Suppression of Nonlinear Interactions in Resonant Macroscopic Quantum Devices : the Example of the Solid-State Ring Laser Gyroscope Sylvain Schwartz1, Franc¸ois Gutty2, Gilles Feugnet1, Philippe Bouyer3 and Jean-Paul Pocholle1 1Thales Research and Technology France, Route D´epartementale 128, F-91767 Palaiseau Cedex, France 2Thales Avionics, 40 rue de la Brelandi`ere, BP 128, F-86101 Chˆatellerault, France 3Laboratoire Charles Fabry de l’Institut d’Optique, CNRS and Universit´e Paris-Sud, Campus Polytechnique, RD 128, F-91127 Palaiseau Cedex, France∗ (Dated: February 2, 2008) We study the suppression of nonlinear interactions in resonant macroscopic quantum devices in thecaseofthesolid-stateringlaser gyroscope. Thesenonlinearinteractionsaretunedbyvibrating 8 the gain medium along the cavity axis. Beat note occurrence under rotation provides a precise 0 measurement of the strength of nonlinear interactions, which turn out to vanish for some discrete 0 values of the amplitude of vibration. Our theoretical description, in very good agreement with 2 the measured data, suggests the use of a higher vibration frequency to achieve quasi-ideal rotation n sensingoverabroadrangeofrotation speeds. Wefinallyunderlinetheanalogy betweenthisdevice a and some other macroscopic quantum rotation sensors, such as ring-shaped superfluid configura- J tions, where nonlinear interactions could be tuned for example by the use of magnetically-induced 7 Feschbach resonance. PACSnumbers: 42.65.Sf,42.62.Eh,06.30.Gv,42.55.Rz ] s c i t The use of macroscopic quantum effects for rotation between the counterpropagating modes induced by the p sensing in ring-shaped configurations has been exten- population inversion grating established in the amplify- o sivelystudiedinthe caseofbothopticalsystems[1,2,3] ing medium [3, 14]. Using the quantitative information . s and superfluids [4, 5], either liquid helium [6, 7] or Bose- on the strength of the nonlinear interactions provided c i Einstein condensed gases [8, 9, 10, 11, 12]. As pointed by the beat note between the counterpropagating laser s out in [13, 14], nonlinear interactions play a crucial role beams [14], we demonstrate experimentally the possibil- y h in the dynamics of such devices, as they can hinder or ityofsuppressingtheseinteractionsforsomediscreteval- p affect their ability to sense rotation, even when coun- ues of the amplitude of the crystal movement. We even- [ teracted by other coupling sources. Consequently, the tually derive, in the limit of high vibration frequencies, 1 possibility of tuning or even suppressing nonlinear inter- a very simple condition for rotation sensing and point v actions is of great importance for using these devices as out the similarity with the equivalent condition for a 0 rotation sensors. toroidalBose-Einsteincondensed gas, resulting from the 3 Several systems offer the possibility of controlling the toy model of [13] where the effects of scattering length 9 0 strength of their nonlinearities. For example, in the tuning described in [15] are included. . case of gas ring laser gyroscopes, one can considerably 1 The solid-state ring laser gyroscope can be described lower mode competition by tuning the cavity out of res- 0 semiclassically, assuming one single identical mode in 8 onance with the atoms at rest, resulting in the quasi- each direction of propagation (something which is guar- 0 suppression of nonlinear interactions [1]. In the case of anteed by the attenuation of spatial hole burning effects : atomic systems, it is also possible to tune and even sup- v thanks to the gain crystal movement [19]), one single i pressnonlinearinteractions,byusingFeshbachresonance X identical state of polarization and plane wave approxi- [15, 16, 17]. As regards solid-state ring lasers, we have mation. Theelectricalfieldinside the cavitycanthen be r recently demonstrated [14] the possibility of stable rota- a written