Abstract

The emission and synchronization of mutually-coupled semiconductor lasers with short cavities has been already recorded, with transversely unstable solutions existing within the chaotic synchronization manifold. Noise and laser-mismatch induced instabilities cause short de-synchronization events within the overall generalized synchronization, influencing the pragmatism of using these signals in secure data exchange applications. However, such operation can be functional for user authentication and sensing applications by assessing a time-averaged performance of synchrony. Until now, this has not been examined either in large-scale laser network configurations or in large transmission coupling paths, as real network implementations oblige. Here we present the first implementation of a fully-coupled fiber network with up to 16 semiconductor lasers, independently controlled and coupled through long interacting cavities. High level of consistent global or cluster synchrony via chaotic signals is demonstrated among all devices of the same origin and under appropriate operation. Devices that are not identical fail to synchronize at any condition, when coupled to the network. Under multiplexed operation, groups of lasers that emit at spectral distances as low as 50pm are shown to preserve intra-cluster synchronization when transmitted in the same fiber-optic channel, despite their large bandwidth of emitted signals.

© 2016 Optical Society of America

1. Introduction

A parallel growing work during the last decades in biological [1,2], socio-economical [3,4] and physical [5,6] sciences has identified deterministic chaos as a common mode of behavior in nonlinear dynamical systems. These systems that obey physical laws or follow mathematical models have been the medium, not only to understand the chaotic operation of various oscillating units, but also to monitor how coupled interactions affect them [7,8]. Synchronization [9–11] among such counterparts under coupling has revealed extended modes of operation, including interplay dynamics, instabilities within synchronization manifolds, bubbling effects, etc [2,12,13]. Synchronization phenomena become even more diverse when oscillating units interact in complex [14–16] or adaptive [17] large-scale network topologies. Extended theoretical works have investigated the relationship between the network structure of various coupling motifs and the multiple dynamical states of operation [18–24]. Such investigations could establish ground-breaking approaches in existing technological applications that employ coupled nonlinear units.

Semiconductor lasers (SLs) have been commonly used as nonlinear units in mutually-coupled systems. Specifically, SLs under time-delayed mutual optical injection [25–29] or electro-optical feedback loops [30] have been extensively investigated in terms of their synchronization potential and their operational instabilities. Such use has been demonstrated in the past for data encryption in communication applications [31]. Theoretical works have lately showed the potential of such systems to upgrade to larger scales [24,32–34]. Especially in [33], it has been shown that only a large number of nodes can contribute to a network synchrony, considering however significant node frequency detuning. The network realizations that investigate and substantially support controllable operation of coupled SLs are still very few. Short-range implementations with mutual coupling, using liquid crystals [35] or multiple lasing elements [36] and spatial light modulation units with short feedback loops have shown chimera states or complex dynamics through global locking in their emissions. Phase-dependent locking and synchrony has been demonstrated using Nd-YAG laser arrays [37,38] or fiber lasers arrays [39,40] through spatiotemporal profile analysis. Only lately, generalized synchrony has been reported in a small network of 4 independent single-longitudinal mode SLs, coupled through long interacting cavities [41].

In the present work we extend the generalized synchrony investigations in a network of up to 16 mutually-coupled identical SLs, connected through similar - yet unmatched - distant optical paths. We show that each unit's properties and operating parameters establish it as a member of the overall synchronized network, a member of intra-network synchronized clusters or just an outlier unit. Strict frequency matching (<200MHz) of the optical emitted signals allow synchrony at configurations with even a few identical SLs. In contrast, when non-identical SLs couple with the network they fail to synchronize at any operational condition. Moreover, when shifting identical SLs from a common emission wavelength (global operation) to multiplexed wavelengths (cluster operation), it is shown that the network can maintain intra-cluster synchrony. The latter property is validated for ultra-dense wavelength multiplexing of the coupled units, with chaotic carrier spectral distance of only 50pm. All the above properties of the coupled-SL networks could be exploited towards multi-channel hardware authentication protocols.

2. Network configuration

The coupling topology follows the fully-connected SL architecture shown in Fig. 1(a). Each laser is selected from a pool of identical devices - as described in paragraph 2.1 - and emits to the network, while receiving from all counterparts - including its own signal - through a common tunable reflector. For up to 16 lasers (or else referred as network nodes) and no long-haul transmission path, one amplification stage provides sufficient power to the injected signals for laser synchrony. However, in the presented investigation two amplification stages are used so that a larger range of coupling strengths can be tested among the laser nodes. Optical filtering with 0.36nm (~40GHz) 3dB-bandwidth is used to reduce erbium-doped fiber amplifiers’ (EDFA) spontaneous emission, without imposing frequency-selective feedback conditions. Inline fiber power monitors (PM) display the circulating average optical power. The total round trip time of the cavities formed between pairs of lasers is 117.92 ± 0.12m. Topologies with an additional long transmission path (fiber spool) or larger path mismatches have been also tested, as discussed later. Each laser's optical output is monitored through isolated ports that eliminate any residual feedback. These outputs are used to screen the optical and microwave properties of the emitted signals through appropriate monitoring instrumentation, as shown in Fig. 1(b). The microwave properties are obtained through a 4-channel 10GHz AC-coupled photoreceiver array. Operational conditions of the lasers are suitably selected so that identical wavelength and optical power emission is achieved, as shown in Fig. 1(c).

 figure: Fig. 1

Fig. 1 Full-mesh-type network with optically-coupled 16 SLs. (a) Laser network topology: PC: Polarization controller, 1x2 and 1x8: optical couplers, EDFA: 25dB-gain Erbium-doped fiber amplifier, OF: Optical filter, PM: Inline optical power monitor, ATT: Optical attenuator. (b) Monitoring stage: It includes optical signal detection from laser's output port in an optical spectrum analyzer (OSA), as well as electrical signal detection in a four-channel real-time oscilloscope (OSC) - by employing equal number of digital photoreceivers - and in a radio-frequency analyzer (RFA). (c) Biasing/temperature conditions of the uncoupled solitary 16 SLs (#: laser identification number) for identical wavelength emission (λ = 1549.600) and for different levels of near-threshold optical power emission (black rectangles: PL#em = −15dBm, red circles: PL#em = −20dBm, blue triangles: PL#em = −25dBm).

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2.1. Laser selection and operating conditions

The distributed feedback (DBF) semiconductor lasers (SLs) used in the coupled network are selected among forty (40) devices provided by the company Xiamen Guang Te Communication Technology Co. These devices came from the same fabrication run, with butterfly housing and without isolation. All SLs have an optical emission line at 1549.50 ± 0.25nm (20°C and 40mA bias), with a slope efficiency of 0.2-0.23mW/mA, while their threshold operation at 20°C ranges from 9.4mA to 10mA. The wavelength dependence on temperature for all devices is 0.100 ± 0.008 nm/°C, while the wavelength dependence on biasing current is negligible. In our setup, precise wavelength matching is tuned for all SLs (at 1549.600 ± 0.004nm). A common level of optical power emission is also selected for all SLs (−20dBm, −15dBm or −10dBm). The level of biasing, and thus emission, also determines the chaotic properties of the emitted signals. In the current study these lasers are considered as identical lasers.

In another investigation, two (2) DFB SLs provided by Fraunhofer HHI have been employed to investigate the synchronization potential of different lasers in the initial network. These SLs have an emission line at 1549.5nm at 26°C and 40mA bias, with a slope efficiency of 0.16mW/mA, while their threshold operation at the same temperature is 9.4mA. Intrinsic properties of these devices, such as alpha-factor, device length, gain coefficient, etc., are not the same with the first batch of 40 lasers. Thus, even if their operational characteristics (wavelength and optical power emission) are matched with the initial coupled network, their intrinsic properties define them as non-identical lasers.

The lasers' operational parameters control is performed through a modular temperature/current Newport 9016 LD Controller, with 16 independent channels and accuracy of 0.01°C/ 0.01mA respectively.

2.2. Timeseries acquistion and statistical analysis

Timetrace acquisition is performed through a 4-channel 40GSa/s DSO81204B Agilent oscilloscope. When recording simultaneously 4 channels, a reduced sampling of 20GSa/s (8GHz signal bandwidth detection) is available. Each channel's single acquisition process captures 218 samples. These 4 channels are serially used to monitor the synchronization level between a given laser (L#1) and the rest of the lasers in the network. Initially, in the remaining three channels cross-correlation of L#1with L#2, L#3 and L#4 is evaluated. For each pair, we calculate the pairwise cross-correlation that is maximized at their path difference time-lag. Average cross-correlation (average-CC) between two lasers is obtained by taking 25 independent acquisitions of 218 samples. This results in duration of 13.1072μs for each trace. Especially in strong coupling conditions, the average-CC value may exhibit some significant variance in independent acquisitions. This is attributed to the lasers' emission instabilities with slowly varying complexity dynamics and expressed with large standard deviations of average-CC values. At the next step, the last three monitoring channels are connected to a new set of lasers (L#5, L#6 and L#7) in order to evaluate their synchrony with L#1. This process continues until all-with-all laser emissions are evaluated. Since one does not interfere with the operating network, all laser pairs exhibit their particular cross-correlation which remains constant over time.

