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Based on a linear chain composed of a central semiconductor laser and two outer semiconductor lasers, chaos synchronization and bidirectional communication between two outer lasers have been investigated under the case that the central laser and the two outer lasers are coupled mutually, whereas there exists no coupling between the two outer lasers. The simulation results show that high-quality and stable isochronal synchronization between the two outer lasers can be achieved, while the cross-correlation coefficients between the two outer lasers and the central laser are very low under proper operation condition. Based on the high performance chaos synchronization between the two outer lasers, message bidirectional transmissions of bit rates up to 20 Gbit/s can be realized through adopting a novel decoding scheme which is different from that based on chaos pass filtering effect. Furthermore, the security of bidirectional communication is also analyzed.
©2011 Optical Society of America
1. Introduction
Chaos synchronization has become a hot topic since the pioneering work of Pecora and Carroll [1] for its potential application in secure communication, neural network, and public channel cryptography [2–37]. In recent years, chaos synchronization and secure communication based on semiconductor lasers (SLs) have attracted much more attention because the chaotic carrier generated by SLs possesses some unique advantages such as wide bandwidth, high complexity, and easy to implement. It is well known that a necessary condition to realize optical chaos secret communication is to achieve good chaos synchronization between the transmitted and received SLs. Previous works concentrated mostly on the unidirectional scheme [4–13]. The feasibility of this method has been confirmed by a field experiment in Athens [5]. However, such a unidirectional secret communication is inadequate, and bidirectional or multidirectional secret communications are always highly expected. As a result, some chaos synchronization configurations allowing for bidirectional chaotic secure communication have been proposed and investigated [14–32].
Generally, chaos synchronization between two SLs in secure communication system can be realized by adopting two kinds of different methods. One is by coupling unidirectionally or mutually one SL with another SL to synchronize them, and the other is by the aid of a third SL or optical injection of a common noise (or chaotic signal) to synchronize two independent SLs (i. e., there exist no direct mutual action between the two synchronized SLs). For the chaotic secure communication based on the first synchronization method, the message is transmitted between two lasers (transmitted and received laser) with direct coupling, and the encoding and decoding of message is accomplished as follows: the message is modulated onto or into the chaos carrier of the transmitted laser, and the received laser synchronizes with the chaos produced in the transmitter but the masked message amplitude is much reduced in the receiver. This effect is termed the chaos pass filtering (CPF) [6]. Thus, when the difference is taken between the input of the receiver and the output of the receiver, the transmitted message may be extracted. Especially, in a mutually coupled system, the two lasers play simultaneously both roles of transmitter and receiver. For the chaotic secure communication based on the second synchronization method, the message is transmitted between two independent SLs. Due to no direct coupling between two independent SLs, a second path (named as the message channel) should be introduced for messages communication besides the coupling path (named synchronized channel) used to synchronize two independent SLs. Since message decoding is accomplished in the message channel and do not need to utilize CPF effect, communication rate of the message is not limited by CPF. It should be specially pointed out that, for the second method, the cross-correlation coefficients between the chaotic carriers in message channel and the chaotic signal in synchronized channel are expected as low as possible for seeking high security of communication.
In this paper, based on previous relevant investigations on the synchronization scheme between two independent SLs [28–32], a system configuration allowing for bidirectional chaos communication is proposed. After having determined the optimal operation parameters, we emphasize on exploring the bidirectional communication performance and the security of this system.
