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We experimentally and numerically investigated the chaos synchronization characteristics of mutually coupled semiconductor lasers (MCSLs) with asymmetrical bias currents. Experimental results show that, asymmetrical bias current level of two MCSLs has obvious influence on chaos synchronization between them, and stable leader-laggard chaos synchronization can be realized under relatively large asymmetrical bias current levels. Moreover, the influences of frequency detuning and mutually coupling strength between the two lasers on chaos synchronization performance have also been discussed. Theoretical simulations basically conform to our experimental observations.
©2011 Optical Society of America
1. Introduction
Synchronization phenomenon exists widely in physics, chemistry, biology, social science and many other fields, and has attracted much attention for years. In particular, synchronization of chaotic oscillations in coupled nonlinear laser systems has become a hot issue since the first prediction of chaos synchronization by Pecora and Carroll in 1990 [1]. Chaos synchronization between two lasers has been experimentally confirmed in solid state lasers [2], CO2 lasers [3] and semiconductor lasers (SLs) [4,5]. Generally, SLs easily generate rich nonlinear dynamics including chaos under one or more external perturbations such as current modulation, optical injection, optical feedback and optoelectronic feedback [6–9]. Up to now, SLs have been proven to be an excellent candidate to investigate the synchronization properties of coupled nonlinear systems.
At present, synchronization based on SLs system has been extensively studied [10–13]. Especially, chaos synchronization in a SL based system has exhibited tremendous application potential in high-speed secret communication because chaos oscillations based on SLs have some unique virtues such as high dimension and broad bandwidth. It is well known that a necessary condition to realize optical chaos secret communication is to achieve good chaos synchronization between the transmitter and receiver SLs. Previous works concentrated mostly on the unidirectional master-slave scheme. In such a configuration, a message is modulated onto or into the chaos carrier of a transmitter laser and recovered in virtue of chaos pass filter (CPF) effect at the receiver laser. The feasibility of this method has been confirmed by a field experiment in Athens [14]. However, such a unidirectional secret communication is dissatisfactory, and high-speed bidirectional or multidirectional secret communications always are highly expected.
In recent years, the mutually coupled semiconductor lasers (MCSLs) system has become a hot topic for its potential application in bidirectional chaos secret communication [15–26]. Compared with a unidirectional system, in a MCSLs system, two lasers is mutually injected and will interact on each other, which is a more complicated case. So far, it has been confirmed that two lasers of a MCSLs system can exhibit complex dynamics and can realize chaos synchronization. Generally, there exist two types of chaos synchronization in the face-to-face MCSLs configuration, namely isochronal synchronization and leader-laggard synchronization. For isochronal synchronization, it is usually difficult to achieve a stable synchronization due to the existence of random noise [20,21], and therefore much effort has been conducted for overcoming this drawback. For example, Klein et al. demonstrated that stable isochronal chaos synchronization can be obtained by adding feedback to two symmetrically mutually coupled lasers [22]. Englert et al. obtained stable zero lag synchronization by adding one or more coupling delay in the face-to-face configuration [23]. Jiang et al. theoretically obtained a stable isochronal synchronization by using a third driving laser in the MCSLs system [24]. Although above schemes can realize stable isochronal synchronization, some additional conditions are introduced into the system, which inevitably complicates the system structure. Compared with isochronal chaos synchronization, the leader-laggard chaos synchronization has good robustness to mismatched parameters. Michael et al. theoretically investigated the synchronization transitions from phase to lag synchronization in a system of two symmetrical coupled nonidentical oscillators [27]. As for leader-laggard chaos synchronization of MCSLs system, by introducing a partially transparent mirror between two lasers, high-quality leader-laggard chaos synchronization and bidirectional information exchange have been theoretically demonstrated [15]. Zhang et al. proposed an extremely asymmetrically mutually coupled system by adjusting two directional coupling coefficients, stable leader-laggard chaos synchronization was achieved and bidirectional chaos secret communication can be realized in theory [25]. Jiang et al. theoretically investigated the properties of the leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers and its application in chaos secret communication [26]. We have noticed that, on one hand, above mentioned schemes usually need extra elements such as external cavities [26] or a partially transparent mirror [15], which inevitably compromise the system practicability. On the other hand, most relevant investigations focused on the leader-laggard chaos synchronization only from a theoretical aspect, and the detailed experimental investigations are scarce. Usually, purely theoretical investigations can deal with various configurations if the feasibility of experimental realization is not taken into consideration. However, in practice, the experimental investigation may be more complicated and difficult compared with theoretical simulations due to equipments’ limitations and other influences from many unknown factors.