3. Network synchrony

The polarization state of each laser’s path is individually aligned following the process described next. After all lasers are physically connected in the network, only the first laser is nearly-threshold biased (power emission around −20dBm) in order to operate in the low frequency fluctuation (LFF) regime. External cavity modes at the MHz range are monitored in the RFA and are used to precisely measure the external cavity path length. At this dynamic regime, the polarization alignment of the path is straightforward since it leads to maximization of the LFF amplitude. Then, the first laser is set unbiased, and the above procedure of optical path measurement and polarization alignment is repeated for each single laser without intervening with the physical setup. All laser paths of the system are now polarization aligned. The birefringence accumulated through the total optical path has been assessed to be invariable in the laboratory controlled conditions for many hours operation; thus the polarization conditions are considered constant during the experimental evaluation of the system. Lasers can now be biased under the desired operational conditions, sustaining the same wavelength and optical power emission. In the coupled operation, wavelength emission of each laser is red-shifted up to ~0.3nm - depending on coupling strength - due to the injected optical power from the network. This shift is not exactly identical for all lasers due to small loss asymmetries of the optical components of the network and can be as high as 0.03nm. By applying temperature fine tuning all SLs are again wavelength-matched, under the coupled operation.

The coupling strength among the SLs determines not only the synchrony performance but also shapes the emitted laser dynamics through which synchrony is achieved. The injection ratio RL# is a measure of the coupling strength among the SLs and expresses the ratio of the optical power inserted into a laser divided by the optical power emitted by the same laser. Thus, for a given laser L# that participates in the coupled network, it is defined as:

RL#=C2PinjtotPemL#
where C is the laser-fiber coupling loss, Pinjtot is the total optical power reaching laser L# through the associated fiber path and PemL# is the optical emitted power of laser L# measured at its output fiber tip. Coupling loss C between laser facet and fiber is a parameter which cannot be verified directly for each device, since all devices are fiber pigtailed. For the injection ratio estimation, a value of C = 0.5 is used for all lasers, according to the specifications given by the laser manufacturer. When identical injection ratios are set for all laser elements, RL# = R. Throughout the experiments, different injection ratios R shift the central emission wavelength of the lasers, as expected. Thus, fine tuning of optical filters in the transmission path is always applied in order to maintain the highest possible SNR values.

The level of R should be such as to trigger correlated emission. As presented in Fig. 2, optical coupling affects signal emission from very low R values (as low as 0.001), forcing the 16 lasers to deviate from the continuous wave emission and oscillate in various dynamical states. Only when R>0.05 correlated chaotic emission for the overall network (average-CC>0.8) is observed among all coupled lasers (gray-marked region). Even slight mismatches in SLs' internal parameters, operational characteristics, optical emission frequencies, as well as small deviations from polarization alignment, may result in different levels of synchrony. In the example of Fig. 2, the laser pair L#1 - L#2 shows an average-CC above 0.93, while the laser pair L#6-L#7 shows an average-CC close to 0.86. The variance of each average-CC value is explained by the transversal instabilities of the synchronization manifold imposed by the overall network operation. The appearance of de-synchronization events, albeit always present, shows a dependence on the coupling strength among the laser nodes. Their duration and occurrence frequency shape the correlation and variance level for each coupling strength condition. In Fig. 3 such de-synchronization events are shown between two SLs (L#1 and L#2) emissions. Usually when power dropouts arise in the emitted dynamics, de-synchronization for a small period of time - of the order of ns - is present. The fact that this duration is significantly shorter than the period for which the two lasers preserve high-level of synchronization deems the change in the statistical metric of averaged-CC insignificant. This behavior refers to an optimally coupled and operated 16-SL network. If SLs are biased to favor LFF emission or unmatched operational conditions apply, the de-synchronization events last longer, affecting the overall synchronization level. Finally, for very strong injection ratios (above 1), increased instabilities are observed, accompanied by longer de-synchronization events or lower complexity attractors, periodic oscillations and even continuous wave operation.

 figure: Fig. 2

Fig. 2 Effect of coupling strength on the correlated emission of a 16-laser coupled network. Averaged-CC values between two pairs of SLs (L#1-L#2 and L#6-L#7) vs. the applied injection ratio, when the emitted power from the lasers is set to −15dBm. Timetraces in insets show the dynamics of the emitted signals for different coupling strengths. Gray region indicates synchronized network coupling conditions through chaotic signals.

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 figure: Fig. 3

Fig. 3 Temporal evolution of L#1 and L#2 emission, as well as their difference in a coupled 16-laser network, when R = 0.2dB. De-synchronziation events appear when power dropouts occur and are minimized for optimal operating and coupling conditions. In the right column, a detail in the temporal region where a de-synchronization event takes place is provided.

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For very strong injection conditions, increased instabilities are observed, accompanied by much longer de-synchronization events (R>2) or lower complexity attractors, periodic oscillations and even continuous wave operation (e.g. R~5). The co-existence of multiple dynamical states results in reduced average-CC values with large standard deviation. The equal injection ratios for all laser interactions generate a consistent synchrony mapping, as presented here. Contrariwise, asymmetric coupling interactions or partially connected topologies are expected to result in mixed emission dynamics and various synchronization levels, affected by the sum of the non-weighted signals coupled in each laser.

In order to generate an average cross-correlation mapping of the 16-node coupled network, we follow the procedure developed in section 2.2. In Fig. 4(a) we show that all 16 SLs succeed to befall in a globally synchronized network, under strong coupling conditions, with the lowest average-CC value obtained between L#3 and L#12 (~0.836) and the highest average-CC value obtained between L#11 and L#15 (~0.965). Deviation from perfect synchrony and robust synchronization is attributed to several reasons, including de-synchronization events phenomena [42,43], optical and electrical noise by amplification and detection units, mismatch of intrinsic and operational laser parameters, non-automated polarization control of the network paths, etc. The signal-to-noise ratio (SNR) of the 16-lasers' recorded signals lies between 26.2dB and 32.5dB. In the present investigation the SNR is defined as:

 figure: Fig. 4

Fig. 4 Cross-correlation mapping of the experimentally built 16-laser coupled network under synchronized conditions. Max-CC among pairs of lasers, considering (a) the detection-limited bandwidth emitted signals (8GHz), and (b) the filtered emitted signals with 2GHz bandwidth. The power of lasers' emissions is −15dBm, frequency detuning is minimized and R = 0.2dB.

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SNR=σchaoticsignal2σsystemnoise2

It is the ratio of the chaotic signal's variance versus the variance of the system's noise sources, which includes the detector's noise, the measurement unit's noise and the optical noise from the amplification units. A notable improvement in the recorded global network synchrony is observed when increasing the SNR of the detected signals; the latter is achieved by reducing the signals’ detected bandwidth at 2GHz, through the use of low-pass 2nd-order Butterworth filters. Within this bandwidth, powerful spectral components of the emitted signals are preserved while noise is reduced. This leads to a SNR improvement of the detected signals by up to 4.6dB. Additionally, the most efficient synchronization potential in such systems is observed in frequencies around the lasers' relaxation frequency (fRO~2.2GHz, in the examined case). The above factors result in the improved synchrony performance of Fig. 4(b). Under the same coupling conditions considered in Fig. 4(a), the lowest average-CC value is obtained now between L#7 and L#16 (~0.925) and the highest average-CC value is obtained between L#4 and L#15 (~0.987). Another justification one could consider for better synchrony could be a reduced signal complexity due to electrical filtering; however, this is proved invalid through normalized permutation entropy measurements of the limited-bandwidth signals, as presented in paragraph 4.

An effort to verify the above experimental behavior in the examined topology by numerical simulations has been performed. Such topologies with very long coupling delays have not been simulated to the best of our knowledge, probably due to the large computational requirements imposed by the large time-delays in the rate equation model. For this reason, relevant numerical simulations based on the Lang-Kobayashi model [44] with much shorter coupling delays (10 ± 1ns) have been performed, excluding any transmission and amplification impairments. Propagation factors are considered not to affect the emitted dynamics and synchrony in the specific topology. Even these delay values are beyond the coherence length of the SLs, meaning that we deal with an incoherent type of coupling interaction. The SLs’ parameters values approach the actual SLs operation, while the model includes frequency detuning terms among oscillators, as shown in Eqs. (3) to (5):

dEi(t)dt=jΔωiEi(t)+12(1+ja)[Gi(t)tph1)Ei(t)+κmEm(tτim)eiω0τim+Dξ(t)
dNi(t)dt=IeNi(t)tsGi(t)|Ei(t)|2
Gi(t)=gn[Ni(t)N0](1+s|Ei(t)|2)1

The time delay and coupling between m and i nodes of the network are τim and κi = κ, respectively. The coupling κ is normalized in ns−1 values. The biasing current for all lasers is set to I = 10.8mA, while the solitary lasing emission threshold is Ith = 10.6mA and the rest of the parameters hold the following values: α = 3, s = 5·10−7, N0 = 1.25·10−8, gn = 2.2·10−5 ns−1, ts = 1.54ns and tph = 2ps. Each laser is detuned (Δωi) with respect to the reference laser frequency ω0 = 193.548THz, with equal spacing from −100MHz (L#1) to + 100MHz (L#16). The uncorrelated complex Gaussian white noise is represented by ξ(t) with amplitude D = 10−4ns−1.

A dynamic mapping of the pairwise average-CC of the 16-SLs' network is performed for a range of coupling conditions from 1ns−1 up to 400ns−1 (see Visualization 1). The κ values considered here cover from continuous wave operation (uncoupled conditions) to strong injection conditions. κ parameter is equal to the square route of R, normalized by the characteristic time of the SLs internal cavity (tn = 10ps). Up to κ = 8ns−1, continuous wave operation is observed for all SL nodes. By increasing κ up to 25ns−1, periodic and quasi-periodic oscillations that are easy to synchronize lead the overall network to almost ideal synchrony. As κ increases further, up to 130ns−1, higher complexity attractors on the emitted signals destroy synchrony. Only higher coupling can bring the emitted chaotic signals to synchrony. Especially, over κ = 300ns−1, synchrony is sealed at values near 1, where signals exhibit lower complexity. The experimental results of the 16-SLs average-CC mapping, as presented in Fig. 4, prove to be in line with the above simulations. The much shorter delay coupling paths used in our simulations do not seem to modify the general picture of intensity synchronization performance.

