Asymptotic Approach To The Analysis Of Mode-hopping In Semiconductor .
PHYSICAL REVIEW A 80, 013823 共2009兲Asymptotic approach to the analysis of mode-hopping in semiconductor ring lasersS. Beri,1,2 L. Gelens,1 G. Van der Sande,1 and J. Danckaert1,21Department of Applied Physics and Photonics, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium2Department of Physics, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium共Received 22 May 2009; published 23 July 2009兲We investigate the directional mode-hopping in semiconductor ring lasers in the limit of small noise intensity. We show how only long time scales are involved in the hopping process. In such limit, the initialstochastic rate equations can be reduced to an auxiliary Hamiltonian system, and the dependence of thehopping on the laser parameters can be investigated. The predictions made from the reduced model agree wellwith the results obtained by simulating the full rate-equation system.DOI: 10.1103/PhysRevA.80.013823PACS number共s兲: 42.55.Px, 42.60.MiI. INTRODUCTIONSemiconductor ring lasers 共SRLs兲 are currently the subject of many experimental and theoretical investigations,ranging from fundamental studies of its nonlinear dynamicalbehavior to multiple practical applications 关1–9兴. In particular, the bistable character of SRLs makes them attractive forthe purpose of encoding information in the direction of emission 关1,3,6兴.Recently, highly performing all-optical flip-flops based ontwo coupled microrings have been fabricated 关7兴. The information is stored in the direction of emission and a counterpropagating signal could be used to switch the operation 关7兴.On the other hand, a device based on a single ring wouldbenefit of a smaller footprint and could take advantage of therecently demonstrated switching schemes based on one-sideinjection 关8兴. However, single-ring devices present a reducedstability when compared to coupled rings designs. In particular, undesired dynamical regimes such as alternate oscillations 关3兴 have been reported, as well as noise-induced modehopping 关9兴. The presence of noise-induced hopping isespecially detrimental for applications such as the realizationof optical memories.In particular, the results of a full bifurcation analysis performed in Ref. 关10兴 suggest that the shape of the manifoldsseparating the basins of attraction of the two counterpropagating modes is controlled by the phase of the backscatteringparameters. However, an investigation of the stability versusstochastic fluctuation has not been addressed yet. Especially,the important role of the linear backscatter phase between thetwo counterpropagating modes remains an open question. Anasymptotic approach to the activation problem consists in theuse of an auxiliary Hamiltonian system and in the solution ofan associate boundary value problem 关11兴. However, thelarge dimensionality of the system makes this analysis extremely intricate in the case of SRLs. On the other hand, theuse of a phenomenological model such as a double-well potential requires the fitting of a large number of parametersand loses a direct connection with the physically meaningfulparameters of the system. A similar problem has been investigated in solid-state ring lasers 关12兴 to reveal stochastic resonance, but it relied on a phenomenological double-well potential which kept hidden the real parameters of the system.In this paper we propose an asymptotic approach to theproblem of stochastic mode-hopping by focusing on the limit1050-2947/2009/80共1兲/013823共7兲of vanishing noise intensity. Two independent limits are considered: first the full rate-equation set are asymptotically reduced to a two-dimensional reduced model as described in关13,14兴; second, the asymptotic limit of vanishing noise intensity is considered 关15–17兴. As a result, we obtain an auxiliary Hamiltonian system which describes the modehopping process and contains the dependence on the originalparameters of the system without any fitting or any furtherphenomenology. The role of the bias current and the important backscattering phase are investigated. The limit of smallbackscattering amplitude is addressed numerically and analytically.The paper is structured as follows. In Sec. II we introducethe rate-equation model