## Chaos synchronization of unidirectionally injected vertical-cavity surface-emitting lasers with global and mode-selective coupling

Optics Express, Vol. 14, Issue 8, pp. 3138-3151 (2006)

http://dx.doi.org/10.1364/OE.14.003138

Acrobat PDF (204 KB)

### Abstract

Chaos synchronization in two unidirectionally coupled vertical-cavity surface-emitting lasers (VCSELs) with open-loop configuration is studied numerically. We consider two low-order transverse modes for weakly index-guided VCSELs and investigate the generalized type of chaos synchronization for both global coupling and mode-selective coupling configurations. It is found that the synchronization can be obtained between each mode of the transmitter and the receiver lasers when all modes of the transmitter are injected into the receiver equally and coupled with the corresponding modes of the receiver laser. For mode-selective cases, the modes with strong optical injection can achieve the synchronization; while the rest modes without coupling can not obtain the synchronization any more even though the injection is very strong. The results afford an opportunity to multichannel optical secure communications.

© 2006 Optical Society of America

## 1. Introduction

01. K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Phys. Rev. Lett. **71**, 65–68 (1993). [CrossRef] [PubMed]

02. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum. Electron. **16**, 347–355 (1980). [CrossRef]

10. G. D. Van Wiggeren and R. Roy, “Communication with chaotic lasers,” Science **279**, 1198–1200 (1998). [CrossRef]

## 2. Theoretical model

31. A. Valle, J. Sarma, and K. A. Shore, “Spatial holeburning effects on the dynamics of vertical cavity surface-emitting laser diodes,” IEEE J. Quantum Electron. **31**, 1423–1431 (1995). [CrossRef]

31. A. Valle, J. Sarma, and K. A. Shore, “Spatial holeburning effects on the dynamics of vertical cavity surface-emitting laser diodes,” IEEE J. Quantum Electron. **31**, 1423–1431 (1995). [CrossRef]

33. X. F. Li, W. Pan, B. Luo, D. Ma, and G. Deng, “Theoretical analysis of multi-transverse-mode characteristics of vertical-cavity surface-emitting lasers”, Semicond. Sci. … Technol. **20**, 505–513 (2005). [CrossRef]

*d*

_{w}for each. Along the longitudinal direction (

*z*direction), the distance between two distributed Bragg reflectors defines the length of laser cavity. The radiuses of the active region and the cladding layer are taken to be

*R*

_{core}and

*R*

_{clad}, respectively. The refractive index is

*n*(

*r*) =

*n*

_{core}for

*r*<

*R*

_{core}and

*n*(

*r*) =

*n*

_{clad}for

*R*

_{core}<

*r*<

*R*

_{clad}. The injection current

*I*(

*r*) is taken to be of a disc form, i.e.

*I*(

*r*) =

*I*

_{0}for

*r*<

*R*

_{d}and

*I*(

*r*)=0 otherwise, where

*R*

_{d}is the radius of the current injection region.

*m*and

*n*denote the corresponding azimuthal and radial orders;

*s*and

*c*denote the sine and cosine azimuthal modes, respectively. In this paper, to simplify the calculations, only two low-order cosine azimuthal modes (LP

_{01}and LP

_{11}) are taken into account. However, the more transverse mode cases can still be studied in a straight forward manner.

*E*

_{i}(

*r, φ, t*) of the ith transverse mode can be written as

*i*= 1 and

*i*= 2 correspond to the LP

_{01}and LP

_{11}modes, respectively.

*E*

_{i}(

*t*) and

*ψ*

_{i}(

*r, ψ*) are the spatially independent average electric field and its corresponding transverse distribution function.

*t, r*and

*φ*are the temporal, radial, and azimuthal variables, respectively. The electric distribution is normalized in both the longitudinal and the transverse directions so that |

*E*

_{i}(

*t*)| is proportional to the photon density

*P*

_{i}(

*t*) of the ith mode. For the special weakly index-guided structure, the transverse mode profiles can be easily obtained by using the optical waveguide theory

*c*

_{i}is the normalization coefficient satisfying following relationship

*J*

_{i-1}and

*K*

_{i-1}are the i-1th order Bessel functions of the first and second kinds, respectively.