4. Sensitivity vs. consistency

The presented network synchrony performance pertains to case of optical paths among nodes with differences of several cm and for minimized frequency detuning among the lasers. However, this behavior is remarkably consistent for larger variances of the optical paths. Specifically, by introducing diverse optical path extensions in the network, we increase the length variance of cavities formed between SLs to ± 0.56m. In another scenario, we insert additional optical delays of a few meters only in laser’s #1 optical path. In both cases no notable change in the synchrony performance that was presented in Fig. 4 is observed. As expected, the cross-correlation maxima are met at the new time-lag differences of the evaluated pairs, defined by their new optical path differences. Finally, we have added a non-zero dispersion transmission fiber spool of 3,499.2m before the reflector end, without changing the absolute length differences among the paths. Again, no effect has been observed in the synchronization or in the complexity of the emitted dynamics, as expected due to the dominant incoherent nature of interaction. The dispersive properties of the transmission path (estimated for the emitted chaotic optical carriers to be around 44ps) also act transparently with respect to network synchrony.

Frequency mismatch of any SL coupled to the 16-SL network affects synchrony to a great extend and is influenced decisively by the level of injection ratio R. This is experimentally demonstrated in Fig. 5(a), where the wavelength emission of L#16 is slightly shifted by fine adjustment of its temperature. The study considers two different injection ratio levels. In Fig. 5(a1) we show that lower injection values (R = −12dB) establish a sensitive network operation, capable of injection pulling frequency-mismatched lasers well below the GHz range. In Fig. 5(a2) we show that higher injection values (R = 0.2dB) set a consistent network operation, by injection locking all SLs that optically emit with a tolerance of several GHz. In both scenarios, minimization of frequency detuning of all SLs to zero results in a slightly increased averaged-CC value (by up to 0.012). This improvement can be potentially used to evaluate the matching conditions among any frequency-detuned laser. However, when a laser is frequency-detuned resulting in loss of synchronization with all other SLs, this does not influence the overall synchrony among the rest nodes in the network. This is shown by the unchanged averaged-CC performance between L#14 and L#15, versus any L#16 frequency detuning.

 figure: Fig. 5

Fig. 5 Effects of a frequency detuned laser (L#16) in correlated emission and NPE of emitted signals, in a 16-laser coupled network. (a) Pairwise average-CC evaluation of L#14, L#15 and L#16, for (a1) low-level (R = −12dB) and (a2) high-level (R = 0.2dB) injection ratios. Low injection conditions establish a highly sensitive system to wavelength emission mismatches, while high injection conditions establish a consistently synchronized system. (b) Normalized permutation entropy mapping of a laser with fixed conditions in the network (L#14) vs. ordinal pattern and applied delay, for: (b1) R = −12dB and frequency matched L#16, (b2) R = −12dB and L#16 with frequency detuning of 3.5GHz, (b3) R = 0.2dB and frequency matched L#16, (b4) R = 0.2dB and L#16 with frequency detuning of 17.6GHz. Contour plots are used for better transient screening.

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The complexity of the chaotic signals through which synchrony is achieved is also dependent on the injection ratio and the frequency mismatch. In order to build a network with consistent operation, we target on SLs’ emission of stable complexity. We focus on the gray-marked region of Fig. 2, in order to evaluate the complexity of the emitted signals by measuring normalized permutation entropy (NPE), noted by H. In principle, frequency mismatch of a single SL can potentially affect the complexity of other lasers that are locked in synchrony. The H metric is used as a measure of disorder and uncertainty [45] and is estimated through the normalized Shannon entropy S associated with a probability distribution P{pi}, with i = 1,...,M, as described in Eq. (6). It has been derived from Bandt and Pompe's work, who constructed the probability distribution using ordinal patterns from time series [46]. It exploits ordinal patterns with different lengths D, as well as different sampling rate - through time delay parameter τd - which allows investigating the complexity of a timeseries at various spectral regions. The ordinal pattern probability distribution P{p(πi)}, with i = 1,...,D, is constructed by determining the relative frequency of all the D! possible permutations πi.

H=H(P)=S(P)Smax=i=1D!p(πi)lnp(πi)lnD!

The above formalism gives values for H between 0 and 1; complete predictability is expressed by zero and randomness with uniform probability distribution is expressed by one. In our calculations, high complexity is expressed by H~1. Since experimentally photodetected signals are initially sampled at 20GSa/s and the photodetection bandwidth is 8GHz, the bandwidth by sampling-rate ratio is BW/S = 0.4GSa−1. When signal bandwidth is reduced by electrical filtering, the sampling-rate should be reduced by such a value of permutation entropy delay that preserves the same BW/S ratio. This results in a consistent comparison in terms of complexity. For example, for a 2GHz bandwidth signal, one should employ a delay equal to τd = 4 in order to obtain the previous BW/S value of 0.4GSa−1.

The NPE measurements on the L#14 emission in a synchronized network are shown in Fig. 5(b), for different conditions of L#16 frequency detuning and for the two R values considered in Fig. 5(a). In a sensitive network coupled with low R values, frequency detuning of even one out of the 16 lasers causes reduction of the NPE compared with the non-mismatched case [Fig. 5(b1)-5(b2)]. This is evident at high delay values and large ordinal patterns. On the other hand, in a network coupled with large R values, frequency detuning of L#16 does not impose changes to the NPE [Fig. 5(b3)-5(b4)]. Under the latter conditions, the complexity maintenance contributes to a consistent synchrony operation.

5. Cluster synchrony

In a smaller network, we optically couple 8 SLs and examine the potential of the network SL nodes to obtain cluster synchrony by imposing frequency emission grouping. Initially, for small frequency detuning (<200MHz) of all 8 SLs - as in the 16-laser network case - optimized conditions lead to highly correlated emission. Specifically, pairwise averaged-CC values of at least 0.88, and as high as 0.97, are recorded for full-bandwidth detected signals, as shown in Fig. 6(a). It becomes clear that by controlling frequency-matching conditions we can drastically reduce the least number of coupled nodes required for synchronized operation, overcoming the limitations presented in the theoretical work in [33]. By thermally shifting the emission wavelengths of the second quartet of lasers (L#5-L#8) by 50pm we obtain the correlation mapping of Fig. 6(b). Synchrony is observed between lasers that participate in each cluster (L#1-L#4 and L#5-L#8) while correlated emission is always at very low levels when comparing inter-cluster nodes (averaged-CC<0.61). Thus, the same configuration can also lead to cluster synchronization.

 figure: Fig. 6

Fig. 6 Cluster synchronization in an 8-SL coupled network configuration. Cross-correlation mapping with (a) zero-detuned wavelength laser emission, and (b) with cluster synchronization among two quartets of lasers (L#1-L#4 and L#5-L#8) that are 50pm spaced in wavelength, when R = 0.8dB. (c) Timeseries of laser emission (L#1: yellow, L#2: green, L#6: purple, L#7: pink) for the cases of (a) (top) and (b) (bottom). Y-axis shift on timeseries is applied only for screening purposes. (d) Averaged-CC between lasers vs. the injection ratio level (two lasers from each cluster are monitored), for the cluster synchronization conditions of case (b). The optical power of all lasers' emission is −15dBm.

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The shift of 50pm which has been applied for cluster synchrony is much lower than the bandwidth of the optical chaotic signals circulating in the network, which are also constrained by the 3-dB optical filters' bandwidth (360pm). This inter-cluster wavelength distance is also smaller than the dense wavelength division multiplexed (DWDM) technology used in optical transmission. From the difference in the timeseries of Fig. 6(c), we conclude that each cluster's dynamics are influenced by neighboring clusters but eventually do not disturb intra-cluster synchrony of nodes. This noteworthy finding indicates that such configurations can operate in very dense, multiplexed, optical coupling conditions. A wide range of coupling strengths allows efficient cluster network synchrony, as shown in Fig. 6(d), by calculating the averaged-CC level versus the injection ratio R. At very strong coupling conditions (R>5dB) frequency pulling through injection locking occurs. Consequently, cluster synchronization degenerates into a global low-performance correlated emission, with significant dynamical instabilities and varying complexity over time. This is expressed in Fig. 6(d) through increased averaged-CC values between inter-cluster lasers, yet with large variances.

