standing waves, associated with the two directions of rotations, are formed and co-exist in the annular cavity of the laser. The time of flight difference is converted into different frequencies of the two waves, and in turn the frequency difference gives rise to a beat note. The frequency of the beat can be read analysing the power spectrum of the signal extracted at any point of the ring. The beat frequency fb is

fb = c2 δτ2Pλ

= − cPλ

√g00

"g0φ

g00r sin θ dφ (5)

where P is the length of the perimeter of the ring and λ is the wavelength of the radiation.


2.1. A laboratory on Earth

So far we have assumed an observer at rest with respect to the fixed stars, which is a quite unphysical situation. In practice, the experiment we want to perform will be located within a laboratory fixed to the solid body of the Earth. If so, we should update our choice of the coordinate system. There are various possibilities; the simplest, probably, is to still choose a global reference frame, but let its axes rotate together with the Earth. In this way, basically, the coordinates remain the same, but colatitude and longitude are terrestrial rather than celestial.

Now, from the viewpoint of the fixed stars, the paths followed by the light beams we want to use are no longer closed in space, because of the motion of the laboratory, but they still turn out to be closed in the corotating terrestrial reference frame. The general form of the line element still is like Eq. (1), but now the functions have different forms. Under the same assumptions as before we may work out the new metric elements applying first a kinematical rotation of the axes at the angular speed of the Earth Ω , then a physical Lorentz boost at the peripheral speed of the Earth in correspondence of the location of the laboratory [10].

The formal result of these two steps is a bit complicated, but for practical purposes, we may approximate the result considering that:

Ω Rc

∼ 10− 6

GM

c2 R= µ

R∼ 10− 9

GJ

c3 R2 = jR2 ∼ 10− 15 (6)

G is Newton’s constant; M is the mass of the Earth; R is its radius at the location of the laboratory and J is the angular momentum of our planet.

The highest order to which we are interested is the lowest non-zero term containing j/R2. Under this condition the final beat frequency fb turns out to be:

fb ≃ 2A

λPΩ(ua · un) + c A

λP R

#2#

Ωµ

csin θ − j

R2 cos θ

$(ur · un) − j

R2 (uθ · un) sin θ

$(7)

where A is the area of the ring; the u’s are unit vectors in the directions, respectively, of the axis of the Earth (a), the normal to the ring (n), the direction of the local meridian (θ ). The ratio A

λP is called the scale factor S of the instrument. The second term on the right of the formula is approximately 10− 9 times the first one.

3. The GINGER project: Gyrolasers for fundamental relativity

Considering the orders of magnitude (6) and formula (7), we see that, in order to reveal general relativistic effects depending on the mass and the angular momentum of the Earth, we need a device endowed with a sensitivity at least nine orders of magnitude better than the one required for measuring the only angular velocity of the Earth, through the classical Sagnac effect. In fact, in formula (7), the first term is the classical Sagnac term, whereas the second one contains both the Lense–Thirring drag, depending on the angular momentum J (whose norm appears in the equation in its geometrized form j), and the de Sitter or geodetic term expressing the interaction of the local Newtonian force with the angular velocity of the Earth. The latter two contributions turn out to have the same order of magnitude on the surface of the planet.

Is the needed sensitivity available or attainable today? Commonly, in navigation applications, ring lasers are based on single longitudinal mode He–Ne lasers operating at a wavelength of 632.8 nm. Inertial navigation devices usually have an area < 0.02 m2 corresponding to a perimeter of 30 cm or less. The typical sensitivity of such devices is around 5 ×10− 7 rad/s/

√Hz and the drift is as low as 0.0001 deg/h. This performance level is fully sufficient for navigational demands,

but falls short by several orders of magnitude for most geophysical applications; even more for fundamental physics.The Gross Ring (G) in Wettzell is a square ring, 4 m in side, mounted on an extremely rigid and thermally stable mono-

lithic Zerodur slab, located under an artificial 35-m thick mound. The most recent performance of G, expressed in terms of measured equivalent angular velocity, has a lower boundary below 1 prad/s (picoradian/second) at 1000-s integration time [11]. This sensitivity is above the requirement for the measurement of the GR effects, but various improvements in technologies, global design and signal cleaning should fill the remaining gap.

Actually at the level of prad/s and less, many delicate problems arise, besides the ones already mentioned, concerning the stability and behavior of the laser and the mirrors. Formula (7) has been obtained under the hypothesis that the rotational speed of the Earth is a constant, but this is not the case because of the coupling of the moment of inertia of the planet with the gravitational influence of the moon and the sun, which in turn changes with the configuration of the two celestial bodies. Furthermore, the non-rigidity of the Earth appears, influencing the instantaneous moment of inertia of the planet. Even the angles appearing in (7) are not stable at the required accuracy of the nrad or less, because of the non-rigidity of the crust of the Earth and because of mechanical and thermal instabilities of the measuring device.


Fig. 1. (Color online.) Octahedral configuration of GINGER. Six mirrors give rise to three mutually perpendicular square rings. The active control of the geometry may be achieved by laser cavities along the three diagonals connecting the mirrors.

A specific difficulty that has to be faced in the LT measurement is that the sought-after effect is a tiny time indepen-dent quantity superposed to a comparatively huge signal (the kinematical Sagnac term), so that the calibration is quite demanding. For this reason, an accurate investigation of the systematics of the laser is needed, and different techniques for extracting the signal need to be considered and evaluated. The result could in principle be validated repeating the measure-ment with different techniques and operating the laser at a different working point. In fact, one could also use a passive cavity, i.e. the measurement could be repeated with the same apparatus, but using an external laser source to interrogate the array of cavities. The technique of the passive cavity Sagnac is however not as mature as the active one, but in general a ring-laser system allows one to repeat the measurement with two different methods, having different systematics thus making the detection of a small constant effect easier.

G has reached remarkable sensitivity and stability, which makes the goal of using this kind of instrumentation for fundamental physics experiments a demanding but reasonable objective.

However, the different contributions to the beat frequency correspond to effective rotations along different directions. In practice, in order to discriminate the various terms, it is necessary to have a three-dimensional device able to measure the three components of the rotation vectors. The monolithic design cannot easily be extended to such a three-axial ring-laser system. Not considering mechanical difficulties, the monolithic solution would have prohibitive costs.

To overcome the above weaknesses and difficulties, we have conceived the idea of a three-dimensional array of square rings (each of which bigger than the present G ring), mounted on a heterolitic structure. Since the control of the shape of the rings is vital at the level of accuracy required, the rigidity of the ‘monument’ carrying the mirrors and cavities would be replaced by a dynamical control of each perimeter (like, at a smaller scale, for G-Pisa). In practice, the size and shape of any loop can be stabilized by piezoelectric actuators applied to the holders of the mirrors. The control loop, that will drive the piezos, will also exploit the passive optical cavities installed along the square diagonals. In order to develop and test the above-said controlled ringlaser, GP2, a new prototype, has been realized; it is equipped with six piezos. The GP2 experimental set-up has been recently completed in the laboratory of S. Piero a Grado, close to Pisa.

In addition, the final location of the laboratory could not be as close to the surface of the Earth as in Wettzell, because of the limits imposed by the top soil slow motion due to atmospheric pressure changes, rain, wind, etc. The location could be underground at the Gran Sasso laboratory (LNGS) facility in Italy, in a cavern under an average rock coverage 1400 m thick. This arrangement will insure a very good shielding against all kinds of surface noise.

A possible configuration for GINGER is shown in Fig. 1. Actually the octahedral structure is the most compact, and, in principle, easy to control, configuration; being the control obtained by means of laser cavities along the three main diagonals of the octahedron. The side of each of the three square loops would be not less than 6 m.

4. The GINGER roadmap

The actual building of GINGER requires a number of preliminary steps and phases related to the technologies and mea-surement strategies to be deployed. For this reason we have devised a roadmap to GINGER.

4.1. Goals and needs

We assume G as a benchmark for our project. Its intrinsic structural stability and a careful work to control the cavity length and laser discharge parameter made it possible to obtain a stability performance very close to the shot noise limit up to ≈ 104 s integration time. This corresponds to a statistical error in the angular velocity evaluation at a level of 10− 8 ×


Fig. 2. (Color online.) The picture compares the Allan deviation of G in Wettzell (years 2012 and 2010, courtesy of K.U. Schreiber) with the most relevant geodetic signals. The green parts show the region of interest for the geodetic precession and the Lense–Thirring effect; on the left of the picture it is possible to see the present level of test obtained by Gravity Probe B and Lageos. The two dotted lines show the shot noise of the G (16-m perimeter), and of a ring with perimeter 24 m.

ΩEarth, a factor of 5 above ΩLT (the Lense–Thirring contribution to the Earth’s rotation); Fig. 2 compares the results of G in Wettzell with what is necessary in order to be sensitive to the relativistic signals.

Such an impressive long-term stability has been obtained by an accurate modelling of all the environmental effects of geodetic, geophysical, or meteorological origin. G lacks, however, absolute accuracy: it is sensitive to one component of the angular velocity vector, and the absolute orientation of the laser cavity with respect to the fixed stars inertial frame cannot be measured with the required degree of precision. In order to arrive to a measurement of the Lense–Thirring effect, we need to improve the instrumental apparatus with respect to the following issues.

1. The signal-to-noise ratio (SNR), where noise is the shot noise of the instrument, should be increased. This can be obtained by:a increasing the size (the increase in SNR is more than quadratic with the size);b improving as much as possible the quality of the mirrors, and with a careful choice of the reflectivity and transmis-

sion;c investigating new techniques of laser operation (multimode locked operation, split mode, etc.);d increasing as much as possible the integration time with a suitable location of the apparatus.

2. The laser’s long-term stability has to be improved in order to allow longer integration times. This can be accomplished by actively controlling its operative parameters.

3. The scale factor stability and the accuracy with which it is known, for each ring of the array, must be improved. This requires:a active control of the geometry of the rings;b active control of the relative size of the different rings and of their relative orientation.

The main research activities and tasks can be grouped in five major areas.

i) The scale factor of each ring must be known and kept constant at the level of 10− 10. This can be achieved by controlling the geometry of each ring and the wavelength of the laser, by metrology techniques. “Heterolithic” ring lasers, i.e. based on a mechanical design, are cheaper, the mirrors support can be implemented with suitable translators (usually piezoelectric), and they are flexible enough to develop complex structures to support rings with different orientations. We are developing a method that uses information from the ring itself and the length of its diagonals (which are as well-resonant optical cavities), in order to drive the actuators of the mirrors and keep the ring, from a geometrical point of view, stable at the required level of 10− 10. To this aim, the heterolithic prototype GP2 has been developed. It has six


piezoeletric actuators, and will be the test bench for the above-mentioned control strategy. At present this work is in progress [12].

ii) The Lense–Thirring measurement requires to recover the angular velocity vector, with errors in the relative alignment of the planes of the rings of the order of 1 nrad. Large-frame ring lasers have been working with different orientations with respect to the Earth’s axis, but a multi-axis system of this size has not been implemented so far. As explained above, the heterolithic ring laser can “easily” be expanded to hold rings with different orientations in order to recover the full angular velocity vector; the very demanding issue is the nrad relative accuracy between different rings. The octahedron arrangement, in principle a very elegant design, could be a solution, since the cavities and diagonals can effectively provide information on the relative angles. The three diagonals of the octahedron are resonant Fabry–Pérot linear cavities that can monitor the relative angle between two different rings of the octahedron. Alternative config-urations, other than octahedral, and different strategies for the relative angle monitoring must be investigated, as for example the use of interferometry with 3D retro-reflectors, which has reached prad-level accuracy [13].

iii) Identify and refine the estimate of the Lamb parameters that regulate the non-linear dynamics of the ring-laser it-self [14]. The identification procedure will consist of two parts: i) identification and monitoring of cavity losses from mono-beam amplitudes and phase; ii) estimation and monitoring of the laser single pass gain and the remaining Lamb parameters from the measured plasma dispersion function of the He–Ne mixture. The scale associated with the non-linear dynamics permits to perform the absolute calibration of the instrument. To this aim, it is important to select the most convenient working point of the ring laser. In addition, the knowledge of laser dynamics enables us to run a non-linear Kalman filter that, a posteriori, can remove a large fraction of the backscattering contributions from the rotation rate measurements. In this way, we can improve the long-term stability of our rings, which represent a key issue of the GINGER project.

iv) Top-quality mirrors are adequate for ring lasers; however, mirrors will always be a point of concern. The mirrors used for ring-lasers are standard 1–2 inches substrate, but the quality of the substrate, the uniformity and quality of the coating are important issues, so that the mirror birefringence at the sub-ppm level and the possible related problems must be investigated as well. Any non-reciprocal effect induced by the mirrors has repercussions in unbalance in the two counter-propagating beams. There are few factories in the world able to provide this kind of mirrors; we are in touch with all of them.

v) Quantify environmental disturbances in order to have a proper assessment of the experimental site. This is of paramount importance, since the ring-laser is an inertial sensor, which is operated to deduce a global measurement quantity. Therefore the properties of the monument connecting the ring lasers to the Earth are critical. The operation of the G ring laser has shown that a near-Earth’s surface installation is subject to seasonal changes and noise generated by local wind patterns [15,16]. Therefore deep underground locations such as the LNGS have inherent advantages. In general, a deep underground installation within solid rock is almost insensitive to external environmental perturbations. This suggests that LNGS is potentially a good site for installing an array of actively stabilized large ring lasers, but dedicated measurements are necessary for site validation.

4.2. Work-plan

At the time of writing, two experimental areas are under construction: GP2, which will be used to develop the geometry control, has just been completed, and GINGERino, the 3.6-m-side ringlaser which will qualify the LNGS site for GINGER, is under construction. GINGERino will be located in a part of the laboratory away from daily activity, it will be acoustically protected and mounted on top of a granite structure well connected to the ground; it should start taking data in the second half of 2014.

Further steps will be taken to complete the characterization of the site, to test the technologies, and to collect information for geophysics.

The scientific work-plan toward the GINGER operation can be summarized as follows.

1) 2014: 3.6 m horizontal ring (in principle we should have an improvement of a factor 7.8 in sensitivity with respect to our first prototype) obtained using the mirror holders of our first prototype and longer tubes. The size is limited by the room presently available in the specific location within LNGS. A different positioning will allow a larger ring. In any case, as long as the present mirrors will be used, the side of the ring cannot exceed 6 m, because the mirrors have a 4-m curvature radius.With GINGERino, the systematics of the laser will be reduced, in particular backscattering noise should be reduced. In fact, we expect larger biases from the Earth’s rotation, the larger distance between mirrors and gas discharge, higher Q of the optical cavity (keeping the quality of the mirror constant). Acoustic shielding and a high-quality reference laser for the perimeter control are required. The task for GINGERino will be to observe the Allan deviation of the measurement, in order to understand the environmental disturbances. We expect to record some relevant seismic events, and, because of the improvement in sensitivity, also geodetic signals should be detected. Correlation with G Wettzell measurements should be possible (common tele-seismic events etc.).

2) 2015: One or two smaller rings should be added to GINGERino in order to reconstruct the angular velocity vector, with µrad precision (at least) in the vector direction. In particular, with a ring aligned with the Earth’s axis, a good measurement of the Length of the Day can be pursued.


Fig. 3. (Color online.) Workplan for the GINGER roadmap.

3) 2015–2016: The geometry and relative orientations of the rings should be defined, and it should as well be defined how to monitor the relative angle between the different rings.

4) 2015–2016: Construction of the octahedral arrangement of the full GINGER experiment.5) 2017–2018: GINGER in operation.

At the end of 2015, it should be possible to qualify the LNGS installation, and to understand at which level of precision the full installation of GINGER can be qualified. Fig. 3 shows the above outlined roadmap.

5. Discussion and conclusions

Summing up, we have started a number of practical steps towards the implementation of an experiment of fundamental physics based on a three-dimensional array of advanced ring lasers, named GINGER. The first objective of the experiment is to measure general relativistic effects due to the rotation of the Earth. In order to overcome the difficulties — implicit in the extreme sensitivity required by the measurement — we have built an international collaboration, involving two more lab-oratories in the world. A framework agreement is being signed between INFN, the University of Canterbury (Christchurch, New Zealand), the Technische Universität and the Maximilian University of Munich, Germany. We have also designed a roadmap (which we have already started to follow) aimed at testing and solving many technological and methodological problems. Step-by-step various intermediate facilities are in use and will be built: GP2 to develop the control of the ge-ometry, and GINGERino, based on our first prototype G-Pisa, to qualify the possible installation inside the underground laboratory of LNGS. In 2015, after the first set of measurements taken inside LNGS, the feasibility of GINGER will be clearer, and its time schedule as well; the construction of the GINGER apparatus per se is rather simple, it should not take more than one or two years. The use of a facility as LNGS, which is a very large and well-equipped laboratory, will facilitate the construction and the start up as well.

We should mention that techniques similar to the ones using lasers could be envisaged, such as atomic beams interfer-ometry [17], or long fibers loops [18]. Atoms would have an advantage with respect to light due to the fact that, for equal rings, the phase difference between the clockwise and the counter-clockwise circulation turns out to be proportional to the rest mass of an atom, whereas for light it is proportional to the energy of a photon. The former can easily be bigger than the latter. For the moment, this type of approach has very high potential for atomic ‘gyroscopes’, but it cannot (as yet) compete with advanced, large-scale ring laser technology, because the areas of atomic gyrometers are much smaller than those of ring lasers; furthermore in the atomic devices the signal-to-noise ratio is generally more unfavorable than with light.

The terrestrial detection of the Lense–Thirring effect is the main, but not the only purpose of GINGER. The main difficulty of the Lense–Thirring measurement is that it corresponds to a constant signal and the calibration is quite demanding. This is the reason why we are investigating as deeply as possible the systematics of the laser, and different techniques to extract the signal: the result could be validated repeating the measurement with different techniques and operating the laser at a different working point. The Sagnac effect works as well for a passive cavity, i.e. the measurement could be repeated with the same apparatus, but using an external laser source to interrogate the array of cavities. The technique of the passive cavity Sagnac, however, is not as mature as the active one. In summary: any measurement of constant effects is in principle difficult, but a ring-laser system allows one to repeat the measurement with two different methods, which have different systematics.

Beyond LT, we should mention that measurement methods from modern Space Geodesy perform at about the 10− 9 error level. Lunar Laser Ranging for example provides precise round trip optical travel times between a geodetic observatory and


cube corner retro-reflectors placed on the moon by the American APOLLO and the Russian LUNA landers [19]. With a long time-series of observations and continuous technical improvements, which reached a range precision of several millimeters in recent years, the error margin has reached a level of 10− 9 ÷ 10− 11. As one of many results according to [19], this led to improved constraints for the gravitational constant and its spatial and temporal variation of G/G = (2 ± 7) × 10− 13 yr− 1

and G/G = (4 ± 5) × 10− 15 yr− 2.Apart from actually measuring the Lense–Thirring effect with a ground-based gyroscope, also high-precision tests of

metric theories of gravity in the framework of the PPN formalism come within reach. With J = I⊕Ω⊕ according to [20], one obtains

ΩG = − (1 + γ )GMc2 R

Ω⊕ sinϑ uϑ (8)

for the geodetic (de Sitter, index G) precession rate and

ΩB = − 1 + γ + α14

2G I⊕c2 R3

%Ω⊕ − 3(Ω⊕ · ur)ur

&(9)

for the gravitomagnetic (Lense–Thirring, index B) precession rate; the sum of the two terms gives the contribution of the gravitational field to the angular velocity. Then there is the dominant kinematical term which is the classical Sagnac precession rate. In Eqs. (8) and (9), α1 and γ represent the PPN parameters that account for the effect of a preferred reference frame and the amount of space curvature produced by a unit rest mass. So, high-precision ring laser measurements performed by GINGER should be able to access α1 and γ . As already stated, being the apparatus on Earth, it should be possible in the future to envisage improvements and upgrading. With improvements of the order of 100–1000, it will be possible to set constraints on the PPN parameters competitive with space experiments.