2. System setup
The schematic of system setup is illustrated in Fig. 1 . Three SLs including two outer semiconductor lasers (OSLs) and one central semiconductor laser (CSL) are used. The two OSLs are mutually coupled with the CSL whereas there is no coupling between two OSLs. Under proper condition, all the three SLs can be operated at chaos states. The encoding and decoding process of two messages are accomplished as follows: message 1 (m1) is encoded into the chaotic carrier emitted from OSL1 by using chaos modulation (CM) method [33]. For this modulation scheme, the modulated signal is equal to (1 + η × m(t)) times of the transmitter output, where η is the amplitude modulation index and m(t) is the normalized signal waveform. The modulated signal is divided into two parts by a beam splitter (BS1) at first. One part is coupled into CSL, and the other part is divided into two parts again by BS2 after passing through an optical isolator (OI1). One part output from BS2 is injected into photodetector 2 (PD2) to assist the decoding of message 2 (m2), while the rest part is transmitted toward the OSL2 side and injects into PD3 through message channel F12 but does not enter into OSL2 because of the existence of OI2. Meantime, a part of signal including m2 and chaotic output from OSL2 injects into PD4. Through comparing the output of PD3 and PD4, the difference between m1 and m2 can be obtained under the case that high-quality chaos synchronization between two OSLs has been achieved. Based on this difference, the message m1 can be decoded as long as the message m2 is known. The encoding and decoding process of m2 is the same as that of m1. As shown in Fig. 1, the messages do not enter into OSLs, therefore the principle of messages decoding is different from that based on chaos pass filtering (CPF) effect.
Fig. 1 Schematic of system setup. OSL: outer semiconductor laser; CSL: central semiconductor laser; M: mirror; BS: beam splitter; OI: optical isolator; PD: photodetector; F: communication channel; m: message.
Dynamics of mutually coupled SLs can be modeled by well known Lang–Kobayashi rate equations after taking mutual coupled terms into account [16,29]. For OSL1 and OSL2:
For CSL: where the subscripts C, 1 and 2 stand for CSL, OSL1 and OSL2, respectively, E is the complex amplitude of the optical field and N represents the carrier number. |…| denotes the amplitude of the complex field. KC1, KC2 are the coupling strengths from CSL to OSL1 and to OSL2, respectively, τC1, τC2 are the corresponding coupling delay. K1C, K2C are the coupling strengths from OSL1 to CSL and from OSL2 to CSL, respectively, and τ1C, τ2C is the corresponding coupling delay time. I is the pump current, τp is the photon lifetime, τe is the carrier lifetime, β is the spontaneous emission rate, α is the line width enhancement factor, f is the oscillated frequency of laser at free-running, and e is the electronic charge constant. The nonlinear gain function G is given by G = g(N-N0)/[1 + ε |E |2], where g is the differential gain, ε is the saturation coefficient, and N0 is the carrier number at the transparency. The frequency detuning between CSL and OSLs is defined as ΔfC1, C2 = fC-f1, 2, and the frequency detuning between OSL1 and OSL2 defined as Δf12 = f1-f2, χ is Gaussian white noise sources with zero-mean and correlation [34].The quality of chaos synchronization and its time shift can be quantified by calculating the cross-correlation coefficient Cij(Δt) [35]:
where the subscripts i, j ( = 1, 2, c) stand for different lasers. The bracket < > denotes the time average, Δt is the time shift, and P(t) is the output intensity and is equals to |E(t)|2. |Cij|ranges from 0 to 1, |Cij| = 1 enables a perfect synchronization. If the corresponding time shift of the maximum of cross-correlation Δtmax is positive, laser i is leading to laser j by Δtmax and vice versa.The performance of a communication system is quantitatively valued by using Q-factor. Here, the Q-factor is defined as [36]:
where 〈P1 〉 and 〈P0 〉 are the average optical powers of bits “1”and “0”, respectively, while σ1 and σ0 are the corresponding standard deviations.3. Results and discussion
Equations (1)–(4) can be numerically solved by using the fourth-order Runge-Kutta algorithm. During the calculations, the initial conditions of three SLs are set differently to investigate if the synchronization solution is physically relevant. Unless explicit mention, all parameters used in the simulations are chosen as: f1 = f2 = 1.94 × 1014Hz, fC = 1.9398 × 1014Hz, gC = g1 = g2 = 1.5 × 104s−1, εC = ε1 = ε2 = 5 × 10−7, αC = α1 = α2 = 5, τp = 2ps, τe = 2ns, β = 1 × 103s−1, N0 = 1.5 × 108, I = 36.75mA, e = 1.602 × 10−19C. The coupling delay times are assumed to be identical in the two coupling branches: τC1 = τC2 = τ1C = τ2C = 5ns, and for any pair of mutual coupled lasers, the coupling strength was such that KC1 = KC2, K1C = K2C.
