1. Introduction

In spite of the extreme sensitivity of chaotic oscillators to small perturbations, it has recently been demonstrated theoretically and experimentally that a network of mutually time-delay coupled identical chaotic units can synchronize. A variety of synchronization types have been observed for coupled chaotic oscillators including leader-laggard type synchronization and the more interesting case of zero-lag synchronization (ZLS) in which the chaotic oscillation are simultaneously synchronized in spite of an arbitrarily large physical distance between the oscillators. There are two classes of zero-lag synchronization: ZLS of all units within the network, and sub-lattice synchronization, where the network units are split into two clusters, each in ZLS, with no synchronization between the clusters [1–3]. Similar classes of synchronization were recently experimentally observed for phase dynamics of laser networks as well as for neuronal circuits [4, 5]. Here we introduce a generalization of zero-lag synchronization, generalized ZLS, for which the synchronized signals are as highly correlated with each other as in ZLS, despite allowing different feedback delay times for the individual oscillators. In generalized ZLS the correlation as a function of delay time has the same symmetry and features as in ZLS, but with the origin of the time axis shifted, as will be shown here.

Chaotic semiconductor (SC) lasers have recently been shown to be an extremely useful system for the generation of broadband physical random numbers [6, 7]. In addition, synchronization of networks of coupled chaotic lasers, separated by distances much greater than the individual laser coherence length, is the basis of several recently proposed broadband secure communication protocols in private and public channel schemes [8–10]. There are two fundamental issues, however, that need to be resolved before such lasers can be used in a communication network. ZLS and sublattice synchronization have to be experimentally demonstrated for small networks configured in various geometries, where the chaos emerges due to optical feedback, as opposed to the recently demonstration emergence of chaos via external optoelectronic feedback [11–14]. The second issue is the constraining limitation of having to use precise and specific ratios among delay times of the network [15, 16] for synchronization to occur. Here we demonstrate various generalized types of synchronization for classes of geometric configurations in small networks for which the precise delay-time requirements have been eliminated.

2. Generalized zero-lag synchronization

In our most basic small network block, two Fabry-Perot semiconductor (SC) diode lasers, LD_A and LD_B, operating near 656 nm, are mutually coupled as well as possessing self feedback as shown in Fig. 1(a) . Signals from each of the lasers are coupled using a 50/50 beam splitter (BS), where τ_a (green) and τ_b (red) denote the optical delay times to the BS from LD_A and LD_B, respectively. The combined signal is coupled into an optical fiber which extends the optical delay by τ=1005 ns (brown) and is retro reflected by mirror, M. A BS near each laser samples a small portion of the beam into a fast (50 GHz bandwidth) photodetector. The dc output, measured via a 40 GHz bandwidth bias-T, measures the average dc power of the lasers, while the ac currents are digitized by a 12 GHz bandwidth, 40 GS/s digital oscilloscope (Tektronix TDS 6124C). Two natural density (ND) filters, consisting of a half-wave plate and two polarizing BSs are used to compensate unavoidable deviations from ideal 50/50 splitting and to ensure that each laser receives the same feedback intensity. The two lasers are temperature tuned to nearly identical wavelengths while their injection current to threshold current ratios are maintained nearly equal at p = I/I_th = 1.2. When subjected to feedback, the lasers display chaotic behavior, consisting of very short and random spiking of the laser intensity [17].

 

Fig. 1 (a) Experimental setup: two mutually coupled SC lasers, LD_A and LD_B. PD, photodiode; M, mirror. τ_a and τ_b are the optical delay times between LD_A and LD_B and the coupling beam splitter, BS, respectively. τ indicates the optical delay between the BS and mirror, M. (b) Schematic diagram of the experimental configuration. (c) Shifted cross correlation between the chaotic intensities of LD_A and LD_B. ZLS for identical self feedback and mutual coupling delays, τ_a = τ_b (light blue), and generalized ZLS for τ_a≠τ_b (dark blue; τ_a-τ_b~-1.95 ns) (d) Shifted cross correlation around the mutual delay time between the two lasers. (e) A generalized star configuration of n SC lasers. The output intensity of the lasers with unequal optical delay times, {τ_k}, is combined with equal weight and reflected by a passive mirror.

