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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 15 — Jul. 20, 2009
  • pp: 12619–12626
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Dual-channel chaos synchronization and communication based on unidirectionally coupled VCSELs with polarization-rotated optical feedback and polarization-rotated optical injection

Jiao Liu, Zheng-Mao Wu, and Guang-Qiong Xia  »View Author Affiliations


Optics Express, Vol. 17, Issue 15, pp. 12619-12626 (2009)
http://dx.doi.org/10.1364/OE.17.012619


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Abstract

A novel dual-channel chaotic synchronization configuration is proposed. This system is constructed on the basis of two unidirectionally coupled vertical-cavity surface-emitting lasers (VCSELs), where a VCSEL subjected to polarization-rotated optical feedback is used as a transmitter and the other VCSEL subjected to polarization-rotated optical injection is used as a receiver. The synchronization and communication performances of such a system are numerically investigated. The results show that, similar to polarization-preserved coupled system with polarization-preserved optical feedback at the T-VCSEL port and polarization-preserved optical injection at the R-VCSEL port, such polarization-rotated coupled system can also realize complete synchronization between each pair of linear polarization (LP) modes and the total output of T-VCSEL and R-VCSEL. Compared with the polarization-preserved coupled system, this proposed system has higher tolerance to mismatched parameters. Furthermore, the average intensities of two orthogonal LP modes are almost the same so that this framework may be used to realize dual-channel chaos communication. Under the additive chaos modulation (ACM) encryption scheme, the encoded messages can be successfully extracted for both of orthogonal LP modes.

© 2009 OSA

1. Introduction

Since the first demonstration of chaos synchronization by Pecora and Carroll [1

1. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990). [CrossRef] [PubMed]

], different chaotic synchronization schemes and their applications have been reported. Chaotic synchronization based on semiconductor lasers (SLs) and their applications in secure communication have attracted considerable interests during the past decades [2

2. C. Masoller, “Anticipation in the synchronization of chaotic semiconductor lasers with optical feedback,” Phys. Rev. Lett. 86(13), 2782–2785 (2001). [CrossRef] [PubMed]

24

24. S. Sivaprakasam and K. A. Shore, “Signal masking for chaotic optical communication using external-cavity diode lasers,” Opt. Lett. 24(17), 1200–1202 (1999). [CrossRef]

]. As one of the microchip lasers, vertical-cavity surface-emitting lasers (VCSELs) exhibit many advantages over the conventional edge-emitting semiconductor lasers (EELs), such as single longitudinal-mode operation, low threshold current, circular output beam with narrow divergence, low cost and easy large-scale integration into two-dimensional arrays etc. Generally, the output of the VCSEL includes two orthogonal linear polarization (LP) modes (i. e., x LP mode and y LP mode) due to weak material and cavity anisotropies, which in our opinions may afford a possibility to realize dual-channel communication by separately using the two LP modes. There were many experimental and theoretical reports on the synchronization and communication characteristics of two unidirectionally coupled VCSELs [7

7. M. S. Torre, C. Masoller, and K. A. Shore, “Synchronization of unidirectionally coupled multi-transverse-mode vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 21(10), 1772–1780 (2004). [CrossRef]

16

16. D. Z. Zhong and Z. M. Wu, “Complete chaotic synchronization mechanism of polarization mode of VCSEL with anisotropic optical feedback,” Opt. Commun. 282(8), 1631–1639 (2009). [CrossRef]

], where different systematical constructions, such as polarization-preserved coupled system with polarization-preserved optical feedback at the transmitter VCSEL (T-VCSEL) port and polarization-preserved optical injection at the receiver VCSEL (R-VCSEL) port [8

8. I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, “Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers,” Opt. Lett. 32(12), 1629–1631 (2007). [CrossRef] [PubMed]

10

10. Y. H. Hong, M. W. Lee, J. Paul, P. S. Spencer, and K. A. Shore, “Enhanced chaos synchronization in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers with polarization-preserved injection,” Opt. Lett. 33(6), 587–589 (2008). [CrossRef] [PubMed]

], the system with polarization-preserved optical feedback and orthogonal optical injection [11

11. Y. H. Hong, M. W. Lee, P. S. Spencer, and K. A. Shore, “Synchronization of chaos in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers,” Opt. Lett. 29(11), 1215–1217 (2004). [CrossRef] [PubMed]

14

14. M. Sciamanna, I. Gatare, A. Locquet, and K. Panajotov, “Polarization synchronization in unidirectionally coupled vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. E 75, 056213–1-10 (2007).

