Hybrid chaos-based communication system consisting of three chaotic semiconductor ring lasers Nianqiang Li,* Wei Pan, Shuiying Xiang, Bin Luo, Lianshan Yan, and Xihua Zou Center for Information Photonics and Communications, Southwest Jiaotong University, Chengdu, Sichuan 610031, China *Corresponding author:
[email protected] Received 27 November 2012; revised 1 February 2013; accepted 1 February 2013; posted 1 February 2013 (Doc. ID 180587); published 28 February 2013
We report on the realization of a hybrid chaos-based communication scheme using three chaotic semiconductor ring lasers (SRLs). In this scheme, two slave SRLs (S-SRLs) are identically driven by a master SRL (M-SRL) subject to delayed optical feedback. Under proper conditions, the S-SRLs are completely synchronized with each other due to the symmetric operation, and they are also synchronized with the M-SRL through the injection-locking effect. The results also show that a message encrypted through chaos shift keying at the M-SRL end can be successfully decrypted by the two S-SRLs, while the two uncoupled S-SRLs allow for dual-channel chaos communication when both counterpropagating modes of one S-SRL are encoded with a message. © 2013 Optical Society of America OCIS codes: 140.1540, 140.5960, 060.4510.
1. Introduction
There is significant interest in investigating the chaotic synchronization properties of transmitter– receiver pairs because of their potential applications in secure communication systems. Chaotic optical communication systems have attracted considerable attention because chaotic carriers with broader spectra are generally available, and they have the advantage of higher-speed message transmission [1–5]. Among these systems, chaos-based communications using semiconductors lasers (SLs) are very interesting; for example, message transmission can be realized using either the two-laser scheme [6,7] or the three-laser scheme [8,9]. Moreover, the two linear polarization modes of vertical-cavity surface-emitting lasers (VCSELs), a special class of SLs, can be developed for polarization-division-multiplexed chaotic carriers [10,11]. On the other hand, another class of SLs for which chaos-based communications have not yet 1559-128X/13/071523-08$15.00/0 © 2013 Optical Society of America
been investigated in depth is semiconductor ring lasers (SRLs). SRLs have many important advantages from different viewpoints [12–15]. One of the advantages of such lasers is the existence of two counterpropagating modes, which have been developed for all-optical switching [16], optical memories [17,18], and random-number generators [19,20]. Another advantage of SRLs is that they are promising sources in photonic integrated circuits because such devices do not require cleaved facets or mirrors to form the laser cavity [12]. Furthermore, SRLs with delayed optical feedback are also interesting. It has been demonstrated that injection only one directional mode back into the counterpropagating mode leads to squarewave oscillations [21]. However, asymmetric cross feedback makes SRL outputs more complex; i.e., hyperchaos optical signals are generated in large parameter regions when the feedback is asymmetric [14,22]. More recently, a primary demonstration of chaos-based communication using two chaotic SRLs has been reported, and the ON/OFF phase shift keying encryption method was adopted [23]. 1 March 2013 / Vol. 52, No. 7 / APPLIED OPTICS