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The experimental setup is depicted schematically in Fig. 1(b), where the mutual coupling delay time is T_M = 2τ + τ_a + τ_b and self-feedback delay times are T_A = 2(τ + τ_a) and T_B = 2(τ + τ_b). Such a geometry obeys the simplest necessary sum rule for synchronization, where the sum of the two self-feedback delay times is equal to twice the mutual delay time while τ_a, τ_b and τ can be independently varied [14, 18, 19].

The intensity correlation between the two lasers is calculated from the correlation of, 10 ns long time segments from each detector, averaged over all such segments within an observation time of 4 μs. The sampling time and step size in the cross correlation calculation was 25 ps. Low frequency fluctuations (LFFs) [20 ,21], which result in temporal synchronization breakdowns, occur on time scales greater than the overall feedback time, which in this case was ~2 μs. We could thus choose data streams which did not contain LFFs. By tuning the delay times between the lasers and coupling BS to be equal, τ_a = τ_b, ZLS with cross correlation, ρ = 0.96, is achieved, Fig. 1(c) (light blue). The correlation repeaks at integer multiples of the round trip time, 2(2τ + τ_a + τ_b), the first pair of which are shown by the light blue trace in Fig. 1(d).

Theoretical analysis of our experimental configuration under the constraint of Eq. (1), for any distance of the two lasers to the BS, predicts a generalized ZLS correlation peak at Δτ = τ_b-τ_a [18]. In other configurations, where the constraint of Eq. (1) is fulfilled by tuning all three delays, it is difficult to observe a generalized ZLS because each detuning has to be precise with a precision approximately equal to the coherence length of the lasers [15, 16], as well as the constraint of equal feedback intensity to the two lasers [1, 17]. By measuring the cross correlation of the lasers when the self-feedback is blocked, we determine τ_a-τ_b~-1.95 ns. Indeed, when the two lasers are coupled as in Fig. 1(a), the cross correlation between the two lasers revealed a generalized ZLS, ρ = 0.95, at Δτ~-1.95 ns, Fig. 1(c) (blue). The notable oscillations with period ~0.3 ns in the cross correlation are attributed to the relaxation oscillation frequency of the SC lasers. Similar oscillations were also observed in simulations of the Lang-Kobayashi equations [22], especially at low coupling intensities. In the experiment the feedback is weak, due to various losses in the optics and the fiber, and this leads to the strong relaxation oscillation shown in Fig. 1(c). Note that synchronization is achieved for delays much larger than internal time scales of the semiconductor laser as the network is operating in the weak chaos regime [3, 23–25].

A laser with self-feedback is quasiperiodic in its intensity fluctuations with a period equal to the feedback delay time. For lasers A and B this occurs at times T_A,n = 2(τ+τ_a)n and T_B,n = 2(τ+τ_b)n, where n is an integer. Because the two lasers are synchronized, both periodicities must be present in the common synchronized fluctuations of the lasers. As a consequence the correlation has additional peaks at time shifts ±T_A,n and ±T_B,n, with respect to the generalized ZLS peak (Fig. 1(d))_. The quasi-periodicity at these two frequencies produces sidebands at ±(T_A,n-T_B,n) = ±2(τ_a-τ_b), which are observed surrounding both the generalized ZLS peak (Fig. 1(c)) and the recurring correlations at ±T_A,1 and ±T_B,1 (Fig. 1(d)). Twice the mutual feedback repetition time is precisely equal to the sum of the two self-feedback times, as in Eq. (1), which insures that all correlation peaks are at the same time shifts.

