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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 8 — Apr. 17, 2006
  • pp: 3138–3151
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Chaos synchronization of unidirectionally injected vertical-cavity surface-emitting lasers with global and mode-selective coupling

Xiaofeng Li, Wei Pan, Dong Ma, and Bin Luo  »View Author Affiliations


Optics Express, Vol. 14, Issue 8, pp. 3138-3151 (2006)
http://dx.doi.org/10.1364/OE.14.003138


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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

In this paper, considering the spatial dependences of the rate equations, the chaos synchronization of two unidirectionally coupled VCSELs with open-loop configuration is studied by establishing and solving the master-slave-type multi-transverse-mode rate equations. In this study, both the transmitter and the receiver VCSELs are assumed to have a weakly index-guided structure and can operate with two low-order transverse modes simultaneously, and only the generalized type of synchronization is taken into account since the other type of synchronization (complete synchronization or anticipating synchronization) is very difficult to be realized in practical system. For the generalized synchronization, the synchronization performance, the system correlation degree, and the influences of the injection strength on chaos synchronization degree are investigated for two different cases: global injection (all-to-all coupling) and mode-selective injection.

2. Theoretical model

Normally, for weakly step-index-guided VCSELs, the potential optical mode profiles are determined by the build-in index guiding introduced by the transverse refractive step in the surrounding region. According to the optical waveguide theory, the linear polarized LPmns,c modes are proper to represent the corresponding transverse modes, where the superscripts 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 (LP01 and LP11) are taken into account. However, the more transverse mode cases can still be studied in a straight forward manner.

By using the method of separating variables, the spatially dependent electric field Ei (r, φ, t) of the ith transverse mode can be written as

Ei(r,φ,t)=Ei(t)ψirφ
(1)

where i= 1 and i = 2 correspond to the LP01 and LP11 modes, respectively. Ei (t) and ψi (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 |Ei (t)| is proportional to the photon density Pi (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

ψirφ=ci[Ji1(uirRcore)Ji1(ui)]cos[(i1)φ]r<Rcore
(2)
ψirφ=ci[Ki1(wirRcore)Ki1(wi)]cos[(i1)φ]r>Rcore
(3)

where ci is the normalization coefficient satisfying following relationship

1πRclad202π0Rcladψirφ2rdrdφ=1
(4)

and J i-1 and K i-1 are the i-1th order Bessel functions of the first and second kinds, respectively. ui=Rcore(ncorek0)2βi2 and wi=Rcoreβi2(ncorek0)2 -(ncladk 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]

]

uiJi(ui)Ji1(ui)=wiKi(wi)Ki1(wi)
(5)

here βi is the propagation constant and k 0 is the vacuum wave number.

Introducing the spatial dependences of the transverse modes into the rate equations, we obtain a set of equations including the transverse mode characteristics of VCSELs. Normally, the mathematical model of external-cavity or optical-injection semiconductor lasers is taken to be of the well-known Lang-Kobayashi form [2

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

]. Besides, as shown in many past papers, two sets of rate equations are required to describe respectively the dynamics of the transmitter and the receiver lasers. In a word, the master-slave rate equations of VCSELs including the spatial effects and carrier diffusion are given by

dEi(m)(t)dt=1jβc2(ΓzGi(m)1τp)Ei(m)(t)+kEi(m)(tτ)exp[jωi(m)τ]
(6)
dEi(s)(t)dt=1jβc2(ΓzGi(s)1τp)Ei(s)(t)+kcEi(m)(tτc)exp[jωi(m)τc]exp(j2πΔυt)
(7)
Nmsrtt=Dn2Nmsrt+I(r)qVNmsrtτeivgg0[NmsrtN0]Eims(t)212π02πψirφ2
(8)

here j = √-1 , βc is the linewidth broadening factor, Γz is the longitudinal field confinement, τp is the photon lifetime, Dn is the carrier diffusion coefficient in the active region, q is the electric charge, V = πRcore23dw is the volume of the active layer, τe is the carrier lifetime, vg is the group velocity, g0 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]

]. During the lasing process, the total energy in the model is conserved, i.e. the number of the photons created is balanced with the number of the carriers lost.