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In this study, we propose and numerically demonstrate a hybrid chaos-based communication scheme consisting of three chaotic SRLs. Two slave SRLs (S-SRLs) are identical (or nearly identical), and both are rendered chaotic by a common driving signal from a chaotic master SRL (M-SRL) with cross feedback; i.e., the delayed signal of the clockwise (CW) mode is injected back to the counterclockwise (CCW) mode and vice versa. In the numerical experiments, the two modes of the M-SRL are simultaneously injected into the two S-SRLs. Thus, when a message is encrypted using chaos shift keying (CSK) at the M-SRL, the two S-SRLs can simultaneously recover the message via generalized synchronization. Furthermore, the pair of S-SRLs allows for dualchannel chaos communication based on complete synchronization; that is, the two counterpropagating modes are separately used to transmit two independent messages by using chaos modulation (CMO) even if the two S-SRLs are not directly coupled. The natural features of SRLs make the hybrid chaos-based communication system beneficial in the implementation of optical chaos communication networks. 2. Model
The system architecture under study is illustrated in Fig. 1. It contains three chaotic SRLs, where SRLA stands for the M-SRL, and SRLB and SRLC represent the two S-SRLs. Here we only consider the open-loop configuration; i.e., only SRLA is subject to delayed optical feedback, while SRLB and SRLC are solitary lasers when no optical injection is introduced. For the sake of simplification, the three lasers are identical or almost identical, i.e., lasers with the same internal parameters. Both modes of SRLA are unidirectionally injected into SRLB and SRLC so as to obtain better intensity synchronization. For message transmission
from the M-SRL to the S-SRLs (m1 t), message encryption in SRLA will be achieved via the CSK method. Message decryption is then performed by comparing the difference between the transmitted signal and the locally generated chaos by SRLB (SRLC ). Note that the message recovery module for the CSK method is not shown in Fig. 1. For message transmission between the two S-SRLs [m2 t and m3 t], two different messages are added to the counterpropagating modes of SRLB, respectively, and recovered by an indirectly coupled SRLC based on complete synchronization (symmetric operation condition). Note that for such dual-channel chaos-based communication, a standard CMO scheme is adopted; messages are extracted by an oscilloscope (OSC) when comparing the difference between the signals from the photodetectors (PDs). Figure 2 shows a schematic of an SRL device with delayed optical feedback [19]. The device is composed of a circular laser cavity, a cross-feedback waveguide, and an output waveguide. The cavity sustains two counterpropagating modes, i.e., CW and CCW modes. The feedback waveguide ensures the delayed signal of the CW mode is reinjected in the CCW mode and vice versa. Here FR stands for optical fiber reflector or cleaved facets of the chip in photonic integrated circuits. Note that only SRLA has the self-optical feedback loop, whereas the feedback part should be removed from Fig. 2 for describing SRLB and SRLC . In the numerical experiments, we use a rate equation model that accounts for the dynamics of the three SRLs, which are assumed to operate in a single transverse and single longitudinal mode and can sustain two counterpropagating modes. For each SRLX (X fA; B; Cg), our model consists of two equations accounting for two slowly varying envelopes of the counterpropagating fields E1X (CW) and E2X (CCW) and a third equation for the carrier population inversion N X [14,21–23]: _ 1A κ1 iαg1A N A − 1E1A − keiϕk E2A E η2A E2A t − τ2A e−iωA τ2A ;
(1)
_ 2A κ1 iαg2A N A − 1E2A − keiϕk E1A E η1A E1A t − τ1A e−iωA τ1A ;
Fig. 1. (Color online) Architecture for hybrid chaos-based communication scheme. SRL, semiconductor ring laser; CW, clockwise; CCW, counterclockwise; M, modulation module; PD, photodetector; OSC, oscilloscope; mt, message. 1524
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(2)
Fig. 2. (Color online) Schematic of an SRL subject to delayed optical feedback. FR, optical fiber reflector.
_ 1B κ1 iαg1B N B − 1E1B − keiϕk E2B E kc E1A t − τc e−iωA τc i2πΔf AB t ;
Table 1.