] or system with orthogonal optical feedback and orthogonal optical injection [15

15. W. L. Zhang, W. Pan, B. Luo, X. F. Li, X. H. Zou, and M. Y. Wang, “Separate polarization modes synchronization and synchronization switches between vertical-cavity surface-emitting lasers,” Opt. Rev. 13(6), 443–448 (2006). [CrossRef]

], are taken into consideration respectively. However, we have noticed that most of relevant works usually focused on the synchronization properties of total output of VCSELs. In this paper, a novel dual-channel chaos synchronization system based on two unidirectionally coupled VCSELs subject to polarization-rotated optical feedback at the T-VCSEL port and polarization-rotated optical injection at the R-VCSEL port is presented, and the synchronization performances of such configuration and the influence of mismatched parameters on synchronization for each LP mode and the total output are investigated numerically. Also, communication performances between each pair of LP modes of two VCSELs have been preliminarily examined.

2. System model and theory

Figure 1
Fig. 1 Schematic diagram for the dual-channel chaos communication system based on the unidirectionally VCSELs subject to polarization-rotated optical feedback and polarization-rotated optical injection. T-VCSEL: transmitter-VCSEL; R-VCSEL: receiver-VCSEL; PBS: polarization beam splitter; HWP: half-wave plate; BS: beam splitter; ISO: optical isolator; M: mirror; PD: photodetector.
is the schematic diagram of the polarization-rotated coupled system. The output of T-VCSEL is divided into x and y LP modes by a polarization beam splitter (PBS). Then, different messages can be independently encoded on each LP mode. These two orthogonal LP modes are recombined by PBS2 before passing through a half-wave plate (HWP). Here, the angles between the x direction and the fast axis of the HWP are 45°. As a result, the x LP mode is rotated into the y direction polarization, while the y LP mode is rotated into the x direction polarization before being fed back into the T-VCSEL and being injected into the R-VCSEL. Optical isolator (ISO) is used to ensure the light unidirectional transmission.

Based on the spin-flip model (SFM) [25

25. J. M. Regalado, F. Prati, M. S. Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997). [CrossRef]

], for such a proposed system, the rate equations for the T-VCSEL with polarization-rotated optical feedback and the R-VCSEL with polarization-rotated optical injection can be described by:

dEx,yTdt=k(1+iα)[(NT1)Ex,yT±inTEy,xT]±(γaiγp)Ex,yT+fEy,xT(tτ)eiωTτ+Fx,yT
(1)
dEx,yRdt=k(1+iα)[(NR1)Ex,yR±inREy,xR]±(γaiγp)Ex,yR+ηEy,xT(tτc)e-iωTτc+iΔωt+Fx,yR
(2)
dNT,Rdt=γ[NT,Rμ+NT,R(|ExT,R|2+|EyT,R|2)]iγnT,R(EyT,RExT,R*ExT,REyT,R*)
(3)
dnT,Rdt=γsnT,RγnT,R(|ExT,R|2+|EyT,R|2)iγNT,R(EyT,RExT,R*ExT,REyT,R*)
(4)

where superscripts T and R stand for T-VCSEL and R-VCSEL, respectively, and subscripts x and y represents x and y LP modes, respectively. E is the slowly varied complex amplitude of the field, N is the total carrier inversion between the conduction and valence bands, n accounts for the difference between carrier inversions for the spin-up and spin-down radiation channels, k is the decay rate of field, α is the line-width enhancement factor, γ is the decay rate of total carrier population, γs is the spin-flip rate, γa and γp are the linear anisotropies representing dichroism and birefringence, respectively, μ is the normalized injection current (μ takes the value 1 at threshold), τ is the feedback delay time and τc is the propagation delay time between the T-VCSEL and R-VCSEL, f is the feedback rate and η is the injection rate. ωT and ω R are respectively the optical frequencies of T-VCSEL and R-VCSEL at the solitary laser threshold in the absence of linear anisotropies, Δω = ω TR is the frequency detuning, and spontaneous emission noises are modeled by following Langevin sources:

FxT,R=βsp/2(NT,R+nT,Rξ1T,R+NT,RnT,Rξ2T,R)
(5a)
FyT,R=iβsp/2(NT,R+nT,Rξ1T,RNT,RnT,Rξ2T,R)
(5b)

where ξ1 and ξ2 indicate independent Gaussian white noise with zero mean and unitary variance, and βsp is spontaneous emission rate [8

8. I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, “Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers,” Opt. Lett. 32(12), 1629–1631 (2007). [CrossRef] [PubMed]

].