In this paper, we experimentally and theoretically focus on stable leader-laggard chaos synchronization in a simple MCSLs system by introducing different bias levels to two lasers, while no extra elements or other additional conditions are introduced. Such an asymmetrical bias current injection is relatively easily realized in experiments. We have investigated experimentally and numerically the effects of asymmetrical bias current level, frequency detuning and coupling strength on synchronization performance. Theoretical calculations are basically consistent with experimental results.
2. Experimental setup
Figure 1 is the experimental setup. Two InGaAsP/InP distributed feedback semiconductor lasers (DFB-SLs) with side-mode suppression ratio (SMSR)>50dB are used in this experiment. The two lasers are elaborately selected so that they have similar properties. They are driven by ultra-low-noise current sources (ILX-Lightwave, LDX-3620). Using temperature controllers (ILX-Lightwave, LDT-5412 with 0.01 K accuracy), the free-running wavelength of DFB-SLs can be adjusted. The output of SL1 is firstly collimated by an aspheric lens (AL1), and then is split into two parts via by a beam splitter (BS1). One part is injected into SL2 via a neutral density filter (NDF) and BS2, and the other part is sent to the signal detection part via an optical isolator (OI1) and a fiber coupler (FC1). In the signal detection part, a 12 GHz bandwidth photo-detector (PD, New Focus 1544-B) is used to convert the optical signal into an electrical signal. A 6 GHz digital oscilloscope (Agilent 54855A, 20 GS/s sample rate) is used to record the time series of the SL1 output. An optical spectrum analyzer (Ando AQ6317C) with a wavelength resolution of around 0.01nm is used to detect the optical spectra. Meanwhile, the output of SL2 experiences a similar process. During this experiment, the asymmetrical bias currents level of the two lasers is easily realized just by adjusting the bias currents of two lasers. The coupled delay time is fixed at about 3.5 ns. It should be pointed out that the coupling strength between the two lasers is very difficult to calibrate accurately in experiment. Because a NDF is used to adjust the coupling strength during experiment and its transmissivity is in direct proportion to the coupling strength, we use the transmissivity T of NDF to experimentally characterize the coupling strength and T is maintained at about 0.9 except particular description during experiment.
Fig. 1 Experimental setup. SL: semiconductor laser; AL: aspheric lens; BS: beam splitter; OI: optical isolator; PD: photo-detector; NDF: neutral density filter; FC: fiber coupler. The solid lines indicate optical paths, while the dashed lines indicate electronic paths.
3. Experimental results and discussion
The synchronization quality between two lasers is evaluated by following correlation coefficient C(Δt):
where P1 and P2 is the output power of SL1 and SL2, respectively, < > denotes the time average, Δt is the time shift. The correlation coefficient is bounded as −1C(Δt)1, where a larger value of |C(Δt)| means a better synchronization quality.When temperatures of two lasers are respectively stabilized at 24.30 °Cand 25.16 °C, free-running optical wavelengths of the two lasers are identical to be 1548.601 nm, and the threshold currents of solitary SL1 (Ith1) and SL2 (Ith2) are respectively measured to be 13.88 mA and 13.20 mA. It should be pointed out that during following experiments, the temperatures of the two lasers will be changed to adjust their oscillation frequencies. Under these circumstances, the thresholds of the two lasers will take according changes though such variations are relatively small within the varied range of the temperatures in experiments. For the convenience of characterizing the system asymmetrical extent, we use above two fixed Ith1 ( = 13.88mA) and Ith2 ( = 13.20mA) as reference thresholds in latter parts.
Figure 2 displays the power spectra of the two lasers and the correlation coefficient between the two lasers under three different asymmetrical bias levels, where Fig. 2(a) corresponds to I1 = 1.8Ith1 and I2 = 1.2Ith2, Fig. 2(b) corresponds to I1 = 1.6Ith1 and I2 = 1.55Ith2 and Fig. 2(c) corresponds to I1 = 1.2Ith1 and I2 = 1.8Ith2. From these power spectra (Figs. 2(a1)–2(c1)), one can observe that the two lasers have comparably dynamical bandwidths. From these correlation coefficient diagrams (Figs. 2(a2)–2(c2)), one can see that the asymmetrical bias level of two lasers have obvious influence on system synchronization performance. When the bias currents of two lasers have relatively large differences, this system can achieve stable leader-laggard chaos synchronization as shown in Figs. 2(a2) and 2(c2). For I1 = 1.8Ith1 and I2 = 1.2Ith2 (see Fig. 2(a2)), the maximal correlation coefficient Cmax = 0.91 locates at 3.5 ns, which indicates that the two lasers achieve a high-quality chaos synchronization and SL1 plays a role of leader to SL2 (laggard). This can be explained as follows. On one hand, the output power of SL1 is far larger than that of SL2. On the other hand, the coupling coefficients are approximately identical for two different coupling directions. Thus, this system becomes asymmetrical and SL2 is injection locked by a strong injection from SL1. Similarly, for the case of I1 = 1.2Ith1 and I2 = 1.8Ith2 (see Fig. 2(c2)), the maximal correlation coefficient locates at −3.5 ns, SL1 is injection locked by a strong injection from SL2 and plays a role of laggard to SL2 (leader). As for the approximately symmetrical bias level case of I1 = 1.6Ith1 and I2 = 1.55Ith2 (see Fig. 2(b2)), the correlation coefficient distributes approximately symmetrically at both sides of the zero time delay, which is easily comprehended.
