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-)∓vvaFF=∓∓wa nnn¶wn2 ¶w modifications of the Sagnac interferometer where oooo C (nngo- ) =ao Vv relative motion between the frame and the medium ∓∓MF2 no no introduces a Doppler shift. As we will show, under this (5) condition, light drag resulting from dispersion affects where no is the phase index at the input frequency of w, the final fringe shift. Such a system, however, can only 2 a F = (1-1/ n o ) is the Fresnel drag coefficient, and be used in relative rotation sensing. Fig.2A shows is the group index, defined as the n g (= n o + w¶n ¶w) another MZI considered for Sagnac interferometry, ratio of the free space velocity of light to the group allowing relative motion of the medium with respect to velocity at w. As a specific example, consider a the interferometer. Note first that the connection with situation where the medium is non-comoving, as the first BS is established using a flexible fiber, thus making it possible to make the MZI rotate without illustrated in Fig. 2B. and 2C., giving rise to VM @ (-v). atomic and solid media using EIT [1,2] have shown

8 The relativistic velocities in this case are given by group index ng to be as large as 10 . Thus one can

conceivably design an ultra-precise rotation sensor for

sensing relative rotation of a local system using a C é 1 (n g - n o )ù V ± » o ± v a , a = 1- + (6) R L L ê 2 2 ú n o ëê n o n o ûú uniformly spinning slow light medium in the

interferometric beam path.

This is essentially the result presented in ref. 6. where aL is the so-called Laub drag coefficient, which has been experimentally measured with great precision However, as noted earlier, ref. 6 does not point out that using a passive ring resonator technique [15]. The this enhancement can not be achieved when measuring expressions for time delay and phase shift are obtained absolute rotation. On the other hand, the results in the as paper by Matsko et al. [10] are simply incorrect, due to

the fact that they do not use special relativity in their

2 2 derivation. For example, in the limit of no dispersion, Dt » (n (1- a L ))Dt o , Df » (n (1- a L ))Dfo (7) the expression they derive for the phase shift is

proportional to the square of the index (see eqns. 16 Unlike the Doppler-shift free situation, the and 17 in ref 10). The correct result in this limit is the coefficients of time delay and phase shift now depend expression in eqn. 2 above, which is independent of the on the refractive property of the medium. Of course, in index [8,11]. the absence of dispersion (i.e., ng=no), one recovers the same results as in eqn. 2. In the presence of a highly As mentioned earlier, one of the drawbacks of the dispersive medium with ng>>no (i.e. ¶n ¶w >> n o w ), slow-light enhanced interferometric configuration characteristic of extreme normal dispersion in a above is that it can not be employed in sensing absolute medium that produces slow-light, the time delay and rotation. This constraint can be overcome by the phase shift are given by considering the Sagnac effect in a passive ring cavity

or resonator. Unlike the interferometer where the

Dt » n g Dt o , Df » n g Df o (8) frequency of light remains unchanged by rotation, the

frequencies of the cavity modes in a resonator get Thus, the rotation sensitivity of a slow-light based modified as a result of rotation. Figure 3 shows the gyroscope scales directly with the group index n g. schematic of a passive ring cavity (PRC) along with the Experimental observation of slow light reported in necessary detection mode and servo mechanism to be It is important to note that even though the basic used in a gyroscope. For analytic convenience, we mechanisms for the MZI and the PRC based assume the light path in the resonator to be circular gyroscopes are the same, the Sagnac frequency with a radius R and an enclosed area A (the effect splitting of the cavity modes explicitly depends on the occurs even when the paths are rectilinear, as shown in refractive index of the medium. In this derivation, we the diagram). In the absence of rotation, the light in the have implicitly assumed the refractive index to be cavity resonates in both directions with a frequency wo independent of frequency (i.e. no dispersion). When given by the effect of dispersion is taken into account, the result

C 2pN changes significantly. Without any loss of generality, w = o (9) o n P o one can write where N is an integer, and P = 2pR is the perimeter. In the presence of rotation, the resonance frequencies are Dw 2pN w± = w ± = V ± × (11) o 2 E P different for the CW(+) and CCW(-) directions, and are given by where Dw is a parameter whose amplitude is to be

± 2Np determined. The effective velocities, VE can be w±@V±×=;VVv±± EP ER∓ (10) written as -+ wwoo4AW Dwo (=w-w) =(2RW) =× ConoCoonP C éùv VV±±=v=±o .1 (12) ER∓ ±±êú n(ww)ëûêúCon()

± where VR are the relativistic velocities given in eqn. 3,

± 2 Expanding the value of n(w) around no, we get VE are the effective velocities and A=pR is the area enclosed by the resonator. The frequency difference

± Co éùv Dw Dwo which is proportional to the rotation rate W, can be VEo=×êú1±∓ n¢¢,nº[¶n/¶w]/n noëûCoon2 experimentally measured from heterodyne beat note of (13) transmitted signals by adjusting the frequency of Substituting eqn. 13 in eqn. 11, one gets a self-

AOM1(AOM2) to maximize the cavity transmission in consistent expression involving Dw that yields

CCW(CW) direction.