(3)
_ 2B κ1 iαg2B N B − 1E2B − keiϕk E1B E kc E2A t − τc e−iωA τc i2πΔf AB t ;
(4)
_ 1C κ1 iαg1C N C − 1E1C − keiϕk E2C E kc E1A t − τc e−iωA τc i2πΔf AC t ;
(6)
_ A;B;C γμ − N A;B;C − g1A;1B;1C N A;B;C jE1A;1B;1C j2 N − g2A;2B;2C N A;B;C jE2A;2B;2C j2 :
Parameter α μ s c κ γ k ϕk λ
Description
Value
Linewidth enhancement factor Injection current Self-saturation coefficient Cross-saturation coefficient Field decay rate Carrier decay rate Backscattering amplitude Backscattering phase shift Wavelength
3.5 1.7 0.005 0.01 100 ns−1 0.2 ns−1 0.44 ns−1 1.5 850 nm
(5)
_ 2C κ1 iαg2C N C − 1E2C − keiϕk E1C E kc E2A t − τc e−iωA τc i2πΔf AC t ;
Values of Parameters Used in Our Simulations [21,23]
(7)
Note that we include here a delayed feedback term of Lang–Kobayashi type in Eq. (1) [Eq. (2)] with feedback strength η2A η1A and delay time τ2A τ1A . Similarly, we introduce the corresponding optical injection terms in Eqs. (3)–(6), where kc is the coupling strength, τc is the coupling delay time, and Δf is the frequency detuning. g1X 1 − sjE1X j2 − cjE2X j2 and g2X 1 − sjE2X j2 − cjE1X j2 are differential gain functions in which s and c are phenomenological selfand cross-saturation coefficients (with c 2s). μ is the renormalized injection current (μ 0 at transparency and μ 1 at lasing threshold), κ is the field decay rate, γ is the carrier decay rate, and α is the linewidth enhancement factor. The linear coupling between the two counterpropagating modes is added through the backscattering rate keiϕk modeled by an amplitude k and a phase shift ϕk . In this contribution, we used the fourth-order Runge–Kutta algorithm to solve Eqs. (1)–(7). Unless otherwise stated, the values for some key parameters used in the simulations are listed in Table 1. The values of other control parameters will be specified in the following sections.
where h·i denotes time average, Pt ‖Et‖2 is the intensity time series, and Δt is the time shift. Note that XCorr 1 (the maximum value of the CCF) indicates perfect synchronization, while XCorr 0 means no synchronization. According to the literature [20,22,23], it is widely accepted that an SRL subject to asymmetric optical feedback can yield complex dynamics for very large parameter regions. For this reason, we consider an SRL subject to optical feedback from two fully asymmetric external cavities [23], i.e., η2A 10 ns−1 , η1A 2.5 ns−1 , τ2A 0.8 ns, and τ1A 0.7 ns. Unless stated otherwise, the injection parameters are set as kc 50 ns−1 , τc 10 ns, and Δf 0 GHz. With these parameters, the behaviors of the three SRLs are fully chaotic with broad power spectra. Figure 3 presents the temporal outputs for the CW modes of the three SRLs and their corresponding CCF curves for SRLA and SRLB [Fig. 3(b)], SRLA and SRLC [Fig. 3(c)], and SRLB and SRLC [Fig. 3(d)]. It is clear from these plots that for the chosen values of feedback and injection parameters, the CW modes of SRLB and SRLC are well synchronized to that of SRLA due to the injection-locking effect, similar to that in the conventional master–slave configuration. Moreover, it is obvious that the CW modes of SRLB and SRLC are completely synchronized because the two S-SRLs operate under the symmetric condition.
3. Synchronization Properties
We started our simulations by investigating the properties of chaos synchronization between any two SRLs. To quantitatively evaluate the synchronization quality, we employed the cross-correlation function (CCF) ρij Δt [24]: ρij Δt hPi t − hPi ti · Pj t − Δt − hPj t − Δtii q ; hjPi t − hPi tij2 i · hjPj t − Δt − hPj t − Δtij2 i (8)
Fig. 3. (Color online) (a) CW temporal outputs of SRLA (upper), SRLB (middle), and SRLC (bottom). CCF curves for (b) CW modes of SRLA and SRLB , (c) CW modes of SRLA and SRLC , and (d) CW modes of SRLB and SRLC . 1 March 2013 / Vol. 52, No. 7 / APPLIED OPTICS
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Fig. 4. (Color online) (a) CCW temporal outputs of SRLA (upper), SRLB (middle), and SRLC (bottom). CCF curves for (b) CCW modes of SRLA and SRLB , (c) CCW modes of SRLA and SRLC , and (d) CCW modes of SRLB and SRLC .