Fig. 2 Power spectra of SL1 and SL2 (left column) and correlation coefficient between the two lasers (right column), where (a) I1 = 1.8Ith1, I2 = 1.2Ith2; (b) I1 = 1.6Ith1, I2 = 1.55Ith2; (c) I1 = 1.2Ith1, I2 = 1.8Ith2.
To specially describe the influence of the current asymmetric extent on the chaos synchronization performance, Fig. 3 displays the variation of maximal correlation coefficient Cmax with the bias current of SL1, where the bias current of SL2 is fixed at 1.1Ith2, 1.2Ith2, and 1.3Ith2, respectively. From this diagram, one can see that, the maximal correlation coefficient Cmax will firstly decrease with the increase of the bias current of SL1. After that, Cmax exhibits an increasing tendency accompanying with fluctuations and the chaos synchronization performance is improved on the whole. Finally, Cmax is maintained at a high synchronization level. Above variation processes can be explained as follows. For this firstly decreasing and then increasing phenomenon, there may exist in two different cases. One is that the bias current I1 of SL1 is gradually increased from I1< I2 to I1> I2 while the bias current I2 of SL2 is fixed at a relatively high level. Under this circumstance, the injection locked synchronization between the two lasers will undergo a conversion from injection locking of SL1 by SL2 to injection locking of SL2 by SL1, which may result in this firstly decreasing and then increasing process. The other case is that at the beginning, both the bias currents of the two lasers are close to their solitary thresholds. Under this condition, the two lasers may operate at low frequency fluctuations (LFFs) region and can also obtain good synchronization as shown in Fig. 4(a) (where I1 = 1.1Ith1 and I2 = 1.1Ith2) for an example. With gradual increase of the bias current of SL1, the two lasers may undergo a transition from a good LFFs synchronization to a good chaos synchronization as shown in Fig. 4(b) (where I1 = 1.8Ith1 and I2 = 1.1Ith2) for an example, which may induce that Cmax experiences a firstly decreasing and then increasing process. Further increasing the bias current of SL1, on one hand, the output power difference between the two lasers will reinforce the injection locking function of SL2 by SL1 and then is helpful to improve the chaos synchronization between the two lasers. On the other hand, this increasing asymmetric bias current level will enhance some differences in operation parameters including frequency detuning between the two lasers, which inevitably may lower the synchronization performance. Above two actions will maintain the synchronization of the two lasers at a high level though sometimes slight ripples may appear.
Fig. 3 Variation of maximal correlation coefficient Cmax with I1, where I2 is fixed at 1.1Ith2, 1.2Ith2 and 1.3Ith2, respectively.
Fig. 4 Temporal traces of the optical power of SL1 (upper) and SL2 (lower) when two lasers respectively operate at LFFs (a) for I1 = 1.1Ith1 and I2 = 1.1Ith2 and chaos oscillation (b) for I1 = 1.8Ith1 and I2 = 1.1Ith2, where the temporal trace of SL2 is vertically shifted lower for convenient of comparison with that of SL1.
It is well known that the synchronization of two lasers is a very complex process and is affected by many factors including external operation conditions, intrinsic parameters and their differences between two lasers. Moreover, changing external conditions will make some properties of a laser undergo some according variations. For example, changing the operation temperature of a laser will affect output power, oscillation frequency, threshold and some intrinsic parameters of this laser. Also, variations of bias current and external optical coupling strength can take similar effects. Therefore, it is necessary to investigate the influence of asymmetrical bias current level on the chaos synchronization performance under different operation conditions. Figure 5 shows the variation of Cmax with the bias current of SL1 for I2 = 1.2Ith 2, where curves A corresponds to temperatures of SL1 And SL2 are respectively stabilized at 24.80 °C and 25.55 °C, and curve B corresponds to temperatures of SL1 And SL2 are respectively stabilized at 24.30 °C and 25.16 °C. From this diagram, one can observe that the temperatures of the two lasers do affect the synchronization quality.