Georges Sagnac would be surprised to see how far his method has gone after one century from his initial experiment. His purpose was to prove Special Relativity wrong now; under his name, we are preparing the most accurate verification of one of the effects of General Relativity. Maybe we shall not prove it wrong but insufficient. We shall know in few years.

References

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K.U. Schreiber, et al., Phys. Rev. Lett. 107 (2011) 173904.[10] A. Di Virgilio, et al., Int. J. Mod. Phys. D 19 (2010) 2331.[11] K.U. Schreiber, J.-P.R. Wells, Rev. Sci. Instrum. 84 (2013) 041101.[12] J. Belfi, et al., Interferometric length metrology for the dimensional control of ultra-stable ring laser gyroscopes, Class. Quantum Gravity (2014), in

press.[13] M. Shao, B. Nemati, Sub-microarcsecond astrometry with SIM-Lite: a testbed-based performance assessment, arXiv:0812.1530v1, 2008;

The Micro-Arcsecond Metrology Testbed, NASA’s Jet Propulsion Laboratory, Pasadena, California, http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20110023976.pdf.

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C. R. Physique 15 (2014) 875–883





The Sagnac effect: 20 years of development in matter-wave

interferometry

L’effet Sagnac : 20 ans de développements des interféromètres à ondes de

matière

Brynle Barrett a, Rémy Geiger b, Indranil Dutta b, Matthieu Meunier b, Benjamin Canuel a, Alexandre Gauguet c, Philippe Bouyer a, Arnaud Landragin b,∗

a LP2N, IOGS, CNRS and Université de Bordeaux, rue François-Mitterrand, 33400 Talence, Franceb LNE-SYRTE, Observatoire de Paris, CNRS and UPMC, 61, avenue de l’Observatoire, 75014 Paris, Francec Laboratoire Collisions Agrégats Réactivité (LCAR), CNRS, Université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse cedex 09, France


Article history:Available online 12 November 2014

Keywords:Matter-wave Sagnac interferometerLight-matter interactionsStimulated Raman transitionsCold atomsPrecision measurementsInertial navigationGeophysics

Mots-clés :Interféromètre Sagnac à ondes de matièreInteraction lumière-matièreTransitions Raman stimuléesAtomes froidsMesures de précisionNavigation inertielleGéophysique

Since the first atom interferometry experiments in 1991, measurements of rotation through the Sagnac effect in open-area atom interferometers have been investigated. These studies have demonstrated very high sensitivity that can compete with state-of-the-art optical Sagnac interferometers. Since the early 2000s, these developments have been motivated by possible applications in inertial guidance and geophysics. Most matter-wave interferometers that have been investigated since then are based on two-photon Raman transitions for the manipulation of atomic wave packets. Results from the two most studied configurations, a space-domain interferometer with atomic beams and a time-domain interferometer with cold atoms, are presented and compared. Finally, the latest generation of cold atom interferometers and their preliminary results are presented.


r é s u m é

Depuis les premières expériences d’interférométrie atomique en 1991, les mesures de rotation basées sur l’effet Sagnac dans des interféromètres possédant une aire physique ont été envisagées. Les études expérimentales ont montré de très bons niveaux de sensibilité rivalisant avec l’état de l’art des interféromètres Sagnac dans le domaine optique. Depuis le début des années 2000, de tels développements ont été motivés par de possibles applications dans les domaines de la navigation inertielle et de la géophysique. La plupart des interféromètres à ondes de matière qui ont été étudiés depuis sont basés sur des transitions Raman à deux photons pour la manipulation des paquets d’ondes atomiques. Nous présentons et comparons ici les résultats portant sur les deux configurations les plus étudiées : un interféromètre dans le domaine spatial utilisant un jet atomique et un interféromètre dans le domaine temporel utilisant des atomes froids. Finalement, la

* Corresponding author.E-mail address: [email protected] (A. Landragin).


876 B. Barrett et al. / C. R. Physique 15 (2014) 875–883

dernière génération d’interféromètres à atomes froids et leurs résultats préliminaires sont présentés, ainsi que les perspectives d’évolution du domaine.


1. Introduction

Rotation sensors are useful tools in both industry and fundamental scientific research. Highly accurate and precise ro-tation measurements are finding applications in inertial navigation [1], studies of geodesy and geophysics [2], and tests of general relativity [3]. Since the early 1900s, there have been many manifestations of Georges Sagnac’s classic experiments [4] that utilize the “Sagnac” interference effect to measure rotational motion, both with light and with atoms [5]. Gyroscopes based on this effect measure a rotation rate, Ω , via a phase shift between the two paths of an interferometer. The Sagnac phase shift, for both photons and massive particles, can be written as:

ΦSagnac = 4πEhc2 A · ! (1)

where A is the area vector of the Sagnac loop (normal to the plane of the interferometer and equal to the area enclosed by the interferometer arms) and E is the energy of the particle (E = hω for a photon of angular frequency ω and E = Mc2

for a particle of rest mass M). Eq. (1) shows that the Sagnac phase for a matter-wave interferometer is larger by a factor of Mc2/hω compared to an optical one with equivalent area. This scale factor is ∼ 1011 when comparing the rest energy of an atom to that of an optical photon in the visible range—emphasizing the high sensitivity of atom-based sensors to rotations. In this article, we will review some of the key developments that have taken place over the last 20 years regarding matter-wave Sagnac interferometers.

There has been dramatic progress in the field of atom interferometry in recent history. During the late 1980s, various types of atom interferometers were proposed as sensitive probes of different physical effects [6–9], and by the early 1990s the first experimental demonstrations had been realized [10–13]. As a result of their intrinsically high sensitivity to inertial effects, atom interferometers are now routinely used as tools for studies of fundamental physics and precision measurements [14]. The first experiments that exploited the rotational sensitivity of atom interferometers were carried out by Riehle et al. [13] using optical Ramsey spectroscopy with a calcium atomic beam. Fig. 1 shows their interferometer configuration and experimental results. By rotating their entire apparatus at various rates Ω , and recording the fringe shift of a Ramsey pattern, they were the first to demonstrate the validity of Eq. (1) for atomic waves.

In 1997, two other research groups [15,16] simultaneously published results pertaining to rotation sensing with atom interferometers.1 Although both experiments relied on atomic beams, they each employed a different method to generate matter-wave interference.

In Ref. [15], a beam of sodium atoms (longitudinal velocity ∼ 1030 m/s) was sent through three nano-fabricated trans-mission gratings (200 nm period, 0.66 m separation) which acted to split, reflect and recombine atomic wave packets taking part in the interferometer. By precisely controlling the applied rotation of their apparatus, they measured rotation rates of the same magnitude as that of the Earth (Ωe = 73 µrad/s), with a short-term sensitivity of about 3 × 10−6 rad/s/

√Hz.

Furthermore, they showed agreement with theory at the 1% level over a relatively large range of ± 2Ωe—corresponding to an improvement by a factor of 10 over the first measurements of Ref. [13]. Some of their experimental results are shown in Fig. 2a.

In contrast to Ref. [15], counter-propagating light pulses were used in Ref. [16] to manipulate a beam of cesium atoms (longitudinal velocity ∼ 290 m/s). In this work, the atoms entered a ∼ 2-m-long interrogation region where they traversed three pairs of counter-propagating laser beams that drove a π/2 − π − π/2 sequence of velocity-selective two-photon Raman transitions between long-lived hyperfine ground states. We explain in detail this interferometer scheme in Section 2.1. Each pair of Raman beams was separated by 0.96 m and aligned perpendicular to the atomic trajectory. By rotating the Raman beams at different rates, an interference pattern was constructed in the number of |F = 4, mF = 0⟩ atoms at the output of the interferometer, as shown in Fig. 2b. The resulting short-term sensitivity of their rotation measurements was 2 × 10−8

rad/s/√

Hz.Comparing the short-term sensitivity achieved by these two experiments, there seems to be a clear advantage to using

light pulses over nano-fabricated transmission gratings to split and recombine the atomic wave packets (although some gain in sensitivity can be attributed the difference in the enclosed area between the two interferometers). The main advantage of using light pulses to interact with the atoms is their versatility and precision. One can easily modify the strength, bandwidth and phase of the light–matter interaction through precise control of the laser parameters. In comparison, nano-fabricated gratings are passive objects that must be carefully handled and placed within the vacuum system—making their modifica-tion or replacement much more challenging. For example, to change the phase of the gratings in Ref. [15] by π/2 requires a

1 In 1996, Oberthaler et al. [17] also carried out sensitive rotation measurements with atoms using a Moiré deflectometer—a device consisting of an atomic beam and three mechanical gratings that can be considered the classical analog of a matter-wave interferometer. No quantum interference was involved in these measurements.

B. Barrett et al. / C. R. Physique 15 (2014) 875–883 877

Fig. 1. a) Ramsey–Bordé configuration of a state-labeled atom interferometer based on single-photon transitions. Here, a beam of atoms traverses two pairs of traveling wave fields. The laser fields within each pair are separated by a distance D , while the two pairs are separated by d and are counter-propagating with respect to each other. b) Optical Ramsey fringes measured for the apparatus standing still (curves labeled a, c, and e), for the apparatus rotating at a rate of Ω = −90 mrad/s (curve b), and for a rate Ω = +90 mrad/s (curve d). The center of the Ramsey patterns for Ω = ± 90 mrad/s are clearly shifted to the right and left, respectively, relative to those for which Ω = 0. Both figures were taken from Ref. [13].2

Reprinted with permission from F. Riehle, T. Kisters, A. Witte, J. Helmcke, C.J. Borde, Phys. Rev. Lett. 67 (1991) 177.© 1991 by the American Physical Society.

Fig. 2. a) Experimental results from Ref. [15]. Here, the rotation rate inferred from the interferometer, Ωmeas, is plotted with respect to the applied rate, Ω , inferred from accelerometers attached to the apparatus. The slope of the linear fit was measured to be 1.008(7). The residuals of the fit are shown below. b) Atomic interference pattern as a function of applied rotation rate from Ref. [16]. The horizontal offset from zero rotation provides a direct measurement of the Earth’s rotation rate, Ωe. (See footnote 2.)Reprinted with permission from A. Lenef, T. Hammond, E. Smith, M. Chapman, R. Rubenstein, D.E. Pritchard, Phys. Rev. Lett. 78 (1997) 760 and from T.L. Gustavson, P. Bouyer, M.A. Kasevich, Phys. Rev. Lett. 78 (1997) 2046.© 1997 by the American Physical Society.

physical displacement of only 50 nm perpendicular to the atomic trajectory. Modifying the phase of the light–matter inter-action requires no moving parts, and can be done electro-optically with high precision. Furthermore, the use of two Raman lasers allows the use of state-labeling techniques to address the diffracted and undiffracted pathways of the interferometer [9]. Usually, one detects the number of atoms remaining in either state by scattering many photons per atom, and inferring the phase shift from the ratio of state populations. This technique, which is not possible with transmission gratings, is less sensitive to fluctuations in total atom number and exhibits a high signal-to-noise ratio.

2 Readers may view, browse, and/or download material for temporary copying purposes only, provided these uses are for noncommercial personal pur-poses. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.


With the conclusion of these proof-of-principle experiments, the study of atomic gyroscopes entered a new phase which focused primarily on developing them as rotation sensors. This meant understanding and reducing sources of noise and systematic error, as well as improving the short-term sensitivity, linearity, long-term stability and accuracy of the devices. Furthermore, there remained a question regarding the type of coherent matter-wave source to design the sensor around: atomic beams or cold atoms. Modern (since the year 2000) atomic rotation sensors and the improvement of their perfor-mances will be described below.

The remainder of this article is organized as follows. In Section 2.1, we review the basic operation principles of Sagnac interferometers based on two-photon Raman transitions, which represents the key experimental technique used in modern atomic gyroscopes. Section 2.2 presents two examples of experiments using respectively atomic beam and cold atoms, and the comparison of their performances. Section 2.3 describes the most recent experiments of cold atom Sagnac interferome-ters. Finally, we give some perspectives for future improvements to these sensors and conclude in Section 3.

2. Atomic rotation sensors

2.1. Principles of atomic Sagnac interferometers

In this section, we describe the basic operation principles of an atom-based gyroscope based on optical two-photon Raman transitions. All light-pulse interferometers work on the principle of momentum conservation between atoms and light. When an atom absorbs (emits) a photon of momentum hk, it undergoes a momentum impulse of hk (−hk). In the case of Raman transitions, the momentum state of the atom is manipulated between two long-lived electronic ground states. Two laser beams with frequencies ω1 and ω2, respectively, are tuned such that their frequency difference, ω1 − ω2, is resonant with a microwave transition between two hyperfine ground states, which we label |1⟩ and |2⟩. When the Raman beams are counter-propagating (i.e. when the wave vector k2 ≈ −k1), a momentum exchange of approximately twice the single photon momentum accompanies these transitions: h(k1 − k2) ≈ 2hk1. This results in a strong sensitivity to the Doppler frequency, keff · v, associated with the motion of the atoms, where keff = k1 − k2 is the effective k-vector of the light field. Under appropriate conditions, a Raman laser pulse can split the atom into a superposition of states |1, p⟩ and |2, p + hkeff⟩ (with a pulse area of π/2), or it can exchange these two states (with a pulse area of π). With these tools, it is possible to coherently split, reflect and recombine atomic wave packets such that they enclose a physical area—forming an interferometer that is sensitive to rotations.

Fig. 3 shows the most common matter-wave interferometer configuration, which consists of a π/2 − π − π/2 sequence of Raman pulses, each separated by a time T (analog to an optical Mach–Zehnder interferometer). If there is a phase shift between the wave packets associated with each internal state at the output of the interferometer, it manifests as a simple sinusoidal variation between the state populations:

N2

N1 + N2= 1 − cosΦtot

2(2)

Here, N1 and N2 are the number of atoms in states |1, p⟩ and |2, p + hkeff⟩, respectively, and Φtot is the total phase shift of the interferometer given by:

Φtot =!φ1 − φA

2"−

!φB

2 − φ3"

(3)

The individual phases, φi , in this expression are imprinted on the atom by each Raman pulse. They take the form φi =k(i)

eff · r(ti) + φ(i)L , based on the orientation of the effective k-vector, k(i)

eff , the position of the center of mass of the wave packet, r(ti), and the relative phase between the two Raman lasers, φ(i)

L , at the time of the ith pulse, t = ti . The superscripts “A” and “B” on φ2 indicate the upper and lower pathways of the interferometer, respectively, as shown in Fig. 3.

In general, there are two types of interferometer signals that can be detected within the realm of inertial effects: changes in absolute velocity (i.e. accelerations) and changes in the velocity vector (i.e. rotations). For accelerations, the sensitivity axis of the interferometer is along the propagation axis of the Raman lasers, while for rotations the interferometer is sensitive along an axis perpendicular to the plane defining the enclosed area. The evaluation of interferometer phase shifts in a non-inertial reference frame (accelerating or rotating) has been described in detail in previous publications [8,18–21]. Here, we give an intuitive calculation of the phase shift for an atom interferometer in a frame rotating at a constant rate. Fig. 3 illustrates the situation from the atom’s perspective, where the Raman lasers are rotating at a rate Ω . At t = 0, the orientation of the Raman beams is rotated by an angle θ1 = −ΩT relative to the propagation axis of the atomic trajectory. Provided that |θ1| ≪ 1, this imprints a phase shift on the atoms of φ1 = keffθ1L. At t = T , the Raman beam is perpendicular to the atomic trajectory, thus the rotation-induced phase shift is zero and, in the center-of-mass coordinate frame, it can be shown that φA

2 = −φB2 . Similarly, at t = 2T , the phase is φ3 = −keffθ3L, where θ3 = ΩT . Using Eq. (3), the total interferometer

phase shift due to the rotation is Φrot = keff(θ1 + θ3)L = −2 keffvΩT 2. Here, we have used the fact that the separation between Raman pulses is L = vT with v the initial atomic velocity at the entrance of the interferometer. A more general form of this expression, where the rotation vector ! is not necessarily perpendicular to the plane of the interferometer, is given by [18]:


Fig. 3. (Color online.) Schematic of a matter-wave Sagnac interferometer based on two-photon Raman transitions. An atom in state |1⟩, with center-of-mass velocity v = p/M , is subjected to a sequence of counter-propagating laser pulses that are rotating relative to the atomic trajectory at a constant rate Ω .

Φrot = −2(keff × v) · ! T 2 (4)

Clearly, the rotation phase shift scales linearly with v and Ω , and it scales quadratically with T (or L). This implies that the rotation sensitivity of the matter-wave interferometer scales with the enclosed area—in the same manner as an optical Sagnac interferometer. In fact, Eq. (4) can be recast to highlight this area dependence by defining the area vector as A =−(hkeff/M)T × vT . Then Φrot = 2MA · !/h, which is equivalent to the Sagnac phase for matter waves given by Eq. (1).

2.2. Space-domain or time-domain atom interferometers: atomic beams versus cold atoms

Following Eq. (4), two strategies exist for maximizing the sensitivity of the rotation sensor: increasing the atomic velocity, i.e. increasing the distance L = vT between the beam splitters, or increasing the interrogation time T . The former requires an atomic beam source and will be referred to as a space-domain interferometer. In this configuration, the Raman lasers are running continuously and the Sagnac area is defined, in practice, by physical quantities L and v (area proportional to L2/v). The latter will instead work in the time domain and requires the use of cold atoms that can be interrogated for sufficiently long times (typically 100 ms). In this second configuration, the Raman lasers are pulsed in order to define the interaction time with the atoms, and the Sagnac area is defined by physical quantities T and v (area proportional to vT 2). We will give two examples of such experiments.

Space-domain interferometers with an atomic beam: Refs. [22,23] By the early 2000s, Sagnac interferometers based on atomic beams had been significantly improved compared to the first experiments in the 1990s [13,15,16]. Specifically, the work of Gustavson et al. [22] at Yale helped realize short-term sensitivities of ∼ 6 × 10−10 rad/s/

√Hz. This gain in sensitivity

arose mainly due to the implementation of a high-flux atom source. Moreover, it solved for the first time the problem of discriminating between phase shifts from rotation and from acceleration by the implementation of a counter-propagating atomic beam geometry. Since the sign of the rotation-induced phase shift given by Eq. (4) depends on the velocity vector, reversing the direction of the atomic beam results in a phase shift with opposite sign. Thus, by measuring the interference fringes from two separate counter-propagating sources, one can suppress via common-mode rejection parasitic phase shifts arising, for example, from the acceleration due to gravity or vibrations of the Raman laser optics.

In an effort to further reduce systematic effects and improve the long-term accuracy of the gyroscope, an additional technique was later introduced by Durfee et al. [23] at Stanford to eliminate spurious non-inertial phase shifts, such as those produced by magnetic fields or ac Stark effects. This involved periodically reversing the direction of keff between measurements of the two interference signals from each atomic beam, which facilitated a sign reversal of the inertial phase while maintaining the sign of the non-inertial phase. Combining these four signals drastically reduced systematic shifts and long-term drift of rotation phase measurements (stability of ∼ 2.5 × 10−9 rad/s in 15 min), at the cost of the short-term sensitivity. Further correlation analysis with measured environmental variables, such as temperature, indicate that the long-term sensitivity could considerably be reduced to ∼ 3 × 10−10 rad/s in 5 h [23] by a correction proportional to those measurements, as shown in Fig. 4b.