3.1. Chaos synchronization
In Fig. 2 , we have given the maximum of C1C (a) and C12 (b) as a function of K1C and KC1. During the simulations, all the parameters of three SLs are identical except that there exists a frequency detuning ΔfC1 = ΔfC2 = -20GHz between CSL and OSLs. Under this condition, all the outputs of three lasers are rendered into chaotic states when K1C and KC1 vary in the range of 10ns−1 −60ns−1. As seen from Fig. 2 (b), good synchronization between two OSLs is always maintained and the influences of K1C and KC1 on synchronization quality are very weak. However, the synchronization quality between OSL1 and CSL is depended strongly on K1C and KC1 (see Fig. 2 (a)). Thus, the synchronization quality between OSL1 and CSL can be controlled easily through adjusting K1C and KC1. In order to guarantee high security of messages transmission in this system, the maximum of C1C is expected to be as low as possible. Combing the requirement of high-quality synchronization between two OSLs, an optimal point (labeled as point A in Fig. 2 (a)) can be determined in the parameter space, and the corresponding values of K1C and KC1 are 13ns−1 and 16ns−1, respectively. Due to the symmetry of system configuration, the map of the maximum of C2C in the parameter space of K1C and KC1 is similar to Fig. 2 (a).
Figure 3 shows the time series, the power spectra and the cross-correlation coefficient between arbitrary pairs of three lasers for optimal parameter value: KC1 = KC2 = 16ns−1 and K1C = K2C = 13ns−1. As seen from Fig. 3 (a1-a3), zero-lag high-quality synchronization between OSL1 and OSL2 can be obtained, where the maximum C12 = 1 presents at Δt = 0. On the contrary, from Fig. 3(b1-b3) and (c1-c3), it can be seen that the time series of OSLs are quite different from that of CSL, and the maximum of C1C is smaller than 0.2 which is much lower than that obtained in Ref [32].
Fig. 3 Time series (left volume), power spectrum (middle volume) and cross-correlation coefficient (right volume).
3.2. Influences of parameter mismatches between OSLs
Firstly, we discuss the influences of external mismatched parameters such as asymmetric coupling time on the synchronization performance. Figure 4(a) gives the dependence of the maximum of the cross-correlation coefficient on mismatched coupling time Δτ ( = τ1c-τ2c = τc1-τ2c) under τ2c = τc2 = 5ns. From this diagram, it can be seen that the synchronization quality is not sensitive to Δτ, and high-quality chaos synchronization between two OSLs can be maintained while the synchronization quality between the CSL and the OSLs is still low. Theoretically, the CSL can locate at such places that this system is convenience to construct and monitor. However, it should be pointed out that above conclusion is drawn only for the short distance coupling case. In practice, we expect that bidirectional long distance secret communication between the two OSLs can be realized via by fiber. Accordingly, the coupling from the CSL to two OSLs may be performed via by long distance fiber. Under these circumstances, if asymmetrical coupling from the CSL to two OSLs is used, the chaos synchronization performance between the two OSLs may worsen due to two different length fiber channels from the CSL to two OSLs. Therefore, the coupling time difference between τc1 and τc2 should be as small as possible in reality. Figure 4(b-d) are the cross-correction function between arbitrary two SLs under τ1c = τc1 = 3ns and τ2c = τc2 = 5ns. Different from the case for symmetric coupling times (see Fig. 3(a3)), the chaos synchronization between two OSLs is not zero-lag synchronization, and the maximum cross-correction coefficient is located at −2ns, which is just equal to τ1c-τ2c.
Fig. 4 (a) Maximum of the cross-correlation coefficient between arbitrary pairs of three SLs as a function of the coupling time mismatch Δτ; (b)-(d) Cross-correlation function between arbitrary pairs of three SLs under τc1 = τ1c = 3ns and τc2 = τ2c = 5ns.
