Mismatch and synchronization: Influence of asymmetries in ... - CSIC

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K. HICKE et al. PHYSICAL REVIEW E 83, 056211 (2011)for τ ≈ 0 (almost equal delay times) and τ τ (when onedelay is very short), the variance of the intensity decreasesdramatically. Also we notice a significant drop in varianceat around τ = 333. This delay mismatch corresponds to arational ratio of r τ = 2/1 of the delay times. At these values ofthe delay mismatch, the dynamics settles on a stable periodicorbit.The narrowest distances between maxima and minimaindicate a stable single mode emission. For these cases,the autocorrelation and cross correlation show large peaks,however, this does not correspond to chaos synchronization.The stabilization of the dynamics for several delay mismatchesis illustrated in Fig. 10, which shows the laserdynamics projected onto the (ω,n) phase space. Close tothe rational ratio r τ = 2/1 (τ ≈ 333 in this case), we seeoscillations of the dynamics around a single mode [Figs. 10(e)and 10(g)]. The values for the delay mismatch correspondto those of the local peaks in the autocorrelation function inFig. 11(b). At the exact rational ratio τ = 333, however,no such stabilization is observed [Fig. 10(f)]. For small delaymismatches [Figs. 10(b) and 10(c)], the dynamics exhibitsrobust and fast stabilization to a single mode. This stabilizingproperty for a slightly misaligned semitransparent mirror isrobust against the variation of several parameters, such asabsolute delay time, coupling strength, and noise magnitude.Simulations over broad ranges of these respective parametersresulted in stabilization of the dynamics. With a strong enoughnonlinear gain saturation, the dynamics and mode spectrumcan be adjusted in such a way that high-level synchronizationand the stabilization of the dynamics described above occur.(For more details on the effect of gain saturation, see the nextsection).To explain the stabilization at small delay mismatches and atdelay mismatches close to simple delay ratios r τ , we speculateas follows. Consider a laser subject to a delayed feedback fromtwo feedback loops with delays τ + τ and τ − τ. Let usassume that at τ = 0, the laser operates in a chaotic regime,where the chaos is induced by the feedback. Although thelaser output is chaotic, there is, on average, a short-term phasecorrelation of the electric field, i.e., the peak at the origin ofthe autocorrelation function of the field is surrounded by localminima and maxima. The first local minimum and maximumis usually very pronounced because of a characteristic smalloscillation period present in the chaotic signal. If we nowchange τ such that it corresponds to the first minimum, then,on average, the arriving feedback signals will destructivelyinterfere, thus diminishing the amplitude and the fluctuationsof the feedback signal, which may lead to more regularbehavior of the laser. The above usually also holds for thechoice of a small rational ratio of the delay times, i.e., τ ∈ Q.τOf course, shifting τ will result in a different autocorrelationfunction, in particular in the case of successful stabilization.However, it can still be assumed that this kind of destructiveinterference mechanism can, in a self-consistent way, result instabilization of the dynamics.V. NONLINEAR GAIN SATURATIONFor our analysis, we have used a Lang-Kobayashi-typemodel with a nonlinear gain function G(E j ,n). A nonlinearFIG. 13. (Color online) Effect of the nonlinear gain saturationon the alignment and transverse stability of the external cavitymodes in the (ω,n S ) phase space; p = 1.0. The larger the pumpcurrent value is, the more pronounced is the effect. Red (black)circles: transversely unstable antimodes. Green (light gray) circles:transversely stable modes. The dynamics is shown in blue (dark gray).The other parameters are τ = 1000, τ = 0, α = 4.0, T = 200,K = 0.1, β = 10 −5 . Inset: bifurcation diagram with varying μ. Uppergreen points: maxima of intensities; lower red points: minima ofintensities.gain, which saturates for high intensities, is often used toaccount for nonlinear deviations of the characteristics of theoptical power versus injection current far above the threshold.The nonlinear gain saturation is a phenomenologicallyintroduced term that is motivated by nonlinear effects in thesemiconductor gain medium, like spectral hole burning andcarrier heating. A linear gain theory cannot account for thosephenomena.With increasing pump current, the nonlinear gain saturationbecomes more relevant and has an increasing effect on thedynamics of the lasers, e.g., on the precise position in phasespace [see Eq. (19)] and the transverse stability of the externalcavity modes.When operating in the LFF regime, the effect of nonlineargain saturation can often be neglected. In the coherencecollapse (CC) regime, however, the high power means that gainsaturation plays an increasing role. This effect is illustrated inFig. 13. Increasing the nonlinear gain saturation μ leads toa smaller maximum field intensity and a decreased varianceof the intensity and carrier density. The inset in Fig. 13shows this effect via a distribution of intensity extrema forvarying μ.In the coupled laser system, the number of transverselystable modes increases with increasing μ and the modes areshifted so that critical events with antimodes and desynchronizationare less likely—the synchronized state is more stable.However, the dynamics is also less complex.With a model that omits the nonlinear gain (μ = 0), allmodes in the vicinity of the minimum gain mode (aroundω = 0) are transversely unstable above a critical pumpcurrent. Without gain saturation, the lasers experience many056211-8