as follows : tion sensing thanks to the circumvention of mode com- petition by the use of an additional stabilizing coupling. However, nonlinear interactions are still present in this 2 configuration, and can even be quantitatively observed E(x,t)=Re E˜ (t)ei(ωct+µpkx) , p [14, 18]. ( ) p=1 X In this Letter, we report the experimental and theo- retical study of a novel technique intended to tune and suppress nonlinear interactions in a solid-state ring laser where µ = (−1)p and where ω and k are respectively p c gyroscope,similarly to the case of scattering length con- the angular and spatial average frequencies of the laser, trol in an atomic system. This is achieved by vibrating whose longitudinal axe is associated with the x coordi- the gain crystal along the optical axis of the laser cav- nate. In the absence of crystal vibration, the equations ity, considering the fact that nonlinear interactions in a of evolution for the slowly-varying amplitudes E˜ and 1,2 solid-state ring laser result mainly from mutual coupling forthepopulationinversiondensityN havethefollowing 2 expression [3, 14] : Polarizing mirror Beat dE˜ γ m˜ Ω note 1,2 = − 1,2E˜ +i 1,2E˜ +iµ E˜ (1) dt 2 1,2 2 2,1 1,22 1,2 θ(cid:1) Laser Diode σ L L + E˜1,2 Ndx+E˜2,1 Ne−2iµ1,2kxdx , Nd-YAG 2T Z0 Z0 ! VIBRATING DEVICE YAG ∂N N aNE(x,t)2 E2−E 2 =W (1+η)− − , (2) 1 2 th ∂t T T 1 1 where γ are the intensity losses per time unit for each 1,2 Figure 1: Scheme of our experimental setup. The diode- mode, m˜ are the backscattering coefficients, Ω is the 1,2 pumped vibrating Nd-YAG crystal is placed inside a 22-cm difference between the eigenfrequencies of the counter- ringcavityonaturntable. Losses oftheform (4) arecreated propagating modes (including the effect of rotation, see byafeedbackloopactingonaFaradayrotator(anadditional further), σ is the laser cross section, T is the cavity YAGcrystalinsideasolenoid),incombinationwithapolariz- round-triptime,ηistherelativeexcessofpumpingpower ingmirrorandaslightnon-planarityofthecavity(notdrawn above the threshold value W , T is the lifetime of the here). Twophotodiodesareusedforgeneratingtheerrorsig- th 1 population inversion and a is the saturation parameter. nal of the feedback loop. A third photodiode measures the frequency of the beat note between the counterpropagating Throughoutthispaperweshallneglectdispersioneffects, modes. consideringthefactthattheNd-YAGgainwidthismuch larger than the laser cavity free spectral range. The backscattering coefficients, which depend on spatial in- whereE(x,t)referstotheelectricfieldinthecavity(non- homogeneities of the propagation medium [20], have the vibrating) frame. Moreover, N (x,t) can be deduced c following expression [18] : from its equivalent in the cavity frame N(x,t) by the identity N (x,t) = N(x+x (t),t), resulting in the fol- c c m˜ =− ωc L ε(x)− iκ(x) e−2iµ1,2kxdx, (3) lowing expressions : 1,2 ε¯cT ω I0 (cid:20) c (cid:21) L L N(x,t)dx= N (x,t)dx, where ε(x) and κ(x) are respectively the dielectric con- c stant and the fictitious conductivity along the cavity  Z0 Z0 perimeter in the framework of an ohmic losses model  LN(x,t)e2ikxdx=e2ikxc(t) LN (x,t)e2ikxdx. c [21], where c is the speed oflightin vacuum andwhereε¯ Z0 Z0 standsforthespatialaverageofε. Inordertocounteract The backscattering coefficients (3) acquire in the pres- mode competition effects and ensure beat regime opera- enceofthecrystalvibrationthefollowingtime-dependent tionunder rotation,anadditionalstabilizingcoupling as form : described in [14] is introduced, resulting in losses of the following form : m˜ (t)=m˜ c e−2iµ1,2kxc(t)+m˜ m , (7) 1,2 1,2 1,2 γ1,2 =γ−µ1,2Ka(|E˜1|2−|E˜2|2), (4) where m˜ c and m˜ m, which are time-independent, ac- 1,2 1,2 countforthebackscatteringduerespectivelytothecrys- where γ = κ¯/ε¯is the average loss coefficient and where tal at rest and to any other diffusion source inside the K >0representsthestrengthofthestabilizingcoupling. laser