As shown in Fig. 4, we also present the temporal outputs for the CCW modes of the three SRLs and their corresponding CCF curves. Apparently, the results are in line with those shown in Fig. 3. In other words, we have achieved generalized synchronization between SRLA and SRLB (SRLC ) and complete synchronization between SRLB and SRLC , which are valuable for the realization of message transmission between any two SRLs. The transmission bandwidth achievable in our scheme was also evaluated. Figure 5 displays the RF spectra of SRLA and SRLB . As can be seen from these plots, the spectra of the SRLs are broadband. Here we define the effective bandwidth of the RF spectrum as the range between DC and the frequency that contains 80% of the spectral power, and then the bandwidths for the CW and CCW modes of SRLA and the CW and CCW modes of SRLB are
Fig. 5. (Color online) Power spectra for (a) CW mode of SRLA, (b) CCW mode of SRLA, (c) CW mode of SRLB, and (d) CCW mode of SRLB. 1526
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estimated approximately as 2.72, 1.81, 2.41, and 2.93 GHz, respectively. Note that the RF spectra for SRLC are the same as those for SRLB. Generally speaking, we have demonstrated that the power spectra of the three SRLs are broad. As mentioned above, we have assumed the SRLs are identical or almost identical, and therefore SRLB and SRLC should be always completely synchronized under the symmetric operation condition. Even if one considers the spontaneous emission noise, the high-quality complete synchronization can be maintained for moderate values of coupling. Thus, here we only evaluate the chaos synchronization between SRLA and SRLB . We hold the hypothesis that high-quality chaos synchronization between them should be obtained in large parameter regions, and therefore we further looked for the synchronization regions. Figure 6 includes the variations for the synchronization performance versus the feedback strength η2A and coupling strength kc —for both CW [Fig. 6(a)] and CCW [Fig. 6(b)] modes. Note that only η2A is varied and η1A is fixed to a value of 2.5 ns−1 . As is evident, chaos synchronization with XCorr ≈ 1 is maintained over large parameter regions, indicating that broadcasting communications from the single M-SRL to two S-SRLs can be easily achieved. The findings are in close accordance with those in [23], where only a point-to-point configuration composed of one transmitter and one receiver was adopted. To further explore the performance of injectionlocking synchronization, we calculated the values of XCorr as a function of coupling strength and frequency detuning between the SRLA and SRLB . The results are presented in Figs. 7(a) (CW mode) and 7(b) (CCW mode). It can be seen from Fig. 7 that inside this injection-locking region, high-quality chaos synchronization is achieved. The synchronization regions are similar in shape to those reported for VCSELs [25] and edge-emitting lasers (EELs) [26]. This is expected because in the configuration two modes of SRLA are simultaneously injected into SRLB , and the effect of optical injection dominates compared to the linear and nonlinear coupling between the two counterpropagating modes. Thus,
Fig. 6. (Color online) XCorr as a function of feedback and coupling strengths for CW (a) and CCW (b) modes. η2A is varied and η1A 2.5 ns−1 .
Fig. 7. (Color online) XCorr as a function of coupling strength and frequency detuning for CW (a) and CCW (b) modes. Other parameters are as in Fig. 5.
the master–slave configuration is essentially equivalent to the VCSEL or single-mode EEL case. Overall, the results in Figs. 6 and 7 indicate that high-quality chaos synchronization between SRLA and SRLB can be obtained for large parameter regions. This also means that SRLB and SRLC can be completely synchronized for the regions mentioned above under the symmetric operation condition. 4. Message Encoding and Decoding
In this section, our focus will now be turned to the message encoding and decoding process. We have demonstrated a hybrid chaos-based communication scheme. First, SRLA broadcasted a message via the CSK technique, and both the SRLB and SRLC could successfully extract the message. Second, the two indirectly coupled SRLB and SRLC could also exchange messages by adopting dual-channel message transmission, where both counterpropagating modes were used as chaotic carriers and two independent messages were encrypted and decrypted at the same time. When the SRLA serves as the transmitter and SRLB (SRLC ) as the receiver, we just focus on the message encoding/decoding between SRLA and SRLB , since SRLC can extract the same message with high quality comparable to that extracted by SRLB. For CSK technique, the injection current of the transmitter is modulated with a small-amplitude message while that of receiver is kept constant. The modulated injection current is expressed mathematically as μA μ1 mCSK m1 t for SRLA and μB μ for SRLB , where m1 t represents the random sequences and mCSK 0.05 is the modulation depth [27]. At the SRLB end, the message recovery is performed by subtracting the outputs of SRLB from the transmitted signals and then filtering the difference by a fifthorder Butterworth filter with a cutoff frequency equal to 0.6 × Bs (Bs is the bit rate of the message). To illustrate the quality of the recovered message, we use the eye diagram and the quality factor Q I 1 − I 0 ∕σ 1 σ 0 [9,27], where I 1 and I 0 are the average optical power of bits “1” and “0,” and σ 1 and σ 0 are the corresponding standard deviations. Figure 8 shows an example of a 100 Mb∕s message encoding and decoding. The original message m1 t is
Fig. 8. Illustration of encoding and decoding process for SRLA and SRLB . (a) Original message at 100 Mb∕s, (b) recovered message, and (c) corresponding eye diagram.