6. Non-identical nodes

The concept of security of the 8-SLs network is tested by substituting one SL with a device provided from another manufacturer (see paragraph 2.1). In this investigation we study the potential of a user optically coupling with the network with a non-identical SL device to synchronize. Wavelength emission is matched among all SLs, while the biasing current of the different SL and the optical injection level are varied so as to achieve the best synchrony level within the network. When considering moderate optical coupling (R = −9dB), the highest achieved averaged-CC between the non-identical SL and any of the identical SLs group is 0.34 at its most, as shown in Fig. 7(a). It is obtained for the non-identical SL’s near-threshold operation and is greatly lower than the worst synchronized pair within the identical SLs group (averaged-CC~0.82). By enhancing the coupling conditions to R = −0.5dB, as presented in Fig. 7(b), an equivalent behavior is observed. The only difference observed is the improved values of averaged-CC for the different (~0.62) and same manufacturers’ (~0.89) SLs. Consequently, dissimilar hardware SL units hold by unauthenticated users fail to synchronize with the network at any operational condition. On the other hand, users with identical devices can access synchronized emission, as long as they select matched operating conditions and same dynamical regimes. Equivalent findings are also validated for a network containing two out of eight SLs from a different manufacturer.

 figure: Fig. 7

Fig. 7 Synchrony comparison in an 8-node coupled network that includes 7 identical SLs and 1 SL from a different manufacturer. The comparison is made between the worst performance among the 7 identical lasers (black rectangles) and the best performance of synchronization between the different SL and the 7 identical lasers (red circles), versus the different SL’s emitted optical power and for (a) R = −9dBm and (b) R = −0.5dBm. All uncoupled identical lasers emit optical power −15dBm, while all uncoupled lasers operate at λ = 1549.600nm.

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7. Conclusions

Under ideal conditions with no laser mismatches or noise, the master stability function [12] dictates a universal stability standard under an invariant synchronization manifold. Practical implementations with large number of active nodes, however, unavoidably include laser dissimilarities, optical paths' loss asymmetries, and optical and electrical noise sources. All these factors induce local instabilities and shape rigorously the complexity and statistics of the emitted signals from each laser. The global synchrony of such networks is affected as well. Identical SL devices and matched operating conditions are shown to significantly reduce the duration of de-synchronization events between chaotic emissions. Therefore, in such configurations, these event only slightly affect the statistical metrics used to validate network synchrony through averaged-CC.

In this work we report on the first large-scale implementation of 16 optically coupled and independently controlled SLs in a fully-connected synchronized network topology. This proof-of-concept prototype records the incoherently-coupled laser operation at large distances, exceeding the restrictions imposed by free-space topologies with localization restrictions or topologies with dependent active nodes [35,36]. The overall consistency of the synchronized network is profound albeit the presence of mismatched or disparate lasers interacting through optical coupling. Local instabilities causing short de-synchronization events do not annihilate the overall high level of synchrony. An upgrade of the investigated network, with more connected lasers and with even longer transmission links is not expected to alter the network behavior in terms of synchrony.

This work can be the basis on which advanced sensing and authentication protocols in future fiber-optic networks can be proposed. In an envisaged application that exploits the concept of this work, the typical telecom SL emitters can turn into broadband sensing elements of the hosting user, through the optically-coupled network. Instead of being modulated for data transmission, their activated chaotic dynamics supply them with chaotic authentication signatures changing over time. These signatures can set them as legitimate participants in this hardware-type security firewall, as long as they conform to the structural and operational restrictions set by the network and remain synchronized. This is the first step to securely protect a typical transmission channel via preserving user authenticity. At first, the allocation of a hardware fiber network for authentication purposes and not for data transmission seems a luxury. However, the properties shown in cluster synchronization with very dense wavelength allocation - much denser than the actual bandwidth of the chaotic signals - allow the operation of numerous independent authentication channels in the same fiber transmission line. These authentication channels can be set at much denser frequencies compared to conventional communication DWDM channels, serving an even larger number of users, and co-exist with data transmission channels.

Acknowledgments

The authors would like to thank A. Fragkos for his support in the design and implementation of the electronic control of lasers. The authors acknowledge the support and funding of Greek General Secretariat for Research and Technology under the research project “ARISTEIA II - 4750 - CONECT”.

References and links

1. K. Aihara, T. Takabe, and M. Toyoda, “Chaotic neural networks,” Phys. Lett. A 144(6-7), 333–340 (1990). [CrossRef]  

2. D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature 393(6684), 440–442 (1998). [CrossRef]   [PubMed]  

3. J. D. Sterman, “Deterministic chaos in an experimental economic system,” J. Econ. Behav. Organ. 12(1), 1–28 (1989). [CrossRef]  

4. E. Diehl and J. D. Sterman, “Effects of feedback complexity on dynamic decision making,” Organ. Behav. Hum. Decis. Process. 62(2), 198–215 (1995). [CrossRef]  

5. E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990). [CrossRef]   [PubMed]  

6. H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993). [CrossRef]  

7. R. Fitzhugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophys. J. 1(6), 445–466 (1961). [CrossRef]   [PubMed]  

8. Y. Kuramoto, “Scaling behavior of turbulent oscillators with non-local interaction,” Prog. Theor. Phys. 94(3), 321–330 (1995). [CrossRef]  

9. R. Roy and K. S. Thornburg Jr., “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72(13), 2009–2012 (1994). [CrossRef]   [PubMed]  

10. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Phys. Rev. Lett. 76(11), 1804–1807 (1996). [CrossRef]   [PubMed]  

11. L. Kocarev and U. Parlitz, “Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems,” Phys. Rev. Lett. 76(11), 1816–1819 (1996). [CrossRef]   [PubMed]  

12. L. M. Pecora and T. L. Carroll, “Master stability functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998). [CrossRef]  

13. P. Ashwin, J. Buescu, and I. Stewart, “Bubbling of attractors and synchronisation of chaotic oscillators,” Phys. Lett. A 193(2), 126–139 (1994). [CrossRef]  

14. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Critical phenomena in complex networks,” Rev. Mod. Phys. 80(4), 1275–1335 (2008). [CrossRef]  

15. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469(3), 93–153 (2008). [CrossRef]  

16. S. H. Strogatz, “Exploring complex networks,” Nature 410(6825), 268–276 (2001). [CrossRef]   [PubMed]  

17. J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014). [CrossRef]   [PubMed]  

18. O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos 18(3), 037116 (2008). [CrossRef]   [PubMed]  

19. I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011). [CrossRef]  

20. F. Sorrentino and E. Ott, “Network synchronization of groups,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5), 056114 (2007). [CrossRef]   [PubMed]  

21. T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012). [CrossRef]   [PubMed]  

22. D. M. Abrams and S. H. Strogatz, “Chimera states for coupled oscillators,” Phys. Rev. Lett. 93(17), 174102 (2004). [CrossRef]   [PubMed]  

23. A. E. Motter, “Nonlinear dynamics: Spontaneous synchrony breaking,” Nat. Phys. 6(3), 164–165 (2010). [CrossRef]  

24. L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014). [CrossRef]   [PubMed]  

25. M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108(21), 214101 (2012). [CrossRef]   [PubMed]  

26. A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis, “Localized synchronization in two coupled nonidentical semiconductor lasers,” Phys. Rev. Lett. 78(25), 4745–4748 (1997). [CrossRef]  

27. J. Mulet, C. R. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B. Quantum Semiclass Opt. 6(1), 97–105 (2004). [CrossRef]  

28. B. B. Zhou and R. Roy, “Isochronal synchrony and bidirectional communication with delay-coupled nonlinear oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026205 (2007). [CrossRef]   [PubMed]  

29. K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 056211 (2011). [CrossRef]   [PubMed]  

30. C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013). [CrossRef]   [PubMed]  

31. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005). [CrossRef]   [PubMed]  

32. V. Flunkert and E. Schöll, “Chaos synchronization in networks of delay-coupled lasers: role of the coupling phases,” New J. Phys. 14(3), 033039 (2012). [CrossRef]  

33. J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. 105(26), 264101 (2010). [CrossRef]   [PubMed]  

34. M. Bourmpos, A. Argyris, and D. Syvridis, “Sensitivity analysis of a star optical network based on mutually coupled semiconductor lasers,” J. Lightwave Technol. 30(16), 2618–2624 (2012). [CrossRef]  

35. A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat. Phys. 8(9), 658–661 (2012). [CrossRef]  

36. D. Brunner and I. Fischer, “Reconfigurable semiconductor laser networks based on diffractive coupling,” Opt. Lett. 40(16), 3854–3857 (2015). [CrossRef]   [PubMed]  

37. F. Rogister, K. S. Thornburg Jr, L. Fabiny, M. Möller, and R. Roy, “Power-law spatial correlations in arrays of locally coupled lasers,” Phys. Rev. Lett. 92(9), 093905 (2004). [CrossRef]   [PubMed]  

38. M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106(22), 223901 (2011). [CrossRef]   [PubMed]  

39. M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, “Passive phase locking of 25 fiber lasers,” Opt. Lett. 35(9), 1434–1436 (2010). [CrossRef]   [PubMed]  

40. M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, and N. Davidson, “Phase locking of two fiber lasers with time-delayed coupling,” Opt. Lett. 34(12), 1864–1866 (2009). [CrossRef]   [PubMed]  

41. Y. Aviad, I. Reidler, M. Zigzag, M. Rosenbluh, and I. Kanter, “Synchronization in small networks of time-delay coupled chaotic diode lasers,” Opt. Express 20(4), 4352–4359 (2012). [CrossRef]   [PubMed]  

42. V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009). [CrossRef]   [PubMed]  

43. J. Tiana-Alsina, K. Hicke, X. Porte, M. C. Soriano, M. C. Torrent, J. Garcia-Ojalvo, and I. Fischer, “Zero-lag synchronization and bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(2), 026209 (2012). [CrossRef]   [PubMed]  

44. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]  