*n*

_{clad}

*k*

_{0})

^{2}are the eigenvalues, which can be obtained by solving following eigenvalue equations [33

33. X. F. Li, W. Pan, B. Luo, D. Ma, and G. Deng, “Theoretical analysis of multi-transverse-mode characteristics of vertical-cavity surface-emitting lasers”, Semicond. Sci. … Technol. **20**, 505–513 (2005). [CrossRef]

*β*

_{i}is the propagation constant and

*k*

_{0}is the vacuum wave number.

02. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum. Electron. **16**, 347–355 (1980). [CrossRef]

*j*= √-1 ,

*β*

_{c}is the linewidth broadening factor, Γ

_{z}is the longitudinal field confinement,

*τ*

_{p}is the photon lifetime,

*D*

_{n}is the carrier diffusion coefficient in the active region,

*q*is the electric charge,

*V*=

*d*

_{w}is the volume of the active layer,

*τ*

_{e}is the carrier lifetime,

*v*

_{g}is the group velocity, g

_{0}is the gain coefficient, and

*N*

_{0}is the transparent carrier density. We can see that the distribution of carriers shows strongly spatial dependences. Different transverse modes have different spatial distributions; therefore, the optical gains of them are actually provided by different local carriers and these modes can interact through the competition for the carriers [31–33

31. A. Valle, J. Sarma, and K. A. Shore, “Spatial holeburning effects on the dynamics of vertical cavity surface-emitting laser diodes,” IEEE J. Quantum Electron. **31**, 1423–1431 (1995). [CrossRef]

*k*and

*k*

_{c}are the corresponding feedback and injection parameters,

*τ*is the feedback delays time, and

*τ*

_{c}is the propagation time of the carrier signal from the transmitter to the receiver.

*υ*= (

*π*) is the frequency detuning between the two lasers. Besides, the azimuthal distribution is also averaged by azimuthal integrating to avoid the complex three-dimensional integration. We assume that both modes have the same value of ∆

*υ*. The optical gain of the ith transverse mode is obtained by completing following two-dimensional integration

*ps*and 0.04

*μm*, respectively. To better characterize the laser behaviors, denser divisions of the considered domains are required; however, this will inevitably increase the computational efforts greatly. Therefore, there is a typical tradeoff between the model accuracy and the calculation speed. In this study, the bias current

*I*

_{0}and the current injection radius

*R*

_{d}are taken to be 1mA and 2.1

*μm*, respectively, so that both modes (LP

_{01}and LP

_{11}modes) can be excited simultaneously. The other parameters can be found in Ref. [33

33. X. F. Li, W. Pan, B. Luo, D. Ma, and G. Deng, “Theoretical analysis of multi-transverse-mode characteristics of vertical-cavity surface-emitting lasers”, Semicond. Sci. … Technol. **20**, 505–513 (2005). [CrossRef]

*u*and

*w*, which can be obtained by solving the eigenvalue equations; therefore, the propagation constant ultimately decides the spatial distribution of a special transverse mode. However, the eigenvalue equations are some transcendent equations which can not be solved directly. In this paper, by using computer simulation, we numerically calculate the eigenvalue equations and obtain the propagation constants corresponding to these two transverse modes. The values of the obtained propagation constants and eigenvalues are listed in Table 1. In section 3, using these parameters together with the typical device parameters, we will solve the rate equations numerically and investigate the chaos synchronization relating issues under different coupling conditions detailedly.