Dwoon Dw==Dwoo=Dw×x (14) 1+wognn¢ In the limit when x®¥ or (ng ®0), non-zero value where we define an enhancement factor x º no/ ng. For of Q ( Q1<< ) prevents the enhancement factor h in a medium that exhibits slow-light, n >>n [i.e., g o eqn. 15 from diverging. In fact, it saturates to a

¶n / ¶w >> (n o / wo ) , where no is (near-unity) resonant maximum value hmax =2 Q . The CAD condition index of the medium], the result in eqn. 14 implies a can be easily achieved using a bi-frequency pumped reduction in rotational sensitivity. On the other hand, it Raman gain in a L-type atomic alkali vapor medium is equally possible to achieve a condition where 0 < ng [4]. Typical value of enhancement bound for gain << 1, characteristic of the anomalous dispersion assisted anomalous dispersion has been estimated [16] (¶n / ¶w< 0) that leads to superluminal pulse by considering the magnitude of second order propagation in the medium (of course, without dispersion for a frequency splitting of 1 MHz and is violating SR) [4]. found to be O(108). In order to achieve this kind of

sensitivity, ng can be tuned close to zero by controlling To realize large enhancement in sensitivity, one has either the strength or the frequency separation of the to operate close to the limit when the group index ng of pumps. If one considers a ring cavity of R=1 meter the medium approaches zero [i.e. containing such a CAD-type atomic medium with a

¶n / ¶w = -(n o / wo ) ], the so-called CAD condition as mean index no=1.0, and assume a minimum measurable defined earlier. However, the factor x diverges as we beat frequency of 10 KHz, for example, the approach this limit. This is because we have neglected corresponding minimum measurable rotation rate is higher order effects. When the second order dispersion about 100 pico-rad per sec. Without the fast-light is included the frequency shift is found to be enhancement, the corresponding minimum measurable

rotation rate would be about 50 mrad per second.

æö2 8 Dw=Dwoo.h=Dwx.( 2) ç÷1+1Q+x Thus, the effective enhancement factor is nearly 5x10 . èø 222 Q=fn¢¢woo,n¢¢ º[¶n¶w ]/n

(15) Unlike the interferometric technique described earlier, this technique is applicable in absolute rotation Dw where f = o is a dimensionless quantity defined as wo sensing. As such, it represents a key mechanism the fractional bandwidth. through which high anomalous dispersion can enhance

rotation sensitivity to enable ultrahigh precision rotation measurements encountered in navigational, 6. U. Leonhardt and P. Piwnicki, Phys. Rev. A 62, geophysical and relativistic applications. 055801 (2000).

7. E. J. Post, Rev. Mod. Phys, 39, 475 (1967).

We acknowledge useful discussions with P. Pradhan, G. 8. G.B. Malykin, Physics-Uspekhi 43, 1229 (2000).

Cardoso, and S. Ezekiel. This work was supported in part by 9. F. Zimmer and M. Fleischhauer, Phys. Rev. Letts. the ARO grant # DAAD19-001-0177 under the MURI 92, 253201 (2004). program, and by the AFOSR grant # FA9550-04-1-0189. 10. A.B. Matsko et al., Opt. Commun. 233, 107

(2004). 1. L.V. Hau et al., Nature 397, 594 (1999). 11. G. E. Stedman, Rep. Prog. Phys. 60, 615 (1997). 2. A. V. Turukhin, V.S. Sudarshanam, M.S. Shahriar, 12. W. W. Chow et al., Rev. Mod. Phys. 57, 61 J.A. Musser, B.S. Ham, and P.R. Hemmer, Phys. (1985). Rev. Lett. 88, 023602 (2002). 13. S. Ezekiel and S. R. Balsamo, Appl. Phy. Lett. 30, 3. E. Podivilov et al., Phys. Rev. Letts. 91, 083902 478 (1977). (2003). 14. P. Kumar and J. Shapiro, Opt. Lett. 10, 226 (1985). 4. L.J. Wang, A. Kuzmich and A. Dogariu,, Nature 15. G.A. Sanders and S. Ezekiel, J. Opt. Soc. Am. B 5, 406, 277 (2000). 674 (1988). 5. M.S. Bigelow, N.N. Lepeshkin, and R.W. Boyd, 16. M.S. Shahriar et al., Phys. Rev. Letts. (submitted, Science 301, 201 (2003). 2005), http://arxiv.org/abs/quant-ph/0507139.

BS2

Det

CW ¤ W

Source BS1

W CCW

Figure 1 Schematic illustration of a Mach-Zehnder type Sagnac interferometer with counter-propagating beams in a circular optical path (the actual paths shown are rectilinear for simplicity). W W LaserLaser V Det VM Det CW Frame & source stationary; medium rotating B CW VVMM VM W W VM LaserLaser Flexible CCW CCW Laser stationary Fiber Clamp Det A Frame & source rotating; medium stationary C

Figure 2 Interferometric setup for rotation sensing using Doppler effect (a) generalized case corresponding to uniform medium translation (b) stationary source and frame, rotating medium (c) co-rotating source and frame, stationary medium

diff.

Df Beat Det V 1 S

W Laser

VCO1 Q diff.

AOM1 AOM2 VCO2

V S 2

Figure 3 Schematic diagram of a passive resonator gyroscope showing detection mode for beat frequency measurement and active closed-loop control for