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Ultrahigh Precision Absolute and Relative Rotation Sensing Using Fast and Slow Light

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Ultrahigh Precision Absolute and Relative Rotation Sensing Using Fast and Slow Light Ultrahigh Precision Absolute and Relative Rotation Sensing using Fast and Slow Light M.S. Shahriar, G.S. Pati, R. Tripathi, V. Gopal, M. Messall and K. Salit Department of Electrical and Computer Engineering Northwestern University, Evanston, IL 60208 We describe a passive resonator based optical gyroscope whose sensitivity for measuring absolute rotation is enhanced via use of the anomalous dispersion characteristic of superluminal light propagation. The enhancement is given by the inverse of the group index, saturating to a bound determined by the group velocity dispersion. For realistic conditions, the enhancement factor is as high as 108. We also show how normal dispersion used for slow light can enhance relative rotation sensing in a specially designed Sagnac interferometer, with the enhancement given by the slowing factor. PACS numbers: 42.5, 42.50.Ct,42.50.Gy, 42.60.Da Recent experiments have demonstrated very slow as rotation. Recently, Leonhardt and Piwnicki [6] well as superluminal group velocities of light using proposed a slow-light enhanced gyroscope; however, many different processes [1-5]. Extreme dispersion in a they did not point out that this enhancement can not be slow-light medium has been recently considered to achieved when measuring absolute rotation. Later, boost the rotational sensitivity of optical gyroscopes Zimmer and Fleischhauer [9] refuted, incorrectly, the via the Sagnac effect [6]. However, the mechanism by idea of any slow light based sensitivity enhancement in which strong material dispersion affects relativistic rotation sensing, and proposed an alternative hybrid Sagnac effect has remained somewhat controversial. It light and matter-wave interferometer, based on slow is commonly perceived that the Sagnac time delay or light momentum transfer from light to atoms. Matsko phase shift in the presence of a dielectric medium is et al. [10] proposed a variation of the idea presented in independent of the refractive property of the medium ref. 6, where they consider the use of microresonators [7,8]. Although this is true for a co-moving medium instead of atoms to provide the normal dispersion. As and absolute inertial rotation, it is not true if the such, this gyroscope is also unable to achieve the medium is moving with respect to the inertial frame of enhancement when measuring absolute rotation. The authors do not point this out. Furthermore, the medium [11-13]. We discuss the effect of dispersion on analysis in this paper makes use of the classical (non- such a passive resonator gyroscope, and show that the relativistic) model, which yields results that are rotation sensitivity can be increased by many orders of incorrect when a material medium of non-unity index magnitude by using what we call critically anomalous is present, as we show later. dispersion (CAD). The CAD condition corresponds to the group index approaching a null value, and occurs in In an interferometric optical gyroscope that allows experiments demonstrating superluminal light relative motion between the medium and the propagation in a medium. We show how the presence interferometer, the dispersive drag coefficient can of group velocity dispersion prevents divergence in the influence the Sagnac fringe shift. Precise experimental degree of enhancement, limiting it to a value of the measurements of light drag in a moving optical order of 108 in a realistic system. Such a sensor can be medium have been analyzed and interpreted [8,11] implemented using bi-frequency pumped Raman gain long before the inception of resonantly dispersive slow- in an atomic medium [14], for example, and can be light phenomena. Here, we first use axioms of special used in navigational or geophysical applications for relativity (SR) to review briefly the light dragging ultrahigh precision absolute inertial rotation sensing. phenomena in a moving dielectric medium. We then identify clearly the conditions under which the use of Let us consider first waves in a Mach-Zehnder slow light can enhance the rotational sensitivity by a interferometer (MZI), as shown in Fig. 1, that are factor as large as the group index in the medium, as constrained to propagate in a circular path (for well as the conditions under which the enhancement convenience in analysis) with a radius R (the effect does not occur. A slow-light enhanced sensor of this occurs for paths that are rectilinear as well, as shown in type is suitable only for precise sensing of the relative the diagram). In general, the wave could be of any rotation of a