Since one cannot find two SLs with identical internal parameters in reality, and then it is necessary to investigate the influences of mismatched internal parameters on the synchronization between arbitrary two SLs. For convenience, the parameter mismatched degree is obtained through independently changing three typical parameters g, ε and α of OSL2 and fixing the corresponding parameters of OSL1 and CSL. As a result, the relative parameter mismatches are defined as:
Figure 5 shows the maximums of cross-correlation coefficient as functions of parameter mismatches between arbitrary pairs of three lasers. From Fig. 5(a), it can be seen that the synchronization quality between OSL1 and OSL2 is decreased sharply with the increasing of parameter mismatches, and therefore the chaos synchronization between OSL1 and OSL2 is highly sensitive to the parameter mismatches and high-quality synchronization can be maintained only in a relatively small parameter mismatched range. Among the three mismatched parameters, the influence of mismatched α on the synchronization quality between two OSLs is the most severe. These sensitivities of synchronization quality to mismatched parameters are necessary to guarantee the security of communication. At the meantime, as shown in Fig. 5 (b)-(c), one can observe that the cross-correlations between CSL and OSLs always keep smaller than 0.2 for parameter mismatches are varied within the rang of −10%-10%.
Fig. 5 Maximums of cross-correlation coefficient versus three typical internal parameter mismatches between arbitrary two SLs, where (a) is for OSL1 and OSL2; (b) is for OSL1 and CSL; and (c) is for OSL2 and CSL. The circle, triangular, and square denote mismatched ε, g and α, respectively.
3.3. Influences of parameter mismatches between CSL and OSLs
Furthermore, we will focus on the influences of the internal parameter mismatches between CSL and OSLs. Here, for convenience, we assume that the corresponding internal parameters of two OSLs are identical and the internal parameters of the CSL are changed. The relative mismatched parameters of three typical parameters g, ε and α are defined as:
Under this case, the variations of the maximum of cross correlation coefficient between arbitrary two SLs with above defined mismatched parameters are shown in Fig. 6 . The synchronization quality between OSL1 and OSL2 is always close to 1 for these parameters mismatched within −10%~10%. Meanwhile, the low cross synchronization between CSL and two OSLs can also be guaranteed. Therefore, the synchronization performance between arbitrary two SLs is relatively insensitive to the parameter mismatches between CSL and OSLs.
Fig. 6 Maximum of cross-correlation coefficient versus three typical internal parameter mismatches between CSL and OSLs, where (a) is for OSL1 and OSL2, (b) is for OSL1 and CSL, and (c) is for OSL2 and CSL. The circle, triangular and square denote mismatched ε, g and α, respectively.
3.4. Bidirectional chaos communication
Next, based on the isochronal synchronization between two OSLs, we will investigate the performance of bidirectional message transmission between them. As shown in Fig. 1, two pseudo-random data are mixed into the chaotic output of two OSLs, and oppositely transmitted in the channel F12 and F21. At the receiver side, encoded messages are decoded by subtracting the signal of sender from the signal of receiver. It is worth noting that the signals, emitted from one OSL and mixed with message, do not inject into the other OSL, so the principle of messages decoding is not based on chaos pass filtering (CPF) effect. Therefore, the bandwidth limitation induced by CPF is invalid in this system, and then high communication rate may be realized.
Figure 7 shows the bidirectional encoding and decoding processes between two OSLs. Here, as shown in Fig. 7(a1)-(b1), two pseudo-random binary sequences with bit rate 5 Gbits/s are mixed into the chaotic output of OSL1 and OSL2 with the amplitude modulation index η = 0.07. Combining Fig. 7(a2)-(b2) with Fig. 7(c) and (d), it can be seen that messages are efficiently hidden in the chaotic waveform. The message difference (m2-m1) can be extracted by subtracting the output of PD2 from the output of PD1 (as shown in Fig. 7(a3)). Through similar process, the message difference (m1-m2) can also be obtained (as shown in Fig. 7(b3)). Finally, as shown in Fig. 7(a4), the message m2 can be completely recovered by comparing the message difference (m2-m1) (Fig. 7(a3)) with the local message m1 (Fig. 7(a1)). Similarly, the message m1 can be also completely recovered (Fig. 7(b4)) by comparing the message difference (m1-m2) (Fig. 7(b3)) with the local message m2 (Fig. 7(b1)). Additionally, it should be pointed out that all above recovered messages are obtained after filtering by a four-order low-pass Butterworth filter with the cutoff frequency of 4GHz.