to describe the operation of a SRL关3兴, and we perform the two asymptotic limits of slowdynamics and vanishing-noise intensities. In Sec. III we discuss the optimal transition paths that are the results of theHamiltonian approach. Changes in the qualitative features ofthe optimal paths are discussed when the model parametersare changed.II. THEORYA semiconductor ring laser is a semiconductor laserwhose active cavity have a circular geometry 关1,3,6兴 as schematically depicted in Fig. 1. Two directional mode are supported by the cavity geometry: one which propagates clockwise 共CW兲 and a counterclockwise 共CCW兲 one. A buswaveguide is integrated on the same chip in order to coupleout power from the ring cavity.We consider a SRL operating in a single-longitudinalsingle-transverse mode. The rate-equation model is formuWGOUT CCWOUT CWCWCCWSRLFIG. 1. A schematic view of the semiconductor ring laser. SRL:semiconductor ring laser cavity; WG: bus waveguide; CW andCCW: clockwise and counterclockwise modes propagating in thecavity; OUT CW and CCW: output beams.013823-1 2009 The American Physical Society
PHYSICAL REVIEW A 80, 013823 共2009兲BERI et al.Ė1,2 共1 i 兲关N共1 s兩E1,2兩2 c兩E2,1兩2兲 1兴E1,2 kei kE2,1, 1,2共t兲,1.40.81.20.60.410.2Powerlated mathematically in terms of two complex equations forthe slowly varying amplitudes E1,2 and one real equation forthe carrier number N. In what follows, in order to fix thenotation we will assume that mode 1 is rotating clockwise,whereas mode 2 is rotating counterclockwise. The equationsfor E1,2 and N read 关1,3,13兴 as0.800.20.25time [ns]0.60.4共1兲0.15τFPT0.2Ṅ 关 N N共1 s兩E1兩2 c兩E2兩2兲兩E1兩2 N共1 s兩E2兩2 c兩E1兩2兲兩E2兩2兴,0共2兲where the dot represents differentiation with respect to t, isthe field decay rate, is the carrier decay rate, is thelinewidth enhancement factor, and is the renormalized injection current with 0 at transparency and 1 at lasingthreshold. The two counterpropagating modes are consideredto saturate both their own and each other gain due to, e.g.,spectral hole burning effects. Self- and cross-saturation effects are added phenomenologically and are modeled by sand c. For a realistic device the cross saturation term is largerthat the self-saturation.In Eq. 共1兲, the spontaneous emission noise has been introduced phenomenologically as complex uncorrelated zeromean stochastic terms described by the correlation terms:具 i共t 兲 ⴱj 共t兲典 2D ij 共 兲, where i , j 1 , 2 and D is thenoise intensity. Carrier noise has been disregarded as its relevance in directional mode-hopping was proved negligible关18兴.For a small size ring laser, reflection of the counterpropagating modes occurs where the ring cavity and couplingwaveguide meet and can also occur at the end facets of thecoupling waveguide. These localized reflections result in alinear coupling between the two fields characterized by anamplitude k and a phase shift k 关19兴.The set of stochastic rate Eqs. 共1兲 and 共2兲 can be solvednumerically using a predictor-corrector scheme 关31兴 for different values of the model parameters and the noise intensityD . Examples of stochastic time series of the modal intensities 兩E1兩2 and 兩E2兩2, as well as the total emitted intensity兩E1兩2 兩E2兩2 are shown in Fig. 2It is clear from Fig. 2 that the SRL emits almost steadilyin one of the two modes until an abrupt transition leads tolasing in the counterpropagating mode. However, duringsuch mode-hoppings, the total intensity 兩E1兩2 兩E2兩2 is conserved. Such conservation of power has also been observedexperimentally 关1兴. A further analysis of the time seriesshown in Fig. 2 reveals that 共for the parameters chosen兲 twoqualitatively different transitions are possible. The first kindof transition is characterized by a long permanence in one ofthe states until the fluctuations induce a hopping to the opposite mode. The same time series reveal short excursions tothe opposite mode; however, as shown in the inset of Fig. 2the system does not settle in the opposite mode, and it always returns to lase in the initial state.012time [µs]34FIG. 2. Example of a stochastic trajectory solution of Eqs. 