3. Results and discussion

The rate Eqs. (1)-(4) can be numerically solved with the fourth-order Runge-Kutta method. During the calculations, the parameters are used as follows [8

8. I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, “Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers,” Opt. Lett. 32(12), 1629–1631 (2007). [CrossRef] [PubMed]

,25

25. J. M. Regalado, F. Prati, M. S. Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997). [CrossRef]

]: α = 3, k = 300GHz, γ = 1GHz, γa = 0.1GHz, γp = 10GHz, γs = 50GHz, ωT = 2.2176 × 1015rad/s (the corresponding central wavelength is 850 nm), Δω = 0, βsp = 10−6GHz, τ = 3ns, τc = 5ns, μ = 1.3, f = η = 15GHz.

3.1 LP mode intensity

Figure 2(a)
Fig. 2 P-I curves for (a) polarization-rotated optical feedback and (b) polarization-preserved optical feedback.
gives P-I curve for each LP mode and the total output of a VCSEL subject to external polarization-rotated optical feedback. For comparison, the corresponding result for polarization-preserved optical feedback system is given in Fig. 2(b). In the numerical simulation, the intensities are averaged over the time window of 1.5μs. As shown in this diagram, the averaged intensity of x LP mode is always equivalent to that of y LP mode for the case of polarization-rotated optical feedback. However, for the case of polarization-preserved optical feedback, the average intensities of these two LP modes are very different.

3.2 Complete chaotic synchronization

Chaos synchronization can be divided into complete synchronization and injection-locking synchronization. The time delay between the two chaotic waveforms is a key to distinguish from these two types of synchronization. The delay time is Δτ = τc-τ for complete synchronization, while it is τc for injection-locking synchronization. In this paper, we only focus on the complete synchronization. The time series of x LP, y LP and total output intensities of the two lasers are shown in Fig. 3(a)
Fig. 3 (a) Time series of x LP, y LP and total output, and (b) corresponding cross-correlation coefficients.
. As shown in these figures, the temporal waveforms for each pair of corresponding LP modes and the total output between the two lasers are almost the same but for a time lag Δτ.

To specifically describe the synchronization quality between the two lasers, the quality of chaos synchronization and its time shift can be quantified by calculating the following shifted correlation coefficient C(Δt):

Cx,y(Δt)=[Ix,yT(tΔt)Ix,yT(tΔt)][Ix,yR(t)Ix,yR(t)][Ix,yT(tΔt)Ix,yT(tΔt)]21/2[Ix,yR(t)Ix,yR(t)]21/2
(6)

where Δt is the time shift between these two lasers output, the bracket 〈 〉 denote temporal average, I = |E|2 is the output intensity of the laser. A large value of C indicates that good synchronization has been achieved. For perfect chaotic synchronization, C equals to 1. Figure 3(b) gives the cross correlation coefficients corresponding to Fig. 3(a) as a function of time shift. It can be seen that the quality of chaotic synchronization is very relevant to time shift. The peaks are very close to 1 for Δt = 2ns, which means that complete synchronization can be achieved with 2ns time lag for each pair of corresponding LP modes and the total output, respectively.

3.3 Effects of internal mismatched parameters on synchronization quality

In order to achieve complete synchronization, it is important to match corresponding parameters of the T-VCSEL and the R-VCSEL. Though external parameters such as current can be controlled easily, internal parameters are difficult to be accurately controlled. Therefore, it is important to investigate the influences of the intrinsic mismatched parameters on the quality of chaos synchronization. For convenience, we fix intrinsic parameters of T-VCSEL, and only change the intrinsic parameters of R-VCSEL. The relative mismatched parameters are defined as:

Δα=αRαTαT,Δk=kRkTkT,Δγ=γRγTγT,Δγp=γpRγpTγpT,Δγs=γsRγsTγsT,Δγa=γaRγaTγaT
(7)