Time-domain interferometers with laser cooled atoms: Refs. [24,25] In contrast to atomic gyroscopes using the propagation of atomic beams over meter-long distances, cold atom interferometers make use of the T 2 scaling of the gyroscope sensitiv-ity by interrogating laser-cooled atoms during ∼ 100 ms. They allow for more compact setups and for a better control of atomic trajectories and thus of systematic effects. A pioneering experiment that started at SYRTE (France) in the early 2000s used two counter-propagating clouds of cesium atoms launched in strongly curved parabolic trajectories. Three single Raman beam pairs, pulsed in time, were successively applied in three orthogonal directions leading to the measurement of the three


Fig. 4. (Color online.) Space-domain interferometer with an atomic beam. a) Interference fringes from the counter-propagating atomic beam experiments in Ref. [22] (figure adapted from [22]). Here, the fringes labeled “N” and “S” are from north- and south-facing beams, respectively, while the difference is labeled “N–S”. A fit to this signal, shown as the solid black line, gives an estimate of the Earth’s rotation rate where the line crosses zero. b) Rotation phase measurements recorded over 14 h from Ref. [23]. The middle plot labeled “(a)” shows the raw measurements compensated with k-reversal, along with a fit to sum of five independent temperature measurements (solid line). Plot “(b)” shows the temperature-compensated phase, and “(c)” is the Allan deviation of the rotation signal (dashed line: Allan deviation of the uncorrected data, solid line: Allan deviation of the corrected data). (See footnote 2.)Fig. 4b is reprinted with permission from D.S. Durfee, Y.K. Shaham, M.A. Kasevich, Phys. Rev. Lett. 97 (2006) 240801.© 2006 by the American Physical Society.

Fig. 5. (Color online.) a) Schematic of the SYRTE atomic gyroscope–accelerometer experiment using two cold atom clouds, from Ref. [25]. b) Interferometer configurations leading to information on the three axes of inertia, from Ref. [24]. Performances of the accelerometer-gyroscope obtained in 2009 by Gauguet et al. [25]. c) The acceleration sensitivity is limited by residual vibrations of the platform (top panel), while the rotation measurement is limited by quantum projection noise (bottom panel).

axes of rotation and acceleration, thereby providing a full inertial base [24]. The SYRTE atomic gyroscope–accelerometer ex-periment is shown in Fig. 5a. Fig. 5b presents the various interrogation configurations that enable extraction of the three components of acceleration and rotation. The short-term acceleration and rotation sensitivity of the instrument (with 1 s of integration) was first 4.7 × 10−6 m/s2 and 2.2 × 10−6 rad/s in the work of Canuel et al. [24], respectively. The setup (in par-ticular the detection system and atom source preparation) was then improved to reach the quantum projection noise limit on the rotation measurement at the level of 2.4 × 10−7 rad/s/

√Hz, and a long-term sensitivity of 1 × 10−8 rad/s at 1000 s

integration time [see Fig. 5c, bottom panel], which was ultimately limited by the fluctuation of the atomic trajectories due


Table 1Comparison of gyroscope properties for systems based on cold atoms and atomic beams. The Yale (2000) experiment demonstrated exceptional short term sensitivity but did not demonstrate long-term stability [22]. The short term sensitivity of the Stanford (2006) experiment is extrapolated back to one second using the long-term stability of 2.5 × 10−9 rad/s and assuming a rotation rate stability scaling as 1/

√τ from Ref. [23].

Domain Time (SYRTE, 2009) Space (Yale, 2000/Stanford, 2006)

Atomic source MOT Atomic beamFlux Low HighSagnac area 4 mm2 24 mm2

Velocity 33 cm/s 290 m/sInterferometer length 2.7 cm 2 mSensor size 0.5 m 2.5 mVelocity control Good (molasses) Poor (Oven)T control Very good Poor (T = L/v)Acceleration sensitivity Very high (large T ) ModerateAcceleration rejection Very good (T symmetric) Moderate (asymmetry in v)Wavefront distortion limited Yes ProbablyShort-term sensitivity (1 s) 2.3 × 10−7 rad/s/Hz1/2 Yale (2000): 6 × 10−10 rad/s/Hz1/2

Stanford (2006): 8 × 10−8 rad/s/Hz1/2

Long-term sensitivity (15 min) 1.0 × 10−8 rad/s Yale (2000): not specifiedStanford (2006): 2.5 × 10−9 rad/s

Fig. 6. (Color online.) a) Atomic trajectories of the SYRTE four-pulse interferometer from Ref. [26]. b) The Allan deviation of rotation rate measurements for an interrogation time 2T = 480 ms. The preliminary results indicate a sensitivity of 4 × 10−9 rad/s with 5000 s of integration time.

to wavefront distortions of the Raman lasers [25]. Two other important features of this device had been tested: the linearity with the rotation rate and the independence of the rotation measurement from the acceleration. First, the evaluation of the non-linearities from a quadratic estimation of the scaling factor evolution has been demonstrated to be below 10−5. Second, the effect of the acceleration on the rotation phase shift is canceled at a level better than 76 dB when adding a well-controlled DC acceleration on the apparatus.

To conclude this section, we present in Table 1 a comparison of the gyroscopes using atomic beams and cold atoms. Al-though the geometries are very different, the final sensitivity levels are similar (atomic beams show increased sensitivity by a factor of ∼ 3). Furthermore, cold atoms offer better control of systematic effects and more compact setups with margins of improvements by an optimization of the geometry. In particular, an improvement of both short-term and long-term stabili-ties should arise from a larger average velocity, which was chosen to be very small in this first experiment (33 cm/s). In the next section, we present the new generation of cold atom experiments since 2009 aiming at improving the performances by more than one order of magnitude.

2.3. Latest generation of cold atom gyroscopes [26–30]

Following the experiments discussed previously, the strategy to enhance the sensitivity of the gyroscope essentially consists in increasing the interferometer area. Two geometries have been developed so far. First, keeping the same three Raman pulses configuration, but with a straighter horizontal trajectory (v = 2.8 m/s), the gyroscope of the University of Hannover [28] has an area five times larger (19 mm2) with preliminary results [30] similar to those of SYRTE.

The second solution is based on four Raman light pulses and an atom cloud following a vertical trajectory [see Fig. 6a]. In that case, the atom interrogation is symmetric with respect to the apogee of the atom trajectory and is not sensitive to the DC acceleration. This new geometry was first demonstrated in Ref. [24] and has shown improved performances in Ref. [27]. Since the interferometer phase shift scales as Φrot ∼ keff g Ω T 3, and the maximum possible area is 300 times larger (11 cm2 with a total interrogation time of 2T = 800 ms), substantial improvements in sensitivity are anticipated. Preliminary results presented in Fig. 6b have already shown a short-term sensitivity similar to the one obtained in the


three-pulse configuration (improved by one order of magnitude compared to the previous four-pulse experiment), as well as an improvement in long-term stability to 4 × 10−9 rad/s in 5000 s of integration time.

The present limit to the sensitivity arises from vibration noise of the Raman beam retro-reflecting mirrors at frequencies higher than those of the rotation signal of interest (which has a characteristic time scale of variation of several hours). The impact of the Raman mirror vibrations is a commonly encountered problem in cold atom inertial sensors and has been addressed in various works, e.g. in atomic gravimeters [31]. The corresponding limit to the sensitivity arises from the dead time between consecutive measurements (due to the cold atom cloud preparation and detection), which results in an aliasing effect when the high-frequency noise is projected onto the measurements. In other words, the dead time corresponds to a loss of information on the vibration noise spectrum, making it difficult to remove from the measurements.

3. Conclusion and perspectives

After the first proof-of-principle experiments in the early 1990s, Sagnac interferometry with matter-waves has benefited from the important progress of atomic physics in the last 20 years. These advances have allowed the continuous improve-ment in performances of atomic gyroscopes in terms of sensitivity, long-term stability, linearity and accuracy, making atomic setups competitive or better than state-of-the-art commercial laser gyroscopes. These improvements are motivated by pos-sible applications in inertial guidance and in geophysics. Both space- and time-domain interferometers have their own advantages. For space-domain interferometers with atomic beams this includes zero dead time between measurements, high dynamic range and a relative simplicity, versus better control of the scaling factor and smaller size for time-domain interferometers with cold atoms.

For applications in inertial navigation, the use of straight horizontal trajectories [28] is more favorable than highly curved parabolic trajectories [25]. On the one hand, using horizontal trajectories with fast atoms reduces the interrogation time T , thereby reducing the acceleration sensitivity (scaling as T 2), while keeping a high Sagnac scale factor (proportional to the atomic velocity), thus optimizing the ratio of rotation sensitivity over residual acceleration sensitivity. On the other hand, as demonstrated in the optical domain by laser-based gyroscopes [32], very-large-area atom interferometers based on highly curved parabolic trajectories [26,29] are of important potential interest in the field of geophysics. In the latter case, the possibility to measure rotation rates and accelerations simultaneously is advantageous in order to distinguish between fluctuations of the Earth’s rotation rate and fluctuations of the projection of this rate on the measurement axis of the gyroscope. Another possibility for enhancing the Sagnac interferometer area could consist in transferring a large momentum to the atoms during the matter-wave diffraction process. Such large momentum transfer beam splitters, studied since 2008 by several groups, could result in more compact Sagnac cold atom gyroscopes of reduced interrogation times.

Nevertheless, the main limitation on increasing the sensitivity of time-domain interferometers in both applications (iner-tial navigation and geophysics) comes from aliasing of high-frequency noise due to measurement dead times. One solution consists in increasing the measurement repetition rate [33], but at the cost of a reduction of the interrogation time and, consequently, the sensitivity. A second method could consist in hybridizing a conventional optical gyroscope with the atom interferometer in order to benefit from the large bandwidth of the former, and the long-term stability and accuracy of the latter. This method has been demonstrated in the case of the measurement of a component of acceleration by hybridiz-ing a classical accelerometer and an atomic gravimeter [31,34]. Another possibility could consist in operating a cold-atom interferometer without dead time between successive measurements in a so-called joint interrogation scheme [35,29].

Besides the improvement of these Sagnac interferometers using atoms in free fall, the development of confined ultra-cold atomic sources opens the way for new types of matter-wave Sagnac interferometers in which the atoms are sustained or guided [36,37]. Under these conditions, the interrogation time should no longer be limited by the free-fall time of the atoms in the vacuum system, and larger interferometer areas can be achieved. The present limitation on long-term stability due to wavefront distortions of the Raman laser could be lifted, since the position of the atoms will be well controlled. In contrast, the interaction with the guide or between ultra-cold atoms should bring new systematic effects that will require further study.

Acknowledgements

This work is supported by Délégation générale pour l’armement grant REI n 2010.34.0005 and the French space agency CNES (Centre national d’etudes spatiales). B. Barrett and I. Dutta also thank CNES and FIRST-TF for financial support. The laboratory SYRTE is part of the Institut francilien pour la recherche sur les atomes froids (IFRAF) supported by the Région Île-de-France.

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C. R. Physique 15 (2014) 884–897




The Sagnac effect: 100 years later / L’effet Sagnac 100 ans après

Large-area Sagnac atom interferometer with robust phase

read out

Interféromètre Sagnac atomique avec une acquisition de signal robuste

Gunnar Tackmann ∗, Peter Berg ∗, Sven Abend, Christian Schubert, Wolfgang Ertmer, Ernst Maria RaselInstitut für Quantenoptik, Gottfried Wilhelm Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany



Keywords:Atom interferometerGyroscopesInertial sensingCold atoms

Mots-clés :Interféromètre atomiqueGyromètresCapteurs inertielsAtomes froids

We report on recent progress on our matter-wave Sagnac interferometer capable of resolving ultra-slow rotations below the µrad s− 1 level with a 1-s measurement time and a repetition rate of 2 Hz. Two Raman interferometers are employed that are susceptible to rotation and acceleration. We demonstrate two read-out schemes exploiting the strict phase correlation of the dual interferometer, the first one locking the interferometer to the mid-fringe position, and the second relying on phase modulation combined with ellipse fitting. In both, the sensitivity to gravity acceleration is employed for controlling the differential interferometer phase without influencing the rotation signal. Furthermore, we discuss errors in the rotation signal arising from atom source instabilities combined with a residual misalignment of the three pulsed light gratings used for atomic diffraction. Monitoring the source position fluctuations allows us to suppress this spurious signal. We achieve stable operation with a sensitivity of 850 nrad s− 1 Hz− 1/2 for a 1-s measurement time, and 20 nrad s− 1 after 4000 s of averaging.


r é s u m é

Nous présentons ici les progrès réalisés avec notre interféromètre Sagnac à ondes de matière capable de résoudre des rotations ultra-lentes de l’ordre du µrad s− 1 avec un temps de mesure de 1 s et un taux de répétition de 2 Hz. Deux interféromètres Raman, sensibles aux rotations et aux accélérations, ont été utilisés. Nous avons développé deux techniques d’acquisition de signal qui exploitent la corrélation de phase du double interféromètre, la première en stabilisant l’interféromètre sur la position à mi-frange, la deuxième se basant sur la modulation de phase en combinaison avec la méthode dite ellipse fitting. Les deux techniques reposent sur la sensibilité à l’accélération gravitationnelle qui est utilisée pour contrôler la phase différentielle sans perturber la phase de rotation. De plus, nous discutons dans cet article des erreurs dans la mesure de la rotation engendrées par des instabilités des sources atomiques croisées, avec un non-alignement résiduel des trois réseaux optiques qui sont utilisés pour la diffraction atomique. L’enregistrement des fluctuations de la position des sources nous permet de réduire ce signal parasite. Nous atteignons ainsi une

* Corresponding authors.E-mail address: [email protected] (E.M. Rasel).


G. Tackmann et al. / C. R. Physique 15 (2014) 884–897 885

sensibilité d’opération stable de 850 nrad s− 1 Hz− 1/2 à une seconde de temps de mesure et 20 nrad s− 1 après 4000 s d’intégration.


1. Introduction

Since the discovery of the rotation-induced phase shift in an interferometer with area-inclining paths, the Sagnac ef-fect [1] has become the basis for high-precision rotation measurements [2]. The first observation of this effect, demonstrated by Georges Sagnac in 1913, was realized with a spectrally refined light field emitted from an electric lamp. Over the last century, the technological advances, in particular the invention of the laser, boosted the sensitivity of light-based Sagnac interferometers. Moreover, the possibility of coherently manipulating neutral atoms with laser light enabled the develop-ment of matter-wave-based Sagnac interferometers, subject of this article. In both, light and atom interferometers rotation measurements rely on the Sagnac phase:

ΦSagnac = 2Ehc2 ! · A (1)

where E is the particle energy of the interfering photons or atoms, respectively. Furthermore, h is the reduced Planck constant, c the speed of light, ! the rotation, and A the area enclosed by the interferometer paths. This yields an advantage in the interferometer scaling factor for atoms featuring much higher particle energy. Comparing the energy E = hω of photons at the helium–neon laser wavelength of λHeNe = 2πcω− 1 ≈ 633 nm with the energy of rubidium-87 atoms at low velocities E = m87Rbc2 yields an increase factor of 1.2 × 1011 in the Sagnac phase for given area and rotation rate. This huge factor is put into perspective by the fact that typical sources for matter-waves have relatively low particle flux leading to high shot-noise [3], and the realization of areas as large as several square centimetres or even many square meters has yet to be demonstrated with matter-waves. Nevertheless, the tremendous boost in the Sagnac scale factor allows the realization of highly compact devices capable of reaching sensitivities comparable with state-of-the-art laser gyroscopes. Furthermore, the absolute value of the area enclosed by a matter-wave Sagnac interferometer is well-controlled through the well-known atomic mass and, in case of atom interferometers involving light gratings for atomic diffraction, the photon-induced atomic recoil velocity, yielding a high accuracy and stability in the rotation measurement.

The most sensitive atom interferometer gyroscopes employ light gratings for atomic beam splitting. Atom interferom-eters based on this technique have rapidly evolved to high-precision tools for measurements of fundamental physical constants [4–7], electromagnetic fields [8], and inertial forces [9] within the last two decades. The cornerstone for these interferometers is the coherent manipulation of the internal and external atomic degrees of freedom using laser light. Pre-cision gyroscopes based on this concept have been pioneered by groups at Stanford University and at SYRTE (“Systèmes de référence temps espaces”) in Paris. The work in Stanford led to the first development of a dual atom interferometer gyroscope [10] using two thermal beams of neutral cesium atoms. It reached sensitivities of down to 0.6 nrad s− 1 Hz− 1/2

inferred from the instrument’s signal-to-noise ratio [11], only one order of magnitude less than today’s best ring–laser gy-roscopes [12]. The group at SYRTE realized a dual atom interferometer capable of measuring all six axes of inertia using two laser-cooled atomic clouds counter-propagating on superimposed trajectories and a single-light grating that was pulsed in time [9]. These works yielded a much more compact and highly sensitive cold-atom gyroscope with a demonstrated sensitivity of 240 nrad s− 1 Hz− 1/2 and 10 nrad s− 1 after integration [13].

In this article, we present advances on our atomic dual Sagnac interferometer, which uses three spatially separated light gratings that are pulsed in time for splitting, redirecting, and recombining the interferometer paths enabling large interfer-ometer areas and therefore high rotation sensitivity. We present two different robust rotation signal read out techniques exploiting the correlation of the two interferometers. The use of three single light gratings sets high demands on the rela-tive alignment of the employed laser beams as well as on the control of the atomic sources position and launch velocities. These demands and their impact on the gyroscope measurement stability are discussed in detail in this article, which is or-ganized as follows: Section 2 describes the operation principle of Raman light-pulse atom interferometers and, in particular, of our atomic gyroscope. In Section 3, the gyroscope measurement procedure and technical setup are described. Section 4introduces the obtained interferometer signals and derives the sensitivity of a rotation measurement using the dual Raman interferometer. The robust read-out techniques are introduced in Section 5, and the long-term stability of the rotation mea-surement as well as the major source for slow fluctuations are discussed in Section 6, before concluding on the achieved results in Section 7.

2. Measuring rotation with atom interferometry

The functional principle of our light-pulse–atom-interferometer-based inertial sensor relies on matter-wave diffraction on light gratings formed by counter-propagating laser beams transferring two photon momenta to the atoms [14,15]. The thereby induced Rabi oscillations occurring between different atomic momentum states enable the realization of coherent

886 G. Tackmann et al. / C. R. Physique 15 (2014) 884–897

Fig. 1. (Colour online.) Raman atom interferometer. (a) We drive Raman transitions between internal atomic states |g⟩ and |e⟩ by shining in two light fields of frequencies ω1 and ω2. Each light field features a large detuning % with respect to an intermediate atomic state |i⟩, so that single-photon transition rates are negligible. (b) By shining in the light fields from opposite sides, we realize an effective wave vector keff, which defines the momentum h kefftransferred to the atoms. (c) An atom undergoing the stimulated Raman transition gains a velocity component vr . (d) The dual Sagnac atom interferometer is realized by launching atomic clouds onto parabolic trajectories and interrogating them with three horizontal counter-propagating pairs of Raman laser beams along their free fall. The enclosed area makes the interferometer susceptible to rotations of the device.

atom-optical elements. Depending on the Rabi angle, being the product of Rabi frequency ΩRabi and pulse duration τ , it is possible to create a coherent superposition of two momentum states, or even to fully transfer the atoms from one momen-tum state to the other. The assigned light pulses that correspond to beam splitters and mirrors are called π/2- and π -pulses, respectively. This process implies a resonance condition on the frequency difference of the two counter-propagating laser beams, which is the equivalent to the Bragg angle condition in the well-known case of an electromagnetic wave impinging on a solid-state crystal structure [16]. Choosing the frequency difference of the counter-propagating laser beams equal to this Bragg resonance frequency plus a frequency corresponding to an atomic transition energy difference allows us to drive a stimulated Raman transition, which coherently couples internal and external atomic degrees of freedom. Hence, the atom performs an internal state transition from ground state |g⟩ to excited state |e⟩ if, and only if it acquires a photon recoil momentum of hkeff corresponding to the effective recoil of the two counter-propagating light fields (see also Fig. 1). This allows us to entangle the internal and external atomic states and thereby to detect the external atomic state population by reading out the internal state population. This has been first introduced by Christian Bordé as state-labelling [17].