Fig. 7 Illustrations of bidirectional message transmission. (a1)-(b1): the original messages; (a2)-(b2): chaotic time series with messages; (a3)-(b3): message difference; (a4)-(b4): recovered messages; (c) and (d) are the power spectra corresponding to the time series shown in (a2) and (b2), respectively.
In order to evaluate the performance of message communication under different bit rates, the Q-factor evolution is given in Fig. 8(a) as a function of message bit rate, and the eye diagrams of message transmission at 5Gbits/s, 10Gbits/s and 15Gbits/s are given in Figs. 8(b)-(d), respectively. It should be noted that the Q-factor always keeps at a high level (above 11) and degrades small with increasing bit rate. The main reason is that the messages do not enter OSLs and the transmission rate of message is not limited by CPF effect, so high bit rate and high Q-factor bidirectional secure communication can be accomplished in this system.
Fig. 8 (a): Q-factor as a function of bit rate of message; (b)-(d): eye diagrams of message with 5Gbits/s, 10Gbits/s and 15Gbits/s, respectively.
Now, let us consider the security of this scheme. Based on above results, the messages cannot be recovered by extracting signal from single channel since the messages are efficiently hidden in the chaotic waveforms (see Fig. 7(a2-b2) and (c-d)). Moreover, through observing carefully the system configuration in Fig. 1, it is easy to understand that the messages cannot be recovered either by extracting two signals from channel F1c and F12 (or F2c and F21, and Fc1 and Fc2), respectively. In the following, we will analyze the security of this system under the other cases. Firstly, it is assumed that both of intercepted signals contain messages, i. e., the two signals are eavesdropped from channel F12 and F21 (or F1c and F2c, F1c and F21, F2c and F12), respectively. Under this condition, it is easy for the eavesdropper to obtain the difference between m1 and m2 (as shown in Fig. 9(a) ) due to high-quality synchronization between OSL1 and OSL2. However, the eavesdropper cannot eavesdrop the real message m1 (or m2) since he does not know the location transmitted message m2 (or m1) [37]. Secondly, it is assumed that one signal is from the channel including message but the other is from another channel without message, i. e., one signal is eavesdropped from Fc1 (or Fc2) whereas the other signal is eavesdropped from F21 (or F12, F1c, and F2c). Under this case, the difference between two signals is given in Fig. 9(b). As seen from Fig. 9(b), the signal difference obtained by the eavesdropper is still chaos, and message m1 or m2 cannot be determined. The reason is that the chaotic carriers in channel Fc1 (or Fc2) are different from that in channel F21 (or F12, F1c, and F2c) due to the cross-correlations coefficient between OSLs and CSL are very low (smaller than 0.2), so the chaotic carrier cannot be removed during the process of subtracting one signal from the other signal. Thirdly, it is assumed that an eavesdropper can obtain a chaos signal synchronized with the chaos carrier, and then the message m1 (or m2) may be captured by acquiring the signal from channel F12 (or F21). Usually, in order to obtain a chaos signal synchronized with the chaos carrier, two schemes may be used by the eavesdropper. One is through changing the coupling strengths K1c and K2c to make all three lasers achieve high-quality chaos synchronization. Under this circumstance, a chaos signal synchronized with the chaos carries is easily obtained from channel Fc1 or Fc2, and then the security of this system may be vulnerable. However, the variation of the synchronization performance between CSL and OSLs, induced by the changed K1c and K2c, can be easily monitored for this system. Once the value of cross synchronization coefficient between CSL and OSLs is detected to fluctuate obviously, the communication can be stopped immediately. Therefore, the security of communication can be guaranteed. The other scheme is through introducing a third OSL into the system. For this attack scenario, the first faced problem for the eavesdropper may be that he must select a special SL whose internal parameters are similar to those of existing two OSLs. Even so, we still assume that the introduced third OSL can highly synchronize with existing two OSLs by adopting some complex methods. However, such attacking processes will change the topological structure of this proposed system and inevitably result in the variation of the synchronization performance between CSL and existing two OSLs, which is also easily monitored through detecting the cross-correction coefficient. Under this case, similar to above analysis, the security of this system can also be guaranteed. In shortly, no matter which attack scenario is used, the security of this proposed system can be insured to certain degree.