Fig. 5 Variation of maximal correlation coefficient Cmax with I1 for I2 = 1.2Ith2, where curve A corresponds to that temperatures of SL1 And SL2 are respectively stabilized at 24.80 °C and 25.55 °C, and curve B corresponds to temperatures of SL1 And SL2 are respectively stabilized at 24.30 °C and 25.16 °C.
Many previous works have demonstrated that frequency detuning Δf ( = f1-f2, where f1 and f2 are free-running frequencies of SL1 and SL2, respectively) between two lasers has an important influence on the synchronization quality [7,10]. Usually, the bias current and temperature of a laser are two major factors to affect oscillation frequency of the laser. In above experiments, temperatures of two SLs are fixed when the bias current of SL1 is varied in a large range. Since the change of bias current of a SL will result in an according variation in free-running frequency of the SL, the evolution process of synchronization performance with the current of SL1 should include the contributions of not only varied bias current of SL1 but also changed frequency detuning between two lasers. By adjusting operation temperature of SL1, the frequency detuning between two lasers can be maintained at a fix value even if the current of SL1 is adjusted in a wide range. In the following parts, we will investigate the synchronization performance in the MCSLs system under a fixed frequency detuning Δf when I 1 is varied continuously. For Δf is respectively fixed at 0 GHz, −5 GHz and −10 GHz via temperature controller, Fig. 6 gives the variation of Cmax with I 1. From the diagram, for a relatively low I1, the smaller Δf is, the better the synchronization quality will be, which is easily understood. While for a large I1, experimental results show that the synchronization quality under no frequency detuning is even worse than that with a frequency detuning. This is an interesting phenomenon and will be specially emphasized on the following.
Fig. 6 Variation of maximal correlation coefficient Cmax with I1, where I2 is fixed at 1.2Ith 2 and Δf is respectively fixed at 0 GHz, −5 GHz and −10 GHz.
To further explore the influence of frequency detuning on synchronization performance, for a determined asymmetrical bias currents level of two lasers, we will investigate the dependence of the maximal correlation coefficient Cmax on frequency detuning through adjusting the temperature of SL1. Figure 7 records the variation of maximal correlation coefficient Cmax with frequency detuning Δf, where SL1 and SL2 are biased at 1.8Ith1 and 1.2Ith2, respectively. From this diagram, one can see that, under such an extremely asymmetrical bias level, positive frequency detuning will degrade synchronization performance, while a suitable negative frequency detuning is helpful to improve synchronization performance. This may be explained as follows. The frequency detuning Δf is measured by the free-running oscillation frequency difference between two SLs, but the external optical injection will make the spectra of a solitary laser undergo a red shift. In this MCSLs structure, two lasers experience mutual injections and interact on each other. Because I1 is far larger than I2 during this experiment, the received injection intensity of SL2 is larger than that of SL1. As a result, Compared with the influence of SL2 on SL1, the influence of SL1 on SL2 plays a major role. For the case without frequency detuning between the two solitary lasers, frequency detuning will appear due to the introduction of mutual injection. Under the case of negative frequency detuning between the two solitary lasers, the frequency detuning value may be compensated due to mutual injection and the synchronization quality may be improved sometimes. While for positive frequency detuning, the introduction of mutual injection will enlarge detuning degree.
Fig. 7 Variation of maximal correlation coefficient Cmax with detuning frequency under I1 = 1.8Ith1, I2 = 1.2Ith2.
Finally, via by changing transmissivity T of NDF, we investigate the influence of mutual injection strength on synchronization performance. Figure 8 shows the variation of Cmax with the NDF transmissivity T for I 1 = 1.8Ith 1, I 2 = 1.2Ith 2 and Δf = −5 GHz. From this diagram, one can see that under above operation conditions, increasing coupling strength will be helpful to improve the synchronization quality. The reason may be due to that increasing coupling injection means to reinforce the injection locked synchronization.
Fig. 8 Variation of maximal correlation coefficient Cmax with transmissivity T of NDF for I1 = 1.8Ith1, I2 = 1.2Ith2 and Δf = −5 GHz.