An interferometer is formed by applying π/2- and π -pulses separated by the interferometer free evolution times Tduring which the atom interferometer paths drift apart from or towards each other when they are in different momentum (and thus also internal) states. While more complex schemes are often used (see, e.g., [9]), we will restrict ourselves to the so-called Mach–Zehnder-like interferometer scheme consisting of one initial beam splitter or π/2-pulse, a mirror or π -pulse that inverts the momenta of the interferometer arms, and a second π/2-pulse that closes the interferometer. The two resulting interferometer output state populations oscillate as a function of the interferometer phase. Assuming the preparation of an atomic ensemble in a well-defined momentum state in the internal ground state, the interferometer phase determines the so-called transition probability, which reads:

Pe = 12 − 1

2 cos (Φ) (2)

This performs a typical interferometer fringe pattern as a function of the total interferometer phase:

Φ = ΦΩ + Φa + Φlaser + ΦEM (3)

containing contributions arising from rotations (ΦΩ ) and accelerations (Φa). In addition, the interferometer phase comprises contributions from the relative phase of the two laser light fields (Φlaser) as well as a phase shift from electromagnetic-field induced atomic level shifts (ΦEM).

2.1. Atomic Sagnac interferometer

In order to create a maximal area for a given interferometer pulse separation time T , the beam-splitting direction is chosen to be perpendicular to the atomic forward velocity (see Fig. 1). Taking into account the atomic velocity vat and the recoil velocity vr gathered by the atom during the beam-splitting process, the enclosed area depicted in this configuration is found to be A = vatom × vrT 2. The recoil velocity vr = h keff m− 1

at contains the atomic mass mat, and the two-photon wave vector keff = k1 − k2. With the Sagnac formula (Eq. (1)), this results in a rotation-induced atom interferometer phase of

ΦΩ = 2(! × vatom) · keff T 2 (4)


This phase shift can also be derived in a slightly different way: according to [14], the laser grating phase φi is imprinted on the matter-wave phase during each atom–light interaction i, leading to a total phase shift of

Φ = φ1 − 2φ2 + φ3 (5)

Here, each phase term φi = φi(t) + keff di (applied during the pulses i = 1, 2, 3) comprises a temporal and a spatial part, where di is the normal distance separating the atom and a well-defined phase–plane position, which is in our case given by the mirror surfaces retro-reflecting the Raman beams (these are introduced in Section 3). This results in an interferometer phase shift in case light-fields are accelerated with respect to the atom in free fall along the interferometer paths. For an effective wave vector of the multi-photon process keff and an acceleration a, this phase shift is found to be first order [14]:

Φa = a · keff T 2 (6)

This also allows us to derive the rotation-induced phase shift by considering the Coriolis acceleration aCoriolis = 2(!× vatom)

reproducing Eq. (4). These considerations yield the inertial phase shifts in first order. Higher-order phase shifts as those considered in [18] can be neglected for the parameter range used in our interferometer.

In order to discriminate between phase shifts induced by acceleration and rotation, we employ two interferometers with atoms travelling on parabolic trajectories with opposite horizontal velocities, as depicted in Fig. 1. The signs of the photon momenta transferred to the atoms in the two clouds are opposite, which will be explained in more detail in Section 3. This causes the acceleration phase to flip sign while the rotation phase is common for both interferometers. The phases measured by the two interferometers can therefore be written as:

Φ1 = ΦΩ + Φa + Φlaser + ΦEM,1 (7)

Φ2 = ΦΩ − Φa + Φlaser + ΦEM,2 (8)

The rejection of the phase shifts induced by electromagnetic fields is then realized in the so-called k-reversal technique, which will be introduced in Section 3.2. The laser-induced phase shift Φlaser enables us to add a phase shift to control the interferometer point of operation, which leaves us with the rotation-induced phase shift ΦΩ .

3. Experimental realization

The gyroscope apparatus containing the two atom interferometers depicted in Fig. 2 was first described in [19]. Each of the two interferometers is created with atomic clouds of 87Rb that are magneto-optically trapped and launched on a parabolic trajectory [20]. The atomic ensembles are launched with a horizontal velocity of 2.79 m s− 1, and a vertical velocity of 0.73 m s− 1. On their flight of about 120 ms, the matter-waves are initially prepared in the interferometer ground state, then split, redirected and recombined with a typical interferometer pulse separation of T = 23 ms, resulting in an area of A = 17.4 mm2. We employ retro-reflected bi-frequency light fields for Raman beam-splitting that are pulsed in time in three parallel beams in the horizontal plane when the atomic clouds travel almost perpendicularly to the beams’ optical axes. The transition probability Pe is then measured in a state-selective fluorescence detection at the end of the free fall.

3.1. Interferometer states

The atoms exit the source in clouds with temperatures around 8 µK and particle numbers of 108 in the hyperfine state |52 S1/2, F = 2⟩, spread over all Zeeman sublevels. A first velocity selective Raman π -pulse selects atoms of a well-defined velocity class [21] that are transferred from the excited state |e⟩ = |52 S1/2, F = 2, mF = 0⟩ to the ground state |g⟩ = |52 S1/2, F = 1, mF = 0⟩. The atoms remaining in the |52 S1/2, F = 2⟩ manifold are pushed out of the measurement zone by applying a circularly polarized laser beam that is resonant to the |52 S1/2, F = 2⟩ → |52 P3/2, F = 3⟩ transition. The two so-called clock states |g⟩ and |e⟩ are chosen as they are sensitive to magnetic fields only to the second order, which strongly reduces the impact of external magnetic fields onto the internal level shift, and thereby on the interferometer phase and signal contrast. In order to discriminate between the Zeeman sublevels and to only address the mF = 0 states, a homogeneous magnetic bias field is applied in the interferometer region (see Fig. 2a). This allows to address a pair of entangled states, which are in our case:

|g⟩ ⊗ |p = 0⟩ = |g,0⟩ (9)

|e⟩ ⊗ |p = h keff⟩ = |e, h keff⟩ (10)

with p being the atomic momentum in the beam-splitting direction.


Fig. 2. (Colour online.) Gyroscope apparatus. (a) The gyroscope apparatus is depicted here including light fields, optics, vacuum chamber, and copper coils for magnetic field generation. (b) The atom interferometer trajectories are depicted in red and blue for atomic paths on which the atoms are in ground and excited state, respectively. The common angle of the Raman light fields with respect to the atomic forward velocity direction is slightly differing from 90

by the Doppler angle αD, which allows the selection of either of the two beam-splitting directions keff,1 and keff,2 via the relative Raman laser detuning (see text). The atom interferometer is depicted here for the case keff,1 in colour, and for keff,2 in grey dashed representation.

3.2. Raman light fields

Raman beam-splitting is enabled through three pairs of spatially separated light fields generated by two power-amplified external cavity diode lasers (ECDL). As the relative phase of these two lasers is transferred to the interferometer states, it is required to be highly stable within the duration of each interferometer sequence. This is realized by stabilizing the beat of the two lasers onto an ultra-stable low-noise microwave reference at about 6.834 GHz, similar to the work presented in [22]. The laser light emitted by both lasers is superimposed and guided to the experimental setup in three polarization-maintaining single-mode fibres. Their output is collimated each to a beam of 19 mm waist, resulting in a triplet of parallel, horizontal beams (see Fig. 2) with centre separations of 65 mm.

The three Raman beams are retro-reflected on single mirrors that form two pairs of counter-propagating light fields in each beam. All three mirrors are mounted on a common aluminium support structure. The two outer mirrors are re-motely controllable in horizontal and vertical tilt using piezo stepper motors. Also the whole support structure itself can be controlled in both tilt directions using pairs of piezo stepper motors and piezo stacks for coarse (stepwise) and fine (continuous) adjustment, respectively. The Raman retro-reflection mirrors feature a common angle αD with respect to the forward velocity of the atomic trajectories (see Fig. 2b). This adds a frequency shift ± ωD = ± vx keff sin(αD) to the Raman transition arising from the two-photon transition Doppler shift for horizontal atomic velocity vx . In the presence of all four light fields in our retro-reflected setup, this results in two resonances in each direction for frequencies +ωD and − ωD, which correspond to momentum transfers keff,1 and keff,2 = − keff,1. Each of the resonances can be addressed by adding


Fig. 3. (Colour online.) Detection signal example. The photodiode signals allow us to deduce the amount of light scattered by the atomic clouds during the travel through the detection zone. The signal is depicted for the two interferometers in red and blue, respectively. The expected signal envelopes for continuously applied detection light is depicted as dashed line. The average height of the four pulses is recorded in order to compute the transition probability Pe for each interferometer (see text).

the Doppler frequency to the frequency difference of the two Raman lasers.1 As the two interferometers feature opposed forward velocities, the interferometer with ± vx is transferred a momentum ± h keff,1 and ∓ h keff,2 as depicted in Fig. 2b. Hence, the two interferometers feature opposite momentum transfer directions and the acceleration-induced phase shifts carry opposite signs in the two interferometers, whereas the phase shift induced by rotations is common for both. This is used for distinguishing accelerational and rotational phase shifts as initially introduced in Section 2.

Whereas acceleration-induced phase shifts can be discriminated from those induced by rotations by the use of two interferometers, phase shifts arising from the influence of electromagnetic fields to the internal atomic structure, such as differential AC-Stark shifts and quadratic Zeeman shifts, are also imprinted onto the rotation phase. However, in case they do not comprise a gradient along the direction of the atomic beam-splitting keff , these shifts can be discriminated from ro-tation phase shifts by employing the so-called k-reversal technique. This consists in alternating the beam-splitting direction in subsequent measurement cycles and computing the interferometer phase pairwise from subsequent measurements em-ploying keff,1 and keff,2, respectively, which corresponds to simultaneously reversing the area of both interferometers [23]. The resulting alternation of the rotation signal allows to suppress the k-invariant parts of the phase shift ΦEM, namely AC-Stark and Zeeman induced shifts, by performing the half-difference of subsequent measurement results.

3.3. Interferometer read out

The interferometer phase is imprinted on the populations of the two interferometer states |g, 0⟩ and |e, h keff⟩. These populations are determined in an internal-state-selective fluorescence detection subsequent to the last beam-splitting pulse when the atom clouds pass through the pulsed detection beams, as depicted in Fig. 2. The internal state population is measured in a sequence of four pulses. During the first pulse a fluorescence signal Ne proportional to the excited state population is measured using circularly polarized laser light to drive a closed transition between the states 52 S1/2, F = 2and 52 P3/2, F = 3. Between these pulses, a pulse with laser light resonant to the 52 S1/2, F = 1 → 52 P3/2, F = 2 transition optically pumps all atoms from the ground to the excited state. The third pulse probes once more the number of atoms in the excited state, which now results in the signal N = Ng + Ne proportional to the total number of atoms. All light fields are set up in a retro-reflected configuration in order to prevent for pushing the atoms out of the detection zone during the pulse sequence. In order to suppress the electronic noise, the detection pulses require a minimal duration of 800 µs.

The fluorescence signal is given by a convolution of the atomic cloud size and of the detection beam shape. Together with the light-pulse sequence, this results in a profile depicted in Fig. 3. As the pulse application takes place at a fixed time in the measurement sequence, the detection signal shape depends on the arrival time of the atomic sample. In particular, Ne is reduced in case of a delayed arrival and rises if the atomic sample arrives early in the detection zone. The second pulse height however remains stable for small arrival time fluctuations as it corresponds to the maximum of the detection signal envelope. This introduced the purpose of the fourth pulse during which the total number of atoms is detected a second time, but on the flank opposite to Ne. The ratio of the signal height of this latter pulse to N allows us to deduce the arrival time variation, and therefore also to correct for this effect on Ne in the detection data treatment.

1 Changing the relative Raman laser detuning also slightly modifies the absolute value of keff , and therefore of the interferometer scaling factor. However, this modification only adds a modification on the ppb level and is therefore negligible here.


Fig. 4. (Colour online.) Interferometer signals. The measured interferometer signals are given by the transition probability Pe depicted as symbols here for interferometers 1 (blue) and 2 (green) with the corresponding fitted sinusoidal curves (lines).

A final pulse is applied after the atoms have left the detection zone in order to determine the background signal that is subtracted from each of the three pulse signals. Furthermore, the detection light intensity is monitored on the output of the detection light fibre in order to correct for intensity fluctuations. All this is then combined in a data treatment that results in a detection signal of the normalized excited state population corrected for errors arising from intensity jitter, background light fluctuations, and (small) arrival time fluctuations.

Since Ne is recorded on the side of the detection profile, the transition probability is not normalized when computing Pe = Ne/(Ne + Ng). As our data treatment does not compensate for this effect, the absolute values of the normalized transition probability are slightly underestimated throughout the article. However, we only consider the signal noise in relation to the signal amplitude. The underestimation therefore has no impact on the noise and stability analyses.

3.4. Gyroscope test bench

The gyroscope apparatus is mounted on a test bench consisting of two MinusK 650BM-1 vibration isolation platforms that are connected using two H-shape steel beams. The breadboard carrying the apparatus rests on 14 Sorbothane® half-spheres in order to prevent vibrations induced by possible acoustic modes of the steel beams from being transferred to the atom interferometer setup. In order to avoid ambient magnetic fields to influence the interferometer measurement, and for reliable realization of the atomic launch, the apparatus is shielded by a mumetal enclosure.

4. Rotation sensitivity

Taking into account the atomic forward velocity, the free evolution time, and the effective wave vector of the beam-splitting process, we deduce a rotation-to-phase scaling factor of 4.7 × 104 rad per rad s− 1 following Eq. (4). In order to deduce the gyroscope measurement performance, we first record interferometer characteristic curves Pe(Φ) by scanning the relative Raman laser phase of the third beam-splitting pulse. Fig. 4 shows typical fringes for both interferometers re-vealing fringe peak-to-valley values of at least 28% and 24%, respectively. A repeated transition probability measurement on mid-fringe then reveals a per-shot phase noise of 210 mrad and 225 mrad for the two interferometers, respectively, 213 mrad in the acceleration phase, and 47 mrad in the rotation phase. Taking into account the interferometer cycle time of 0.4 s, this corresponds to a residual noise in the range of 600 nrad s− 1 Hz− 1/2 for our gyroscope in a 1-s measurement time. Depending on the exact environmental conditions in terms of vibrations, we observe sensitivities between 550 and 660 nrad s− 1 Hz− 1/2 in our gyroscope.2 From independent measurements, we infer that the short-term sensitivity is limited partially by the noise of the detection process (of the order of a few 10− 3), and partially by residual rotation vibrations of the platform.

2 The level of vibrations of the laboratory floor depends on human activities in the our building, which is the reason for the short term sensitivity values to vary throughout this article.


5. Stable gyroscope operation

Whereas the influence of accelerations onto the rotation phase is highly suppressed, it can have a deleterious effect on the read out of the rotation signal in case the acceleration phase causes the signal to leave the so-called mid-fringe position, and to scan over many fringes resulting in ambiguities in the phase read out. This is induced by seismic noise inducing vibrations of the apparatus, and also by varying platform orientations resulting in fluctuations in the projection keff · g. We have implemented two methods that mitigate for these effects.

The first method relies on a closed-loop stabilization of one of the two interferometers to mid-fringe operation by acting on the vertical Raman retro-reflection mirror orientation. This results in changing the effective wave vector projection onto gravity and hence the acceleration phase. By realizing the stabilization for one interferometer, the acceleration phase of the second interferometer is automatically stabilized as the mirror orientation is common to both interferometers with opposite sign since the transferred momenta have opposite sign. The rotation phase can be read out simply by performing the phase sum as before. This technique, which allows the mitigation of slowly fluctuating acceleration phase shifts, will be discussed in Section 5.1.

The second technique is based on an approach complementary to the first. It directly exploits the correlation of the two interferometers in the presence of large acceleration phase variations using an ellipse fitting method for the extraction of the rotation signal. This method is described in [24] for the signal read out in an atomic gravity gradiometer. Although is comes with a slight decrease in sensitivity, this technique holds essential advantages for our present setup as it allows us to suppress transition probability offset drifts of the interferometer fringe signal if these are slow versus the interferometer’s cycle time. Furthermore, this technique is particularly interesting for future applications in environments showing higher vibration levels, as well as for atom gyroscopes with larger areas.3 It will also enable reliable phase read out when scaling up the interferometer area, which would also increase the acceleration phases arising from platform tilting and vibrations. We implement this method by modulating the acceleration phase via the Raman mirror orientation, which is reported in Section 5.2.

5.1. Mid-fringe lock

A prior determination of the fringe parameters from data such as those depicted in Fig. 4 allows one to derive an error signal from the mid-fringe measurement. This error signal is deduced from the interferometer phase, which can be determined via the transition probability Pe on an interval of [0, π) using

Pe,i = P0,i + Ai · cos (Φi) (11)

⇒ Φi = cos− 1!

Pe,i − P0,i

Ai

"(12)

where P0,i and Ai are offset and amplitude of the interferometer fringe of the interferometer i with i = 1, 2. It is then digitally processed in order to be fed back to the piezo stepper motor that vertically tilts the common mount of the three Raman laser retro-reflection mirrors. The projection of the wave vector onto gravity is thereby employed to correct for slow drifts of the acceleration phase.

Figs. 5 and 6 show the interferometer running for 500 shots respectively with open and closed fringe stabilization feedback loop. The effect of the fringe lock becomes clearly visible here: whereas without the fringe lock, the acceleration phase variation induces both interferometers to leave one fringe, it remains in mid-fringe position in the case of the closed loop. In fact, no proper phase calculation is possible in case the interferometer phase is close to nπ (n ∈ Z), corresponding to top and bottom fringe positions. However, locking the interferometer signal to mid-fringe allows the stable interferometer phase to be read out over a long time. The short-term performance of the gyroscope reaches similar values as before. The short-term sensitivity in the locked case is found to be as low as 660 nrad s− 1 Hz− 1/2. Consequently, the stepwise corrected mirror orientation does not induce detrimental rotational vibration modes in the mirror mount of the platform. Whereas this technique has been employed here for signal stabilization, it can more generally be used to keep the interferometer orientation horizontally stable throughout the whole measurement.

5.2. Correlated phase modulation

The modulation-based stable interferometer readout scheme relies on the strict anti-correlation of the acceleration phases. Whereas this technique is particularly well-suited for application in environments with strong acceleration-induced phase noise in the range of π and above, this noise is much smaller in our case. We therefore induce a modulation of the vertical beam splitter mirror orientation using the vertical piezo stack (and thus of the keff orientation) in order to commonly modulate the acceleration phase of both interferometers. An acceleration phase shift of %Φa = 2π is found for

3 When increasing the interferometer area in order to increase the gyroscope sensitivity, also the susceptibility to acceleration increases, leading to a smaller required modulation of the wave vector projection to g.


Fig. 5. (Colour online.) Fringe lock switched off. (a) Interferometer signals (symbols, blue: interferometer 1, green: interferometer 2) and corresponding fits for a Raman laser phase scan (lines). (b) Interferometer signals in a repeated measurement of 500 shots length, (c) inferred interferometer phases, (d) derived rotation (red) and acceleration-like phase (black).