Fig. 9 The signals extracted from two different communication channels by an eavesdropper, where (a) is for the case that both of channels include messages; and (b) is for the case that one channel includes message but the other channel does not.
4. Conclusions
Chaos synchronization and bidirectional communication between two OSLs have been investigated systematically when the two OSLs are mutually coupled with a CSL and there is no direct coupling between them. High quality of isochronal synchronization between the two OSLs, and very low correlated synchronization between two OSLs and CSL are achieved at an optimal operation condition, and the dependences of maximums of cross-correlation coefficient on the parameter mismatches has been given. Furthermore, under a novel decoding scheme, the transmission performances of bidirectional messages with 5Gbits/s between two OSLs are analyzed and the Q-factor evolution as a function of bit rate of message is also given. As a result, high rate and high Q-factor bidirectional secure communication can be accomplished in this system. Finally, the security of system is also analyzed for some possible attack scenarios, and the results indicate that the security of this scheme can be insured no matter which attack scenario is used by the eavesdropper.
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant Nos. 60978003, 61078003, 61178011 and 11004161, the Natural Science Foundation of Chongqing City under Grant Nos. CSTC2010BB9125 and CSTC2011jjA40035, the Fundamental Research Funds for the Central Universities under Grant Nos. XDJK2010C019, XDJK2010C021 and XDJK2011C057, and the Open Fund of the State Key Lab of Millimeter Waves under Grant No. K201109.
References and links
1. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990). [CrossRef] [PubMed]
2. I. Fischer, O. Hess, W. Elsaβer, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994). [CrossRef] [PubMed]
3. G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998). [CrossRef] [PubMed]
4. S. Tang and J. M. Liu, “Message encoding-decoding at 2.5 Gbits/s through synchronization of chaotic pulsing semiconductor lasers,” Opt. Lett. 26(23), 1843–1845 (2001). [CrossRef] [PubMed]
5. 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]
6. J. Paul, M. W. Lee, and K. A. Shore, “Effect of chaos pass filtering on message decoding quality using chaotic external-cavity laser diodes,” Opt. Lett. 29(21), 2497–2499 (2004). [CrossRef] [PubMed]
7. G. Q. Xia, Z. M. Wu, and J. G. Wu, “Theory and simulation of dual-channel optical chaotic communication system,” Opt. Express 13(9), 3445–3453 (2005). [CrossRef] [PubMed]
8. J. Ohtsubo, Semiconductor Lasers, Stability, Instability and Chaos, 2nd ed. (Springer-Verlag, 2008).
9. Y. H. Hong, M. W. Lee, J. Paul, P. S. Spencer, and K. A. Shore, “GHz bandwidth message transmission using chaotic vertical-cavity surface-emitting lasers,” J. Lightwave Technol. 27(22), 5099–5105 (2009). [CrossRef]
10. R. Vicente, J. Daudén, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41(4), 541–548 (2005). [CrossRef]
11. J. G. Wu, G. Q. Xia, and Z. M. Wu, “Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback,” Opt. Express 17(22), 20124–20133 (2009). [CrossRef] [PubMed]
12. J. Liu, Z. M. Wu, and G. Q. Xia, “Dual-channel chaos synchronization and communication based on unidirectionally coupled VCSELs with polarization-rotated optical feedback and polarization-rotated optical injection,” Opt. Express 17(15), 12619–12626 (2009). [CrossRef] [PubMed]
13. A. Jafari, H. Sedghi, Kh. Mabhouti, and S. Behnia, “Slave-master dynamics of semiconductor laser with short external cavity,” Opt. Commun. 284(12), 3018–3029 (2011). [CrossRef]
14. E. Klein, R. Mislovaty, I. Kanter, and W. Kinzel, “Public-channel cryptography using chaos synchronization,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016214 (2005). [CrossRef] [PubMed]