3. Generalized star configuration

The experimental scheme for generalized ZLS, without requiring precise matching of self-feedback and mutual delays, can be generalized to multiple user synchrony. A generalized star configuration [26], of n SC lasers is shown in Fig. 1(e), where the output intensity of all the lasers are combined and mixed with equal weight and upon reflection, redistributed back to the entire network but with unequal optical delay times, {τ_k}. Figures 1(a)-1(b) is a special case of this geometry with n = 2. Each pair (m,k) in the general configuration thus has generalized ZLS at a time shift corresponding to the delay time difference between the pair, τ_m-τ_k. The solutions of the time shifts for all possible pair combinations are consistent with each other even when multiple traverses are allowed. For instance, the time shift for pair (m,k), τ_k-τ_m, is identical to the sum of all possible delays in getting from m to k; (m,m + 1), (m + 1,m + 2), …, (k-1,k).

This multiple user synchrony was confirmed in simulations of the Lang-Kobayashi equations [22], up to n=7 and might be a source for abundant multiple-user communication and secure communication protocols.

4. Generalized sub-lattice synchronization

A modification of the geometry of Fig. 1(a), allows for a sub-lattice configuration, in which four lasers are coupled without self-feedback, as shown in Fig. 2(a) . The mirror, M, of Fig. 1(a) is replaced by two additional SC lasers, LD_C and LD_D, with optical delay to their coupling BS, τ_c and τ_d, respectively. The delay time τ between the two coupling BSs is arbitrary and can be made as long as necessary by insertion of a fiber, though in the experiment described below it was maintained at a free space value of a few ns. The resulting system has quadrilateral geometry and is shown schematically in Fig. 2(b). It is important to note that the four independent delays (τ_a,τ_b,τ_c,τ_d) in this geometry do not form an unconstrained quadrilateral, since the difference between the delays (D,A)-(A,C) has to be equal to (D,B)-(B,C).

 

Fig. 2 (a) Experimental setup of four SC lasers in a quadrilateral geometry. The optical time delays between lasers A and B (C and D) and their coupling beam spliter is τ_a and τ_b (τ_c and τ_d), respectively_. τ is the the optical delay between the two coupling BSs. (b) Schematic diagram of the quadrilateral experimental setup. (c) Shifted cross correlation of the chaotic intensities between lasers A and B (light blue; time shift ~-0.03 ns) and C and D (purple; time shift ~-0.195 ns), indicating generalized ZLS between diagonal pairs.

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For the limiting case, τ_a=τ_b=τ_c=τ_d, the geometry reduces to a square, for which sub-lattice synchronization was theoretically predicted, with the four lasers split into diagonal pairs, (A,B) and (C,D), which are ZLS [27], while the correlation between lasers belonging to adjacent pairs (A,C; C,B; B,D; D,A) is characterized by attenuated correlations at multiples of the delay time between the lasers, ±(2τ_a+τ), similar to a face-to-face configuration of two lasers.

Here we experimentally observe generalized sub-lattice synchronization for the general quadrilateral geometry. Figure 2(c) shows the measured shifted cross correlation, with ρ = 0.93 between LD_A and LD_B at a shifted time τ_a-τ_b ~-0.03 ns and ρ = 0.94 between LD_C and LD_D at a shifted time of τ_c-τ_d ~-0.195 ns. Additional attenuated correlation peaks between the diagonal lasers pairs occur at ±37.28 ns around the central generalized ZLS peaks. The adjacent laser pairs, however, have no correlation near zero time delay but have correlation (not shown) at times corresponding to an effective face to face delay time as described below. Note that the background correlation in Fig. 1(c) is slightly above zero because for these experimental parameters LFFs occur at times > ~100 ns and in order to avoid data segments containing LFFs only the highest 50% of the correlation segments are averaged.