The last terms on the right-hand-side of equations (6) and (7) stand for the feedback and injection effects, respectively, where k and kc are the corresponding feedback and injection parameters, τ is the feedback delays time, and τc is the propagation time of the carrier signal from the transmitter to the receiver. ωim and ωis denote the angular frequencies of the ith mode of transmitter and receiver lasers, respectively. ∆υ = (ωim - ωis )/(2π) 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

Gi(m,s)=1πRclad202π0Rladevgg0[N(m,s)(r,t)N0]ψi(r,φ)2rdrdφ
(9)

Equations (1)–(9) give the detailed description of the unidirectionally coupled multi-transverse-mode VCSELs with open-loop configuration. In fact, the model obtained here is composed of two sets of differential equations with infinite dimension. In stead of the conventional rate equations which solely consider the longitudinal modes, our model includes the temporal, radial, and azimuthal variables. Hence, common numerical methods such as the Runge-Kutta method can not solve this model. From the mathematical viewpoint, the divisions in the considered domains, including the time and the space, are needed. In this paper, the finite difference method is used to obtain the solutions. The solutions can be very close to the accurate ones as long as the integration steps are chosen to be small enough. Since we have averaged the azimuthal distribution though azimuthal integration, hence the division in the azimuthal direction is not needed. This greatly simplifies our calculations. The time and the space integration steps used in this study are 0. 1ps and 0.04μm, 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 Rd are taken to be 1mA and 2.1μm, respectively, so that both modes (LP01 and LP11 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]

].

It is obvious that the spatial distributions of the considered two transverse modes must be obtained firstly. In fact, as shown in Eqs. (2) and (3), the transverse mode distributions for the given parameters are determined by the eigenvalues 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.

Table 1. Propagation constants and eigenvalues of LP01 and LP11 modes.

table-icon
View This Table

3. Results and discussions

Fig. 1. Chaotic series of (a) modal powers, (b) optical phases, and (c) carrier density. Here, k =8ns -1 and Lext =2cm. The solid and the dot lines stand for LP01 and LP11 modes, respectively.

It has been indicated that two different types of chaos synchronization can be obtained in the unidirectionally coupled semiconductor laser system. One is complete synchronization, which is the result of perfect symmetry between the transmitter and the receiver. The synchronization degree of complete type is very high; however, it is very difficult to be realized in practical environment. The other type of chaos synchronization is the generalized type, which originates from the strong optical injection locking and amplification. Although the output strength of the receiver is relatively stronger than that of the transmitter, the variation rules are almost the same. Compared with complete synchronization, the synchronization degree of the second type is relatively low; however, it dose not require a severe symmetry between the two parts of the communication system and has a relatively larger tolerance to the parameter mismatch. Therefore, the generalized chaos synchronization is easy to be realized and most of the chaos synchronizations found experimentally correspond to this type. In following parts we will solely focus on this type of synchronization.

Fig. 2. Output power of the transmitter (first column) and the receiver (second column) in generalized synchronization regime, where k =8ns -1 , Lext =2cm , kc =533ns -1 , and Lc =0cm. The results for the total output power, the LP01 mode power, and the LP11 mode power are given in the first, second and third rows, respectively. The corresponding correlation curves are plotted in the third column.

Fig. 3. Time series [(a) and (c)] and power spectra [(b) and (d)] of the mode powers of the transmitter and receiver lasers. Here, the black and gray lines correspond to the LP01 and LP11 modes, respectively. The feedback and coupling conditions are k = 8ns -1 , τ = 1ns , kc =533ns -1, and τc =3ns.

C(Δt)={Pm,i(tΔt)Pm,i(tΔt)}{Pm,i(t)Ps,i(t)}{Pm,i(tΔt)Ps,i(tΔt)}2{Ps,i(t)Ps,i(t)}2
(10)

Fig. 4. Variations of correlation coefficients C(∆t) with respect to temporal delay ∆t, where (a), (b), and (c) correspond to the total power, the LP01 mode power, and the LP11 mode power, respectively. The feedback and coupling conditions are k = 8ns -1 , τ = 1ns, kc =533ns -1 , and τc = 3ns. Here, the global maximum is observed at ∆t = 1ns , which corresponds to the propagation delay of the chaotic carrier from the transmitter to the receiver.

Fig. 5. As Fig. 2 but only LP01 mode of the transmitter is injected into the receiver laser.
Fig. 6. As Fig. 2 but only LP11 mode of the transmitter is injected into the receiver laser.