shown in Fig. 8(a), the recovered message in Fig. 8(b), and the corresponding message in Fig. 8(c). As is evident, the message is successfully recovered: the eye diagram is very clear and wide open (Q 8.85). On the contrary, we here show that the error-free message cannot be extracted by using only one mode. Figure 9 presents the results for separately using CW [Figs. 9(a) and 9(b)] and CCW [Figs. 9(c) and 9(d)] mode extraction. It can be seen that one cannot successfully recover the message by using only one mode extraction, and the eye diagrams are very closed [Q 2.12 for Fig. 9(b) and Q 1.94 for Fig. 9(d)]. This is attributed to the fact that only one mode does not convey all information about the dynamics of the system and the encoded message. Finally, we briefly examined the performance for the dual-channel chaos-based communication. The message is added to the chaotic carrier by the CMO method, i.e., Eext t E1B;2B t1 mCMO m2;3 t, wherein Eext t denotes the transmitted signal from SRLB to SRLC , E1B;2B t is the chaotic carrier (the outputs of SRLB ), mCMO 0.05 is the modulation depth,
Fig. 9. Illustration of encoding and decoding process for using only one mode. (a), (b) for CW mode; (c), (d) for CCW mode. 1 March 2013 / Vol. 52, No. 7 / APPLIED OPTICS
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Fig. 10. Illustration of encoding and decoding process for dual-channel chaos-based communication. (a)–(c) for CW mode; (d)–(f) for CCW mode. m2 t at 400 Mb∕s; m3 t at 100 Mb∕s.
and m2;3 t represents the random sequences [9]. Note that the decoding process is not based on the chaospass filtering effect because the transmitted signals (the outputs of SRLB plus messages) do not enter into the receiver (SRLC ). Specifically, two independent messages (m2 t, m3 t) with different bit rates are encoded into chaotic waveforms of the CW and CCW modes of SRLB, respectively, and transmitted to SRLC . Due to the symmetric operation condition (complete synchronization), the two sent messages are recovered at the receiver end by comparing the difference between the transmitted signal and the locally generated chaotic carrier. This decoding process is similar to that in the CSK technique. Note that in the simulations the coupling delay times between SRLB and SRLC have not been taken into account. As an example, Fig. 10 displays the results for our dual-channel transmission. Here two different messages with bit rates at 400 and 100 Mb∕s are encoded into the CW and CCW modes of the SRLB , respectively. It is obvious that messages are successfully decoded with Q 6.27 for the case of m2 t [Figs. 10(a)–10(c)] and Q 5.69 for the case of m3 t [Figs. 10(d)–10(f)], and their eye diagrams are quite open. Compared to the communication between SRLA and SRLB where the message cannot be recovered using only one mode, here we demonstrated that messages can be efficiently encrypted/decrypted using each mode. From the physics point of view, the two SRLs operate under a completely symmetrical condition and the transmitted signals do not affect the dynamics of the receiver. On the other hand, it should be mentioned that the chaotic carrier of the CW mode can allow for better performance for message encoding and decoding even if a message at larger bit rate is added (for example, 400 Mb∕s), compared to that of CCW mode. This is because we have selected a larger feedback strength for 1528
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the CW mode in the SRLA , i.e., η2A 10 ns−1 > η1A 2.5 ns−1 , and then the CW modes of the three SRLs have relatively higher optical intensity components, which is important for message encryption. 5. Discussion