共1兲and 共2兲 for the following parameter’s choice: K 0.44 ns 1, k 1.5, 1.72, 3.5. and D 7.7 10 4 ns 1. The intensities ofthe CW mode 共black兲 and the CCW mode 共gray兲 are shown togetherwith the total intensity P1 P2 共light gray兲. The total power isshifted upward for clarity. The arrow indicates an example of firstpassage time. A short excursion to the opposite mode is shown inthe inset.The different features of these two transitions have beeninvestigated in a previous publication 关9兴 using topologicalarguments. In particular, it has been shown that the transitions of the first kind are the result of a noise-induced activation process, whereas the short excursions are a consequence of the finiteness of the noise.In this paper, we aim to address the first kind of modehopping by formulating it in the form of an activation in anonequilibrium bistable system. It is known from the theoryof stochastic systems 关16兴 that activation problems can becharacterized by the mean first passage time 具 FPT典 acrossthe boundary of the basin of attraction of the initial state. Anexample of first passage time is given in Fig. 2. The meanfirst passage time can be expressed to logarithmic accuracyby a generalized nonequilibrium potential 关17,20–23兴具 FPT典 eS/D ,共3兲where S is the nonequilibrium potential and D is the intensityof the stochastic force that drives the fluctuations. It is important to understand the dependence of the quasipotential Son the parameters of systems 共1兲 and 共2兲. However, Eqs. 共1兲and 共2兲 are too complicated for an analytical or quasianalytical approach. Nevertheless, one cannot introduce any ad hocmodel such as a double-well potential as this would introduce phenomenological model parameters which are not related to the physically relevant quantities such as k, k, or .However, from the observation that the escape events arerare when compared to the other time scales of the systemand the quasiconservation of the total power during the hopping, we propose that the hopping problem can be successfully described in the framework of an asymptotically reduced model 关13兴.013823-2
PHYSICAL REVIEW A 80, 013823 共2009兲ASYMPTOTIC APPROACH TO THE ANALYSIS OF MODE- which corresponds to unstable lasing in the out-of-phase bidirectional mode.The basins of attraction of the CW and CCW modes areseparated by the stable manifold of SP. It is clear from Fig. 3that the shape of the basin boundary can be complicated, andits topological features have been used to explain unexpectedbehaviors of SRL 关8,9兴. The unstable manifold of the saddle共not shown in Fig. 3兲 is made up of two branches, each ofthem connecting the saddle with one of the stable states.4.54ψ3.5SpCCW3CW2.521.5-1.5Asymptotic analysis of the Fokker-Planck equation-1-0.50θ0.511.5FIG. 3. Phase space portrait of Eqs. 共5兲 and 共6兲. The two stationary points are marked by circles and correspond to CW andCCW unidirectional operations. The SP is marked by a cross andthe stable manifold of SP is marked as a solid line. The unstablemanifold of SP is not shown. The parameters are as follow k 0.44, k 1.5, 1.65, and 3.5.On time scales slower than the relaxation oscillations, ithas been shown 关13兴 that the total intensity are indeed asymptotically conserved:兩E1兩2 兩E2兩2 1 0.共4兲The long time scale dynamics is then described by thetime evolution of two auxiliary angular variables: K 共 , 兲 2 sin k sin 2 cos k cos sin J sin cos ,共5兲In order to investigate the stochastic properties of Eqs. 共5兲and 共6兲, we study the Fokker-Planck equation 共FPE兲 for theprobability density 共 , 兲. The FPE is written as follows: 2 2 K K D 2 D 2 , t where D is the noise intensity for the reduced system whichis the small parameter in our theory. The value of D is theonly quantity that require a fitting in our analysis.In the limit of vanishing noise intensity D 0, anasymptotic WKB expansion can be performed for the probability density 关24兴. We stress here that we consider twoseparate asymptotic limits in our approach. The first limit共discussed above兲 is related to time scale separation and depends on the laser parameters. Such first asymptotic does notaffect the noise properties of the system. In what follows, weintroduce a second asymptotic limit that depends on thenoise intensity and not on the laser parameters.We consider the following ansatz for :cos cos K 共 , 兲 J sin cos 2 cos k sin 2 sin k cos sin , Ze S/DD 0,共6兲where 2 arctan冑兩E2兩2 / 兩E1兩2 / 2 苸 关 / 2 , / 2兴 represents the relative modal intensity and 苸 关0 , 2 兴 is the phasedifference between the counterpropagating modes. Primenow denotes derivation to the slow time scale kt. Finally,in this reduced model the pump current has been rescaled asJ 共c s兲共 1兲 / k. As the phase space of Eqs. 