Finally, we investigate the effects of injection parameter kc on the system synchronization degree. Figure 7 displays the variations of cross-correlation coefficient C(∆t = τc ) with respect to kc for different coupling configurations, where (a), (b), and (c) correspond to all-to-all injection (global injection), LP01-injection, and LP11-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 kc , however, the correlation coefficients begin to increase. Especially, when kc is larger than 300ns -1 all these coefficients are near 1, which shows that the system achieves an accurate synchronization under strong optical injection. However, for LP01 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 LP01 modes of these two lasers can obtain synchronization, while the rest modes without optical coupling can not achieve synchronization. With the synchronization of LP01 mode, the total power is also synchronized since the LP11 mode is actually suppressed by the mode-selective injection and the total laser power is solely provided by the synchronized LP01 mode at this time. Due to the lack of optical coupling, the correlation degree of LP11 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 LP11 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 LP01 mode is not completely suppressed by the mode-selective injection the correlation coefficient dose not become as small as LP01-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

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]

].

Fig. 7. Variations of correlation coefficient C(∆t = 0ns) with respect to the injection parameter kc for different coupling conditions. Thick solid, thin solid, and dot lines correspond to the global injection, LP01 mode injection, and LP11 mode injection cases, respectively. The parameters are the same to those of Fig. 2.

4. Conclusions

Finally, for different coupling cases, we study the influence of the injection parameter on the synchronization performance. It is found that, with the increase of the injection strength, the correlation level of the modes with optical coupling can be very high; however, the correlation coefficient of the other two modes without optical coupling keeps a relatively small value even though the optical injection is very strong. It is worth pointing out that the multichannel communication basing on multi-mode semiconductor lasers (multi-longitudinal EELs or multi-transverse-mode VCSELs) has already been investigated by several authors numerically [16

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]

, 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

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]

]. From experiment point of view, we can use the frequency-selective filter in the transmission line to select a certain mode for realizing the mode-selective optical injection, since different transverse modes always have different frequencies; besides, utilizing special mode-selection techniques to change the mode distribution of the transmitter laser, the mode-selective injection can also be implemented in practical systems. In the receiver part, as indicated in Ref. [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]

], some band-pass filters and low-pass filters are needed to decode the signals hidden in the corresponding chaotic outputs. In virtue of the multi-mode synchronization, the chaotic optical communication system can achieve multiplexed transmission with higher communication speed. Therefore, it is expected that the relevant results will greatly stimulate the experimental investigations.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 10174057 and 90201011, the Key Project of Chinese Ministry of Education under Grant 2005-105148, and the Doctor Innovation Fund of Southwest Jiaotong University.

References and links

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03.

H. Kakiuchida and J. Ohtsubo, “Characteristics of a semiconductor laser with external feedback,” IEEE J. Quantum. Electron. 30, 2087–2097 (1994). [CrossRef]

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. 34, 1797–1801 (1998). [CrossRef]

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. 37, 329–336 (2001). [CrossRef]

06.

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]

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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]

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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]

09.

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10.

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

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

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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]

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A. Uchida, Y. Liu, and P. Davis, “Characteristics of chaotic masking in synchronized semiconductor lasers,” IEEE J. Quantum Electron. 39, 963–970 (2003). [CrossRef]

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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 63, 031802-1-4(R) (2001). [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]

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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]

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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]

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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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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]

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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. 27, 1402–1408 (1991). [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]

32.

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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]

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. 17, 2612–2632 (1999). [CrossRef]

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]

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

  1. 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]
  2. R. Lang, and K. Kobayashi, "External optical feedback effects on semiconductor injection laser properties," IEEE J. Quantum. Electron. 16, 347-355 (1980). [CrossRef]
  3. H. Kakiuchida, and J. Ohtsubo, "Characteristics of a semiconductor laser with external feedback," IEEE J. Quantum. Electron. 30, 2087-2097 (1994). [CrossRef]
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. 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).
  10. G. D. Van Wiggeren, and R. Roy, "Communication with chaotic lasers," Science 279, 1198-1200 (1998). [CrossRef]
  11. V. Ahlers, U. Parlitz, and W. Lauterborn, "Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers," Phys. Rev. E 58, 7208-7213 (1998). [CrossRef]
  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]
  14. A. Uchida, Y. Liu, and P. Davis, "Characteristics of chaotic masking in synchronized semiconductor lasers," IEEE J. Quantum Electron. 39, 963-970 (2003). [CrossRef]
  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. A63, 031802-1-4(R) (2001). [CrossRef]
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