45. J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express 22(2), 1713–1725 (2014). [CrossRef]   [PubMed]  

46. C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002). [CrossRef]   [PubMed]  

References

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  1. K. Aihara, T. Takabe, and M. Toyoda, “Chaotic neural networks,” Phys. Lett. A 144(6-7), 333–340 (1990).
    [Crossref]
  2. D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature 393(6684), 440–442 (1998).
    [Crossref] [PubMed]
  3. J. D. Sterman, “Deterministic chaos in an experimental economic system,” J. Econ. Behav. Organ. 12(1), 1–28 (1989).
    [Crossref]
  4. E. Diehl and J. D. Sterman, “Effects of feedback complexity on dynamic decision making,” Organ. Behav. Hum. Decis. Process. 62(2), 198–215 (1995).
    [Crossref]
  5. E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
    [Crossref] [PubMed]
  6. H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993).
    [Crossref]
  7. R. Fitzhugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophys. J. 1(6), 445–466 (1961).
    [Crossref] [PubMed]
  8. Y. Kuramoto, “Scaling behavior of turbulent oscillators with non-local interaction,” Prog. Theor. Phys. 94(3), 321–330 (1995).
    [Crossref]
  9. R. Roy and K. S. Thornburg., “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72(13), 2009–2012 (1994).
    [Crossref] [PubMed]
  10. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Phys. Rev. Lett. 76(11), 1804–1807 (1996).
    [Crossref] [PubMed]
  11. L. Kocarev and U. Parlitz, “Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems,” Phys. Rev. Lett. 76(11), 1816–1819 (1996).
    [Crossref] [PubMed]
  12. L. M. Pecora and T. L. Carroll, “Master stability functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
    [Crossref]
  13. P. Ashwin, J. Buescu, and I. Stewart, “Bubbling of attractors and synchronisation of chaotic oscillators,” Phys. Lett. A 193(2), 126–139 (1994).
    [Crossref]
  14. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Critical phenomena in complex networks,” Rev. Mod. Phys. 80(4), 1275–1335 (2008).
    [Crossref]
  15. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469(3), 93–153 (2008).
    [Crossref]
  16. S. H. Strogatz, “Exploring complex networks,” Nature 410(6825), 268–276 (2001).
    [Crossref] [PubMed]
  17. J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014).
    [Crossref] [PubMed]
  18. O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos 18(3), 037116 (2008).
    [Crossref] [PubMed]
  19. I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
    [Crossref]
  20. F. Sorrentino and E. Ott, “Network synchronization of groups,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5), 056114 (2007).
    [Crossref] [PubMed]
  21. T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
    [Crossref] [PubMed]
  22. D. M. Abrams and S. H. Strogatz, “Chimera states for coupled oscillators,” Phys. Rev. Lett. 93(17), 174102 (2004).
    [Crossref] [PubMed]
  23. A. E. Motter, “Nonlinear dynamics: Spontaneous synchrony breaking,” Nat. Phys. 6(3), 164–165 (2010).
    [Crossref]
  24. L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
    [Crossref] [PubMed]
  25. M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108(21), 214101 (2012).
    [Crossref] [PubMed]
  26. A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis, “Localized synchronization in two coupled nonidentical semiconductor lasers,” Phys. Rev. Lett. 78(25), 4745–4748 (1997).
    [Crossref]
  27. J. Mulet, C. R. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B. Quantum Semiclass Opt. 6(1), 97–105 (2004).
    [Crossref]
  28. B. B. Zhou and R. Roy, “Isochronal synchrony and bidirectional communication with delay-coupled nonlinear oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026205 (2007).
    [Crossref] [PubMed]
  29. K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 056211 (2011).
    [Crossref] [PubMed]
  30. C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013).
    [Crossref] [PubMed]
  31. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
    [Crossref] [PubMed]
  32. V. Flunkert and E. Schöll, “Chaos synchronization in networks of delay-coupled lasers: role of the coupling phases,” New J. Phys. 14(3), 033039 (2012).
    [Crossref]
  33. J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. 105(26), 264101 (2010).
    [Crossref] [PubMed]
  34. M. Bourmpos, A. Argyris, and D. Syvridis, “Sensitivity analysis of a star optical network based on mutually coupled semiconductor lasers,” J. Lightwave Technol. 30(16), 2618–2624 (2012).
    [Crossref]
  35. A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat. Phys. 8(9), 658–661 (2012).
    [Crossref]
  36. D. Brunner and I. Fischer, “Reconfigurable semiconductor laser networks based on diffractive coupling,” Opt. Lett. 40(16), 3854–3857 (2015).
    [Crossref] [PubMed]
  37. F. Rogister, K. S. Thornburg, L. Fabiny, M. Möller, and R. Roy, “Power-law spatial correlations in arrays of locally coupled lasers,” Phys. Rev. Lett. 92(9), 093905 (2004).
    [Crossref] [PubMed]
  38. M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106(22), 223901 (2011).
    [Crossref] [PubMed]
  39. M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, “Passive phase locking of 25 fiber lasers,” Opt. Lett. 35(9), 1434–1436 (2010).
    [Crossref] [PubMed]
  40. M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, and N. Davidson, “Phase locking of two fiber lasers with time-delayed coupling,” Opt. Lett. 34(12), 1864–1866 (2009).
    [Crossref] [PubMed]
  41. Y. Aviad, I. Reidler, M. Zigzag, M. Rosenbluh, and I. Kanter, “Synchronization in small networks of time-delay coupled chaotic diode lasers,” Opt. Express 20(4), 4352–4359 (2012).
    [Crossref] [PubMed]
  42. V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
    [Crossref] [PubMed]
  43. J. Tiana-Alsina, K. Hicke, X. Porte, M. C. Soriano, M. C. Torrent, J. Garcia-Ojalvo, and I. Fischer, “Zero-lag synchronization and bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(2), 026209 (2012).
    [Crossref] [PubMed]
  44. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
    [Crossref]
  45. J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express 22(2), 1713–1725 (2014).
    [Crossref] [PubMed]
  46. C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
    [Crossref] [PubMed]

2015 (1)

2014 (3)

J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express 22(2), 1713–1725 (2014).
[Crossref] [PubMed]

J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014).
[Crossref] [PubMed]

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

2013 (1)

C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013).
[Crossref] [PubMed]

2012 (7)

V. Flunkert and E. Schöll, “Chaos synchronization in networks of delay-coupled lasers: role of the coupling phases,” New J. Phys. 14(3), 033039 (2012).
[Crossref]

M. Bourmpos, A. Argyris, and D. Syvridis, “Sensitivity analysis of a star optical network based on mutually coupled semiconductor lasers,” J. Lightwave Technol. 30(16), 2618–2624 (2012).
[Crossref]

A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat. Phys. 8(9), 658–661 (2012).
[Crossref]

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108(21), 214101 (2012).
[Crossref] [PubMed]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

Y. Aviad, I. Reidler, M. Zigzag, M. Rosenbluh, and I. Kanter, “Synchronization in small networks of time-delay coupled chaotic diode lasers,” Opt. Express 20(4), 4352–4359 (2012).
[Crossref] [PubMed]

J. Tiana-Alsina, K. Hicke, X. Porte, M. C. Soriano, M. C. Torrent, J. Garcia-Ojalvo, and I. Fischer, “Zero-lag synchronization and bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(2), 026209 (2012).
[Crossref] [PubMed]

2011 (3)

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106(22), 223901 (2011).
[Crossref] [PubMed]

K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 056211 (2011).
[Crossref] [PubMed]

I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
[Crossref]

2010 (3)

A. E. Motter, “Nonlinear dynamics: Spontaneous synchrony breaking,” Nat. Phys. 6(3), 164–165 (2010).
[Crossref]

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. 105(26), 264101 (2010).
[Crossref] [PubMed]

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, “Passive phase locking of 25 fiber lasers,” Opt. Lett. 35(9), 1434–1436 (2010).
[Crossref] [PubMed]

2009 (2)

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, and N. Davidson, “Phase locking of two fiber lasers with time-delayed coupling,” Opt. Lett. 34(12), 1864–1866 (2009).
[Crossref] [PubMed]

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

2008 (3)

O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos 18(3), 037116 (2008).
[Crossref] [PubMed]

S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Critical phenomena in complex networks,” Rev. Mod. Phys. 80(4), 1275–1335 (2008).
[Crossref]

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469(3), 93–153 (2008).
[Crossref]

2007 (2)

B. B. Zhou and R. Roy, “Isochronal synchrony and bidirectional communication with delay-coupled nonlinear oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026205 (2007).
[Crossref] [PubMed]

F. Sorrentino and E. Ott, “Network synchronization of groups,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5), 056114 (2007).
[Crossref] [PubMed]

2005 (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

2004 (3)

J. Mulet, C. R. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B. Quantum Semiclass Opt. 6(1), 97–105 (2004).
[Crossref]

D. M. Abrams and S. H. Strogatz, “Chimera states for coupled oscillators,” Phys. Rev. Lett. 93(17), 174102 (2004).
[Crossref] [PubMed]

F. Rogister, K. S. Thornburg, L. Fabiny, M. Möller, and R. Roy, “Power-law spatial correlations in arrays of locally coupled lasers,” Phys. Rev. Lett. 92(9), 093905 (2004).
[Crossref] [PubMed]

2002 (1)

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref] [PubMed]

2001 (1)

S. H. Strogatz, “Exploring complex networks,” Nature 410(6825), 268–276 (2001).
[Crossref] [PubMed]

1998 (2)

L. M. Pecora and T. L. Carroll, “Master stability functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature 393(6684), 440–442 (1998).
[Crossref] [PubMed]

1997 (1)

A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis, “Localized synchronization in two coupled nonidentical semiconductor lasers,” Phys. Rev. Lett. 78(25), 4745–4748 (1997).
[Crossref]