Fig. 6. (Colour online.) Fringe lock switched on. (a) Interferometer signals (symbols, blue: interferometer 1, green: interferometer 2) and corresponding fits for a Raman laser phase scan. (b) interferometer signals in a repeated measurement of 500 shots length, (c) inferred interferometer phases, (d) derived rotation (red) and acceleration-like phase (black).

an angle modulation of %α = 75 µrad using Eq. (6). Although this brings along a modulation of the atom interferometer area orientation, and therefore of the rotation phase, this influence is negligible for our experimental parameters. The latter can be seen as a variation of the projection nA · !Earth with nA = A|A|− 1, which results in estimated fluctuations in the measured rotation rate and rotation phase values using Eq. (4) of only %(nA · !Earth) ≈ 1 nrad s− 1. Moreover, the rotation rate modulation is not limited to this value when scaling up the interferometer: If the interferometer area is enlarged by increasing the pulse separation time, the acceleration modulation also only requires smaller angles, leading to a smaller rotation rate modulation.


Fig. 7. (Colour online.) Ellipse fit examples. A set of 15 shots with results (Pe,1, Pe,2) (blue dots) is plotted parametrically and fitted by an ellipse (red line), which allows us to extract the common phase of the two correlated interferometers. The ellipses are shown for both beam splitter momentum transfer directions, k1 (left) and k2 (right).

The employed ellipse fit method (Fig. 7) follows the realization in [24] based on the parametric representation (Pe,1, Pe,2)of the two interferometers with

Pe,1 = P0,1 + A1 cos(Φ1) (13)

Pe,2 = P0,2 + A2 cos(ΣΦ − Φ1) (14)

where ΣΦ = Φ1 + Φ2. The rotation phase is determined from this plot by fitting the conic form

u · x = Ax2 + Bxy + C y2 + Dx + E y + F = 0 (15)

to the data set, where x = (x2, xy, y2, x, y, 1) and u = (A, B, C, D, E, F )T. Fitting is realized using a least-squares (LSQ) fit method yielding the parameter values of the conic representation. From this, we can extract the common phase of the two interferometers, which reads as:

ΣΦ = cos− 1! − B

2√

AC

"(16)

As introduced in Section 2, the acceleration-induced phase shifts in the two interferometers have respectively opposite signs. The ellipse fit method therefore yields the phases that contribute with the same sign to the phases of both interferometers, and by using Eq. (5) we find:

ΣΦ

2= ΦΩ + Φlaser + ΦEM,1 + ΦEM,2

2(17)

ΦEM,1 and ΦEM,2 are then rejected in the k-reversal technique introduced in Section 3.2. Tuning the relative Raman laser phase Φlaser along the interferometer sequence allows us to choose the opening of the ellipse. This leaves us with the rotation-induced phase shift ΦΩ plus possible systematic phase shifts.

The number of points for each fit is chosen by a trade-off of gyroscope spectral resolution and fit uncertainty. A certain number of points is needed to be able to identify the ellipse shape in a fit. Furthermore, the modulation frequency should not be too high in order to minimize vibrations of the platform. We find a number of 15 points for each fit for an effective repetition rate of about 1 Hz in the k-reversal measurement, and a modulation frequency of 100 mHz to optimally meet these demands.

The rotation sensitivity is slightly reduced using the ellipse fit method compared to mid-fringe operation as not both interferometers are necessarily driven at the same optimal operation point. We chose the phases to be in quadrature by tuning the laser phase before the last beam-splitter pulse in order to get an open ellipse, which yields best fit results. Taking into account the typical normalized detection noise in the order of 10− 3, a numerical estimation results in a typical excess factor in short-term noise of approximately 1.5. This is confirmed by our measurements in which we observe a typical short-term sensitivity of 850 nrad s− 1 Hz− 1/2 (compared to 550 nrad s− 1 Hz− 1/2 in the case of a free-running mid-fringe measurement performed before the integration measurement).

On the one hand, this technique shows slightly higher readout noise. On the other hand, it is perfectly suited for usage in the environment of a vibration noise that induces a phase noise exceeding π , where all other commonly used phase-read-out techniques fail. We choose this read-out technique for the analyses presented in Section 6 as it additionally mitigates interferometer fringe offset fluctuations, which are observed to induce spurious rotation signals in the case of the fringe-lock technique.


Fig. 8. (Colour online.) Induced pulse application mismatch. The atomic trajectories and the three interaction zones are shown in top view for optimal timing and optimal alignment (a), optimal timing with imperfect alignment (b), and for a timing delay δt before the application of the Mach–Zehnder sequence with imperfect alignment (c), respectively. (d) The graph depicts rotation phase values as a function of the induced cloud overlap mismatch for three different horizontal tilt angles of one of the outer Raman mirrors. The error bars take into account the phase error between subsequent measurements due to slow fluctuations. The statistical uncertainty on each point is below 20 mrad. A wave-front induced phase %ΦWF,i with i = 1, 2, 3, can be derived for each pulse leading to a total wave front phase, which is observed as a phase variation as shown in (d). The sketches are not to scale.

6. Impact of the use of three pairs of Raman beams

Using three Raman retro-reflection mirrors holds the advantage of scaling up the interferometer size, but it implies high demands of the relative beam alignment. As the two light fields needed for driving the Raman process are guided to the experiment on a common path, the beam shape and wave fronts (thus including the wave front tilt) are identical for both beams at the position of the atoms. However, one of the two light fields is reflected before interacting with the atoms, which is why the spatial part of the phase transferred to the atoms during atom–light interaction in the Raman process is defined by the retro-reflection setup (vacuum chamber windows and retro-reflection mirror). In particular, the mirror orientation (together with the wedges of the chamber windows) defines the effective wave front tilt of each pair of Raman beams. The total interferometer phase (5) therefore comprises a phase that corresponds to the sum of the normal mirror–atom distances in each shot. Hence, in case of a relative tilt of the three Raman beams, the single pulse phases can be different for atoms that interact with the Raman light fields at different positions. If this dephasing is as large as 2πacross the atomic cloud diameter, the integrated interferometer contrast vanishes as the transition probability is recorded for all atoms at once without spatial resolution in the fluorescence detection. This sets strong requirements on the relative tilt alignment of the retro-reflection mirrors, which is found to be in the range of 10–20 µrad. We therefore implemented an alignment procedure involving the interferometer signal contrast in different interferometer configurations to align the three Raman beam retro-reflection mirrors, which is described in detail in [25].

Not only the fringe contrast, but also the interferometer phase offsets are influenced by residual relative tilts of the Ra-man retro-reflection mirrors. This phase shift carries opposite signs for the two interferometers and is therefore suppressed similar to accelerations. However, in case the interferometers do not overlap, a rotation bias occurs (see Fig. 8). This can be observed as a bias on the half-sum of the two interferometer phases and therefore on the measured rotation rate as we deliberately induce an interferometer overlap mismatch. We realize this by adding a variable offset δt to the pulse tim-ing, which changes the distance of the atomic samples of the two interferometers during beam-splitting pulse application. The rotation phase bias is plotted versus the timing mismatch in Fig. 8 for different tilt angles of one of the outer Raman retro-reflection mirrors.

As source position fluctuations are correlated in the vertical and anti-correlated in the horizontal direction in our setup, it is sufficient to consider the latter. In case of a horizontal position fluctuation %x, the overlap of the two interferometers deviates by 2%x. The fixed experimental cycle causes the atomic arrival time in our detection scheme to vary as a function of the position of the source. We can therefore use the arrival time as a measure of the position, which we confirm by measuring the latter using a CCD (charged-coupled device) camera taking a picture of the atom clouds before launch.

Measuring the source position fluctuations allows us to deduce fluctuations of the atom interferometer overlap from the arrival time signals. The scaling from the arrival time to source position fluctuation and finally to rotation phase fluctuations mainly depends on the exact shape of the detection signal and on the respective Raman mirror tilt. A corresponding set of factors is found directly by correlation to the measured rotation signal. Fitting a linear combination of the arrival time


Fig. 9. Rotation rate averaging measurement. (a) The rotation phase (black) is plotted with the time series of expected rotation phase values deduced from the arrival time measurements (red). The coefficients for the latter are found in a fit of a linear combination of the atomic arrival times to the rotation signal. (b) Subtracting the expected phase values results in a suppression of the wave front influence on the rotation phase. Offsets have been removed. (c) The Allan standard deviation of the rotation signals corresponding to the rotation phases on the left side allows us to analyse the gyroscope stability. The dotted line represents the statistical errors of the Allan standard deviation, and the dashed line represents a τ − 1/2 averaging at 850 nrad s− 1 Hz− 1/2.

signals to the rotation phase therefore allows us to deduce the spurious rotation phase induced by fluctuating interferometer overlap coupled to residual effective wave front tilts of the Raman light fields. The rotation signal and the fitted linear combination of the arrival time signals are plotted in Fig. 9.

From the correlation, we can furthermore correct for this influence by subtracting the fitted curve from the rotation phase. This strongly improves the averaging behaviour of our gyroscope. Removing the slow signal oscillations allows us to average to a sensitivity to rotations of 20 nrad s− 1 after 4000 s of integration, which is inferred from the Allan standard deviation also depicted in Fig. 9.

7. Conclusions

We have presented recent progress in rendering our atom gyroscope more stable against both, short and long term acceleration noise, i.e. vibrations and slow platform tilting, respectively. We have employed two different techniques for achieving this goal. The first one is based on an active feedback loop stabilizing the interferometer signal to the mid-fringe position from shot to shot through the control of the orientation of the atom interferometer. It results in a short term sen-sitivity of 660 nrad s− 1 Hz− 1/2. The second stable read out method relies on an ellipse fit of the parametric representation of the two atom interferometer signals. It strikes with simplicity and high immunity against vibrations of the apparatus, which is important when running the gyroscope in noisy environments. Adding a modulation to the gyroscope orientation allows exploiting the interferometer’s sensitivity to accelerations in order to achieve the desired correlated phase modula-tion required for ellipse fitting. This yields a highly robust read-out method that is immune not only against (non-rotational) vibrations acting on the device, but also against slow fluctuations in the interferometer’s signal offset and amplitude in the device itself. We achieve a short-term sensitivity of 850 nrad s− 1 Hz− 1/2 and enable studies on slowly changing spurious rotation signals by enabling robust long-term measurements. This technique is of particular interest not only for environ-ments in which vibrations induce acceleration noise of multiples of 2π , but also for enabling technically more simple atomic gyroscopes, e.g., in cases where the normalization in the detection process is not stable over long times.

Using the ellipse-fit-based phase read out allowed us to identify the major source for long-term fluctuations of the device itself, a coupled effect of residual Raman beam misalignments and atom source position fluctuations. Furthermore, we have implemented a technique that allows us to correlate the observed spurious rotation rate with the atomic sample arrival time, and hence to correct the rotation signal for this influence. In integration of the rotation signal, we reach a resolution for rotations of down to 20 nrad s− 1 after 4000 s of measurement time.

Whereas a detailed study of the instrument noise will be subjected to additional work, we can clearly derive from the work presented here that both the atom source position stability and the relative Raman beam alignment are crucial for a stable rotation measurement. The phase shift arising from the cross-coupling of source position and wave front quality should be suppressed by increasing the rotation-to-phase interferometer scaling factor. In the prospect of an improvement of our sensor or the development of new atom interferometer gyroscopes, this can be achieved by augmenting the free evolu-tion time T . Techniques for enlarging the momentum recoil as double-diffraction technique [26], large-momentum transfer (LMT) beam splitters [27], or sequential beam-splitter pulses similar to the ones presented in [28] for the Bragg case have been proposed and demonstrated for enhancing interferometry scaling factors. Nevertheless, the wave-front induced phase shift is always proportional to the recoil momentum transferred to the atoms in the interferometer and scales up likewise. In contrast to this, an increase in T allows us to effectively suppress position-related phase shifts, which remain constant and are therefore suppressed quadratically when increasing T . Augmenting the free evolution time between two pulses requires either to realize smaller forward velocities, e.g., in spaceborne gyroscopes for measurements of the Lense–Thirring effect such as proposed in [29], or to increase the separation of the single mirrors. The latter necessitates a very good control


on the relative mirror alignment on large setups below the µrad regime.4 Not only the relative mirror tilt and wave-front quality, but also the source position has to be well controlled. In this respect, it will be advantageous to replace molasses-cooled atomic ensembles of several µK temperatures with ultra-cold atoms with temperatures of nK in the next generation of a gyroscope. Recent progress in atom–chip based sources [30] make these highly promising candidates for sources used in atom interferometry with ultra-cold or even Bose–Einstein condensed matter when combined with launching techniques based on accelerated optical lattices [31].

The study of robust read out techniques performed in this paper will be of particular interest for gyroscopes that are en-visaged for, e.g., navigational applications, such the gyroscope recently presented in [32], reaching repetition rates of 100 Hz. The modulation technique presented in our work could also be implemented in such interferometers of high repetition rate, both for modulation of the rotation and acceleration part, when choosing an appropriate mirror mounting in order to dis-tinguish between both moments of inertia. Studies made here on the wave-front-induced phase shifts in the interferometer will also have to be considered in different atom interferometers. In particular when expanding the interferometer baseline and when employing technically demanding concepts as presented in [33] for the case of a space-borne gravity gradiometer, the cross-coupling between atomic trajectories and wave-front imperfections have to be taken into account, especially when striving for highest stabilities.

Acknowledgements

We would like to thank P. Bouyer for providing two Minus-K vibration isolation platforms that strongly helped us to realize the presented work. The research work for this article was supported in part by the Deutsche Forschungsgemein-schaft (SFB407), the European Union (Contr. No. 012986-2 (NEST), FINAQS, Euroquasar, IQS), and the Centre for Quantum Engineering and Space Time Research QUEST. G.T. acknowledges the support by the Max-Planck-Gesellschaft, the INTERCAN network, and the UFA-DFH.

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C. R. Physique 15 (2014) 898–906





Sagnac-based rotation sensing with superfluid helium

quantum interference devices

Capteurs de rotation fondés sur l’effet Sagnac avec interférences quantiques

dans l’hélium superfluide

Yuki SatoRowland Institute at Harvard, Harvard University, Cambridge, MA 02142, USA



Keywords:Sagnac effectRotation sensingMatter–wave interferometrySuperfluid heliumJosephson effectsBose–Einstein condensate

Mots-clés :Effet SagnacCapteurs de rotationInterférométrie à onde de matièreHélium superfluideEffets JosephsonCondensat de Bose–Einstein

The Sagnac effect has played an instrumental role for the fundamental studies of relativity, and various devices that utilize this effect have been applied to many disciplines ranging from inertial navigation to geodesy and to seismology. In this context we present an overview of recent developments related to superfluid helium quantum interference devices. With the discovery of superfluid Josephson phenomena in 4He, the device technology has been rapidly developing in the past 10 years. We discuss the underlying working principles of these interference devices and their applications. We focus on their use as sensitive rotation sensors based on the Sagnac effect coupled with the existence of a macroscopic quantum phase via particle–wave duality.


r é s u m é

L’effet Sagnac a joué un rôle déterminant dans les études fondamentales en relativité, et les dispositifs utilisant cet effet ont trouvé des applications dans des disciplines variées allant, de la navigation inertielle à la géodésie et à la sismologie. Dans ce contexte, nous présentons un aperçu de l’évolution récente des dispositifs d’interférence quantique dans l’hélium superfluide. Avec la découverte de l’effet Josephson dans l’hélium 4 superfluide, cette technologie s’est rapidement développée au cours des dix dernières années. Nous discutons ici les principes sous-jacents à ces dispositifs d’interférences et leurs applications. Nous nous concentrons sur leur utilisation en tant que capteurs de rotation fondés sur l’effet Sagnac couplée avec l’existence d’une phase quantique macroscopique via la dualité particule–onde.


1. Sagnac effect

The Sagnac effect refers to a rotation-induced phase shift in interferometry [1,2]. The most well-known scenario appears in a laser interferometer where a beam of light is split in half and recombined again while enclosing a finite area A. When

E-mail address: [email protected].


Y. Sato / C. R. Physique 15 (2014) 898–906 899

Fig. 1. A schematic depicting two superfluids joined at a junction (labeled X). Wavefunctions overlap and couple weakly across the junction. From Ref. [11].

the device is rotated at angular speed Ω about an axis perpendicular to its plane n, the path difference traveled by light in the general relativistic point of view leads to a phase shift

"φS = 2ω

c2 Ω · A (1)

where A (= An) is the area vector of the device loop, c is the speed of light, and ω is the angular frequency. The equation above relates the rate of rotation applied to the apparatus to the experimental observable of the phase difference, effectively rendering the device a rotation sensor. The effect has been validated, and for optical interferometers it has seen applications in various forms including fiber optical and ring laser gyroscopes in fields ranging from inertial sensing to geodesy and to seismology [3–5].

In addition to the case of photons discussed above, as it became possible to split beams of particles and to have them recombine and interfere, the same general treatment has been applied to “massive” systems of neutrons, atoms, and super-fluids [6–11]. Among these, all but the last category share the common operational characteristic that individual particles traverse the interferometer arms to recombine and interfere with themselves while the applied rotation affects the phase of traveling wavepackets. The superfluid device based on the exploitation of the Josephson phenomena is somewhat unique in that individual particles in the condensate essentially sit idle but they collectively interfere with themselves across the so-called Josephson junctions while the rotation acts on the macroscopic wavefunction phase to perturb this interference. Here we describe the physics of these superfluid helium quantum interference devices and their application and potential as Sagnac-based rotation sensors [11].

2. Superfluid helium and Josephson effects

Superfluid helium is an example of a macroscopic quantum system, a class that also includes superconductors and Bose–Einstein-condensed (BEC) gases. The overarching similarity in these systems is the existence of an “order parameter” with a well-defined overall phase. The existence of such a quantum state can be invoked using particle–wave duality of matter. All matter behaves as a wave with a characteristic wavelength that is inversely proportional to its momentum. Therefore as one cools a collection of atoms, their momentum decreases and the associated wavelength increases. At some point, the wavelength of each particle should become comparable to the interatomic distances, and the spatial extent of their presence starts to overlap with one another. The entire group of atoms then forms a single system that can only be thought of as one coherent probability cloud. The superfluid states of quantum liquids (3He and 4He) are described by a macroscopic wavefunction of the form Ψ = √

ρseiφ , where ρs is the superfluid density and φ is the quantum mechanical phase [12].

The interference device that is the topic of discussion here is based on a phenomenon that appears when two superfluids are weakly coupled together (see Fig. 1). Two superfluid reservoirs are joined at a junction labelled X. If the coupling between the two reservoirs is sufficiently weak, the two superfluids are described by two distinct wavefunctions ΨR =√

ρs,ReiφR and ΨL = √ρs,LeiφL . The time-dependent Schrodinger equation applied to this coupled system gives [13]:

ih∂ΨR

∂t= µRΨR + hχΨL (2)

and

ih∂ΨL

∂t= µLΨL + hχΨR (3)

where µR and µL denote the chemical potentials of the two reservoirs. The term hχ represents the coupling across the junction, and it is defined such that χ has the dimension of frequency to give a measure of ΨR leaking into ΨL and vise versa. Inserting ΨR and ΨL into Eqs. (2) and (3) and defining the phase difference "φ = φR −φL and the chemical potential difference "µ = µR −µL, one obtains two governing equations:

I = I0 sin"φ (4)

and

∂"φ

∂t= −"µ

h(5)

900 Y. Sato / C. R. Physique 15 (2014) 898–906

Eq. (4), often called the dc-Josephson equation [14], describes the nonlinear relation between the quantum phase difference across the junction and the superfluid mass current I that flows through it. This relation applies strictly to weakly coupled quantum systems. Eq. (5), often called the Josephson–Anderson phase evolution equation [15], states that the phase differ-ence evolves in response to the chemical potential difference. Unlike Eq. (4), Eq. (5) is applicable to the time evolution of phase difference between any two locations in superfluids, whether weakly coupled or not.