3.4 Message encoding and decoding

Finally, we will briefly examine the communication performances of this proposed system. Different encoding and decoding techniques such as chaos shift keying (CSK), chaos masking (CMS) and chaos modulation (CM) have been proposed and investigated for secure message transmission based on chaotic synchronization. In this paper, message is assumed to be encoded by means of additive chaos modulation (ACM). For the ACM encryption scheme as shown in Fig. 1, the message is mixed with the chaotic carrier in the nonlinear system, and then a new chaotic state different from the original one will be conformed. Because the symmetry of system can be maintained under the ACM scheme, the complete chaos synchronization can be achieved in the system and an excellent synchronous signal can be obtained at the receiver port in principle. The message encoding is carried out by modulating the chaotic carrier of the T-VCSEL so that Ex,yT'(t)=Ex,yT(t)(1+mx,y(t)) [3

3. A. Sanchez-Diaz, C. R. Mirasso, P. Colet, and P. Garcia-Fernandez, “Encoded Gbit/s digital communications with synchronized chaotic semiconductor lasers,” IEEE J. Quantum Electron. 35(3), 292–297 (1999). [CrossRef]

,17

17. D. Kanakidis, A. Argyris, A. Bogris, and D. Syvridis, ““Influence of the decoding process on the performance of chaos encrypted optical communication systems,” J. Lightwave Technol. 24, 335–341 (2006).

], where the modulation index is 5% and the modulation frequency is 500MHz. By using a Fourth-Butterworth low-pass filter, signals can be demodulated by filtering the intensity difference between T-VSEL output and R-VCSEL output for each LP mode. Figure 5
Fig. 5 Original message mx,y(t) and recovered message mx,y(t), where (a) and (b) correspond to x LP mode and y LP mode, respectively.
displays the encoding mx,y(t) and decoding message mx,y(t) for each LP mode, respectively. From this figure, one can clearly observe that sent message carried by these two LP modes can be extracted satisfactorily.

4. Conclusions

In summary, a novel dual-channel chaos communication system by using the two LP modes of VCSELs is presented. In such a system, a VCSEL with polarization-rotated optical feedback is used as a transmitter and a VCSEL subject to polarization-rotated optical injection is used as a receiver. The synchronization characteristics and the influence of mismatched parameters on synchronization performances have been numerically investigated. Similar to polarization-preserved synchronization system, complete synchronization can also be achieved between each pair of the LP modes and the total output of T-VCSEL and R-VCSEL for this polarization-rotated coupled configuration. Internal mismatched parameters between T-VCSEL and R-VCSEL have influence on synchronization quality, but this system possesses better robustness to mismatched parameters compared with the polarization-preserved synchronization system. Additionally, the communication performances are preliminarily examined by using different message carried separately by two LP modes.

Acknowledgments

This work was supported by the Open Fund of the State Key Lab of Millimeter Waves of China under Grant K200805 and the Natural Science Foundation Project of Chongqing City of China under Grant 2007BB2333.

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.

C. Masoller, “Anticipation in the synchronization of chaotic semiconductor lasers with optical feedback,” Phys. Rev. Lett. 86(13), 2782–2785 (2001). [CrossRef] [PubMed]

3.

A. Sanchez-Diaz, C. R. Mirasso, P. Colet, and P. Garcia-Fernandez, “Encoded Gbit/s digital communications with synchronized chaotic semiconductor lasers,” IEEE J. Quantum Electron. 35(3), 292–297 (1999). [CrossRef]

4.

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]

5.

F. Rogister, A. Locquet, D. Pieroux, M. Sciamanna, O. Deparis, P. Mégret, and M. Blondel, “Secure communication scheme using chaotic laser diodes subject to incoherent optical feedback and incoherent optical injection,” Opt. Lett. 26(19), 1486–1488 (2001). [CrossRef]

6.

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]

7.

M. S. Torre, C. Masoller, and K. A. Shore, “Synchronization of unidirectionally coupled multi-transverse-mode vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 21(10), 1772–1780 (2004). [CrossRef]

8.

I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, “Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers,” Opt. Lett. 32(12), 1629–1631 (2007). [CrossRef] [PubMed]

9.