Last but not least, we should discuss the feasibility and security of the hybrid chaos-based communication system. On one hand, since mismatch will be encountered for a practical implementation, it is valuable to investigate the effects of the device parameter mismatch. The simulations were repeated taking into account mismatch between the internal and external parameters of the SRLs. Although parameter deviations do affect the synchronization performance as well as the communication performance, it is very interesting to observe that the system can operate satisfactorily under small levels of parameter mismatch, e.g., within a 5% accuracy. Besides, other obstacles such as noises, temperature/phase drifts, and fiber links may impair the message recovery performance. However, such technologies for developing optical chaos communications are practically interesting since chaos-based communications have been demonstrated over 120 km of optical fiber in the metropolitan area network of Athens (Greece) [2]. On the other hand, the security of these physical-layer communication techniques is of vital importance. Although we here consider a hybrid chaos-based communication system where message recovery can be achieved between the master and slave lasers as well as between the two slave lasers, security can be enhanced by only considering the communication between the slave lasers as those in [8,9]. More importantly, the time delay signature will be completely eliminated from the intensity and
phase time series on the condition that the parameter values of the M-SRL are carefully chosen as those in [14]. Also, the messages cannot be extracted by employing direct detection and linear filtering because the modulation depth is limited to rather small values [28]. In combination, the security of such chaotic communication systems can be remarkably improved for practical implementation. 6. Conclusions
The numerical investigations reported in this paper have established that chaotic SRLs are promising candidates for generating chaotic carriers in chaosbased communications. We have proposed a threeSRL scheme that allows for hybrid chaos-based optical communication. Two S-SRLs, which are subject to chaotic optical injection from the M-SRL with optical feedback from two different external cavities, can exchange messages using a dual-channel transmission scheme, and they can also recover the same message broadcasted by the M-SRL. For communication between the M-SRL and S-SRLs, the CSK technique has been adopted and the message cannot be successfully decoded by using only one mode. This indicates that the security of chaotic systems can be enhanced when chaotic SRLs are employed. For communication between the two S-SRLs, the two counterpropagating modes can be used as chaotic carries. The communication possibility has been preliminarily demonstrated by using two different messages encoded separately by two modes. In terms of further work, we will pay attention to the bandwidth enhancement for chaotic SRLs since the main advantage of SRLs is their lower bandwidth. We believe that their chaos bandwidth can be effectively enhanced by using chaotic optical injection as in VCSEL- or EEL-based systems. Moreover, the interesting investigations of message transmission considering fiber-optic links in chaotic SRLs should be systematically addressed elsewhere. The authors thank the three reviewers for their helpful comments and suggestions on this paper. This work is partially supported by the National Natural Science Foundation of China (60976039 and 61274042), the Basic Research Foundation of Sichuan Province (2011JY0030), and the funds for the Excellent Ph.D. Dissertation of Southwest Jiaotong University (2011). References 1. S. Donati and C. R. Mirasso, “Feature section on optical chaos and applications to cryptography,” IEEE J. Quantum Electron. 38, 1138–1140 (2002). 2. A. Argyris, D. Syvrids, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437, 343–346 (2005). 3. R. M. Nguimdo and P. Colet, “Electro-optic phase chaos systems with an internal variable and a digital key,” Opt. Express 20, 25333–25344 (2012).
