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K. HICKE et al. PHYSICAL REVIEW E 83, 056211 (2011)For K = 0 or L = 0, the transversal and longitudinal(within the synchronization manifold) stability properties ofthe modes are equal, and since all modes, except for themaximum gain mode, are longitudinally unstable, zero-lagsynchronization is deterministically unstable too [22]. Thecorrelation is thus small (see Fig. 2).In the coherence collapse regime, which occurs for higherpump currents, the dynamics can be described as a chaoticitinerancy between modes and antimodes. Also in this case, thenumber of transversely stable modes decreases with increasingmismatch, leading to more desynchronization events. We alsonote that the number of stable modes decreases for increasingpump current if the gain saturation effect is not accountedfor, i.e., μ = 0. The transverse stability of the modes is higher,taking into account gain saturation, which explains the broaderarea of high synchronization levels in Fig. 2. For a moredetailed discussion of the nonlinear gain, see Sec. V.The correlation between the two lasers at zero lag dependson the difference of self-coupling and cross coupling, but itdoes not depend on the sign of the mismatch, i.e, whether K>L or L>K. This can again be understood by looking at theECMs’ transverse stability, which depends on the magnitudeof L − K, but not on the sign.When investigating the correlation at nonzero lags andthe stability of synchronization of the leader-laggard type,we get different results. Considering the peaks of the crosscorrelationfunction at shifts of multiples of the delay time,kτ, k ≠ 0 [Fig. 4(b)], we find that if L − K increases (positivemismatch), all considered non-zero-lag peaks decrease in thesame way. The corresponding peaks in the autocorrelation andthus the dynamics of the lasers do not change much with apositive mismatch [Fig. 4(a)].For a negative mismatch (K >L), the lasers evolvegradually from identical zero-lag synchronization to generalizedsynchronization of the leader-laggard type: the crosscorrelationpeak at one delay time and other odd numberedpeaks do not change much with the mismatch, but thecorrelation at zero lag and at even multiples of the delay vanish[Fig. 4(b)]. Also the autocorrelation is affected: the even peaksremain and the odd peaks vanish [Fig. 4(a)]. Note that thesum of the autocorrelation and cross-correlation function issymmetric for positive and negative mismatch (not shown).III. DRIVE-RESPONSE CONFIGURATION: OPEN ANDCLOSED LOOPA coupling configuration that is frequently used for chaoscommunication purposes is a drive-response configuration[10,14,47,48]. The drive laser, or transmitter, is subject toits own delayed feedback; the responding laser, or receiver,receives a chaotic input from the transmitter. In a so-calledclosed-loop configuration, which is depicted schematically inFig. 5, the receiver is also a chaotic element subject to its ownfeedback. In the open-loop configuration, the receiver has noself-feedback and therefore exhibits a stable continuous waveoutput when decoupled from the transmitter. Both cases canbe modeled byX˙1 = f (X 1 ) + KCX 1 (t − τ), (9)X˙2 = f (X 2 ) + (1 − ɛ)KCX 1 (t − τ c ) + ɛKCX 2 (t − τ). (10)FIG. 4. (Color online) Evolution of the peaks of the autocorrelationand cross-correlation function C(t), respectively, with themismatch L − K. Both correlations are calculated for the lasers’intensities: (a) autocorrelation and (b) cross-correlation. Black thickline: cross-correlation at zero lag C(0). Blue circles: peak at one delaytime C(τ). Red diamonds: C(2τ). Green squares: C(3τ). Magentatriangles: C(4τ). Parameters: τ = 1000, μ = 0, p = 1, T = 200,β = 10 −5 .If ɛ = 0, we have an open-loop receiver, and if ɛ>0, theconfiguration is a closed loop.The dynamics in the synchronization manifold is the dynamicsof the transmitter laser, S(t) = X 1 (t). Synchronizationwithout any time shift occurs only for τ c = τ. Ifτ>τ c ,the receiver laser’s dynamics anticipates the transmitter’sdynamics, and for τ