MISMATCH AND SYNCHRONIZATION: INFLUENCE OF ... PHYSICAL REVIEW E 83, 056211 (2011)FIG. 14. (Color online) Cross-correlation of the field intensitiesvs K and L with and without delay mismatch. (a) Symmetric delays:τ = 0, zero-lag synchronization; (b) with delay mismatch τ =20, cross-correlation at t = τ. The nonlinear gain saturation isignored for these simulations. Parameters: μ = 0, p = 1.0, α = 4.0,T = 200, τ = 1000, and noise β = 10 −5 .desynchronization events or do not synchronize at all. TheL − K plots in Fig. 14 show therefore significantly smaller areasof high synchronization for μ = 0 than the correspondingplots for μ = 0.26 (Figs. 2 and 9).VI. CONCLUSIONSWe have numerically and analytically investigated thedynamical and synchronization properties of a system oftwo delay-coupled semiconductor lasers in different couplingschemes. Using a model for a setup of two lasers coupledvia a semitransparent mirror, we have studied the influenceof a mismatch between the transmission and reflection ofthe mirror, i.e., a mismatch between coupling and feedbackstrength, as well as a delay mismatch corresponding to amisalignment of the mirror from the middle.We showed that a coupling mismatch deteriorates thestability of the synchronized solution, but does not change thesynchronized dynamics. In both the LFF regime and the CCregime, a larger mismatch results in longer bubbling events,since less modes are transversely stable. However, we stillobserve high-level synchronization if the mismatch |L − K|remains small. The cross-correlation (at zero lag) of the twolasers does not depend on the sign of the mismatch betweencoupling and feedback L − K, but only on the absolute value.The inclusion of a saturable nonlinear gain leads not only toa reduction of complexity in the time series but also to a broaderdomain of high synchronization quality, since it increases thetransverse stability of the ECMs.Our analysis has shown that the synchronization propertiesof a configuration with a semitransparent mirror are the sameas those of a drive-response configuration, at least on thelevel of the ECMs. An open-loop configuration synchronizesbest; the transverse stability of the modes is the same asin a configuration without coupling mismatch. Addingself-feedback to the receiver has the same effect on stabilityas introducing a coupling mismatch in the relay setup.If there is a delay mismatch, we found that the laserscan synchronize with a nonzero lag. The time traces thenexhibit a relative time shift proportional to the misalignmentof the mirror. The stability of the time-shifted identicalsynchronization of the lasers is not much affected by the valueof the delay mismatch. The dynamics, however, undergoesdrastic changes for varying τ, as the alignment of theexternal cavity modes is altered compared to symmetricdelays.Especially for very small mismatch, for rational ratiosof the respective delay times, and if the second delay isvery short compared to the first one, the chaotic dynamicscan be suppressed and stabilized toward either single modeoutput or periodic behavior. These qualitative features do notdepend on the delay time, but merely on the delay mismatchand relative ratios of delay times. They were found robust forbroad ranges of the coupling strengths (i.e., the transmissionand reflection of the relay mirror) and even for high noiselevels. We conjecture that they are caused by destructiveinterference.We also found that a delay mismatch makes the systemmuch more sensitive to a coupling mismatch. The correlationbetween the two laser outputs decreases much more rapidlywith increasing coupling mismatch than for the case ofsymmetric delays.Our results are of high practical value for future(experimental) investigations considering delay-coupledlasers. The ability to target specific stability regions viamismatch adjustments can, for example, be helpful for chaoscommunication systems or random number generation. In addition,due to the broad interest in synchronization phenomenathroughout the scientific community, our results concern notonly the immediate applications to chaotic communication,but may also be relevant to scientists in related fields suchas computational neuroscience, engineering, and biology.The presented analytical results can be transferred to otherdelay-coupled systems with corresponding network topologyand therefore could find applications, e.g., in electro-optic orneuronal systems.ACKNOWLEDGMENTSO.D. acknowledges the Research Foundation Flanders(FWO-Vlaanderen) for support. This research was partiallysupported by MICINN (Spain) under project DeCoDicA(Project No. TEC2009-14101), by DFG (Germany) in theframework of SFB 910, by the Interuniversity AttractionPoles program of the Belgian Science Policy Office, underGrant No. IAP VI-10 (photonics@be), and by the projectPHOCUS. The project PHOCUS, in turn, acknowledges thefinancial support of the Future and Emerging Technologies(FET) program within the Seventh Framework Programmefor Research of the European Commission, under FET-OpenGrant No. 240763.APPENDIX : STABILITY CALCULATION FOR CAVITYMODESAn external cavity mode (ECM) of the Lang-Kobayashiequations is a steady-state solution of the form E j = Ae iωtand n j = n. To calculate its (transverse) stability, we make thetransformation E j → Ee −iωt such that the ECM becomes asteady state, and split the electric field into a real and imaginarypart E = A = x + iy. With this, we calculate the JacobianDf (S) with trajectory S corresponding to the transformedECM solution. We then determine the eigenvalues of the056211-9
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