## 3. Results and discussions

*k*=8

*ns*

^{-1}and

*L*

_{ext}=2

*cm*. In this figure, (a), (b), and (c) correspond to the dynamics of the output power, the optical phase, and the carrier density, respectively. With the parameters under consideration, the free-running powers of the two modes are different, i.e. the strength of LP

_{01}mode is relatively larger than that of LP

_{11}mode due to its stronger coupling with the carrier profile. With strong feedback, as shown in Fig. 1(a), the power of LP

_{01}mode is always larger than that of LP

_{11}mode, and both modes show noise-like oscillations in the time domain. The fluctuation amplitudes are very large, which are beneficial to mask the message efficiently. Similarly, the optical phases of both modes also show chaotic oscillations, that is to say the frequency chirps as well as the laser operating frequencies will vary with the same rule. Due to the noise-like variations of the modal strengths, the consumptions of both modes to the carriers also show randomicity, which leads the carrier density to show similar noise-like fluctuation. The temporal variation of carrier density is plotted in Fig. 1(c), where the chaotic behaviors can be seen clearly. As explained previously, the chaotic output is very complex and ultra-sensitive to the system parameters and initial conditions. It is very difficult to decode the message without a proper receiver due to the high frequencies involved and the large number of dynamic degrees of freedom of the chaotic carriers. In other words, the chaos synchronization can not be achieved without proper system parameters, including the device parameters and operating parameters.

_{01}mode, and the power of LP

_{11}mode, respectively. It is obvious that all they show chaotic oscillations introduced by the strong optical feedback (

*k*=8

*ns*

^{-1}and

*L*

_{ext}=2

*cm*). All these chaotic outputs of M-VCSEL are injected into the resonant cavity of S-VCSEL. The injection strength is chosen to be so strong (

*k*

_{c}= 533

*ns*

^{-1}) that the receiver laser can easily be locked by the transmitter laser. Hence, as shown in the second column of Fig. 2, the total output power and the powers of both modes show the same variation trends with those of the transmitter laser. Due to the amplification effects of the strong optical injection, the output power of S-VCSEL is relatively stronger than M-VCSEL’s. In addition, the injection delay is taken to be zero (

*L*

_{c}= 0

*cm*) in order to estimate the synchronization degree and calculate the correlation curve without shifting any of these two chaotic signals. The correlation curves of the total output, LP

_{01}mode output, and LP

_{11}mode output are given in the third column of Fig. 2, respectively. From these curves, we can get that the synchronization quality is very high.

*τ*=1

*ns*and

*τ*

_{c}=3

*ns*) to investigate the time delay between the chaotic signals of the two lasers. In Fig. 3, the black line denotes the LP

_{01}mode and the gray line stands for the LP

_{11}mode. Since the synchronization considered currently is the generalized type; therefore, according to the previous papers [12

12. J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. **38**, 1141–1154 (2002). [CrossRef]

13. K. Kusumoto and J. Ohstubo, “Anticipating synchronization based on optical injection-locking in chaotic semiconductor lasers,” IEEE J. Quantum Electron. **39**, 1531–1536 (2003). [CrossRef]

20. X. F. Li, W. Pan, B. Luo, and D. Ma, “Effects of unwanted feedback on synchronized chaotic optical communications,” Appl. Opt. **45**, (2006) (to be published) [CrossRef] [PubMed]

*τ*

_{c}, i.e. 3ns for the parameters here, as shown in Fig. 3(a) and (c). Figures 3(b) and (d) give the power spectra corresponding to Fig. 3(a) and (c), respectively, by applying fast Fourier transformation (FFT) to the time series of modal powers. As shown in Fig. 3(b), the power spectra of both modes are broadened greatly compared with the free-running case (the free-running power spectra are not given here). There are so many frequency components in the spectrum of the chaotic signal; therefore, as introduced previously the chaotic carrier used in the communication system is very complex and it is difficult to be predicted or duplicated. Due to the strong optical injection of the chaotic signal from the transmitter, the power spectra of the modes of the slave laser can not keep its original simple forms (free-running power spectrum) any more, but show the similar broadened chaotic spectra with those of the transmitter laser. Moreover, the spectrum intensities are relatively larger than those of the injecting chaotic signals, including LP

_{01}and LP

_{11}modes. From these figures, we can see that with these parameters and working conditions, the output intensities and the corresponding power spectra of the two modes achieve the synchronization with high performance.