MISMATCH AND SYNCHRONIZATION: INFLUENCE OF ... PHYSICAL REVIEW E 83, 056211 (2011)of this system. Instead a time-shifted synchronized solutionX 1 (t) = X 2 (t − τ) is possible [31]. To analyze the stabilityof this time-shifted identically synchronized solution, wedefine the symmetric variable as S(t) = 1 2 [X 1(t + τ) +X 2 (t)] and the antisymmetric variable as A(t) = 1 2 [X 1(t +τ) − X 2 (t)]. The temporal evolution of these two variablesis then given byFIG. 6. Scheme of a bidirectional coupling setup with a delaymismatch of the self-feedback. The distances are given in time for aone-way trip. Both self-feedback strengths have the value L and thecoupling strengths both equal K.When linearizing around A(t) = 0, we find that the linearstability of the zero-lag synchronized solution in an openloopreceiver (ɛ = 0) is the same as the setup of two laserscoupled through a semitransparent mirror, without self- andcross-coupling mismatch (L = K). In contrast, in a closedloopreceiver, the stability of the synchronization manifoldis the same as in a mirror setup with a mismatch betweenself-coupling and cross-coupling.In chaotic communication, the open-loop configuration istraditionally preferred over the closed-loop scheme becauseit is more robust against parameter mismatches and mucheasier to implement. In addition, the resynchronization timein the case of a sudden interruption of the connection ismuch shorter (see Ref. [13] and references therein). It wasshown [37], however, that if the feedback delays are identical,the synchronization quality for the closed-loop rather thanfor the open-loop scheme is much higher. Also Ref. [49]showed that the performance of closed-loop receivers are lesssensible to detuning between the emitter and receiver. Furthermore,several methods considering chaotic communicationshave been proposed that take advantage of specific propertiesof the closed-loop configuration [13,50].Ṡ = 1 2[f (S + A) + f (S − A)]+ 1 2 (L + K)C[S(t − τ 1) + S(t − τ 2 )]+ 1 2 (L + K)C[A(t − τ 1) − A(t − τ 2 )], (15)Ȧ = 1 [f (S + A) − f (S − A)]2+ 1 2 (L − K)C[S(t − τ 1) − S(t − τ 2 )]+ 1 2 (L − K)C[A(t − τ 1) + A(t − τ 2 )]. (16)It is clear that a (time-shifted) synchronized solution [A(t) =0] only exists if the self-coupling and cross-coupling are equal(L = K). This is in contrast to the case without delay mismatch(see Sec. II), where for any values of K and L, a synchronizedsolution exists, although it may be unstable. Thus for the casewith delay mismatch, we expect a strong dependence of thesynchronization quality on the coupling mismatch L − K.For K = L, the synchronized solution corresponds to onelaser subject to two different feedbacks with the same strength12 (K + L) = K and respective delays τ 1 and τ 2 . A linearstability analysis of Eq. (16), orthogonal to the synchronizationmanifold around A(t) = 0, then leads toδA(t) ˙ = Df (S)δA. (17)IV. RELAY CONFIGURATION WITH DELAY MISMATCHA. Synchronization propertiesWe come back to the case of bidirectionally coupled laserswith self-feedback. As discussed above, this coupling schemecan be realized experimentally with a semitransparent mirrorbetween the lasers. We now assume that the mirror is no longerpositioned in the perfect middle between the two lasers. Thelasers are hence subject to a different feedback delay τ 1,2 =τ ± τ, but they still experience the same coupling delayτ = 1 2 (τ 1 + τ 2 ), since the distance between the lasers does notchange. A schematic representation of the setup is shown inFig. 6.We can then model this system via the equationsX˙1 = f (X 1 ) + 1 2 LCX 1[t − (τ + τ)]+ 1 2 KCX 2(t − τ), (13)X˙2 = f (X 2 ) + 1 2 LCX 2[t − (τ − τ)]+ 1 2 KCX 1(t − τ) , (14)where we consider, without loss of generality, τ 0.The first laser then has a larger self-feedback delay(τ 1 = τ + τ) than the second laser (τ 2 = τ − τ). Zerolagsynchronization X 1 (t) = X 2 (t) is no longer a solutionFIG. 7. (Color online) Dynamics and correlation of two delaycoupledlasers with a delay mismatch of τ = 100 in the coherencecollapse regime. Upper panel: time series of laser output intensitiesI 1 (t) [green (dark gray) line] and I 2 (t) (black line). The time tracesexhibit a time lag of τ. For demonstration purposes, the noisewas disregarded in this simulation. Lower panel: cross-correlationfunction of laser intensities. The time shift of the maximumcorrelation peak equals the delay mismatch: t =−τ. Simulatedwith noise β = 10 −5 . Parameters for both simulations are τ = 2000,α = 4.0, p = 1.0, μ = 0.26, T = 200; all coupling strengths wereidentical L = K = 0.1.056211-5
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