1996 (2)

M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Phys. Rev. Lett. 76(11), 1804–1807 (1996).
[Crossref] [PubMed]

L. Kocarev and U. Parlitz, “Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems,” Phys. Rev. Lett. 76(11), 1816–1819 (1996).
[Crossref] [PubMed]

1995 (2)

E. Diehl and J. D. Sterman, “Effects of feedback complexity on dynamic decision making,” Organ. Behav. Hum. Decis. Process. 62(2), 198–215 (1995).
[Crossref]

Y. Kuramoto, “Scaling behavior of turbulent oscillators with non-local interaction,” Prog. Theor. Phys. 94(3), 321–330 (1995).
[Crossref]

1994 (2)

R. Roy and K. S. Thornburg., “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72(13), 2009–2012 (1994).
[Crossref] [PubMed]

P. Ashwin, J. Buescu, and I. Stewart, “Bubbling of attractors and synchronisation of chaotic oscillators,” Phys. Lett. A 193(2), 126–139 (1994).
[Crossref]

1993 (1)

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993).
[Crossref]

1990 (2)

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
[Crossref] [PubMed]

K. Aihara, T. Takabe, and M. Toyoda, “Chaotic neural networks,” Phys. Lett. A 144(6-7), 333–340 (1990).
[Crossref]

1989 (1)

J. D. Sterman, “Deterministic chaos in an experimental economic system,” J. Econ. Behav. Organ. 12(1), 1–28 (1989).
[Crossref]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

1961 (1)

R. Fitzhugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophys. J. 1(6), 445–466 (1961).
[Crossref] [PubMed]

Abarbanel, H. D. I.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993).
[Crossref]

Abeles, M.

I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
[Crossref]

Abrams, D. M.

D. M. Abrams and S. H. Strogatz, “Chimera states for coupled oscillators,” Phys. Rev. Lett. 93(17), 174102 (2004).
[Crossref] [PubMed]

Aihara, K.

K. Aihara, T. Takabe, and M. Toyoda, “Chaotic neural networks,” Phys. Lett. A 144(6-7), 333–340 (1990).
[Crossref]

Annovazzi-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Arenas, A.

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469(3), 93–153 (2008).
[Crossref]

Argyris, A.

M. Bourmpos, A. Argyris, and D. Syvridis, “Sensitivity analysis of a star optical network based on mutually coupled semiconductor lasers,” J. Lightwave Technol. 30(16), 2618–2624 (2012).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Ashwin, P.

P. Ashwin, J. Buescu, and I. Stewart, “Bubbling of attractors and synchronisation of chaotic oscillators,” Phys. Lett. A 193(2), 126–139 (1994).
[Crossref]

Aviad, Y.

Bandt, C.

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref] [PubMed]

Bourmpos, M.

Brown, R.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993).
[Crossref]

Brunner, D.

Buescu, J.

P. Ashwin, J. Buescu, and I. Stewart, “Bubbling of attractors and synchronisation of chaotic oscillators,” Phys. Lett. A 193(2), 126–139 (1994).
[Crossref]

Carroll, T. L.

L. M. Pecora and T. L. Carroll, “Master stability functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

Cohen, D.

I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
[Crossref]

Colet, P.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

D’Huys, O.

K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 056211 (2011).
[Crossref] [PubMed]

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos 18(3), 037116 (2008).
[Crossref] [PubMed]

Dahms, T.

C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013).
[Crossref] [PubMed]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

Danckaert, J.

K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 056211 (2011).
[Crossref] [PubMed]

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos 18(3), 037116 (2008).
[Crossref] [PubMed]

Davidson, N.

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108(21), 214101 (2012).
[Crossref] [PubMed]

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106(22), 223901 (2011).
[Crossref] [PubMed]

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, “Passive phase locking of 25 fiber lasers,” Opt. Lett. 35(9), 1434–1436 (2010).
[Crossref] [PubMed]

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, and N. Davidson, “Phase locking of two fiber lasers with time-delayed coupling,” Opt. Lett. 34(12), 1864–1866 (2009).
[Crossref] [PubMed]

Díaz-Guilera, A.

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469(3), 93–153 (2008).
[Crossref]

Diehl, E.

E. Diehl and J. D. Sterman, “Effects of feedback complexity on dynamic decision making,” Organ. Behav. Hum. Decis. Process. 62(2), 198–215 (1995).
[Crossref]

Dorogovtsev, S. N.

S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Critical phenomena in complex networks,” Rev. Mod. Phys. 80(4), 1275–1335 (2008).
[Crossref]

Erneux, T.

O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos 18(3), 037116 (2008).
[Crossref] [PubMed]

A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis, “Localized synchronization in two coupled nonidentical semiconductor lasers,” Phys. Rev. Lett. 78(25), 4745–4748 (1997).
[Crossref]

Fabiny, L.

F. Rogister, K. S. Thornburg, L. Fabiny, M. Möller, and R. Roy, “Power-law spatial correlations in arrays of locally coupled lasers,” Phys. Rev. Lett. 92(9), 093905 (2004).
[Crossref] [PubMed]

Fischer, I.

D. Brunner and I. Fischer, “Reconfigurable semiconductor laser networks based on diffractive coupling,” Opt. Lett. 40(16), 3854–3857 (2015).
[Crossref] [PubMed]

J. Tiana-Alsina, K. Hicke, X. Porte, M. C. Soriano, M. C. Torrent, J. Garcia-Ojalvo, and I. Fischer, “Zero-lag synchronization and bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(2), 026209 (2012).
[Crossref] [PubMed]

K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 056211 (2011).
[Crossref] [PubMed]

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos 18(3), 037116 (2008).
[Crossref] [PubMed]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

J. Mulet, C. R. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B. Quantum Semiclass Opt. 6(1), 97–105 (2004).
[Crossref]

Fitzhugh, R.

R. Fitzhugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophys. J. 1(6), 445–466 (1961).
[Crossref] [PubMed]

Flunkert, V.

V. Flunkert and E. Schöll, “Chaos synchronization in networks of delay-coupled lasers: role of the coupling phases,” New J. Phys. 14(3), 033039 (2012).
[Crossref]

K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 056211 (2011).
[Crossref] [PubMed]

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

Fradkov, A.

J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014).
[Crossref] [PubMed]

Fridman, M.

Friedman, M.

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106(22), 223901 (2011).
[Crossref] [PubMed]

Friesem, A. A.

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108(21), 214101 (2012).
[Crossref] [PubMed]

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106(22), 223901 (2011).
[Crossref] [PubMed]

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, “Passive phase locking of 25 fiber lasers,” Opt. Lett. 35(9), 1434–1436 (2010).
[Crossref] [PubMed]

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, and N. Davidson, “Phase locking of two fiber lasers with time-delayed coupling,” Opt. Lett. 34(12), 1864–1866 (2009).
[Crossref] [PubMed]

Garcia-Ojalvo, J.

J. Tiana-Alsina, K. Hicke, X. Porte, M. C. Soriano, M. C. Torrent, J. Garcia-Ojalvo, and I. Fischer, “Zero-lag synchronization and bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(2), 026209 (2012).
[Crossref] [PubMed]

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. 105(26), 264101 (2010).
[Crossref] [PubMed]

García-Ojalvo, J.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Gavrielides, A.

A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis, “Localized synchronization in two coupled nonidentical semiconductor lasers,” Phys. Rev. Lett. 78(25), 4745–4748 (1997).
[Crossref]

Goltsev, A. V.

S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Critical phenomena in complex networks,” Rev. Mod. Phys. 80(4), 1275–1335 (2008).
[Crossref]

Grebogi, C.

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
[Crossref] [PubMed]

Hagerstrom, A. M.

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat. Phys. 8(9), 658–661 (2012).
[Crossref]

Heil, T.

J. Mulet, C. R. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B. Quantum Semiclass Opt. 6(1), 97–105 (2004).
[Crossref]

Hicke, K.

J. Tiana-Alsina, K. Hicke, X. Porte, M. C. Soriano, M. C. Torrent, J. Garcia-Ojalvo, and I. Fischer, “Zero-lag synchronization and bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(2), 026209 (2012).
[Crossref] [PubMed]

K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 056211 (2011).
[Crossref] [PubMed]

Hohl, A.

A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis, “Localized synchronization in two coupled nonidentical semiconductor lasers,” Phys. Rev. Lett. 78(25), 4745–4748 (1997).
[Crossref]

Hövel, P.

J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014).
[Crossref] [PubMed]

A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat. Phys. 8(9), 658–661 (2012).
[Crossref]

Kane, D. M.

Kanter, I.

Y. Aviad, I. Reidler, M. Zigzag, M. Rosenbluh, and I. Kanter, “Synchronization in small networks of time-delay coupled chaotic diode lasers,” Opt. Express 20(4), 4352–4359 (2012).
[Crossref] [PubMed]

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108(21), 214101 (2012).
[Crossref] [PubMed]

I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
[Crossref]

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106(22), 223901 (2011).
[Crossref] [PubMed]

Kinzel, W.

I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
[Crossref]

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Kocarev, L.

L. Kocarev and U. Parlitz, “Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems,” Phys. Rev. Lett. 76(11), 1816–1819 (1996).
[Crossref] [PubMed]

Kopelowitz, E.

I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
[Crossref]

Kovanis, V.

A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis, “Localized synchronization in two coupled nonidentical semiconductor lasers,” Phys. Rev. Lett. 78(25), 4745–4748 (1997).
[Crossref]

Kuramoto, Y.