D. Z. Zhong, G. Q. Xia, Z. M. Wu, and X. H. Jia, “Complete chaotic synchronization characteristics of the linear-polarization mode of vertical-cavity surface-emitting semiconductor lasers with isotropic optical feedback,” Opt. Commun. 281(6), 1698–1709 (2008). [CrossRef]

10.

Y. H. Hong, M. W. Lee, J. Paul, P. S. Spencer, and K. A. Shore, “Enhanced chaos synchronization in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers with polarization-preserved injection,” Opt. Lett. 33(6), 587–589 (2008). [CrossRef] [PubMed]

11.

Y. H. Hong, M. W. Lee, P. S. Spencer, and K. A. Shore, “Synchronization of chaos in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers,” Opt. Lett. 29(11), 1215–1217 (2004). [CrossRef] [PubMed]

12.

M. W. Lee, Y. Hong, and K. A. Shore, “Experimental demonstration of VCSEL-based chaotic optical communications,” IEEE Photon. Technol. Lett. 16(10), 2392–2394 (2004). [CrossRef]

13.

R. Ju, P. S. Spencer, and K. A. Shore, “Polarization-preserved and polarization-rotated synchronization of chaotic vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41(12), 1461–1467 (2005). [CrossRef]

14.

M. Sciamanna, I. Gatare, A. Locquet, and K. Panajotov, “Polarization synchronization in unidirectionally coupled vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. E 75, 056213–1-10 (2007).

15.

W. L. Zhang, W. Pan, B. Luo, X. F. Li, X. H. Zou, and M. Y. Wang, “Separate polarization modes synchronization and synchronization switches between vertical-cavity surface-emitting lasers,” Opt. Rev. 13(6), 443–448 (2006). [CrossRef]

16.

D. Z. Zhong and Z. M. Wu, “Complete chaotic synchronization mechanism of polarization mode of VCSEL with anisotropic optical feedback,” Opt. Commun. 282(8), 1631–1639 (2009). [CrossRef]

17.

D. Kanakidis, A. Argyris, A. Bogris, and D. Syvridis, ““Influence of the decoding process on the performance of chaos encrypted optical communication systems,” J. Lightwave Technol. 24, 335–341 (2006).

18.

M. W. Lee, J. Paul, C. Masoller, and K. A. Shore, “Observation of cascade complete chaos synchronisation with zero time lag in laser diodes,” J. Opt. Soc. Am. B 23(5), 846–851 (2006). [CrossRef]

19.

A. Locquet, C. Masoller, and C. R. Mirasso, “Synchronization regimes of optical-feedback-induced chaos in unidirectionally coupled semiconductor lasers,” Phys. Rev. E. 65, 056205–1-12 (2002).

20.

G. Q. Xia, Z. M. Wu, and J. F. Liao, “Theoretical investigations of cascaded chaotic synchronization and communication based on optoelectronic negative feedback semiconductor lasers,” Opt. Commun. 282(5), 1009–1015 (2009). [CrossRef]

21.

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]

22.

C. Mirasso, J. Mulet, and C. Masoller, “Chaos shift keying encryption in chaotic external-cavity semiconductor lasers using a single-receiver scheme,” IEEE Photon. Technol. Lett. 14(4), 456–458 (2002). [CrossRef]

23.

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]

24.

S. Sivaprakasam and K. A. Shore, “Signal masking for chaotic optical communication using external-cavity diode lasers,” Opt. Lett. 24(17), 1200–1202 (1999). [CrossRef]

25.

J. M. Regalado, F. Prati, M. S. Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997). [CrossRef]

OCIS Codes
(060.4510) Fiber optics and optical communications : Optical communications
(140.1540) Lasers and laser optics : Chaos
(060.4785) Fiber optics and optical communications : Optical security and encryption
(140.7260) Lasers and laser optics : Vertical cavity surface emitting lasers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: May 14, 2009
Revised Manuscript: June 28, 2009
Manuscript Accepted: July 1, 2009
Published: July 20, 2009

Citation
Jiao Liu, Zheng-Mao Wu, and Guang-Qiong 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)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12619