cavity (including the mirrors). As regards the dif- We assume the following sinusoidal law to account for ference Ω between the eigenfrequencies of the counter- the gain crystal vibration : propagating modes, it results from the combined effects x of the rotation (Sagnac effect [22]) and of the crystal m x (t)= sin(2πf t), (5) c 2 m movement in the cavity frame (Fresnel-Fizeau drag ef- fect [23]), resulting in the following expression : where x (t) is the coordinate, in the frame of the laser c cavity,of a givenreference point attachedto the crystal, Ω 4A 2x˙ (t)l(n2−1) = θ˙− c , (8) andwherexm andfm arerespectivelytheamplitudeand 2π λL λL the frequency of the vibration movement. The popula- where A is the area enclosed by the ring cavity, λ = tioninversiondensityfunctionintheframeofthevibrat- 2πc/ω is the emission wavelength, θ˙ is the angular ve- ing crystal N (x,t) is ruled by the following equation : c c locityofthecavityarounditsaxis,andlandnarerespec- ∂N N aN E(x+x (t),t)2 tively the length and the refractive index of the crystal c c c c =W (1+η)− − , (6) ∂t th T T (dispersiontermsareshowntobenegligibleinthiscase). 1 1 3 The dynamics of the solid-state ring laser gyroscope 150 No vibrating crystal (exp.) θ(cid:1) with a vibrating gain medium is eventually ruled, in the Vibrating crystal (exp.) 3 frameworkofourtheoreticaldescription,bythefollowing 120 Vibrating crystal (theory) equations : z] H k dE˜1,2 = −γ1,2E˜ +im˜1,2E˜ +iµ ΩE˜ (9) cy [ 90 1,2 2,1 1,2 1,2 n dt 2 2 2 e u θ(cid:1) 50 +2σT E˜1,2Z0LNcdx+E˜2,1e2ikxcZ0LNce−2iµ1,2kxdx! , eat freq 60 1 40 B MAG 30 30 where γ , x , N , m˜ and Ω obey respectively equa- 1,2 c c 1,2 20 tions (4), (5), (6), (7) and (8). It comes out from this analysisthatthesolid-stateringlaserbenefits,asarota- 0 40 60 0 20 40 60 80 100 120 140 160 180 200 tion sensor, from the crystal vibration in three separate Rotation speed [deg/s] and complementary ways : • the contrast of the population inversion grating, Figure 2: Experimental beat frequency as a function of the which is responsible for nonlinear coupling, is re- rotation speed. White and black circles refer respectively to ducedonbothconditionsthattheamplitudeofthe thesituationswherethecrystalisatrestandwherethecrys- movement is of the same order of magnitude than tal is vibrating with a frequency fm ≃ 40 kHz and an am- the step of the optical grating (typically a fraction plitude xm ≃ 0.74 µm. The insert shows a magnification of µm) and that the period of the movement 1/fm around θ˙1 – see eq. (10) –, together with theoretical predic- issignificantlylargerthanthepopulationinversion tionsresulting from numerical simulations with thefollowing measured [18] parameters : γ = 15.34 106 s−1, η = 0.21, response time T ; the atomic dipoles are then no longer confined 1into a nodal or an antinodal area |am˜rg1c(,2m˜|1m=/1m˜.52m1)04=s−π1/,1|7m˜, 1Km,2|==180.57 1s−041.s−I1n,teagrgra(tm˜io1cn/ms˜t2ecp) =is – see eq. (6) –, and become sensitive to the time- 0.1 µs, average values are computed between 8 and 10 ms. averagevalue of the electric field, which can be in- dependent of their position on the crystal (at least when the laser is not rotating) provided the condi- thenonlinearinteractions[14],isconsiderablyreducedin tionJ (kx )=0isobeyed[24],J referringtothe the zone ranging from 10 to 40 deg/s. Some nonlineari- 0 m 0 zero-orderBessel’s function); tiesareobservedaroundthediscretevaluesθ˙ ≃55deg/s and θ˙ ≃ 165 deg/s, in agreement with our theoretical • the light backscattered on the gain crystal from model. As a matter of fact, analyticalcalculationsstart- one mode into the other can be shifted out of res- ing from equation (9) reveal the existence of disrupted onance by the Doppler effect resulting from the zonescenteredondiscretevaluesoftherotationspeedθ˙ q crystal movement in the cavity frame; this phe- obeying the following equation : nomenon, which