Y. Kuramoto, “Scaling behavior of turbulent oscillators with non-local interaction,” Prog. Theor. Phys. 94(3), 321–330 (1995).
[Crossref]

Kurths, J.

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469(3), 93–153 (2008).
[Crossref]

M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Phys. Rev. Lett. 76(11), 1804–1807 (1996).
[Crossref] [PubMed]

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Larger, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Lehnert, J.

J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014).
[Crossref] [PubMed]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

Masoller, C.

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. 105(26), 264101 (2010).
[Crossref] [PubMed]

Mendes, J. F. F.

S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Critical phenomena in complex networks,” Rev. Mod. Phys. 80(4), 1275–1335 (2008).
[Crossref]

Mirasso, C. R.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

J. Mulet, C. R. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B. Quantum Semiclass Opt. 6(1), 97–105 (2004).
[Crossref]

Möller, M.

F. Rogister, K. S. Thornburg, L. Fabiny, M. Möller, and R. Roy, “Power-law spatial correlations in arrays of locally coupled lasers,” Phys. Rev. Lett. 92(9), 093905 (2004).
[Crossref] [PubMed]

Moreno, Y.

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469(3), 93–153 (2008).
[Crossref]

Motter, A. E.

A. E. Motter, “Nonlinear dynamics: Spontaneous synchrony breaking,” Nat. Phys. 6(3), 164–165 (2010).
[Crossref]

Mulet, J.

J. Mulet, C. R. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B. Quantum Semiclass Opt. 6(1), 97–105 (2004).
[Crossref]

Murphy, T. E.

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013).
[Crossref] [PubMed]

A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat. Phys. 8(9), 658–661 (2012).
[Crossref]

Nixon, M.

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108(21), 214101 (2012).
[Crossref] [PubMed]

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106(22), 223901 (2011).
[Crossref] [PubMed]

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, “Passive phase locking of 25 fiber lasers,” Opt. Lett. 35(9), 1434–1436 (2010).
[Crossref] [PubMed]

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, and N. Davidson, “Phase locking of two fiber lasers with time-delayed coupling,” Opt. Lett. 34(12), 1864–1866 (2009).
[Crossref] [PubMed]

Omelchenko, I.

A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat. Phys. 8(9), 658–661 (2012).
[Crossref]

Ott, E.

F. Sorrentino and E. Ott, “Network synchronization of groups,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5), 056114 (2007).
[Crossref] [PubMed]

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
[Crossref] [PubMed]

Parlitz, U.

L. Kocarev and U. Parlitz, “Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems,” Phys. Rev. Lett. 76(11), 1816–1819 (1996).
[Crossref] [PubMed]

Pecora, L. M.

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

L. M. Pecora and T. L. Carroll, “Master stability functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

Pesquera, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Pikovsky, A. S.

M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Phys. Rev. Lett. 76(11), 1804–1807 (1996).
[Crossref] [PubMed]

Pompe, B.

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref] [PubMed]

Porte, X.

J. Tiana-Alsina, K. Hicke, X. Porte, M. C. Soriano, M. C. Torrent, J. Garcia-Ojalvo, and I. Fischer, “Zero-lag synchronization and bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(2), 026209 (2012).
[Crossref] [PubMed]

Reidler, I.

Rogister, F.

F. Rogister, K. S. Thornburg, L. Fabiny, M. Möller, and R. Roy, “Power-law spatial correlations in arrays of locally coupled lasers,” Phys. Rev. Lett. 92(9), 093905 (2004).
[Crossref] [PubMed]

Ronen, E.

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108(21), 214101 (2012).
[Crossref] [PubMed]

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106(22), 223901 (2011).
[Crossref] [PubMed]

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, and N. Davidson, “Phase locking of two fiber lasers with time-delayed coupling,” Opt. Lett. 34(12), 1864–1866 (2009).
[Crossref] [PubMed]

Rosenbluh, M.

Rosenblum, M. G.

M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Phys. Rev. Lett. 76(11), 1804–1807 (1996).
[Crossref] [PubMed]

Roy, R.

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013).
[Crossref] [PubMed]

A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat. Phys. 8(9), 658–661 (2012).
[Crossref]

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. 105(26), 264101 (2010).
[Crossref] [PubMed]

B. B. Zhou and R. Roy, “Isochronal synchrony and bidirectional communication with delay-coupled nonlinear oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026205 (2007).
[Crossref] [PubMed]

F. Rogister, K. S. Thornburg, L. Fabiny, M. Möller, and R. Roy, “Power-law spatial correlations in arrays of locally coupled lasers,” Phys. Rev. Lett. 92(9), 093905 (2004).
[Crossref] [PubMed]

R. Roy and K. S. Thornburg., “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72(13), 2009–2012 (1994).
[Crossref] [PubMed]

Schöll, E.

J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014).
[Crossref] [PubMed]

C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013).
[Crossref] [PubMed]

V. Flunkert and E. Schöll, “Chaos synchronization in networks of delay-coupled lasers: role of the coupling phases,” New J. Phys. 14(3), 033039 (2012).
[Crossref]

A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat. Phys. 8(9), 658–661 (2012).
[Crossref]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 056211 (2011).
[Crossref] [PubMed]

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

Selivanov, A.

J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014).
[Crossref] [PubMed]

Shore, K. A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Sidorowich, J. J.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993).
[Crossref]

Soriano, M. C.

J. Tiana-Alsina, K. Hicke, X. Porte, M. C. Soriano, M. C. Torrent, J. Garcia-Ojalvo, and I. Fischer, “Zero-lag synchronization and bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(2), 026209 (2012).
[Crossref] [PubMed]

Sorrentino, F.

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013).
[Crossref] [PubMed]

F. Sorrentino and E. Ott, “Network synchronization of groups,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5), 056114 (2007).
[Crossref] [PubMed]

Sterman, J. D.

E. Diehl and J. D. Sterman, “Effects of feedback complexity on dynamic decision making,” Organ. Behav. Hum. Decis. Process. 62(2), 198–215 (1995).
[Crossref]

J. D. Sterman, “Deterministic chaos in an experimental economic system,” J. Econ. Behav. Organ. 12(1), 1–28 (1989).
[Crossref]

Stewart, I.

P. Ashwin, J. Buescu, and I. Stewart, “Bubbling of attractors and synchronisation of chaotic oscillators,” Phys. Lett. A 193(2), 126–139 (1994).
[Crossref]

Strogatz, S. H.

D. M. Abrams and S. H. Strogatz, “Chimera states for coupled oscillators,” Phys. Rev. Lett. 93(17), 174102 (2004).
[Crossref] [PubMed]

S. H. Strogatz, “Exploring complex networks,” Nature 410(6825), 268–276 (2001).
[Crossref] [PubMed]

D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature 393(6684), 440–442 (1998).
[Crossref] [PubMed]

Syvridis, D.

M. Bourmpos, A. Argyris, and D. Syvridis, “Sensitivity analysis of a star optical network based on mutually coupled semiconductor lasers,” J. Lightwave Technol. 30(16), 2618–2624 (2012).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Takabe, T.

K. Aihara, T. Takabe, and M. Toyoda, “Chaotic neural networks,” Phys. Lett. A 144(6-7), 333–340 (1990).
[Crossref]

Thornburg, K. S.

F. Rogister, K. S. Thornburg, L. Fabiny, M. Möller, and R. Roy, “Power-law spatial correlations in arrays of locally coupled lasers,” Phys. Rev. Lett. 92(9), 093905 (2004).
[Crossref] [PubMed]

R. Roy and K. S. Thornburg., “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72(13), 2009–2012 (1994).
[Crossref] [PubMed]

Tiana-Alsina, J.

J. Tiana-Alsina, K. Hicke, X. Porte, M. C. Soriano, M. C. Torrent, J. Garcia-Ojalvo, and I. Fischer, “Zero-lag synchronization and bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(2), 026209 (2012).
[Crossref] [PubMed]

Toomey, J. P.

Torrent, M. C.

J. Tiana-Alsina, K. Hicke, X. Porte, M. C. Soriano, M. C. Torrent, J. Garcia-Ojalvo, and I. Fischer, “Zero-lag synchronization and bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(2), 026209 (2012).
[Crossref] [PubMed]

Toyoda, M.

K. Aihara, T. Takabe, and M. Toyoda, “Chaotic neural networks,” Phys. Lett. A 144(6-7), 333–340 (1990).
[Crossref]

Tsimring, L. S.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993).
[Crossref]

Vardi, R.

I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
[Crossref]

Vicente, R.

O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos 18(3), 037116 (2008).
[Crossref] [PubMed]

Watts, D. J.

D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature 393(6684), 440–442 (1998).
[Crossref] [PubMed]

Williams, C. R. S.

C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013).
[Crossref] [PubMed]

Yorke, J. A.

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
[Crossref] [PubMed]

Zamora-Munt, J.

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. 105(26), 264101 (2010).
[Crossref] [PubMed]

Zhou, B. B.

B. B. Zhou and R. Roy, “Isochronal synchrony and bidirectional communication with delay-coupled nonlinear oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026205 (2007).
[Crossref] [PubMed]

Zhou, C.

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469(3), 93–153 (2008).
[Crossref]

Zigzag, M.