15. E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006). [CrossRef] [PubMed]
16. R. Vicente, C. R. Mirasso, and I. Fischer, “Simultaneous bidirectional message transmission in a chaos-based communication scheme,” Opt. Lett. 32(4), 403–405 (2007). [CrossRef] [PubMed]
17. W. L. Zhang, W. Pan, B. Luo, X. H. Zou, M. Y. Wang, and Z. Zhou, “Chaos synchronization communication using extremely unsymmetrical bidirectional injections,” Opt. Lett. 33(3), 237–239 (2008). [CrossRef] [PubMed]
18. M. C. Chiang, H. F. Chen, and J. M. Liu, “Experimental synchronization of mutually coupled semiconductor lasers with optoelectronic feedback,” IEEE J. Quantum Electron. 41(11), 1333–1340 (2005). [CrossRef]
19. 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]
20. T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009). [CrossRef]
21. A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. 104(11), 114102 (2010). [CrossRef] [PubMed]
22. N. Jiang, W. Pan, L. Yan, B. Luo, W. L. Zhang, S. Y. Xiang, L. Yang, and D. Zheng, “Chaos synchronization and communication in mutually coupled semiconductor lasers driven by a third laser,” J. Lightwave Technol. 28(13), 1978–1986 (2010). [CrossRef]
23. T. Deng, G. Q. Xia, Z. M. Wu, X. D. Lin, and J. G. Wu, “Chaos synchronization in mutually coupled semiconductor lasers with asymmetrical bias currents,” Opt. Express 19(9), 8762–8773 (2011). [CrossRef] [PubMed]
24. J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011). [CrossRef]
25. R. V. Jensen, “Synchronization of randomly driven nonlinear oscillators,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(6), R6907–R6910 (1998). [CrossRef]
26. C. S. Zhou, J. Kurths, E. Allaria, S. Boccaletti, R. Meucci, and F. T. Arecchi, “Constructive effects of noise in homoclinic chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(6), 066220 (2003). [CrossRef] [PubMed]
27. A. Uchida, R. McAllister, and R. Roy, “Consistency of nonlinear system response to complex drive signals,” Phys. Rev. Lett. 93(24), 244102 (2004). [CrossRef] [PubMed]
28. I. Fischer, R. Vicente, J. M. Buldú, M. Peil, C. R. Mirasso, M. C. Torrent, and J. García-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006). [CrossRef] [PubMed]
29. R. Vicente, I. Fischer, and C. R. Mirasso, “Synchronization properties of three delay-coupled semiconductor lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066202 (2008). [CrossRef] [PubMed]
30. T. Yamamoto, I. Oowada, H. Yip, A. Uchida, S. Yoshimori, K. Yoshimura, J. Muramatsu, S. I. Goto, and P. Davis, “Common-chaotic-signal induced synchronization in semiconductor lasers,” Opt. Express 15(7), 3974–3980 (2007). [CrossRef] [PubMed]
31. I. Oowada, H. Ariizumi, M. Li, S. Yoshimori, A. Uchida, K. Yoshimura, and P. Davis, “Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback,” Opt. Express 17(12), 10025–10034 (2009). [CrossRef] [PubMed]
32. V. Annovazzi-Lodi, G. Aromataris, M. Benedetti, and S. Merlo, “Private message transmission by common driving of two chaotic lasers,” IEEE J. Quantum Electron. 46(2), 258–264 (2010). [CrossRef]
33. J. M. Liu, H. F. Chen, and S. Tang, “Synchronized chaotic optical communications at high bit rates,” IEEE J. Quantum Electron. 38(9), 1184–1196 (2002). [CrossRef]
34. T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001). [CrossRef] [PubMed]
35. J. M. Buldú, J. García-Ojalvo, and M. C. Torrent, “Multimode synchronization and communication using unidirectionally coupled semiconductor lasers,” IEEE J. Quantum Electron. 40(6), 640–650 (2004). [CrossRef]
36. D. Kanakidis, A. Argyris, and D. Syvridis, “Performance characterization of high-bit-rate optical chaotic communication systems in a back-to-back configuration,” J. Lightwave Technol. 21(3), 750–758 (2003). [CrossRef]
37. R. Mislovaty, E. Klein, I. Kanter, and W. Kinzel, “Public channel cryptography by synchronization of neural networks and chaotic maps,” Phys. Rev. Lett. 91(11), 118701 (2003). [CrossRef] [PubMed]