*P*

_{m,i}and

*P*

_{s,i}are the power of the ith-mode of the transmitter and receiver, respectively. ∆

*t*is the time delay between the two sets of the chaotic signals of these two lasers. The brackets denote the temporal averaging. For the calculation of the correlation function of the total modal power, the powers of the transmitter and the receiver should be the addition of the powers of the LP01 and LP11 modes. The simulation results are plotted in Fig. 4, where (a), (b), and (c) correspond to the total power, the LP

_{01}mode, and the LP

_{11}mode, respectively. We consider that the system is synchronized when the global maximum of the correlation function is over 0.9. At the same time, the time value corresponding to that maximum value indicates the kind of synchronization. From these three small figures in Fig. 4, we can easily find that the maximum of the correlation coefficient happens at the value of ∆

*t*=3

*ns*, which equals to the transmission delay of the chaotic signal emitted from the transmitter by the square. This indicates that the synchronization achieved here is the generalized type, where the slave laser responds the received chaotic signal immediately. In fact, the delay is the result of the propagation of the optical signal in the channel. It is worthy pointing out that the time value corresponding to the maximum of the correlation coefficient should be 2

*ns*(

*τ*

_{c}-

*τ*) if the complete type of synchronization is considered. The detailed discussions can be found in some other papers [12

12. J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. **38**, 1141–1154 (2002). [CrossRef]

13. K. Kusumoto and J. Ohstubo, “Anticipating synchronization based on optical injection-locking in chaotic semiconductor lasers,” IEEE J. Quantum Electron. **39**, 1531–1536 (2003). [CrossRef]

16. J. M. Buldú, J. García-Ojalvo, and M. C. Torrent, “Multimode synchronization and communication using unidirectionally coupled semiconductor lasers,” IEEE J. Quantum Electron. **40**, 640–650 (2004). [CrossRef]

_{01}-injection case. It is more convenient here to compare these results with those of the global coupling case. As shown in Fig. 5, we can see that the synchronization performance gets worse compared with Fig. 2, since the equivalent injection strength without the coupling of LP

_{11}mode is actually decreased for a fixed value of

*k*

_{c}. Simultaneously, it can also be seen that the LP

_{11}mode of the slave laser is completely suppressed by the LP

_{01}-mode-selective injection. In fact, the mode-selective injection can be used to improve the single-mode characteristics, which has attracted much attention already. Figure 6 displays the similar results to those of Fig. 5. In Fig. 6 only the LP

_{11}mode is injected into the slave laser. Unlike the case of Fig. 5, the LP

_{01}mode has not been suppressed thoroughly by the LP

_{01}mode injection due to its relatively larger free-running strength. In addition, the synchronization performance is degraded further compared with LP

_{01}-injection case. From the physical viewpoint, it is ultimately the result of the further decrease of the equivalent injection strength. We have pointed out that the free-running strength of LP

_{11}mode is weaker than that of LP

_{01}mode; therefore, with the same injection parameter the coupling between the transmitter and the receiver is weaker for the LP

_{11}-injection case. From Fig. 5 and Fig. 6 we can also conclude that when the mode-selective injection is considered the other modes without optical injection can not achieve synchronization any more, being different from the modes with optical injection.

*k*

_{c}on the system synchronization degree. Figure 7 displays the variations of cross-correlation coefficient

*C*(∆

*t*=

*τ*

_{c}) with respect to

*k*

_{c}for different coupling configurations, where (a), (b), and (c) correspond to all-to-all injection (global injection), LP

_{01}-injection, and LP

_{11}-injection, respectively. These figures clearly display how the two VCSELs of the secure communication system evolve from free-running state to the synchronization with the increase of the coupling degree. Here, to simplify the calculation of the correlation coefficient, the transmission delay is taken to be 0ns. As shown in Fig. 7(a), for weak injection cases, the correlation coefficients for both modes as well as the total laser power are very small, that is to say the synchronization degree of the system is very low. With the increase of

*k*

_{c}, however, the correlation coefficients begin to increase. Especially, when

*k*

_{c}is larger than 300

*ns*

^{-1}all these coefficients are near 1, which shows that the system achieves an accurate synchronization under strong optical injection. However, for LP