共5兲 and 共6兲 isrestricted to two dimensions, it allows for a clear physicalpicture of the influence of all parameters on the dynamicalevolution of the variables in a plane. For a comprehensiveanalysis of Eqs. 共5兲 and 共6兲, including a complete bifurcationanalysis, we refer to 关10兴.We make here the assumption of uncorrelated whiteGaussian noise terms , for the angular variables too. Thisassumption is indeed unphysical, as multiplicative terms andcross correlations are expected to appear in the noise. However, the agreement with the results of the full model 共seebelow兲 and the benefits in simplicity will justify this assumption.The phase space portrait of Eqs. 共5兲 and 共6兲 when the laseroperates in unidirectional regime is characterized by four stationary solutions as shown in Fig. 3 关8,9兴: an unstable inphase bidirectional state in 共0,0兲 共not shown兲; two symmetricstable states CW and CCW at , both corresponding tounidirectional operation; and a saddle point 共SP兲 in 共0 , 兲共7兲共8兲where S is an auxiliary function. S is the analogous of apotential in a gradient system and it is referred to as quasipotential in nonequilibrium systems such as Eqs. 共5兲 and 共6兲.It is known from the theory of stochastic systems that, at theleading order D1 , the quasipotential S satisfies a HamiltonJacobi equation for a classical action:冉冊 S S S H , , ,, t 共9兲with the Hamiltonian H共 , , S , S 兲 defined asH 冉 冊 冉 冊1 S2 2 1 S2 2 S SK K . 共10兲In what follows, we will refer to the function S indifferentlyas action or nonequilibrium potential. The analogy with thecase of a classical Hamiltonian problem is completed by defining the auxiliary momenta:p S, leading to the Hamiltonian013823-3p S, 共11兲
PHYSICAL REVIEW A 80, 013823 共2009兲BERI et al.p 2 p 2 K p K p .22共12兲With this approach, initial stochastic systems 共5兲 and 共6兲 canbe mapped into a deterministic Hamiltonian system H K p , p 共13兲 H K p , p 共14兲ṗ H K K p p , 共15兲ṗ H K K p p . 共16兲The action S evolves along the solutions of Eqs. 共13兲–共16兲following the equationdS 1 2 1 2 p p .dt 2 2 共17兲The action S calculated along a certain trajectory can beinterpreted as a “cost” for such transition to take place andultimately its probability. Consider a transition between aninitial point xi and a final point x f . A large action indicatestrajectory with low probability whereas a smaller action indicates a more likely trajectory. In the limit of vanishingnoise intensity D 0, the transitions corresponding to theminimum actions become exponentially more likely than anyother transition path, and only trajectories with minimumactions become relevant in the calculation of the probabilitydistribution. In other words, only those trajectories corresponding to a global minimum can be observed in a physicalexperiment in the zero noise-intensity limit 关11,25,26兴.III. OPTIMAL ESCAPE PATHS IN SEMICONDUCTORRING LASERSIn the previous sections, we reviewed the general theoryfor stochastic fluctuation in the limit of vanishing noise intensity. In this section we formulate the problem for themode-hopping in SRLs, and we discuss the general topological features of the escape trajectories that realize the hopping.During the regular operation of SRL, the system spendsthe majority of its time in the close vicinity of one of thestationary states. In order for a mode-hop to take place, thespontaneous emission noise must drag the system outside thebasin of attraction of the original state 共for instance CW兲 tothe basin of attraction of the counterpropagating mode 共forinstance CCW兲. As the basins of attraction of the CW andCCW modes are separated by the stable manifold of thesaddle SP 共see Fig. 3兲, the mode-hop is realized by a trajectory solution of Eqs. 共13兲–共16兲 that connects the stationarystate CW with the stable manifold of the saddle.In general there are infinitely many solutions of Eqs.共13兲–共16兲 that emanates from the initial state and reach theψH共p ,p , , 兲 4.5 (a)43.53CCW2.521.5-1.5 -1 -0.5(b)Sp0θCW0.5 1CCW1.5-1 -0.5Sp0θCW0.5 11.5FIG. 4. MPEP paths for different values of the backscatteringphase: 共a兲 k 1.5; 共b兲 k 1.08. The other parameters are as inFig. 