induces a decrease of the corre- spondingcouplingstrength,haspreviouslybeenre- 4Aθ˙ =qf where q is an integer, (10) q m ported in the case of vibrating mirrors [25, 26]; in λL ourmodel,itarisesfromthetime-dependentphase the size of each disrupted zone being proportional to factor exp(2ikx ) in front of the coupling coeffi- c J (kx ). With our experimental parameters, the first q m cients m˜1c,2 and Ncdx in equations (7) and (9); criticalvelocitycorrespondstoθ˙ =55.5deg/s,thezones 1 • thefrequencynoRn-reciprocitybetweenthecounter- observed on Fig. 2 corresponding to the cases q =1 and q = 3. The numerical simulations shown on the insert propagatingmodesduetotheFresnel-Fizeaudrag- of this figure are in good agreement with our analytical ging effect – eq. (8) – has a similar role as the me- andexperimentaldata. Suchaphenomenonofdisrupted chanicaldithering typically used for circumventing zoneshasbeenreportedpreviouslyinthecaseofgasring the lock-in problem in the case of usual gas ring laser gyroscopes with mechanical dithering. It is some- laser gyroscopes [27]. times designed as ‘Shapiro steps’ [27], in reference to an The solid-state ring laser setup we used in our exper- equivalent effect in the field of Josephson junctions [28]. iment is sketched on Fig. 1. Thanks to the additional The dependence of the beat frequency on the ampli- stabilizing coupling (4), a beat note signal is observed tude of the crystal movement is shown on Fig. 3, for above a critical rotation speed, whose frequency is plot- a fixed rotation speed (200 deg/s). This graph illus- ted on Fig. 2. It can be seen on this figure that the trates the good agreement between our numerical sim- difference between the ideal Sagnac line and the experi- ulationsandourexperimentaldata. Moreover,this isan mental beat frequency, which is a direct measurementof experimental demonstration of the direct control of the 4 147 the first case and the movement amplitude x in the Theory m Experiment second case. 146 In conclusion, we have developed a concrete method Hz] for tuning and suppressing nonlinear interactions in the k y [ 145 caseofasolid-stateringlaser,byvibratingthegaincrys- nc tal along the cavity axis. Our theoretical model shows e u a very good agreement with the experiment. The obser- q at fre 144 vloawtisonthoefdriorteacttiomneasesnusrienmgeinnttohfethsoelisdtr-setnagttehroinfgnolansleinreaalr- e B 143 Ideal frequency interactions, leading to the experimental demonstration response (no coupling) of their fine tuning and even suppression. Furthermore, following the previous work of [14], we have underlined 142 the analogy between our system and other ring-shaped 0,0 0,2 0,4 0,6 0,8 macroscopicquantumconfigurationswhere nonlinearin- Movement amplitude [µm] teractions could be tuned, for example a Bose-Einstein condensedgaswith magnetically-inducedFeshbachreso- nance. This illustrates the richness of such devices, both Figure3: Beatfrequencyasafunctionoftheamplitudeofthe crystal movement for θ˙ = 200 deg/s. The theoretical values from applicative and fundamental perspectives. (crosses)comefromthenumericalintegrationofequation(9) The authors thank M. Defour, M. Mary, E. Bon- with thesame parameters as Fig. 2. neaudetandThalesAerospaceDivisionforconstantsup- port. Theyarealsogratefulto A.Aspect andA. Mignot forfruitful discussions,andto F.Grabischforhis contri- strength of nonlinear interactions in the solid-state ring bution to the numerical simulations. laser. In particular, for some special amplitudes of the crystal movement, the influence of mode coupling van- ishes, resulting in a beat frequency equal to the ideal Sagnac value. This study suggests the use of a higher vibration fre- ∗ Electronic address: [email protected] quency ofthe crystal,inorderto increasethe value ofθ˙1 [1] F. Aronowitz in Laser Applications (Academic, New as much as possible. 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