_{01}mode injection [Fig. 7(b)], the results are different since the two modes have different coupling designs. We can see that, for strong optical injections, only LP

_{01}modes of these two lasers can obtain synchronization, while the rest modes without optical coupling can not achieve synchronization. With the synchronization of LP

_{01}mode, the total power is also synchronized since the LP

_{11}mode is actually suppressed by the mode-selective injection and the total laser power is solely provided by the synchronized LP

_{01}mode at this time. Due to the lack of optical coupling, the correlation degree of LP

_{11}modes keeps very low. Besides, we can also find from Fig. 7(b) that values of the system correlation coefficient for strong optical injection cases are somewhat smaller than those of the global injection case (compared with the maximum shown in Fig. 4). This indicates that the synchronization degree is actually degraded by the single mode coupling. This coincides well with the observation obtained by analyzing the correlation curves shown in Fig. 2 and 5. The case of LP

_{11}mode injection is also examined and the simulation results are given in Fig. 7(c), where similar behaviors are also observed. The differences Fig. 7(b) and Fig. 7(c) are: (1) the LP

_{01}mode is not completely suppressed by the mode-selective injection the correlation coefficient dose not become as small as LP

_{01}-injection case; (2) the system synchronization performance is degraded further due to the further decrease of the equivalent injection level. However, this degradation can easily be compensated by increasing the injection strength. From the utilizing point of view, the multi-mode VCSELs and mode-selective injection configuration provide an opportunity for multichannel optical secure communications [35

35. J. K. White and J. V. Moloney, “Multichannel communication using an infinite dimensional spatiotemporal chaotic system,” Phys. Rev. A **59**, 2422–2426 (1999). [CrossRef]

36. S. F. Yu, P. Shum, and N. Q. Ngo, “Performance of optical chaotic communication system using multimode vertical cavity surface emitting lasers,” Opt. Commun. **200**, 143–152 (2001). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

01. | K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Phys. Rev. Lett. |

02. | R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum. Electron. |

03. | H. Kakiuchida and J. Ohtsubo, “Characteristics of a semiconductor laser with external feedback,” IEEE J. Quantum. Electron. |

04. | H. Lamela, G. Carpintero, and F. J. Mancebo, “Period tripling and chaos in the dynamic behavior of directly modulated diode lasers,” IEEE J. Quantum.Electron. |

05. | S. Tang and J. M. Liu, “Chaotic pulsing and quasiperiodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum. Electron. |

06. | F. Y. Lin and J. M. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum. Electron. |

07. | T. B. Simpson, “Mapping the nonlinear dynamics of a distributed feedback semiconductor laser subject to external optical injection,” Opt. Commun. |

08. | X. F. Li, W. Pan, B. Luo, and D. Ma, “Nonlinear dynamic behaviors of an optically injected vertical-cavity surface-emitting laser,” Chaos, Solitons … Fractals |

09. | X.F. Li, W. Pan, B. Luo, D. Ma, Z. Zhao, and G. Deng, “Theoretical study on controlling nonlinear behaviors of a couple-cavity VCSEL by external optical injection,” Chinese Opt. Lett. |

10. | G. D. Van Wiggeren and R. Roy, “Communication with chaotic lasers,” Science |

11. | V. Ahlers, U. Parlitz, and W. Lauterborn, “Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,” Phys. Rev. E |

12. | J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. |

13. | K. Kusumoto and J. Ohstubo, “Anticipating synchronization based on optical injection-locking in chaotic semiconductor lasers,” IEEE J. Quantum Electron. |

14. | A. Uchida, Y. Liu, and P. Davis, “Characteristics of chaotic masking in synchronized semiconductor lasers,” IEEE J. Quantum Electron. |

15. | Y. Liu, H. F. Chen, J. M. Liu, P. Davis, and T. Aida, “Synchronization of optical-feedback-induced chaos in semiconductor lasers by optical injection,” Phys. Rev. A |

16. | J. M. Buldú, J. García-Ojalvo, and M. C. Torrent, “Multimode synchronization and communication using unidirectionally coupled semiconductor lasers,” IEEE J. Quantum Electron. |