3.basin boundary. According to the discussion in Sec. IIA, inthe limit of small noise intensity D 0, the escape takesplace with overwhelming probability along the trajectory thatminimizes the action S. Such path is know in literature as themost probable escape path 共MPEP兲 关11,15,27–30兴.As the motion along the stable manifold of SP is deterministic and deterministic motion does not increase the action, the minimal action along the stable manifold of thesaddle coincide with the saddle itself 关11,15,27–29兴. Therefore, the MPEP satisfies the following boundary conditions:it emanates from the initial state CW 共CCW兲 at time t and converges to the saddle SP at time t . The transition to the opposite mode is completed deterministically byfollowing the unstable manifold of SP.We calculated the MPEP for a SRL by minimization ofaction functional 共17兲 along the solutions of Eqs. 共13兲–共16兲for different values of the system parameters.Some examples of MPEPs are shown in Fig. 4 for transitions from the CW state to the CCW state. As expected according to the previous discussion, the MPEP connects theCW state to the saddle SP and a deterministic relaxationcompletes the mode-hop to the CCW mode.It is clear from Fig. 4, that the backscattering phase kplays a major role in determining the actual shape of theMPEP. In the next sections, we will discuss the role of kand the other parameters of the system in the mode-hopping.IV. DEPENDENCE OF THE ACTIVATION ENERGYON THE PARAMETERS OF THE SYSTEMIn order to validate our analysis and to provide insight inthe dependence of the mode-hopping versus the parametersof the system, we calculate the activation energy for differentvalues of the current and the backscattering phase using bothnumerical simulation and the Hamiltonian theory.The full set of rate Eqs. 共1兲 and 共2兲 is solved using aMonte Carlo method 关31兴 for different values of the noiseintensity D . The activation energy is calculated by fitting themean-first-passage-time versus 1 / D . The dependence of theactivation energy versus current and phase k is shown inFig. 5 by markers.The theoretical activation energy is calculated by solutionof the boundary value problem associated with Eqs.共13兲–共16兲 and by minimization of the function S 关11兴. Theparameters are the same as in Eqs. 共1兲 and 共2兲, excluding thenoise intensity D which is fitted. The results of this calcula-013823-4
PHYSICAL REVIEW A 80, 013823 共2009兲ASYMPTOTIC APPROACH TO THE ANALYSIS OF MODE- 0.7800Φ 1.08Φ .651.71.75µ1.81.851.9FIG. 5. Activation energy as a function of the current fordifferent values of the backscattering phase. Markers are used toindicate the results of the integration of the full rate-equation model关Eqs. 共1兲 and 共2兲兴, whereas the solid lines represent the solution ofauxiliary Hamiltonian systems 共13兲–共16兲.tion are shown in Fig. 5 with lines. The agreement betweentheoretical and numerical values of the activation energy isgood.In Fig. 6, we show the dependence of the activation energy calculated by solution of the boundary value problem asa function of the backscattering phase k for different valuesof the bias current . We observe that the activation energyincreases with in accordance with the results of Fig. 5 andthe intuitive argument that increasing the bias current stabilizes the laser operation. On the other hand, the dependenceof S on the backscattering phase is nonmonotonous, indicating that the stability of the directional modes versus randomfluctuations decreases exponentially from a maximum at maxk 0.85. The loss of stability of the directional modescan be understood by considering the bifurcation scenario ofa SRL 关10兴. According to 关10兴, for the physically relevantparameters considered in Fig. 6, stable unidirectional lasingis possible between a pitchfork bifurcation taking place at k 0.2 and a subcritical Hopf bifurcation taking place at k 1.6. Therefore, the decrease in S for k maxcorreksponds to a loss of stability of the system when approachingthe pitchfork bifurcation line, whereas the decrease in S for k maxis consistent with a loss of stability of the direck0.6µ 1.70µ 1.80µ 1.900.5S0.40.30.20.100.40.60.81ΦK1.21.41.6FIG. 6. Activation energy as a function of the phase of thebackscattering K for different values of the bias current .0200 400 600 800 1000-1 1200 1400 1600 1800 2000KFIG. 7. Activation energy S as a function of the inverse backscattering 1 / 兩K兩 obtained as a solution of Eqs. 