If a fixed (i.e. constant) chemical potential difference is established between two weakly coupled superfluids, the phase difference increases linearly in time according to Eq. (5). Plugging this into Eq. (4) gives:

I = I0 sin"µh

t (6)

Hence counterintuitively an oscillating mass current appears across the junction in response to a constant "µ, and its oscillation frequency is proportional to the applied "µ. We refer to this phenomenon as a Josephson oscillation and its frequency f J = "µ/h as the Josephson frequency. The junction in this context is often called a “weak link” to imply the nature of coupling although the term Josephson junction is also common. For superfluid helium, "µ = m∗("P/ρ − s"T )

where ρ is the fluid density, s is the specific entropy, "P and "T are pressure and temperature differentials, and m∗ is the effective mass of superfluid constituents, either the atomic mass of 4He or twice the 3He atomic mass [12].

3. Superfluid helium weak link

In superconducting systems a thin film of normal metal or insulator can be used as a weak link between two super-conducting materials as electrons quantum-tunnel through such a barrier to establish the weak coupling required for the Josephson phenomena to emerge [14]. However the same approach cannot be employed for superfluid helium as helium atoms are too large to exhibit appreciable tunneling. To make a superfluid weak link, one needs to construct the equivalent of a Dayem Bridge [16], a constricted connection passage whose dimensions are on the order of the superfluid healing length ξ . This is the characteristic minimum length scale over which the wavefunction can change significantly, and super-fluidity is effectively suppressed within this distance away from the wall. Two superfluid reservoirs joined through a passage of this length scale are therefore neither completely disconnected (and independent of each other) nor strongly connected (and virtually a single reservoir). Two reservoirs then form a weakly coupled quantum system. For 3He, the healing length is given by [17]

ξ3 = 65 nm(1 − T /Tc)1/2 (7)

where the superfluid transition temperature Tc = 1 mK. In contrast the healing length of 4He is represented by [18]

ξ4 = 0.3 nm(1 − T /Tλ)0.67 (8)

where the superfluid transition temperature Tλ = 2.17 K. While the temperature-dependent healing length for 3He is typi-cally on the order of tens of nanometers, it is typically two orders of magnitude smaller for 4He. Naturally, when the field of micro/nano-fabrication advanced to feature sizes of tens of nanometers, the era of Josephson physics in superfluids began with 3He. The first reports of a flow signature consistent with the Josephson current-phase relation came in 1988 from the team of Éric Varoquaux and Olivier Avenel in Saclay, France [19], followed by the direct observation of Josephson oscillations a decade later [20].

As for superfluid 4He, early evidence of sinφ current-phase behavior was finally reported in 2001 [21], followed by the direct observation of Josephson oscillations four years later [22]. Instead of attempting to fabricate sub-nanometer size apertures to match the zero-temperature healing length, these works took advantage of the diverging behavior of ξ4 very close to Tλ as given in Eq. (8). This variation predicts ξ4 ∼ 60 nm when Tλ − T ∼ 1 mK, which allowed the use of channels that could be fabricated with e-beam lithography. The 4He Josephson work has progressed rapidly since then given the ease of cryogenics involved at 2 K compared to 1 mK required for superfluid 3He.

4. Superfluid helium quantum interference device

A superfluid helium quantum interference device is a tool that exploits the Josephson oscillation phenomena discussed above (see Fig. 2 for a simplified device configuration). A torus filled with superfluid helium is interrupted by two weak links. Each weak link here is an array of apertures, and the array is used to amplify the overall oscillation signal to a detectable level. Here "φ1 and "φ2 denote the phase differences across the two weak links. For the wavefunction to be single-valued, the phase integral

!φ · dl has to equal 2πn for the closed path taken along the torus. If a constant "µ is

applied across these weak links, mass currents I1 sin "φ1 and I2 sin"φ2 appear across the junctions according to Eq. (4),


Fig. 2. (Color online.) A schematic view of the superfluid helium quantum interference device. A torus filled with superfluid helium is interrupted by two weak links. The phase differences between the two weak links are defined as "φ1 and "φ2 respectively. Each weak link is an array of nanoscale apertures.

while both "φ1 and "φ2 evolve in time according to Eq. (5). The total mass current I = I1 sin "φ1 + I2 sin "φ2 can be written as

I = I∗ sin"φ = I∗ sin"µh

t (9)

where "φ = ("φ1 + "φ2)/2. If there exists some external phase shifting influence, the phase integral condition reads !φ · dl = "φ1 −"φ2 + "φext = 2πn. Setting n = 0, one finds that the overall oscillation amplitude modulates according to

I∗c = (I1 + I2)

"cos2 "φext

2+ γ 2 sin2 "φext

2(10)

with γ = (I1 − I2)/(I1 + I2). This is the operating principle of the superfluid helium quantum interference device. Quantum coherence within the superfluid leads to Josephson oscillations whose combined amplitude from two separated weak links provides a direct measure of the phase gradients present.

In the case of superconducting quantum interference devices [23], the phase shift "φext is induced by the change in the magnetic flux threading through the superconducting loop, making the device an extremely sensitive magnetometer. We show in the following section that, for the neutral superfluid counterpart, the role of magnetic flux can be played by the so-called rotation flux: Ω · A, rendering the device a sensitive rotation sensor.

5. Phase shift by rotation

Without much loss of generality, we first compute for massive particles a rotation-induced phase shift. Following Ref. [24], for a particle at location r, the velocity in the inertial frame is related to that in the rotating frame via v ′ = v + Ω × r. Hence the Lagrangian can be written as

L(r, v) = 12

m(v + Ω × r)2

= 12

mv2 + mΩ · (r × v) + 12

m(Ω × r)2 (11)

Neglecting the last higher-order term, the second term (Coriolis term) can be treated as the sole perturbation to the La-grangian compared to the free-particle scenario. The extra phase accumulated can then be computed by the action integral:

"φ = 1h

#"Ldt = m

hΩ ·

#(r × v)dt = m

hΩ ·

#r × dr = 2m

hΩ · A (12)

where A represents the area swept by the particle trajectory r. This phase shift, obtained pseudo-classically above, corre-sponds to the Sagnac phase shift for massive particles. See Refs. [25,26] for a more relativistic motivation for the effect and the derivation of the rotation-induced Lagrangian terms for matter waves via fully general-relativistic equations.

With a general solution in hand, we now turn to the case specific to a superfluid helium quantum interference device. Let’s first consider the simplest scenario of rotating the device about the area vector of the torus so that Ω ∥ A as depicted in Fig. 3. Due to the existence of partitions containing weak links, fluid flow similar to solid-body motion |vs| ≈ ΩR (where R is the major radius of the torus) will be induced in the region away from the weak link junctions. Applying the current operator J = −(ih/2m)(Ψ ∗Ψ −Ψ Ψ ∗) to a wavefunction of the form Ψ = √

ρseiφ , the phase gradient in these regions is related to superfluid velocity via φ = (m/h)vs [12]. Using these relations, a closed path integral of the phase gradient along the torus can be divided into four parts and computed as


Fig. 3. (Color online.) A schematic view of the superfluid helium quantum interference device with a sensing area A. The device is rotated about the area vector of the torus with angular speed Ω . In this example, Ω ∥ A, and both vectors point out of the page. Symbols a, b, c, and d denote the starting and ending locations for sectional path integrals defined in the text. The dotted line depicts the integral path.

#φ · dl =

b$

a

φ · dl +c$

b

φ · dl +d$

c

φ · dl +a$

d

φ · dl

= mh

b$

a

vs · dl + "φ1 + mh

d$

c

vs · dl −"φ2

= 2mh

Ω A + "φ1 −"φ2 (13)

where A is the area enclosed by the integrals over vs along the two main arms of the interferometer, which equals the device area. In a more general scenario where Ω ∦ A, the product of the angular speed and the sensing area needs to be generalized by the vector dot product such that

#φ · dl = 2m

hΩ · A + "φ1 −"φ2 (14)

Comparing the above result with the phase integral contributions for the non-rotating case: !

φ · dl = "φ1 −"φ2, it can be seen that the Sagnac phase shift introduced by the application of finite rotation to the superfluid interferometer is given by

"φS = 2mh

Ω · A (15)

which agrees with Eq. (12).It is instructive to compare this result with the optical Sagnac phase shift given by Eq. (1). One sees that the two are

identical except that the effective photon mass hω/c2 is replaced by the helium atomic mass, which is 1010 times heavier. Thus for the same change in the rotation rate, the signal from the helium device would be 10 orders of magnitude larger compared to optical devices with the same sensing area. Although it is nontrivial to imagine constructing a superfluid device that spans hundreds of meters across in footprint as have been done with ring laser and fiber optical gyros, the signal size ratio of ten billion for the same device dimensions suggests immense potential for compact and yet ultrasensitive rotation sensors.

Setting !

φ · dl = 2πn with n = 0 in Eq. (14), the total mass current can be written as in Eqs. (9) and (10) where "φext is now given by the Sagnac phase shift "φS. Quantum interference has been observed in both 3He and 4He, and the underlying principle has been tested by inducing and detecting the Sagnac phase shift arising from the spinning Earth. Fig. 4 shows such interference pattern obtained with a 4He quantum interference device [10]. Through the Sagnac effect acting on the macroscopic wavefunction phase with the phase integral requirement, a superfluid interferometer becomes a rotation sensor.

We note that the discussion on the novel oscillation phenomenon and its utilization for interferometry has thus far focused on the regime where two superfluid reservoirs are weakly coupled. For superfluid 4He, as the temperature is lowered and the healing length becomes much smaller than the aperture dimensions, Josephson oscillations cease to exist. However it turns out that a different type of oscillation (based on the physics of “phase slippage”) emerges at the Josephson frequency, enabling the operation of superfluid interferometry even in the regime where two superfluid reservoirs are strongly connected. For details on the nature of fluid oscillations in these two distinct regimes as well as the intermediate cross-over regime, we refer readers to Ref. [27].


Fig. 4. (Color online.) Modulation of the Josephson oscillation amplitude as a function of rotation flux Ω · A in a superfluid 4He matter-wave interferometer. The Earth is the source of rotation flux, and its magnitude is varied by reorienting the interferometer with respect to the north–south axis of the Earth. From Ref. [10].

Fig. 5. (Color online.) a) Experimental apparatus depicting a flux-locked rotation sensor. The inside is filled with superfluid 4He and the entire apparatus is immersed in a bath of liquid helium. The heat injected at R induces a phase shift "φheat across the top arm of interferometer loop. S represents a heat sink. Diaphragm (D) and electrode (E) form a sensitive sensor for fluid motion in the weak links. b) Equivalent SQUID circuit. "φext is the overall phase shift that the interferometer is being used to measure. "φ1 and "φ2 are the phase differences across the two weak links and "φheat is the phase shift due to injected heater power. When the device is rotated, the Sagnac phase shift appears in addition to the above phase shifts. From Ref. [28].

6. Flux modulation and feedback circuit for rotation sensing

Interferometers almost invariably have a transfer function wherein the device output is a nonlinear function of some variable of interest. A cosinusoidal transfer function such as Eq. (10) is one of the most typical cases encountered. The device sensitivity then varies depending on the size of the signal that it detects, which significantly hinders the utility of the device. Moreover due to the periodicity of the device output exemplified by the data shown in Fig. 4, large changes in signal can be determined only by tracing the whole interference pattern in real time, which is temporally impractical. For these reasons, it is critical to be able to linearize the device to have an output that is directly proportional to the variable of interest. Although such linearization of the instrument response remains nontrivial for most interferometric devices, a novel scheme has been developed for a superfluid helium quantum interference device [28].

Fig. 5a displays an interferometer loop in which one arm is a straight tube of length l and cross-sectional area σcontaining a heater (R) at one end and a heat sink (S) at the other. According to the two-fluid model [12], when power Q is applied to the heater, the normal component flows away from the heat source with velocity vn while the superfluid counterflow is established with velocity:

|vs| =ρn

ρρsT sσQ (16)


where ρ , ρs, and ρn are the total, super, and normal fluid densities respectively. Through the relation φ = (m/h)vs, the application of heat corresponds to an injection of phase shift into the interferometer arm [29]:

"φheat = lσ

mh

ρn

ρρsT sQ (17)

Various phase contributions in the absence of rotation are depicted schematically in Fig. 5b. When there exists external rotation, both the Sagnac phase shift and the heat current phase shift are picked up by the device. In that scenario, any change in rotation flux can be canceled by injecting heater power so that the overall phase shift "φext = "φS + "φheatremains constant. A heat-induced phase shift works as a flux modulator, and the interferometer output can be maintained at a fixed amplitude and the flux is locked. This allows one to bias the device at the point of the greatest slope and operate there without having to trace many interference cycles arising from large changes in rotation flux. Furthermore, the amount of power needed for this feedback provides a linear measure of the change in rotation flux: |Ω · A| = aQ + const. where a ≡ lρn/(2σρρsT s).

Fig. 6 demonstrates this flux-locked rotation sensing. When the interferometer is reoriented about the vertical, changing the angle between the device’s area vector and the Earth’s spin axis, the instrument’s output exhibits a signature Sagnac interference shown in Fig. 6a. In panel b, the feedback is turned on to nullify the Sagnac phase shift with the injected heat current. The device shows a constant mass current amplitude even when it is reorientated with respect to the Earth’s rotation vector. When the heater power required for this feedback is plotted against the rotation flux (Fig. 6c), it can be clearly seen that the technique provides a linear measure of the variable of interest while the sensitivity is now constant and independent of the rotation flux.

7. Rotational sensitivity

With several proof-of-concept devices already implemented, it is natural to ask how sensitive these rotation sensors can be. In 2011, a large-area multi-turn superfluid 4He interferometer was reported [30]. The device’s arm consisted of four turns in astatic configuration to make gradiometric phase shift measurements, and the interfering path length was 0.5 m, enclosing the total area of 200 cm2. The intrinsic rotational sensitivity of this particular device (if the path of the same length was wound in one direction) is reported to be ∼ 10−8 rad/s in 1 second measurement time. Although this sensitivity is already quite high, there is much room for significant improvement as the rotational sensitivity of a superfluid helium quantum interference device depends on several parameters including the number of apertures in a junction, sensing area per turn of the interferometer arm, and the number of the overall turns. For example a single superfluid weak link typically contains roughly 5000 nanoscale apertures in a 200 µm by 200 µm SiN membrane. An increase in the number of apertures by an order of magnitude is easily feasible with the current e-beam technology, and it is projected that the signal increase just from that alone could make this device one of the most sensitive matterwave-based rotation sensors reported.

The parameters mentioned above in relevance to the device sensitivity are those that have to do with the instrument configuration and geometry. Much sensitivity improvement is also expected from exploiting nonlinear Josephson dynamics. Flux-locking discussed in the previous section is an example of making an effective use of a novel physical phenomenon to enhance the utility of the instrument. Similarly many nonlinear phenomena have been reported that could be used to enhance the sensitivity significantly [31,32]. The so-called Fiske effect which occurs in the mixing of Josephson oscillations with the local acoustic resonances of the apparatus has been demonstrated to increase the signal to noise ratio by a factor as much as 30 [33]. With further work it may be possible to even engineer sharp acoustic resonances to further take advan-tage of this robust amplification mechanism. A phenomenon called bifurcation has also been reported in the resonance of superfluid Josephson systems with a potential to create a novel amplifier based on hysteretic switching of signal amplitudes [34].

Configured as a phase meter, the ultimate phase resolution limit of superfluid-based interferometers may be quantum fluctuations embodied in the uncertainty principle. With ∼ 1023 atoms involved in the device and a large area made possible by the macroscopic nature of the quantum liquid, superfluid devices may be advantageous in comparison to those based on BEC gases [35,36]. However, that ultimate resolution will always be masked by the vibrational noise from the environment. Although a simple translation will not correspond to signal changes, any small tilting of the instrument can lead to rotational noises. In applications such as geodetic gyroscopy, placing the devices in an underground low vibration laboratory may be a necessity to mitigate any background signals as the ultimate sensitivity is explored. On the other hand in scenarios such as inertial navigation where the noisy environment is expected, the flux-locking technique would have to be developed further with a slew rate high enough to track nuisance signals.

As mentioned previously the exquisite sensitivity per device size raises an interesting potential as a compact rotation sensor that can be moved from one location to another. As a fixed instrument, it will be fascinating to see whether or not the sensitivity can be raised further to complement the world’s most sensitive ring laser gyroscopes in their study of minute variations in the Earth’s rotation as well as their quest for terrestrial tests of frame dragging from the spinning Earth [5]. Besides simple bench-marking between different platforms, it will be fascinating to tackle an experiment where beams of light and matter respond quite differently to not only learn about a particular phenomenon of interest but also to gain insights into various similarities and dissimilarities that exist among these instruments.


Fig. 6. a) Interferometer output as a function of the Earth’s rotation flux in the absence of heater-induced phase biasing. b) The modulation in the interfer-ometer signal is compensated by injected heater current, making the amplitude independent of the rotation flux. The amplitude is maintained constant at the bias point circled in panel a. c) Feedback heater power used for a given value of the rotation flux to flux-lock the interferometer. From Ref. [28].

8. Conclusion

In 1913 George Sagnac split a beam of light, counter-propagated the beams along the same optical path, and demon-strated that the shift in the interferometry pattern was directly proportional to the rate of rotation applied to the apparatus. This observation is now known as the Sagnac effect, and over the past hundred years it has seen applications in many disciplines ranging from inertial navigation to seismology. In this context we have presented an overview of recent devel-opments related to superfluid helium quantum interference devices. In these novel devices, atoms in the condensate sit idle but collectively interfere with themselves across two weak links to produce Josephson oscillations. Through the Sagnac effect the externally applied rotation shifts the macroscopic wavefunction phase to affect this interference, altering the am-plitude of the mass current oscillations and rendering the device a sensitive matter-wave rotation sensor. With quantum phase fluctuations being the ultimate limiting factor in the gyroscope’s performance, these superfluid devices form viable candidates for high precision rotation sensing along with state-of-the-art laser and atom interferometers. See Ref. [11] for more details related to the physics and applications of superfluid helium quantum interference devices. For discussion on related superfluid gyrometers based on a single junction and a large parallel path, see Ref. [8].

References

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[10] E. Hoskinson, et al., Phys. Rev. B 74 (2006) 100509(R).[11] Y. Sato, R.E. Packard, Rep. Prog. Phys. 75 (2012) 016401.[12] D.R. Tilley, J. Tilley, Superfluidity and Superconductivity, Institute of Physics, Bristol, UK, 1990.[13] R.P. Feynman, et al., The Feynman Lectures in Physics, vol. 3, Addison-Wesley, Reading, UK, 1963.[14] K.K. Likharev, Dynamics of Josephson Junctions and Circuits, Academic Press, New York, 1986.[15] P.W. Anderson, Rev. Mod. Phys. 38 (1966) 298.[16] P.W. Anderson, A.H. Dayem, Phys. Rev. Lett. 13 (1988) 195.