17. | H. F. Chen and J. M. Liu, “Open-loop chaotic synchronization of injection-locked semiconductor lasers with gigahertz range modulation,” IEEE J. Quantum Electron. |

18. | R. Vicente, T. Pérez, and C. R. Mirasso, “Open- versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers,” IEEE J. Quantum Electron. |

19. | A. Murakami, “Phase locking and chaos synchronization in injection-locked semiconductor lasers,” IEEE J. Quantum Electron. |

20. | X. F. Li, W. Pan, B. Luo, and D. Ma, “Effects of unwanted feedback on synchronized chaotic optical communications,” Appl. Opt. |

21. | Y. Takiguchi, H. Fujino, and J. Ohtsubo, “Experimental synchronization of chaotic oscillations in external cavity semiconductor lasers in low-frequency fluctuation regime,” Opt. Lett. |

22. | H. Fujino and J. Ohtsubo, “Experimental synchronization of chaotic oscillations in external-cavity semiconductor lasers,” Opt. Lett. |

23. | M. Peil, T. Heil, I. Fischer, and WW. Elsäßer , “Synchronization of chaotic semiconductor laser systems: a vectorial coupling-dependent scenario,” Phys. Rev. Lett. |

24. | G. Q. Xia, Z. M. Wu, and J. G. Wu, “Theory and simulation of dual-channel optical chaotic communication system,” Opt. Express |

25. | N. Fujiwara, Y. Takiguchi, and J. Ohtsubo, “Observation of the synchronization of chaos in mutually injection vertical-cavity surface-emitting semiconductor lasers,” Opt. Lett. |

26. | S. Sivaprakasam, E. M. Shahverdiec, P. S. Spencer, and K. A. Shore, “Experimental demonstration of anticipating synchronization in chaotic semiconductor lasers with optical feedback,” Phys. Rev. Lett. |

27. | Y. 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. |

28. | P. S. Spencer, C. R. Mirasso, P. Colet, and K. A. Shore, “Modeling of optical synchronization of chaotic external-cavity VCSEL’s,” IEEE J. Quantum Electron. |

29. | P. S. Spencer and C. R. Mirasso, “Analysis of optical chaos synchronization in frequency-detuned external-cavity VCSEL’s,” IEEE J. Quantum Electron. |

30. | C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. Von Lehmen, L. T. Florez, and N. G. Stoffel, “Dynamic, polarization and transverse mode characteristics of VCSEL’s,” IEEE J. Quantum Electron. |

31. | A. Valle, J. Sarma, and K. A. Shore, “Spatial holeburning effects on the dynamics of vertical cavity surface-emitting laser diodes,” IEEE J. Quantum Electron. |

32. | H. B. Zhang, G. Mrozynski, A. Wallrabenstein, and J. Schrage, “Analysis of transverse mode competition of VCSELs based on a spatially independent model,” IEEE J. Quantum Electron. |

33. | X. F. Li, W. Pan, B. Luo, D. Ma, and G. Deng, “Theoretical analysis of multi-transverse-mode characteristics of vertical-cavity surface-emitting lasers”, Semicond. Sci. … Technol. |

34. | P. V. Mena, J. J. Morikuni, S. M. Kang, A. V. Harton, and K. W. Wyatt, “A comprehensive circuit-level model of vertical-cavity surface-emitting lasers,” IEEE J. Lightwave Technol. |

35. | J. K. White and J. V. Moloney, “Multichannel communication using an infinite dimensional spatiotemporal chaotic system,” Phys. Rev. A |

36. | S. F. Yu, P. Shum, and N. Q. Ngo, “Performance of optical chaotic communication system using multimode vertical cavity surface emitting lasers,” Opt. Commun. |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(140.1540) Lasers and laser optics : Chaos

(250.7260) Optoelectronics : Vertical cavity surface emitting lasers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 21, 2005