共13兲–共16兲. The current is set at 1.8 and the phase at k 1.3.tional modes when approaching the subcritical Hopf bifurcation.The asymptotic limit J can be treated analytically,which corresponds to either or K 0. In this limit, wecan neglect the terms depending on the backscattering phase K in Eqs. 共5兲 and 共6兲 and obtain J sin cos ,共18兲cos J sin cos .共19兲Therefore, the dynamics of the variable is decoupled by thevariable which becomes a follower of . The activationenergy S can be calculated by noticing that ⵜU, withU 21 J cos共2 兲. Therefore S J / 2. In this limit, the action islinearly dependent on J and inversely proportional to 兩K兩. InFig. 7 the dependence of the activation energy versus theinverse of the backscattering magnitude for 1.8 and k 1.3 is shown as the solution of Eqs. 共13兲–共16兲. The linearityis evident.Finally, we investigate the effect of a change in the linewidth enhancement factor in the laser medium. In Fig. 8共a兲,the activation energy is calculated numerically by MonteCarlo simulations of the full rate-equation model 关Eqs. 共1兲and 共2兲兴. A monotonic increase in the activation energy with is evident. An unexpected feature of the dependence of Sversus is the abrupt change in the gradient of S. Such achange in scaling can be interpreted by investigating the topology of the MPEP in the reduced model. A typical dependence of S on calculated using the Hamilton’s Eqs.共13兲–共16兲 is exemplified in Fig. 8共b兲. The same change inscaling as in Fig. 8共a兲 is observed. Moreover, this can beexplained by studying the different topology of the MPEP forparameters corresponding to the different branches of theS- curve.For values of lower than a critical value c, the MPEPconnects the initial stationary state with the saddle directly关Fig. 8共c兲兴, whereas, for c the MPEP surrounds the unstable in-phase bidirectional mode 关Fig. 8共d兲兴. Therefore, forlow values of a small excursion in the phase different isexpected, whereas, for values of larger than c an excursion in larger that 2 is expected. We conjecture the exis-013823-5
PHYSICAL REVIEW A 80, 013823 共2009兲BERI et )SpCWCCW-1.5ψψ0(b)SS FIG. 8. Activation energy S as a function of the linewidth enhancement factor when calculated 共a兲 by fitting the MFPT obtained from Monte Carlo simulations of Eqs. 共1兲 and 共2兲 and 共b兲from the Hamiltonian formalism. Both approaches reveal the“knee.” The current is set at 1.8, the backscattering amplitude tok 0.44 and the phase at k 1.2. 共Bottom兲 Qualitatively differentMPEP for 共c兲 5 and 共d兲 6 as calculated with Hamiltoniansystems 共13兲–共16兲.tence of other critical values of corresponding to excursions in 4 , 6 , . . . and so forth, leading tocorresponding changes in the scaling of the activation energy. At the critical value c, the MPEP is degenerate andtwo qualitatively different trajectories reach the saddle withthe same probability.We remark here that simpler double-well models cannotpredict such change in the gradient of the activation energy.dimensional rate-equation system, and we proved that themain features of the mode-hopping can be understood in theframework of a reduce two-dimensional model.Stochastic planar systems 共5兲 and 共6兲 was subsequentlymapped into a four-dimensional Hamiltonian system 关Eqs.共13兲–共16兲兴 and the optimal escape paths could be calculated.As expected from the general theory of escape in nonlinearsystems 关11,25,30兴, the most probable escape paths connectsthe initial stationary lasing operation with a saddle in thephase space. The results of the reduced model were confirmed by the numerical simulations of the full rate-equationmodel 关Eqs. 共1兲 and 共2兲兴.Opposite to the adoption of a phenomenological bistablesystem such as a double-well potential, the use of theasymptotic reduction Eqs. 共5兲 and 共6兲 does not require theintroduction of phenomenological parameters to be matchedwith Eqs. 