4. Theoretical simulations
The mutually coupled semiconductor lasers system can be modeled by following well known Lang–Kobayashi rate equations:
where the subscripts 1, 2 stand for the two lasers, respectively. E is the slowing varying complex electric field, N is the carrier number, α is the linewidth enhancement factor, τp is the photon lifetime, τs is the carrier lifetime, f is the free-running laser frequency, Δf ( = f 1-f 2) is the detuned frequency between two lasers, τin is the round-trip time in the laser cavity, k is the coupling coefficient, τ is the coupling delay time, I is the bias current, e is the charge of an electron, β is the spontaneous emission factor, ξe is the gauss white noise, G is the optical gain in the laser cavity and can be expressed aswhere g is the optical gain coefficient, s is the gain saturation coefficient, N0 is the transparent carrier number. The rate Eqs. (2)–(4) can be numerically solved by the fourth-order Runge-Kutta method. During the calculations, the used data are: α1 = 3.4, α2 = 3.5, g1 = 9.4 × 103s−1, g2 = 8.4 × 103 s−1, τp1 = 2.4 ps, τp2 = 2.7 ps, τs1 = 1.8 ns, τs2 = 2 ns, τin = 8 ps, s = 1 × 10−7, Ν01 = 1.1 × 108, Ν02 = 1.2 × 108, β = 1 × 103 s−1, τ = 3.5 ns. Based on above data, the threshold currents of solitary SL1 and SL2 are respectively calculated to be 13.8 mA and 13.1 mA.Figure 9 simulates the correlation coefficient between two lasers under different bias currents. From this diagram, one can see that theoretical simulation basically accords to experimental observation of Fig. 2. Figure 10 calculates the variation of Cmax with the bias current I 1 for I 2 = 1.2Ith 2 and different Δf. Compared this calculated results with experimental results of Fig. 6, a similar tendency can also be observed.
Fig. 9 Simulated correlation coefficient between SL1 and SL2, where (a) I1 = 1.8Ith1, I2 = 1.2Ith2, k = 0.4; (b) I1 = 1.6Ith1, I2 = 1.55Ith2, k = 0.4; (c) I1 = 1.2Ith1, I2 = 1.8Ith2, k = 0.4.
Fig. 10 Variation of maximal correlation coefficient Cmax with I1 for I2 = 1.2Ith2, where the detuning frequency between two lasers are respectively fixed at 0 GHz, −5 GHz and −10 GHz.
Above experiments demonstrated that frequency detuning between two lasers extensively affects the synchronization performance. Figure 11 simulates Cmax under different frequency detuning for I 1 = 1.8Ith 1, I 2 = 1.2Ith 2 and k = 0.4. Theoretical results confirm our experimental observations as shown in Fig. 7.
Fig. 11 Variation of maximal correlation coefficient Cmax with frequency detuning between two lasers for I1 = 1.8Ith1, I2 = 1.2Ith2 and k = 0.4.
For establishing a comparison to Fig. 8, Fig. 12 shows the variation of Cmax with coupling coefficient k for I1 = 1.8Ith1, I2 = 1.2Ith2 and Δf = −5 GHz. Compared this diagram with Fig. 8, a similar variation tendency can be observed except that the theoretical results appear a saturation process. Such a saturation process may be explained as follows. On one hand, increasing injection will reinforce the injection locking, which is helpful to improve the chaos synchronization between the two lasers. On the other hand, this increasing injection will enhance the gain saturation effect, which has been considered in rate equations. The joint action of above two effects may result in our theoretical results. Experimentally, due to the limitations of experimental setup, mutual coupling between two lasers may not attain a relatively high degree.
Fig. 12 Variation of maximal correlation coefficient Cmax with coupling coefficient k for I1 = 1.8Ith1, I2 = 1.2Ith2 and Δf = −5 GHz.
To further specify the influences of frequency detuning and coupling strength on synchronization performance, Fig. 13 integrally simulates Cmax in the parameter space of detuning frequency and coupling coefficient under I1 = 1.8Ith1 and I2 = 1.2Ith2. From this diagram, an optimum region with good synchronization quality can be defined.
Fig. 13 Map of maximal correlation coefficient Cmax as function of detuning frequency and coupling coefficient.
5. Conclusions
In summary, the chaos synchronization characteristics of MCSLs with asymmetrical bias currents have been experimentally and numerically investigated by respectively adjusting the bias currents of two lasers. Experimental results demonstrate that such an asymmetrical system can realize good stable leader-laggard chaos synchronization due to the injection locking mechanism. Increasing the mutual injection strength will reinforce the injection locked synchronization and then improve the system synchronization performance. Under some certain circumstances, positive frequency detuning will degrade the synchronization quality, while suitable negative frequency detuning is helpful to improve the synchronization performance to some extent. Theoretical simulations basically conform to our experimental results.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant Nos. 60978003, 61078003 and 11004161, and the Open Fund of the State Key Lab of Millimeter Waves.
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