4. X. F. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injectionlocking chaos synchronization,” IEEE J. Quantum Electron. 42, 953–960 (2006). 5. Q. C. Zhao, H. X. Yin, and X. L. Chen, “Long-haul dense wavelength division multiplexing between a chaotic optical secure channel and a conventional fiber-optic channel,” Appl. Opt. 51, 5585–5590 (2012). 6. S. Sivaprakasam and K. A. Shore, “Signal masking for chaotic optical communications using external-cavity diode lasers,” Opt. Lett. 24, 1200–1202 (1999). 7. V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-lasers systems and its applications to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996). 8. 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, 258–264 (2010). 9. N. Q. Li, W. Pan, L. S. Yan, B. Luo, M. F. Xu, Y. L. Tang, N. Jiang, S. Y. Xiang, and Q. Zhang, “Chaotic optical cryptographic communication using a three-semiconductor-laser scheme,” J. Opt. Soc. Am. B 29, 101–108 (2012). 10. 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, 12619–12626 (2009). 11. N. Jiang, W. Pan, B. Luo, S. Y. Xiang, and L. Yang, “Bidirectional dual-channel communication based on polarization-divisionmultiplexed chaos synchronization in mutually coupled VCSELs,” IEEE Photon. Technol. Lett. 24, 1094–1096 (2012). 12. L. Chrostowski and W. Shi, “Monolithic injection-locked high-speed semiconductor ring lasers,” J. Lightwave Technol. 26, 3355–3362 (2008). 13. S. T. Kingni, G. van der Sande, L. Gelens, T. Erneux, and J. Danckaert, “Direct modulation of semiconductor ring lasers: numerical and asymptotic analysis,” J. Opt. Soc. Am. B 29, 1983–1992 (2012). 14. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett. 37, 2541–2543 (2012). 15. L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009). 16. T. Perez, A. Scire, G. Van der Sande, P. Colet, and C. R. Mirasso, “Bistability and all-optical switching in semiconductor ring lasers,” Opt. Express 15, 12941–12948 (2007). 17. S. Zimmermann, A. Wixforth, J. P. Kotthaus, W. Wegscheider, and M. Bichler, “A semiconductor-based photonic memory cell,” Science 283, 1292–1295 (1999). 18. M. T. Hill, H. J. S. Dorren, H. T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004). 19. S. Sunada, T. Harayama, K. Arai, K. Yoshimura, K. Tsuzuki, A. Uchida, and P. Davis, “Random optical pulse generation with bistable semiconductor ring lasers,” Opt. Express 19, 7439–7450 (2011). 20. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bits generation based on a single chaotic semiconductor ring laser,” Opt. Express 20, 28603–28613 (2012). 21. L. Mashal, G. van der Sande, L. Gelens, J. Danckaert, and G. Verschaffelt, “Square-wave oscillations in semiconductor ring lasers with delayed optical feedback,” Opt. Express 20, 22503–22516 (2012). 22. I. V. Ermakov, G. van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17, 4767–4779 (2012). 23. I. V. Ermakov, S. T. Kingni, V. Z. Tronciu, and J. Danckaert, “Chaotic semiconductor ring lasers subject to optical feedback: applications to chaos-based communications,” Opt. Commun. 286, 265–272 (2013). 1 March 2013 / Vol. 52, No. 7 / APPLIED OPTICS
1529
24. N. Q. Li, W. Pan, L. S. Yan, B. Luo, X. H. Zou, and S. Y. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top. Quantum Electron., doi:10.1109/JSTQE.2012.2210394 (2012). 25. R. Ju, P. S. Spencer, and K. A. Shore, “Polarization-preserved and polarization-rotated synchronization of chaotic verticalcavity surface-emitting lasers,” IEEE J. Quantum Electron. 41, 1461–1467 (2005). 26. A. Locquet, C. Masoller, and C. R. Mirasso, “Synchronization regimes of optical-feedback-induces chaos in unidirectionally
1530
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coupled semiconductor lasers,” Phys. Rev. E 65, 056205 (2002). 27. S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, N. Q. Li, and H. N. Zhu, “Message encoding/decoding using unpredictability-enhanced chaotic VCSELs,” IEEE Photon. Technol. Lett. 24, 1267–1269 (2012). 28. N. Q. Li, W. Pan, B. Luo, L. S. Yan, X. H. Zou, and S. Y. Xiang, “Multiuser optical communication system based on generalized and complete synchronization,” Optik, doi:10.1016/j.ijleo.2012.09.008 (2012).