Revised Manuscript: March 22, 2006

Manuscript Accepted: April 8, 2006

Published: April 17, 2006

**Citation**

Xiaofeng Li, Wei Pan, Dong Ma, and Bin Luo, "Chaos synchronization of unidirectionally injected vertical-cavity surface-emitting lasers with global and mode-selective coupling," Opt. Express **14**, 3138-3151 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3138

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### References

- K. M. Cuomo, and A. V. Oppenheim, "Circuit implementation of synchronized chaos with applications to communications," Phys. Rev. Lett. 71, 65-68 (1993). [CrossRef] [PubMed]
- R. Lang, and K. Kobayashi, "External optical feedback effects on semiconductor injection laser properties," IEEE J. Quantum. Electron. 16, 347-355 (1980). [CrossRef]
- H. Kakiuchida, and J. Ohtsubo, "Characteristics of a semiconductor laser with external feedback," IEEE J. Quantum. Electron. 30, 2087-2097 (1994). [CrossRef]
- H. Lamela, G. Carpintero, and F. J. Mancebo, "Period tripling and chaos in the dynamic behavior of directly modulated diode lasers," IEEE J. Quantum. Electron. 34, 1797-1801 (1998). [CrossRef]
- S. Tang, and J. M. Liu, "Chaotic pulsing and quasiperiodic route to chaos in a semiconductor laser with delayed opto-electronic feedback," IEEE J. Quantum. Electron. 37, 329-336 (2001). [CrossRef]
- F. Y. Lin, and J. M. Liu, "Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback," IEEE J. Quantum. Electron. 39, 562-567 (2003). [CrossRef]
- T. B. Simpson, "Mapping the nonlinear dynamics of a distributed feedback semiconductor laser subject to external optical injection," Opt. Commun. 215, 135-151 (2003). [CrossRef]
- X. F. Li, W. Pan, B. Luo, and D. Ma, "Nonlinear dynamic behaviors of an optically injected vertical-cavity surface-emitting laser," Chaos, Solitons & Fractals 27, 1387-1394 (2006). [CrossRef]
- X. F. Li, W. Pan, B. Luo, D. Ma, Z. Zhao, and G. Deng, "Theoretical study on controlling nonlinear behaviors of a couple-cavity VCSEL by external optical injection," Chinese Opt. Lett. 2, 278-281 (2004).
- G. D. Van Wiggeren, and R. Roy, "Communication with chaotic lasers," Science 279, 1198-1200 (1998). [CrossRef]
- V. Ahlers, U. Parlitz, and W. Lauterborn, "Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers," Phys. Rev. E 58, 7208-7213 (1998). [CrossRef]
- J. Ohtsubo, "Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback," IEEE J. Quantum Electron. 38, 1141-1154 (2002). [CrossRef]
- K. Kusumoto, and J. Ohstubo, "Anticipating synchronization based on optical injection-locking in chaotic semiconductor lasers," IEEE J. Quantum Electron. 39, 1531-1536 (2003). [CrossRef]
- A. Uchida, Y. Liu, and P. Davis, "Characteristics of chaotic masking in synchronized semiconductor lasers," IEEE J. Quantum Electron. 39, 963-970 (2003). [CrossRef]
- Y. Liu, H. F. Chen, J. M. Liu, P. Davis, and T. Aida, "Synchronization of optical-feedback-induced chaos in semiconductor lasers by optical injection," Phys. Rev. A63, 031802-1-4(R) (2001). [CrossRef]
- J. M. Buldú, J. García-Ojalvo, and M. C. Torrent, "Multimode synchronization and communication using unidirectionally coupled semiconductor lasers," IEEE J. Quantum Electron. 40, 640-650 (2004). [CrossRef]
- H. F. Chen, and J. M. Liu, "Open-loop chaotic synchronization of injection-locked semiconductor lasers with gigahertz range modulation," IEEE J. Quantum Electron. 36, 27-34 (2000). [CrossRef]
- R. Vicente, T. Pérez, and C. R. Mirasso, "Open- versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1197-1204 (2002). [CrossRef]
- A. Murakami, "Phase locking and chaos synchronization in injection-locked semiconductor lasers," IEEE J. Quantum Electron. 39, 438-447 (2003). [CrossRef]
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