[17] D. Vollhardt, P. Wolfle, The Superfluid Phases of Helium-3, Taylor and Francis, New York, 1990.[18] R.P. Henkel, et al., Phys. Rev. Lett. 23 (1969) 1276.[19] O. Avenel, E. Varoquaux, Phys. Rev. Lett. 60 (1988) 416.[20] S.V. Pereverzev, et al., Nature 388 (1997) 449.[21] K. Sukhatme, et al., Nature 411 (2001) 280.[22] E. Hoskinson, et al., Nature 433 (2005) 376.[23] J. Clarke, A.I. Braginski, The SQUID Handbook: Fundamentals and Technology of SQUIDs and SQUID Systems, Wiley-VCH, Weinheim, Germany, 2004.[24] P. Storey, C.C. Tannoudji, J. Phys. II 4 (1994) 1999.[25] B.H.W. Hendriks, G. Nienhuis, Quantum Opt. 2 (1990) 13.[26] E. Varoquaux, G. Varoquaux, Phys. Usp. 51 (2008) 205.[27] E. Hoskinson, et al., Nat. Phys. 2 (2005) 23.[28] Y. Sato, et al., Appl. Phys. Lett. 91 (2007) 074107.[29] Y. Sato, et al., Phys. Rev. Lett. 98 (2007) 195302.[30] S. Narayana, Y. Sato, Phys. Rev. Lett. 106 (2001) 255301.[31] E. Hoskinson, et al., AIP Conf. Proc. 850 (2006) 117.[32] A. Joshi, R.E. Packard, J. Low Temp. Phys. 172 (2013) 162.[33] Y. Sato, Phys. Rev. B 81 (2010) 172502.[34] S. Narayana, Y. Sato, Phys. Rev. Lett. 105 (2010) 205302.[35] K.C. Wright, et al., Phys. Rev. Lett. 110 (2013) 025302.[36] C. Ryu, et al., Phys. Rev. Lett. 111 (2013) 205301.

C. R. Physique 15 (2014) 909–910




Index des auteurs / Author indextome 15, 2014

Abdeddaim, R. 448Abend, S. 884Adrien, J. 674Allegrini, M. 866Allouche, J.-P. 6Alloul, H. 519Altantzis, T. 140Aoki, D. 616, 630Apperley, M.H. 269Arita, R. 587Audier, M. 12

Backov, R. 761Baillis, D. 683Bals, S. 140Barrett, B. 875Bartout, J.-D. 705Bastien, G. 630Béché, A. 190Beghi, A. 866Beleggia, M. 126Belfi, J. 866Bérenger, J.-P. 393Berg, P. 884Bertaina, M.E. 300Beverini, N. 866Bienvenu, Y. 705, 719Blandin, J.-J. 662Blasi, P. 329Blazit, J.D. 158Bommier, V. 430Bosi, F. 866Bouchaud, É. 527Bouhadef, B. 866Bouyer, P. 875Boyes, E.D. 200Bréchet, Y. 481, 649Burteau, A. 705

Caciuffo, R. 553Cahn, J.W. e1Caillard, D. 82Calamai, M. 866Callegari, T. 468

Canuel, B. 875Carelli, G. 866Castin, Y. 285Cecconi, B. 441Clark, L. 190Colas, L. 421Coleman, P. 557Colineau, É. 599Colliex, C. 101, 158Collignon, G. 458Collino, F. 403Cuccato, D. 866

Darces, M. 415Darrigol, O. 789da Silva, F. 421Dautray, R. 481de Boissieu, M. 1, 58de Fornel, F. 385de Jonge, N. 214de Laissardière, G.T. 70Deligny, O. 367Deloudi, S. 40Dendievel, R. 662Destribats, M. 761Di Virgilio, A. 866Dollet, B. 731Dunin-Borkowski, R.E. 126Dutta, I. 875

El Badaoui, N. 841Épicier, T. 276Ertmer, W. 884

Fameau, A.-L. 748Farrar, G.R. 339Faudot, É. 421Feugnet, G. 841Flint, R. 557Flouquet, J. 630Forest, S. 705Fournée, V. 48Friedel, J. 3, 481

Gai, P.L. 200Gauguet, A. 787, 875Gebauer, A. 859Geiger, R. 875Georget, É. 448Gloter, A. 158Gómez, C.P. 30Goris, B. 140Gourgout, A. 630Gratias, D. 1, 18Griveau, J.-C. 599Gusakov, E. 421Gutty, F. 841Guyot, P. 12Guzzinati, G. 190

Hacquin, S. 421Hafdallah Ouslimani, H. 458Haga, Y. 616Harari, D. 376Hawkes, P. 110Heidari, H. 140Hélier, M. 415Herzog, C. 285Heuraux, S. 421Honda, F. 616Hurst, R.B. 859

Igel, H. 859Ikeda, H. 587Izawa, K. 616

Jacquot, J. 421Janssen, T. 58Joly, P. 403Juchtmans, R. 190

Kallel, A. 468Kambe, S. 563Kampert, K.-H. 318Kanane, H. 458Kasama, T. 126Kasmi, C. 415Knebel, G. 630Koch, C.T. 119Kociak, M. 158Kolorenc, J. 640

http://dx.doi.org/10.1016/S1631-0705(14)00169-81631-0705/2014 Publié par Elsevier Masson SAS pour l’Académie des sciences.

910 Index des auteurs / Author index / C. R. Physique 15 (2014) 909–910

Lacotte, G. 458Landragin, A. 875Lecouvez, M. 403Ledieu, J. 48Lefèvre, H.C. 851Legros, M. 224Letessier-Selvon, A. 297Lhuissier, P. 696Liard, L. 468Lili, T. 509Lipari, P. 357Lo, J. 468Losquin, A. 158Lubk, A. 190

Maccioni, E. 866Machida, Y. 616Magnani, N. 580Mahfoud, Z. 158Maire, É. 674Mangler, C. 241March, K. 158Marmottant, A. 662Martin, G. 662Martin, P. 841Marzougui, H. 509Mayou, D. 70Maystre, D. 387Meunier, M. 875Meuret, S. 158Meyer, J.C. 241Meyer, Y. 6Mompiou, F. 82Monteux, C. 775Morbieu, B. 841Mosseri, R. 90

Olshanii, M. 285Onuki, Y. 616Oppeneer, P.M. 580Ortolan, A. 866

Pascal, O. 468Pascaud, R. 468Passeggio, G. 866Petit, C. 674Pizarro, F. 468Pocholle, J.-P. 841Porzio, A. 866Pourret, A. 630Pozzi, G. 126Priou, A. 458Prouff, E. 415

Quiquandon, M. 18

Randrianalisoa, J. 683Rasel, E.M. 884Raufaste, C. 731Ringer, S.P. 269Rouchon, P. 841Ruggiero, M.L. 866

Sabouroux, P. 448Sadoc, J.-F. 90Saint-Jalmes, A. 649Salonen, A. 748Salvo, L. 649, 662Santagata, R. 866Santini, P. 573Sato, Y. 898Schmitt, V. 761Schreiber, K.U. 859Schubert, C. 884Schuh, T. 214Schwartz, S. 841Serfaty, S. 539Settai, R. 616Shick, A.B. 640Snoeck, É. 276, 281Sokoloff, J. 468Spencer Baskin, J. 176Stanev, T. 349

Stéphan, O. 158Steurer, W. 40Stupfel, B. 403Su, D.S. 258Suard, M. 662Suenaga, K. 151Suzuki, M.-T. 580, 587Syseova, K. 421

Tackmann, G. 884Takimoto, T. 587Tartaglia, A. 866Tencé, M. 158Teplova, N. 421Tian, H. 190Tinyakov, P. 318Tizei, L.H.G. 158Tokunaga, Y. 563Tsai, A.P. 30

Van Aert, S. 140Van Boxem, R. 190Van den Broek, W. 119Vanhove, P. 547Van Tendeloo, G. 140, 190, 281Verbeeck, J. 190Villain, J. 479

Walstedt, R.E. 563Wassermann, J. 859Wells, J.-P.R. 859

Yoshida, S. 309Yuan, T. 458

Zabler, S. 653Zagonel, L.F. 158Zewail, A.H. 176Zhang, B. 258

C. R. Physique 15 (2014) 911–913




Index des mots cléstome 15, 2014

3D – Maire É., 674

A

Accélération – Blasi P., 329Acquisition comprimée – Koch C.T., 119Actinides – Santini P., 573Adsorption – Monteux C., 775Adsorption de particules – Schmitt V.,

761Agrégats de protéines – Fameau A.-L.,

748Agregats de tensioactifs – Fameau A.-L.,

748Alliages intermétalliques complexes –

Quiquandon M., 18 – de Laissardière G.T., 70

Amas atomique – Gómez C.P., 30Amas polyatomiques – Guyot P., 12Américium – Dautray R., 481AMMRF – Ringer S.P., 269Analyse d’images – Maire É., 674Analyse dispersive en énergie – Suenaga

K., 151Anisotropie – Bertaina M.E., 300 – Kam-

pert K.-H., 318Antenne – Sokoloff J., 468Antenne large-bande – Hafdallah Ousli-

mani H., 458Antenne monopole – Hafdallah Ousli-

mani H., 458Antenne « sabre » – Hafdallah Ouslimani

H., 458Antenne ultra-compacte – Hafdallah

Ouslimani H., 458Appariement composite – Flint R., 557Applications – Bienvenu Y., 719Approximants – Gómez C.P., 30Atomes froids – Herzog C., 285 – Barrett

B., 875 – Tackmann G., 884

B

Bruit – Hawkes P., 110

C

Calcul ab-initio – de Laissardière G.T., 70Capteur inertiel – Di Virgilio A., 866Capteurs de rotation – Sato Y., 898Capteurs inertiels – Tackmann G., 884Caractérisation électromagnétique –

Georget É., 448Carbone – Mangler C., 241Cartographie de champs – Pozzi G., 126Catalyseur – Zhang B., 258Cathodoluminescence – Kociak M., 158Cellule coaxiale – Georget É., 448Cellule eukaryote – Schuh T., 214Céramique – Salvo L., 662Champ proche – Spencer Baskin J., 176Champs électriques – Pozzi G., 126Champ magnétique – Pozzi G., 126 –

Bommier V., 430Champs magnétique galactique – Farrar

G.R., 339Charge topologique – Verbeeck J., 190Chauffage – Heuraux S., 421Cheville – Bertaina M.E., 300 – Deligny

O., 367Coalescence limitée – Schmitt V., 761Communication – Bérenger J.-P., 393Composés de Hume Rothery – Friedel J.,

3Composition chimique – Zhang B., 258Composition en masse – Bertaina M.E.,

300 – Kampert K.-H., 318 – Lipari P., 357

Condensat de Bose–Einstein – Sato Y., 898

Correction d’aberrations – Colliex C., 101 – Hawkes P., 110

Corrélations électroniques fortes – Shick A.B., 640

Corrosion sous contrainte – Bouchaud É., 527

Cristal – Cahn J.W., e1Cristallographie N-dimensionnelle –

Quiquandon M., 18Cuprates – Alloul H., 519

D

Défauts – Mosseri R., 90Déflexions magnétiques – Farrar G.R.,

339Déformation plastique – Legros M., 224Détermination des structures atomiques

en trois dimensions – Koch C.T., 119Diagnostic – Heuraux S., 421Diffusion – Lecouvez M., 403Dioxydes d’actinides – Magnani N., 580Dislocations – Mompiou F., 82Dispositifs à phase – Hawkes P., 110Dissipation – Dollet B., 731Domaine de décomposition – Lecouvez

M., 403Drainage – Zabler S., 653Dynamique – Janssen T., 58

E

Échantillon liquide – Schuh T., 214Écoulement turbulent en décroissance

libre – Marzougui H., 509Effet Lense–Thirring – Di Virgilio A., 866Effet Sagnac – Darrigol O., 789 – Lefèvre

H.C., 851 – Schreiber K.U., 859 – Di Virgilio A., 866 – Sato Y., 898

Effets Josephson – Sato Y., 898Elaboration – Salvo L., 662Élasticité – Dollet B., 731Électromagnétisme – Maystre D., 387Electrons corrélés – Alloul H., 519

http://dx.doi.org/10.1016/S1631-0705(14)00171-61631-0705/2014 Publié par Elsevier Masson SAS pour l’Académie des sciences.

912 Index des mots clés / C. R. Physique 15 (2014) 911–913

Émulsions de Pickering – Schmitt V., 761

Endommagement – Bouchaud É., 527Énergie nucléaire – Dautray R., 481Ensembles modèles – Allouche J.-P., 6Équations de Ginzburg–Landau – Serfaty

S., 539Équations de Maxwell – Maystre D., 387ESTEEM – Snoeck E., 281Europe – Snoeck E., 281Excitons – Kociak M., 158

F

Fabrication – Bienvenu Y., 719Faisceaux vorticiels – Hawkes P., 110FDTD – Bérenger J.-P., 393Fermions lourds – Onuki Y., 616 – Aoki

D., 630Ferromagnétisme – Aoki D., 630Films minces – Ledieu J., 48Flambage – Burteau A., 705Fluage – Burteau A., 705Fluctuations de multipôle – Ikeda H.,

587Fluorescence – Darrigol O., 789Fond diffus cosmologique – Stanev T.,

349Fond diffus de photons – Stanev T., 349Forme des particules – Zhang B., 258Frustrations – Mosseri R., 90

G

Galactique–extragalactique – Blasi P., 329

Gaz de bosons unidimensionnel – Her-zog C., 285

Gaz de Coulomb – Serfaty S., 539Génération et détection des signaux –

Colliex C., 101Genou – Bertaina M.E., 300 – Blasi P., 329Genou de fer – Deligny O., 367Géophysique – Barrett B., 875Goniopolarimétrie – Cecconi B., 441Graphène – Mangler C., 241Grossissement – Zabler S., 653Gyromètre – Tackmann G., 884Gyromètre à fibre optique – Lefèvre H.C.,

851Gyromètre laser – Lefèvre H.C., 851Gyroscope laser à anneau – Schreiber

K.U., 859GZK – Harari D., 376

H

Hélium superfluide – Sato Y., 898Holographie électronique – Pozzi G.,

126

I

Infrastructure – Ringer S.P., 269 – Épi-cier T., 276 – Snoeck E., 281

Interaction lumière-matière – Barrett B., 875

Interactions atomes–surface solide – Boyes E.D., 200

Interactions hadroniques – Lipari P., 357Interface – Monteux C., 775Interféromètre atomique – Tackmann

G., 884Interféromètre Sagnac à ondes de ma-

tière – Barrett B., 875Interférométrie – Darrigol O., 789Interférométrie à onde de matière – Sato

Y., 898Inversion directe de diffusion multiple

des électrons – Koch C.T., 119

L

Laser à anneaux – Di Virgilio A., 866LF – Bérenger J.-P., 393Liaisons atomiques d’électrons de tran-

sition – Friedel J., 3Liaisons atomiques d’électrons presque

libres – Friedel J., 3Limites sur les flux de neutrinos – Stanev

T., 349Liquide quantique – Herzog C., 285Lois d’échelle – Lhuissier P., 696Longueur du jour – Di Virgilio A., 866

M

Magnétisme gravitationnel – Di Virgilio A., 866

Magnétohydrodynamique (MHD) – Bommier V., 430

Marché – Bienvenu Y., 719Marches atomiques – Spencer Baskin J.,

176Matériau souple – Georget É., 448Matériaux à basse dimension – Suenaga

K., 151Matériaux cellulaires – Maire É., 674 –

Randrianalisoa J., 683MEB – Maire É., 674Mécanique des mousses – Zabler S., 653Mesures de précision – Barrett B., 875MET – Mompiou F., 82Métal – Salvo L., 662Métamatériau – Hafdallah Ouslimani H.,

458 – Sokoloff J., 468METSA – Épicier T., 276Micro-ondes – Sokoloff J., 468Microanalyse – Ringer S.P., 269Microscopie – Ringer S.P., 269

Microscopie électronique – Spencer Bas-kin J., 176 – Zhang B., 258 – Épicier T., 276

Microscopie électronique à haute réso-lution – Mangler C., 241

Microscopie électronique en transmis-sion – Colliex C., 101 – Pozzi G., 126 – Verbeeck J., 190 – Snoeck E., 281

Microscopie électronique à transmission en balayage – Kociak M., 158

Microscopie électronique in situ – Boyes E.D., 200

Microscopie MET in situ – Legros M., 224Microtomographie – Burteau A., 705Milieux stratifiés – Bommier V., 430Miroirs – Hawkes P., 110Mise en réseau – Épicier T., 276Modèle de Hubbard – Alloul H., 519Modèle des excès – Kasmi C., 415Moment angulaire – Verbeeck J., 190Mosaïque de Voronoï – Randrianalisoa J.,

683Mosaïque de Voronoï–Laguerre – Ran-

drianalisoa J., 683Mousse – Zabler S., 653 – Salvo L., 662 –

Lhuissier P., 696 – Fameau A.-L., 748 – Monteux C., 775

Mousse à cellules ouvertes – Burteau A., 705

Mousse de nickel – Burteau A., 705Mousses aqueuses – Dollet B., 731Mousses céramiques – Randrianalisoa J.,

683Mousses et matériaux cellulaires inorga-

niques – Bienvenu Y., 719Mousses métalliques – Zabler S., 653 –

Randrianalisoa J., 683Mousses polymères – Randrianalisoa J.,

683Multiéchelle – Dollet B., 731

N

Nano-optique – Kociak M., 158Nanolaboratoire – Colliex C., 101Nanomanipulation – Verbeeck J., 190Nanoparticule d’or – Schuh T., 214Nanostructures – Bals S., 140Nanotube de carbone – Mangler C., 241Navigation inertielle – Lefèvre H.C., 851

– Barrett B., 875Neutrino – Yoshida S., 309Nombre de Reynolds magnétique faible

– Marzougui H., 509NpPd5Al2 – Onuki Y., 616

O

Onde de fuite – Sokoloff J., 468Ondes – Heuraux S., 421Ondes électromagnétiques – Maystre D.,

387Optique – Darrigol O., 789

Index des mots clés / C. R. Physique 15 (2014) 911–913 913

Optique électromagnétique – Maystre D., 387

Ordre caché – Flint R., 557 – Ikeda H., 587

Ordre chimique – Gómez C.P., 30Ordre multipolaire – Walstedt R.E., 563

– Magnani N., 580 – Ikeda H., 587Oxydes d’actinides – Walstedt R.E., 563

P

Particule – Fameau A.-L., 748Pavage apériodique – Cahn J.W., e1Périodicité – Mosseri R., 90Permittivité relative – Georget É., 448Phase décagonale – Steurer W., 40Phases cristallines approximantes –

Guyot P., 12Phases icosaédriques – Quiquandon M.,

18Phases icosaédriques AlMnSi, AlLiCu –

Guyot P., 12Phases intermétalliques complexes –

Steurer W., 40 – Mosseri R., 90Phasons – Janssen T., 58Photoémission – Shick A.B., 640Physique à l’avant – Lipari P., 357Physique aurorale – Cecconi B., 441Physique Kondo – Flint R., 557Physique spatiale – Cecconi B., 441PINEM – Spencer Baskin J., 176Plasma – Heuraux S., 421 – Bommier V.,

430Plasma à un constituant – Serfaty S., 539Plasma froid – Sokoloff J., 468Plasmons de surface – Kociak M., 158Plasticité – Dollet B., 731Plutonium – Dautray R., 481Polymère – Salvo L., 662 – Monteux C.,

775Porte-échantillon – Georget É., 448Potentiel de Riesz – Lecouvez M., 403Processus dynamique – Zhang B., 258Production de neutrinos et de rayons

gamma – Stanev T., 349Produits de fission – Dautray R., 481Propagation – Bérenger J.-P., 393Propriétés magnétiques – Verbeeck J.,

190Propriétés mécaniques – Mompiou F., 82Propriétés structurales – Lhuissier P.,

696Pseudo-gap – Alloul H., 519Ptychographie – Hawkes P., 110

Q

Quadrupoles – Santini P., 573Quasicristal – Allouche J.-P., 6 – Qui-

quandon M., 18 – Gómez C.P., 30 – Steurer W., 40 – Ledieu J., 48 – Jans-sen T., 58 – de Laissardière G.T., 70 – Mompiou F., 82 – Cahn J.W., e1