共1兲 and 共2兲. In this way we investigated the dependence of the activation energy on the principal parameters asshown in Figs. 5–8. In particular, our asymptotic methodallowed us to understand the change in scaling of the dependence of S on which cannot be explained with other phenomenological models such as a double-well potential.Our approach can be straightforwardly extended to analyze the problem of fluctuations in other bistable optical systems 关22,32–35兴 or in the recently discovered multistableregimes of operation of SRLs 关36兴.ACKNOWLEDGMENTSIn conclusion, we have investigated the stochastic modehopping in semiconductor ring lasers. We started with a five-This work has been funded by the European Communityunder Project No. IST-2005-34743 共IOLOS兲 and the BelgianScience Policy Office under Grant No. IAP-VI10. G.V., S.B.,and L.G. are supported by the Research Foundation-Flanders共FWO兲.关1兴 M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P.J. R. Laybourn, IEEE J. Quantum Electron. 39, 1187 共2003兲.关2兴 V. R. Almeida and M. Lipson, Opt. Lett. 29, 2387 共2004兲.关3兴 M. Sorel, J. P. R. Laybourn, A. Sciré, S. Balle, G. Giuliani, R.Miglierina, and S. Donati, Opt. Lett. 27, 1992 共2002兲.关4兴 Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu,IEEE Photonics Technol. Lett. 20, 1228 共2008兲.关5兴 S. Furst and M. Sorel, IEEE Photonics Technol. Lett. 20, 366共2008兲.关6兴 J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, Appl.Phys. Lett. 70, 1192 共1997兲.关7兴 M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H.den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe,and M. K. Smit, Nature 共London兲 432, 206 共2004兲.关8兴 L. Gelens, S. Beri, G. Van der Sande, J. Danckaert, N. Calabretta, H. J. S. Dorren, R. Nötzel, E. A. J. M. Bente, and M. K.Smit, Opt. Express 16, 10968 共2008兲.关9兴 S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Ver-schaffelt, A. Scirè, G. Mezosi, M. Sorel, and J. Danckaert,Phys. Rev. Lett. 101, 093903 共2008兲.L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, Phys.Rev. E 79, 016213 共2009兲.S. Beri, R. Mannella, D. G. Luchinsky, A. N. Silchenko, and P.V. E. McClintock, Phys. Rev. E 72, 036131 共2005兲.B. McNamara, K. Wiesenfeld, and R. Roy, Phys. Rev. Lett.60, 2626 共1988兲.G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, J. Phys. B 41, 095402 共2008兲.T. Erneux, J. Danckaert, K. Panajotov, and I. Veretennicoff,Phys. Rev. A 59, 4660 共1999兲.D. Ludwig, SIAM Rev. 17, 605 共1975兲.A. D. Wentzel and M. I. Freidlin, Fluctuations in DynamicSystems: Effects of Small Random Perturbations 共Nauka, Moscow, 1979兲.P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62,251 共1990兲.V. 关15兴关16兴关17兴013823-6
ASYMPTOTIC APPROACH TO THE ANALYSIS OF MODE- PHYSICAL REVIEW A 80, 013823 共2009兲关18兴 P. Lett and L. Mandel, J. Opt. Soc. Am. B 2, 1615 共1985兲.关19兴 R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel,and J. P. Woerdman, Phys. Rev. A 42, 4315 共1990兲.关20兴 R. S. Maier and D. L. Stein, Phys. Rev. E 48, 931 共1993兲.关21兴 V. I. Mel’nikov, Phys. Rep. 209, 1 共1991兲.关22兴 B. Nagler, M. Peeters, J. Albert, G. Verschaffelt, K. Panajotov,H. Thienpont, I. Veretennicoff, J. Danckaert, S. Barbay, G.Giacomelli, and F. Marin, Phys. Rev. A 68, 013813 共2003兲.关23兴 B. Nagler, M. Peeters, I. Veretennicoff, and J. Danckaert, Phys.Rev. E 67, 056112 共2003兲.关24兴 M. I. Dykman, P. V. E. McClintock, V. N. Smelyanski, N. D.Stein, and N. G. Stocks, Phys. Rev. Lett. 68, 2718 共1992兲.关25兴 V. N. Smelyanskiy, M. I. Dykman, and R. S. Maier, Phys. Rev.E 55, 2369 共1997兲.关26兴 D. G. Luchinsky, P. V. E. McClintock, and M. I. Dykman, Rep.Prog. Phys. 61, 889 共1998兲.关27兴 A. N. Silchenko, S. Beri, D. G. Luchinsky, and P. V. E. McClintock, Phys. Rev. Lett. 91, 174104 共2003兲.关28兴 M. I. Dykman and M. A. Krivoglaz, Sov. Phys. JETP 50, 30共1979兲.关29兴 A. N. Silchenko, S. Beri, D. G. Luchinsky,
Asymptotic approach to the analysis of mode-hopping in semiconductor ring lasers S. Beri,1,2 L. Gelens,1 G. Van der Sande,1 and J. Danckaert1,2 1Department of Applied Physics and Photonics, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium 2Department of Physics, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium Received 22 May 2009; published 23 July 2009
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Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. 3 Crawford M., Marsh D. The driving force : food in human evolution and the future.