local system. We also consider a passive kind: optical, matter or acoustic. For the clockwise resonator based gyroscope for absolute rotation CW(+) and counter-clockwise CCW(-) directions, the sensing where the Sagnac effect manifests itself in the ± relativistic velocities VR of the phase fronts (PFs), the form of the frequency splitting of cavity modes. Unlike time T ± of PFs to travel from BS1 to BS2, and the an interferometer, frequency-splitting in a resonator effective distances L± from the BS1 to BS2 are related gyroscope is governed by optical path lengths that are as follows always dependent on the refractive property of the V ± v L± Df=4pWmA/h, can be obtained from eqn. 2 by V ± = P , L± = p R ± vT ± , T ± = (1) R 2 ± 1± VP v / C o VR 2 inserting the Compton frequency w=(mco /)h . Note that the Sagnac effect can be explained fully within SR, where VP is the velocity of each PF in the absence of without invoking general relativity, if the degree of rotation, Co is the velocity of light in vacuum, v=WR is space-time curvature is insignificant, which is the case the tangential velocity of rotation, and W is the rotation for optical loops typically used in Sagnac rate. interferometers [8]. Thus, the time delay between the propagating For the case of optical waves in a Sagnac waves and their relative phase shift at the detector are interferometer consisting of a refractive medium, the given by phase shift Dfo in eqn. 2 is independent of the - + + - é (VR - VR ) + 2v ù Dt o = T - T = pRê - + ú refractive index n. This result is somewhat ë(VR + v)(VR - v)û counterintuitive, since it is expected that the light 2AW 2AW æ v ö = » , b çº ÷ << 1 2 2 2 ç ÷ C o (1- b ) C o è C o ø beams experience index-dependent drag in the moving medium. Thus the travel time delay around the optical 4pAW Df = wDt = o o circuit could be expected to depend on n. However, this l o C o (2) is not true for a co-rotating medium in the Sagnac where A=pR2 is the enclosed area, w is the angular interferometer. One can explicitly show this by re- expressing the velocities V ± of the CW and CCW frequency, lo is the vacuum wavelength of the wave R and b is the boost parameter, assumed to be much less phase fronts in terms of the Fresnel drag: than unity for typical rotations. The form of the time delay in eqn. 2 due to Sagnac effect attests to the fact C 1 V ± = o ± v a , a = (1 - ) (3) R n F F n 2 that the time delay is simply a geometric effect, attributable to relativistic time dilations, and does not where Co/n is the velocity Vp of the phase fronts in the depend at all on the velocity of the wave. It only absence of rotation, and the term ‘aF’ is the Fresnel involves the free space velocity of light, even if drag coefficient. Using eqns. 2, one can write the time acoustic waves or matter waves are used. The familiar delay and the phase shift in a generalized form result of matter-wave Sagnac phase shift, 2 2 rotating the source. If the source is stationary while the Dt » (n (1- a F ))Dt o , Df = (n (1- a F ))Dfo (4) frame and the medium co-rotate, a Doppler shift is produced at the first BS. However, this shift does not Substituting ‘aF’ from eqn. 3 in eqn. 4, one gets the affect the phase shift, since the BS effectively serves as same results as in eqn. 2. This is because the derivation the source for both arms. Therefore, the fringe shift using SR already includes the effect of the medium described in eqn. 2 still holds. Thus, even in the most motion on light propagation. However, the magnitude general case, a relative motion between the first BS and of this effect is independent of the Fresnel drag the source has no effect on the Sagnac interference. coefficient, as well as the refractive index, for a Sagnac Next, consider the case where the medium inside the interferometer that uses a common frame of rotation MZI paths is allowed to move at a velocity V with for light source, interferometer as well as the medium. M respect to the MZI frame (i.e. the mirrors and beam Such a system is commonly employed for absolute splitters). As seen by medium, the CW(+) and CCW(-) rotation sensing in navigational systems. This system beams are now Doppler shifted by equal and opposite does not possess (to first-order) Doppler shift of any ± kind between the medium and the source or the amounts, given by Dw = ±wVM / C o . Expressions for interferometer, since they are all co-rotating at the relativistic velocities can be given by including first- same rate W. order dispersion in n(w) as One can relax this constraint and allow several ± ± CCooDw¶¶nnVM VR=(1-)mvvaFF=mmwa nnn¶wn2 ¶w modifications of the Sagnac interferometer where oooo C (nngo- ) =ao Vv relative motion between the frame and the medium mmMF2 no no introduces a Doppler shift.
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