In order to understand the emergence of the generalized sub-lattice synchronization we performed simulations using the Lang-Kobayashi equations where the explicit equations and their parameters are given in [20, 22], and p=I/I_th=1.2, similar to the experimental operating parameters. The time delays for the simulated quadrilateral geometry were arbitrarily selected to be τ_a=2 ns, τ_b=4 ns, τ_c=3 ns, τ_d=7 ns and τ=11 ns, thus yielding a ZLS delay between laser A and B, τ_a-τ_b=2 ns002C and between lasers C and D, τ_c-τ_d=4 ns, Fig. 3(a) . To achieve a better understanding of this type of generalized synchronization, the quadrilateral geometry is transformed into a square, Fig. 3(b), in the following way: Since we know that laser D has a generalized ZLS with laser C of 4 ns we move along the laser emission line of laser D by 4 ns towards A and B, to create a virtual laser, that would be in ZLS with C. Similarly we follow the laser B emission to a virtual position by moving it 2 ns towards C and D so that virtual laser B is now in ZLS with laser A. We have thus created a square configuration, indicated by the dashed line in Fig. 3(b). From sub-lattice synchronization in a square geometry [27], we can easily determine that the revival of the correlation will take place at a time shift of ±32 ns around the central generalized ZLS peaks, corresponding to the total delay between pairs (A,B) and (C,D). Since our two virtual lasers lie along the space-timeline of the real laser configuration, the shifted correlations are the same for both the constructed square and real configurations.

 

Fig. 3 (a) Configuration for numerical simulation with τ_a=2 ns, τ_b=4 ns, τ_c=3 ns, τ_d=7 ns and τ=11 ns. Pairs of lasers with the same color (light blue and purple) are generalized ZLS as indicated by the relative time in ns units. (b) The transformation from a quadrilateral geometry to a square. Dashed lines represent the effective time delays after compensating for the synchronization time shift. (c) Calculated shifted cross correlation of the chaotic intensities of the quadrilateral geometry, Fig. 3(a).

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The generalized ZLS between pairs (A,B) and (C,D) was confirmed in the simulations as shown in Fig. 3(c). The simulations also show the attenuated revival correlation peaks between diagonal pairs at a time shift of ±32 ns around the central generalized ZLS peaks. The cross correlation between adjacent lasers, exemplified by A and D in Fig. 3(c), is similar to a face-to-face configuration where attenuated peaks are observed at ± 16 ns around the 4 ns generalized ZLS of laser D.

5. Equilateral triangle – smallest network with ZLS

To complete the set of experimentally examined small networks, we turn to the smallest network which exhibits ZLS in the absence of self-feedback; an equilateral triangle, which was previously examined in the presence of self-feedback [28]. The experimental setup, Fig. 4(a) , is similar to that of Fig. 1(a), where the mirror is replaced by a third laser and for the case of an equilateral triangle, the optical fiber is removed. The system is schematically presented in Fig. 4(b) and the delay times are tuned to be equal, τ_a=τ_b=τ_c. Experimental results, with p = 1.1, show ZLS between all three pairs of lasers, ρ_AB = 0.93, ρ_BC = 0.92, ρ_AC = 0.91 at zero time shift, as shown in Fig. 4(c), consistent with simulations.

 

Fig. 4 (a) Experimental setup of three SC lasers in an equilateral triangle. Each laser is bidirectionally coupled to two adjacent lasers where τ_a=τ_b=τ_c. (b) Schematic diagram of the experimental setup. (c) Shifted cross correlation of the chaotic intensities of all three pairs of lasers indicating ZLS.

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6. Conclusion

In conclusion, we experimentally demonstrate generalized ZLS in small networks consisting of time-delay coupled chaotic SC lasers, without the limiting requirement of precisely matched delay times. For a quadrilateral geometry, sub-lattice synchronization is observed in agreement with numerical results. Depending on the specific geometric configuration, generalized ZLS and ZLS can be observed. Numerical results and theoretical arguments suggest the applicability of these experimental results to other configurations and larger networks and open a route to new multi-user communication schemes which can be of great significance to modern communication networks.