Quasiparticules – Santini P., 573Quasipériodicité – Guyot P., 12

R

Radioastronomie – Cecconi B., 441Radioélectricité – Maystre D., 387Rapport électron sur atome – Gómez

C.P., 30Rayons cosmiques – Bertaina M.E., 300 –

Yoshida S., 309 – Kampert K.-H., 318 – Lipari P., 357 – Deligny O., 367 – Ha-rari D., 376

Rayons cosmiques de ultra-haute éner-gie – Stanev T., 349

Rayons X – Darrigol O., 789RCUHE – Farrar G.R., 339Réactions catalytiques – Boyes E.D., 200Recherche collaborative – Ringer S.P.,

269 – Épicier T., 276 – Snoeck E., 281Reconstruction en trois dimensions –

Bals S., 140Réseau d’Abrikosov – Serfaty S., 539Résolution atomique – Bals S., 140 –

Boyes E.D., 200Résonance magnétique nucléaire – Wal-

stedt R.E., 563Rhéologie – Dollet B., 731Rhéologie interfaciale – Monteux C., 775Rotation de la Terre – Schreiber K.U., 859Rupture quasi fragile – Bouchaud É., 527

S

Sagnac – Darrigol O., 789Science des matériaux – Snoeck E., 281Science des surfaces – Ledieu J., 48Sécurité – Dautray R., 481Sécurité électromagnétiques – Kasmi C.,

415Simulation – Heuraux S., 421Sismologie – Schreiber K.U., 859Soleil : magnétisme de surface – Bom-

mier V., 430Soleil : photosphère – Bommier V., 430Solides isotropes – Cahn J.W., e1Solides ordonnés – Cahn J.W., e1Soliton brillant – Herzog C., 285Sonde atomique – Épicier T., 276Sources des rayons cosmiques – Yoshida

S., 309Spectre en énergie – Bertaina M.E., 300Spectroscopie d’atome unique – Suenaga

K., 151Spectroscopie de perte d’énergie – Sue-

naga K., 151 – Kociak M., 158Stabilité – Fameau A.-L., 748STEM – Suenaga K., 151 – Schuh T., 214STEM résolue en temps – Schuh T., 214Stimulable – Schmitt V., 761Structure – Salvo L., 662Structure de type Cd–Yb – Gómez C.P., 30

Structure électronique – Shick A.B., 640Structure et dynamique des dislocations

– Legros M., 224Structures commensurables (cristalline)

et incommensurables (quasicristal-lines) – Friedel J., 3

Suites automatiques – Allouche J.-P., 6Supraconductivité – Alloul H., 519 – Ser-

faty S., 539 – Griveau J.-C., 599Supraconductivité anisotrope – Onuki

Y., 616Supraconductivité non conventionnelle

– Aoki D., 630 – Shick A.B., 640Surface – Zhang B., 258Surface de Fermi – Onuki Y., 616

T

Taches solaires – Bommier V., 430TEMPEST – Kasmi C., 415Théorie de la résolution – Schuh T., 214Théorie LSDA + U – Magnani N., 580Tomographie aux rayons X – Maire É.,

674Tomographie électronique – Bals S., 140Transition galactique/extragalactique –

Deligny O., 367Transitions Raman stimulées – Barrett

B., 875Transmission (conditions) – Lecouvez

M., 403Transport électronique – de Laissardière

G.T., 70Transuraniens – Griveau J.-C., 599Turbulence MHD – Marzougui H., 509

U

UCoGe – Aoki D., 630UGe2 – Onuki Y., 616 – Aoki D., 630UHECR – Blasi P., 329Ultra-haute énergie – Yoshida S., 309Ultra-hautes énergies – Harari D., 376UO2 – Santini P., 573UPt3 – Onuki Y., 616URhGe – Aoki D., 630URu2Si2 – Ikeda H., 587

V

Valeurs extrêmes – Kasmi C., 415Verre – Cahn J.W., e1Verre de silice – Bouchaud É., 527VHECR – Blasi P., 329Vieillissement – Fameau A.-L., 748Viscoplasticté – Burteau A., 705VLF – Bérenger J.-P., 393Vortex – Verbeeck J., 190 – Serfaty S.,

539

C. R. Physique 15 (2014) 914–916




Keyword indexVol. 15, 2014

A

Ab initio DFT calculation – de Lais-sardière G.T., 70

Aberration correction – Colliex C., 101Aberration correctors – Hawkes P., 110Abrikosov lattice – Serfaty S., 539Acceleration – Blasi P., 329Actinide dioxides – Magnani N., 580Actinide oxides – Walstedt R.E., 563Actinides – Santini P., 573Adsorption – Monteux C., 775Aging – Fameau A.-L., 748Americium – Dautray R., 481AMMRF – Ringer S.P., 269Angular momentum – Verbeeck J., 190Anisotropic superconductivity – Onuki

Y., 616Anisotropy – Bertaina M.E., 300 – Kam-

pert K.-H., 318Ankle – Bertaina M.E., 300 – Deligny O.,

367Antenna – Sokoloff J., 468Aperiodic tilings – Cahn J.W., e1Aperiodicity – Mosseri R., 90Applications – Bienvenu Y., 719Approximant crystalline phases – Guyot

P., 12Approximants – Gómez C.P., 30Aqueous foams – Dollet B., 731Atom interferometer – Tackmann G.,

884Atom probe – Épicier T., 276Atomic bounds with nearly free elec-

trons – Friedel J., 3Atomic bounds with transitional elec-

trons – Friedel J., 3Atomic resolution – Bals S., 140 – Boyes

E.D., 200Atomic resolution tomography – Koch

C.T., 119Atomic steps – Spencer Baskin J., 176Auroral physics – Cecconi B., 441Automatic sequences – Allouche J.-P., 6

B

Bose–Einstein condensate – Sato Y., 898Bright soliton – Herzog C., 285Broadband and ultra-compact antenna

– Hafdallah Ouslimani H., 458Buckling – Burteau A., 705

C

Carbon – Mangler C., 241Carbon nanotube – Mangler C., 241Catalyst – Zhang B., 258Catalytic reactions – Boyes E.D., 200Cathodoluminescence – Kociak M., 158Cd–Yb type structure – Gómez C.P., 30Cellular materials – Maire É., 674 – Ran-

drianalisoa J., 683Ceramic – Salvo L., 662Ceramic foams – Randrianalisoa J., 683Chemical composition – Zhang B., 258Chemical order – Gómez C.P., 30Cluster – Gómez C.P., 30Coarsening – Zabler S., 653Coaxial cell – Georget É., 448Cold atoms – Barrett B., 875 – Tackmann

G., 884Cold plasma – Sokoloff J., 468Collaborative research – Ringer S.P., 269

– Épicier T., 276 – Snoeck E., 281Commensurate (crystallines) and in-

commensurate (quasicrystalline) structures – Friedel J., 3

Communication – Bérenger J.-P., 393Complex intermetallic alloys – Quiquan-

don M., 18Complex intermetallic phases – Mosseri

R., 90Complex metallic alloys – de Laissardière

G.T., 70Composite pairing – Flint R., 557Compressed sensing – Koch C.T., 119Correlated electrons – Alloul H., 519Cosmic ray sources – Yoshida S., 309

Cosmic rays – Bertaina M.E., 300 – Yoshida S., 309 – Lipari P., 357 – Deligny O., 367 – Harari D., 376

Coulomb gas – Serfaty S., 539Creep – Burteau A., 705Crystals – Cahn J.W., e1Cuprates – Alloul H., 519

D

3D – Maire É., 674Damage – Bouchaud É., 527Decagonal phases – Steurer W., 40Decaying homogeneous turbulence –

Marzougui H., 509Defects – Mosseri R., 90Diagnostic – Heuraux S., 421Dislocation structure and dynamics –

Legros M., 224Dislocations – Mompiou F., 82Dissipation – Dollet B., 731Domain decomposition – Lecouvez M.,

403Drainage – Zabler S., 653Dynamic process – Zhang B., 258Dynamics – Janssen T., 58

E

Earth rotation – Schreiber K.U., 859EDX – Suenaga K., 151EELS – Suenaga K., 151Elasticity – Dollet B., 731Electric fields – Pozzi G., 126Electromagnetic characterization –

Georget É., 448Electromagnetic optics – Maystre D.,

387Electromagnetic security – Kasmi C., 415Electromagnetic waves – Maystre D.,

387Electromagnetics – Maystre D., 387Electron energy loss spectroscopy – Ko-

ciak M., 158

http://dx.doi.org/10.1016/S1631-0705(14)00173-X1631-0705/2014 Published by Elsevier Masson SAS on behalf of Académie des sciences.

Keyword index / C. R. Physique 15 (2014) 914–916 915

Electron holography – Pozzi G., 126Electron microscopy – Spencer Baskin J.,

176 – Zhang B., 258 – Épicier T., 276Electron per atom ratio – Gómez C.P., 30Electron tomography – Bals S., 140Electronic structure – Shick A.B., 640Electronic transport – de Laissardière

G.T., 70Energy spectrum – Bertaina M.E., 300 –

Kampert K.-H., 318ESTEEM – Snoeck E., 281Eukaryotic cell – Schuh T., 214Europe – Snoeck E., 281Excess model – Kasmi C., 415Excitons – Kociak M., 158Extreme tendencies – Kasmi C., 415

F

FDTD – Bérenger J.-P., 393Fermi surface – Onuki Y., 616Ferromagnetism – Aoki D., 630Fiber-optic gyroscope – Lefèvre H.C., 851Field mapping – Pozzi G., 126Fission products – Dautray R., 481Flexible material – Georget É., 448Fluorescence – Darrigol O., 789Foam – Zabler S., 653 – Salvo L., 662 –

Lhuissier P., 696 – Fameau A.-L., 748 – Monteux C., 775

Foam mechanics – Zabler S., 653Forward physics – Lipari P., 357Frustrations – Mosseri R., 90

G

Galactic magnetic field – Farrar G.R., 339Galactic/extragalactic transition – Blasi

P., 329 – Deligny O., 367Geophysics – Barrett B., 875Ginzburg–Landau equations – Serfaty S.,

539Glasses – Cahn J.W., e1Gold nanoparticle – Schuh T., 214Goniopolarimetry – Cecconi B., 441Graphene – Mangler C., 241Gravito-magnetism – Di Virgilio A., 866Gyroscopes – Tackmann G., 884GZK – Harari D., 376

H

Hadronic interactions – Lipari P., 357Heating – Heuraux S., 421Heavy fermion – Onuki Y., 616 – Aoki D.,

630Hidden order – Flint R., 557 – Ikeda H.,

587High-resolution electron microscopy –

Mangler C., 241Hubbard model – Alloul H., 519Hume Rothery and Raynor compounds –

Friedel J., 3

I

Icosahedral AlMnSi, AlLiCu phases – Guyot P., 12

Icosahedral phases – Quiquandon M., 18Image analysis – Maire É., 674In situ electron microscopy – Boyes E.D.,

200In situ TEM – Legros M., 224Inertial navigation – Lefèvre H.C., 851 –

Barrett B., 875Inertial sensing – Tackmann G., 884Inertial sensor – Di Virgilio A., 866Infrastructure – Ringer S.P., 269 –

Épicier T., 276 – Snoeck E., 281Inorganic foam and cellular materials –

Bienvenu Y., 719Interactions atom–solid surface – Boyes

E.D., 200Interface – Monteux C., 775Interferometry – Darrigol O., 789Intermetallics – Steurer W., 40Inverse dynamical electron scattering –

Koch C.T., 119Iron knee – Deligny O., 367Isotropic solids – Cahn J.W., e1

J

Josephson effects – Sato Y., 898

K

Knee – Bertaina M.E., 300 – Blasi P., 329Kondo physics – Flint R., 557

L

Laguerre–Voronoi tessellation – Randri-analisoa J., 683

LDA + U theory – Magnani N., 580Leaky wave – Sokoloff J., 468Length of the day – Di Virgilio A., 866Lense–Thirring effect – Di Virgilio A.,

866LF – Bérenger J.-P., 393Light–matter interactions – Barrett B.,

875Limited coalescence – Schmitt V., 761Limits on the neutrino fluxes – Stanev T.,

349Liquid specimen – Schuh T., 214Low magnetic Reynolds number – Mar-

zougui H., 509Low-dimension materials – Suenaga K.,

151Low-voltage electrons – Koch C.T., 119

M

Magnetic deflections – Farrar G.R., 339Magnetic fields – Pozzi G., 126 – Bom-

mier V., 430Magnetic properties – Verbeeck J., 190

Magnetohydrodynamics (MHD) – Bom-mier V., 430

Markets – Bienvenu Y., 719Mass composition – Bertaina M.E., 300 –

Kampert K.-H., 318 – Lipari P., 357Materials science – Snoeck E., 281Matter-wave interferometry – Sato Y.,

898Matter-wave Sagnac interferometer –

Barrett B., 875Maxwell equations – Maystre D., 387Mechanical properties – Mompiou F., 82Metal – Salvo L., 662Metallic foams – Zabler S., 653 – Randri-

analisoa J., 683Metamaterial – Hafdallah Ouslimani H.,

458Metamaterials – Sokoloff J., 468METSA – Épicier T., 276MHD turbulence – Marzougui H., 509Microanalysis – Ringer S.P., 269Microscopy – Ringer S.P., 269Microtomography – Burteau A., 705Microwave – Sokoloff J., 468Mirrors – Hawkes P., 110Model sets – Allouche J.-P., 6Monopole antenna – Hafdallah Ousli-

mani H., 458Multipolar fluctuations – Ikeda H., 587Multipolar order – Walstedt R.E., 563 –

Ikeda H., 587Multipolar ordering – Magnani N., 580Multiscale – Dollet B., 731

N

N-dim crystallography – Quiquandon M., 18

Nano-optics – Kociak M., 158Nanolaboratory – Colliex C., 101Nanomanipulation – Verbeeck J., 190Nanostructures – Bals S., 140Near-field – Spencer Baskin J., 176Networking – Épicier T., 276Neutrino – Yoshida S., 309Neutrino and gamma ray production –

Stanev T., 349Nickel foam – Burteau A., 705Noise – Hawkes P., 110Non-conventional superconductors –

Griveau J.-C., 599NpPd5Al2 – Onuki Y., 616Nuclear magnetic resonance – Walstedt

R.E., 563Nuclear power – Dautray R., 481

O

One-component plasma – Serfaty S., 539One-dimensional Bose gas – Herzog C.,

285Open cell foam – Burteau A., 705Optics – Darrigol O., 789Ordered solids – Cahn J.W., e1

P

Particle adsorption – Schmitt V., 761Particle shape – Zhang B., 258

916 Keyword index / C. R. Physique 15 (2014) 914–916

Particles – Fameau A.-L., 748Phase plates – Hawkes P., 110Phasons – Janssen T., 58Photoemission – Shick A.B., 640Pickering emulsions – Schmitt V., 761PINEM – Spencer Baskin J., 176Plasma – Heuraux S., 421 – Bommier V.,

430Plastic deformation – Legros M., 224Plasticity – Dollet B., 731Plutonium – Dautray R., 481Polyatomic clusters – Guyot P., 12Polymer – Salvo L., 662 – Monteux C.,

775Polymer foams – Randrianalisoa J., 683Precision measurements – Barrett B.,

875Processing – Salvo L., 662Processing routes – Bienvenu Y., 719Propagation – Bérenger J.-P., 393Protein aggregates – Fameau A.-L., 748Pseudo-gap – Alloul H., 519Ptychography – Hawkes P., 110

Q

Quadrupoles – Santini P., 573Quantum gravity – Vanhove P., 547Quantum liquid – Herzog C., 285Quasi-brittle fracture – Bouchaud É., 527Quasicrystal – Allouche J.-P., 6 – Qui-

quandon M., 18 – Gómez C.P., 30 – Steurer W., 40 – Ledieu J., 48 – Janssen T., 58 – de Laissardière G.T., 70 – Mompiou F., 82 – Cahn J.W., e1

Quasiparticles – Santini P., 573Quasiperiodicity – Guyot P., 12

R

Radio electricity – Maystre D., 387Radioastronomy – Cecconi B., 441Relative permittivity – Georget É., 448Resolution theory – Schuh T., 214Rheology – Dollet B., 731Riesz potential – Lecouvez M., 403Ring laser gyroscope – El Badaoui N., 841

– Schreiber K.U., 859Ring-laser – Di Virgilio A., 866Ring-laser gyroscope – Lefèvre H.C., 851Rotation sensing – Sato Y., 898

S

“Sabre” antenna – Hafdallah Ouslimani H., 458

Safety – Dautray R., 481Sagnac – Darrigol O., 789Sagnac effect – Darrigol O., 789 – El

Badaoui N., 841 – Lefèvre H.C., 851 – Schreiber K.U., 859 – Di Virgilio A., 866 – Sato Y., 898

Sample holder – Georget É., 448Scaling laws – Lhuissier P., 696Scanning transmission electron mi-

croscopy – Kociak M., 158Scattering – Lecouvez M., 403Scattering amplitudes – Vanhove P., 547Seismology – Schreiber K.U., 859SEM – Maire É., 674Signal generation and recording – Col-

liex C., 101Silicate glass – Bouchaud É., 527Simulation – Heuraux S., 421Single atom spectroscopy – Suenaga K.,

151Solid-state laser – El Badaoui N., 841Space physics – Cecconi B., 441Stability – Fameau A.-L., 748STEM – Suenaga K., 151 – Schuh T., 214Stimulated Raman transitions – Barrett

B., 875Stimuli-responsive – Schmitt V., 761Stratified media – Bommier V., 430Stress corrosion cracking – Bouchaud É.,

527String theory – Vanhove P., 547Strong electron correlations – Shick A.B.,

640Structural properties – Lhuissier P., 696Structure – Salvo L., 662Sun: photosphere – Bommier V., 430Sun: surface magnetism – Bommier V.,

430Sunspots – Bommier V., 430Superconductivity – Alloul H., 519 – Ser-

faty S., 539 – Griveau J.-C., 599Superfluid helium – Sato Y., 898Surface – Zhang B., 258Surface plasmons – Kociak M., 158Surface rheology – Monteux C., 775Surface science – Ledieu J., 48Surfactant aggregates – Fameau A.-L.,

748Surfactants – Zabler S., 653

T

TEM – Mompiou F., 82TEMPEST – Kasmi C., 415

Thin film growth – Ledieu J., 48Three-dimensional reconstruction –

Bals S., 140Time-lapse STEM – Schuh T., 214Topological charge – Verbeeck J., 190Transmission conditions – Lecouvez M.,

403Transmission Electron Microscopy –

Colliex C., 101 – Pozzi G., 126 – Ver-beeck J., 190 – Snoeck E., 281

Transuranium – Griveau J.-C., 599

U

UCoGe – Aoki D., 630UGe2 – Onuki Y., 616 – Aoki D., 630UHECR – Kampert K.-H., 318 – Blasi P.,

329 – Farrar G.R., 339Ultra high energy – Yoshida S., 309 –

Harari D., 376Ultracold atoms – Herzog C., 285Ultrahigh energy cosmic rays – Stanev T.,

349Unconventional superconductivity –

Aoki D., 630 – Shick A.B., 640Universal photon backgrounds – Stanev

T., 349UO2 – Santini P., 573UPt3 – Onuki Y., 616URhGe – Aoki D., 630URu2Si2 – Ikeda H., 587

V

VHECR – Blasi P., 329Viscoplasticity – Burteau A., 705VLF – Bérenger J.-P., 393Voronoi tessellation – Randrianalisoa J.,

683Vortex beam – Verbeeck J., 190Vortex beams – Hawkes P., 110Vortices – Serfaty S., 539

W

Wave – Heuraux S., 421

X

X-ray tomography – Maire É., 674X-rays – Darrigol O., 789

Documents

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