Y. Aviad, I. Reidler, M. Zigzag, M. Rosenbluh, and I. Kanter, “Synchronization in small networks of time-delay coupled chaotic diode lasers,” Opt. Express 20(4), 4352–4359 (2012).
[Crossref] [PubMed]

I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
[Crossref]

Biophys. J. (1)

R. Fitzhugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophys. J. 1(6), 445–466 (1961).
[Crossref] [PubMed]

Chaos (1)

O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos 18(3), 037116 (2008).
[Crossref] [PubMed]

Eur. Phys. Lett. (1)

I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
[Crossref]

IEEE J. Quantum Electron. (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

J. Econ. Behav. Organ. (1)

J. D. Sterman, “Deterministic chaos in an experimental economic system,” J. Econ. Behav. Organ. 12(1), 1–28 (1989).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. B. Quantum Semiclass Opt. (1)

J. Mulet, C. R. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B. Quantum Semiclass Opt. 6(1), 97–105 (2004).
[Crossref]

Nat. Commun. (1)

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

Nat. Phys. (2)

A. E. Motter, “Nonlinear dynamics: Spontaneous synchrony breaking,” Nat. Phys. 6(3), 164–165 (2010).
[Crossref]

A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat. Phys. 8(9), 658–661 (2012).
[Crossref]

Nature (3)

D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature 393(6684), 440–442 (1998).
[Crossref] [PubMed]

S. H. Strogatz, “Exploring complex networks,” Nature 410(6825), 268–276 (2001).
[Crossref] [PubMed]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

New J. Phys. (1)

V. Flunkert and E. Schöll, “Chaos synchronization in networks of delay-coupled lasers: role of the coupling phases,” New J. Phys. 14(3), 033039 (2012).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Organ. Behav. Hum. Decis. Process. (1)

E. Diehl and J. D. Sterman, “Effects of feedback complexity on dynamic decision making,” Organ. Behav. Hum. Decis. Process. 62(2), 198–215 (1995).
[Crossref]

Phys. Lett. A (2)

K. Aihara, T. Takabe, and M. Toyoda, “Chaotic neural networks,” Phys. Lett. A 144(6-7), 333–340 (1990).
[Crossref]

P. Ashwin, J. Buescu, and I. Stewart, “Bubbling of attractors and synchronisation of chaotic oscillators,” Phys. Lett. A 193(2), 126–139 (1994).
[Crossref]

Phys. Rep. (1)

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469(3), 93–153 (2008).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (7)

F. Sorrentino and E. Ott, “Network synchronization of groups,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5), 056114 (2007).
[Crossref] [PubMed]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

B. B. Zhou and R. Roy, “Isochronal synchrony and bidirectional communication with delay-coupled nonlinear oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026205 (2007).
[Crossref] [PubMed]

K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 056211 (2011).
[Crossref] [PubMed]

J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014).
[Crossref] [PubMed]

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

J. Tiana-Alsina, K. Hicke, X. Porte, M. C. Soriano, M. C. Torrent, J. Garcia-Ojalvo, and I. Fischer, “Zero-lag synchronization and bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(2), 026209 (2012).
[Crossref] [PubMed]

Phys. Rev. Lett. (13)

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref] [PubMed]

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. 105(26), 264101 (2010).
[Crossref] [PubMed]

R. Roy and K. S. Thornburg., “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72(13), 2009–2012 (1994).
[Crossref] [PubMed]

M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Phys. Rev. Lett. 76(11), 1804–1807 (1996).
[Crossref] [PubMed]

L. Kocarev and U. Parlitz, “Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems,” Phys. Rev. Lett. 76(11), 1816–1819 (1996).
[Crossref] [PubMed]

L. M. Pecora and T. L. Carroll, “Master stability functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
[Crossref] [PubMed]

C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013).
[Crossref] [PubMed]

F. Rogister, K. S. Thornburg, L. Fabiny, M. Möller, and R. Roy, “Power-law spatial correlations in arrays of locally coupled lasers,” Phys. Rev. Lett. 92(9), 093905 (2004).
[Crossref] [PubMed]

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106(22), 223901 (2011).
[Crossref] [PubMed]

D. M. Abrams and S. H. Strogatz, “Chimera states for coupled oscillators,” Phys. Rev. Lett. 93(17), 174102 (2004).
[Crossref] [PubMed]

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108(21), 214101 (2012).
[Crossref] [PubMed]

A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis, “Localized synchronization in two coupled nonidentical semiconductor lasers,” Phys. Rev. Lett. 78(25), 4745–4748 (1997).
[Crossref]

Prog. Theor. Phys. (1)

Y. Kuramoto, “Scaling behavior of turbulent oscillators with non-local interaction,” Prog. Theor. Phys. 94(3), 321–330 (1995).
[Crossref]

Rev. Mod. Phys. (2)

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993).
[Crossref]

S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Critical phenomena in complex networks,” Rev. Mod. Phys. 80(4), 1275–1335 (2008).
[Crossref]

Supplementary Material (1)

NameDescription
» Visualization 1: MP4 (2579 KB)      Mapping of the pairwise max-CC of the 16-SLs' network

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Figures (7)

Fig. 1
Fig. 1 Full-mesh-type network with optically-coupled 16 SLs. (a) Laser network topology: PC: Polarization controller, 1x2 and 1x8: optical couplers, EDFA: 25dB-gain Erbium-doped fiber amplifier, OF: Optical filter, PM: Inline optical power monitor, ATT: Optical attenuator. (b) Monitoring stage: It includes optical signal detection from laser's output port in an optical spectrum analyzer (OSA), as well as electrical signal detection in a four-channel real-time oscilloscope (OSC) - by employing equal number of digital photoreceivers - and in a radio-frequency analyzer (RFA). (c) Biasing/temperature conditions of the uncoupled solitary 16 SLs (#: laser identification number) for identical wavelength emission (λ = 1549.600) and for different levels of near-threshold optical power emission (black rectangles: PL#em = −15dBm, red circles: PL#em = −20dBm, blue triangles: PL#em = −25dBm).
Fig. 2
Fig. 2 Effect of coupling strength on the correlated emission of a 16-laser coupled network. Averaged-CC values between two pairs of SLs (L#1-L#2 and L#6-L#7) vs. the applied injection ratio, when the emitted power from the lasers is set to −15dBm. Timetraces in insets show the dynamics of the emitted signals for different coupling strengths. Gray region indicates synchronized network coupling conditions through chaotic signals.
Fig. 3
Fig. 3 Temporal evolution of L#1 and L#2 emission, as well as their difference in a coupled 16-laser network, when R = 0.2dB. De-synchronziation events appear when power dropouts occur and are minimized for optimal operating and coupling conditions. In the right column, a detail in the temporal region where a de-synchronization event takes place is provided.
Fig. 4
Fig. 4 Cross-correlation mapping of the experimentally built 16-laser coupled network under synchronized conditions. Max-CC among pairs of lasers, considering (a) the detection-limited bandwidth emitted signals (8GHz), and (b) the filtered emitted signals with 2GHz bandwidth. The power of lasers' emissions is −15dBm, frequency detuning is minimized and R = 0.2dB.
Fig. 5
Fig. 5 Effects of a frequency detuned laser (L#16) in correlated emission and NPE of emitted signals, in a 16-laser coupled network. (a) Pairwise average-CC evaluation of L#14, L#15 and L#16, for (a1) low-level (R = −12dB) and (a2) high-level (R = 0.2dB) injection ratios. Low injection conditions establish a highly sensitive system to wavelength emission mismatches, while high injection conditions establish a consistently synchronized system. (b) Normalized permutation entropy mapping of a laser with fixed conditions in the network (L#14) vs. ordinal pattern and applied delay, for: (b1) R = −12dB and frequency matched L#16, (b2) R = −12dB and L#16 with frequency detuning of 3.5GHz, (b3) R = 0.2dB and frequency matched L#16, (b4) R = 0.2dB and L#16 with frequency detuning of 17.6GHz. Contour plots are used for better transient screening.
Fig. 6
Fig. 6 Cluster synchronization in an 8-SL coupled network configuration. Cross-correlation mapping with (a) zero-detuned wavelength laser emission, and (b) with cluster synchronization among two quartets of lasers (L#1-L#4 and L#5-L#8) that are 50pm spaced in wavelength, when R = 0.8dB. (c) Timeseries of laser emission (L#1: yellow, L#2: green, L#6: purple, L#7: pink) for the cases of (a) (top) and (b) (bottom). Y-axis shift on timeseries is applied only for screening purposes. (d) Averaged-CC between lasers vs. the injection ratio level (two lasers from each cluster are monitored), for the cluster synchronization conditions of case (b). The optical power of all lasers' emission is −15dBm.
Fig. 7
Fig. 7 Synchrony comparison in an 8-node coupled network that includes 7 identical SLs and 1 SL from a different manufacturer. The comparison is made between the worst performance among the 7 identical lasers (black rectangles) and the best performance of synchronization between the different SL and the 7 identical lasers (red circles), versus the different SL’s emitted optical power and for (a) R = −9dBm and (b) R = −0.5dBm. All uncoupled identical lasers emit optical power −15dBm, while all uncoupled lasers operate at λ = 1549.600nm.

Equations (6)

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R L# = C 2 P inj tot P em L#
SNR= σ chaoti c signal 2 σ syste m noise 2
d E i (t) dt =jΔ ω i E i (t)+ 1 2 (1+ja)[ G i (t) t ph 1 ) E i (t)+κ m E m (t τ im ) e i ω 0 τ im + D ξ(t)
d N i (t) dt = I e N i (t) t s G i (t) | E i (t) | 2
G i (t)= g n [ N i (t) N 0 ] (1+s | E i (t) | 2 ) 1
H=H(P)= S(P) S max = i=1 D! p( π i )lnp( π i ) lnD!

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