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References

  1. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990). [CrossRef] [PubMed]
  2. C. Masoller, “Anticipation in the synchronization of chaotic semiconductor lasers with optical feedback,” Phys. Rev. Lett. 86(13), 2782–2785 (2001). [CrossRef] [PubMed]
  3. A. Sanchez-Diaz, C. R. Mirasso, P. Colet, and P. Garcia-Fernandez, “Encoded Gbit/s digital communications with synchronized chaotic semiconductor lasers,” IEEE J. Quantum Electron. 35(3), 292–297 (1999). [CrossRef]
  4. 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]
  5. F. Rogister, A. Locquet, D. Pieroux, M. Sciamanna, O. Deparis, P. Mégret, and M. Blondel, “Secure communication scheme using chaotic laser diodes subject to incoherent optical feedback and incoherent optical injection,” Opt. Lett. 26(19), 1486–1488 (2001). [CrossRef]
  6. 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]
  7. M. S. Torre, C. Masoller, and K. A. Shore, “Synchronization of unidirectionally coupled multi-transverse-mode vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 21(10), 1772–1780 (2004). [CrossRef]
  8. I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, “Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers,” Opt. Lett. 32(12), 1629–1631 (2007). [CrossRef] [PubMed]
  9. D. Z. Zhong, G. Q. Xia, Z. M. Wu, and X. H. Jia, “Complete chaotic synchronization characteristics of the linear-polarization mode of vertical-cavity surface-emitting semiconductor lasers with isotropic optical feedback,” Opt. Commun. 281(6), 1698–1709 (2008). [CrossRef]
  10. Y. H. Hong, M. W. Lee, J. Paul, P. S. Spencer, and K. A. Shore, “Enhanced chaos synchronization in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers with polarization-preserved injection,” Opt. Lett. 33(6), 587–589 (2008). [CrossRef] [PubMed]
  11. Y. H. Hong, M. W. Lee, P. S. Spencer, and K. A. Shore, “Synchronization of chaos in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers,” Opt. Lett. 29(11), 1215–1217 (2004). [CrossRef] [PubMed]
  12. M. W. Lee, Y. Hong, and K. A. Shore, “Experimental demonstration of VCSEL-based chaotic optical communications,” IEEE Photon. Technol. Lett. 16(10), 2392–2394 (2004). [CrossRef]
  13. R. Ju, P. S. Spencer, and K. A. Shore, “Polarization-preserved and polarization-rotated synchronization of chaotic vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41(12), 1461–1467 (2005). [CrossRef]
  14. M. Sciamanna, I. Gatare, A. Locquet, and K. Panajotov, “Polarization synchronization in unidirectionally coupled vertical-cavity surface-emitting lasers with orthogonal optical injection,” Phys. Rev. E 75, 056213–1-10 (2007).
  15. W. L. Zhang, W. Pan, B. Luo, X. F. Li, X. H. Zou, and M. Y. Wang, “Separate polarization modes synchronization and synchronization switches between vertical-cavity surface-emitting lasers,” Opt. Rev. 13(6), 443–448 (2006). [CrossRef]
  16. D. Z. Zhong and Z. M. Wu, “Complete chaotic synchronization mechanism of polarization mode of VCSEL with anisotropic optical feedback,” Opt. Commun. 282(8), 1631–1639 (2009). [CrossRef]
  17. D. Kanakidis, A. Argyris, A. Bogris, and D. Syvridis, ““Influence of the decoding process on the performance of chaos encrypted optical communication systems,” J. Lightwave Technol. 24, 335–341 (2006).
  18. M. W. Lee, J. Paul, C. Masoller, and K. A. Shore, “Observation of cascade complete chaos synchronisation with zero time lag in laser diodes,” J. Opt. Soc. Am. B 23(5), 846–851 (2006). [CrossRef]
  19. A. Locquet, C. Masoller, and C. R. Mirasso, “Synchronization regimes of optical-feedback-induced chaos in unidirectionally coupled semiconductor lasers,” Phys. Rev. E. 65, 056205–1-12 (2002).
  20. G. Q. Xia, Z. M. Wu, and J. F. Liao, “Theoretical investigations of cascaded chaotic synchronization and communication based on optoelectronic negative feedback semiconductor lasers,” Opt. Commun. 282(5), 1009–1015 (2009). [CrossRef]
  21. 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]
  22. C. Mirasso, J. Mulet, and C. Masoller, “Chaos shift keying encryption in chaotic external-cavity semiconductor lasers using a single-receiver scheme,” IEEE Photon. Technol. Lett. 14(4), 456–458 (2002). [CrossRef]
  23. 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]
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