Nonlinear dynamic behaviors of an optically injected vertical-cavity surface-emitting laser

Transcription

1 Chaos, Solitons and Fractals 27 (2006) Nonlinear dynamic behaviors of an optically injected vertical-cavity surface-emitting laser Xiaofeng Li *, Wei Pan, Bin Luo, Dong Ma, Yong Wang, Nuohan Li School of Computer and Communication Engineering, Southwest Jiaotong University, Chengdu , PRChina Accepted 6 May 2005 Abstract Nonlinear dynamics of a vertical-cavity surface-emitting laser (VCSEL) with external optical injection are studied numerically. We consider a master slave configuration where the dynamic characteristics of the slave are affected by the optical injection from the master, and we also establish the corresponding Simulink model. The period-doubling route as well as the period-halving route is observed, where the regular, double-periodic, and chaotic pulsings are found. By adjusting the injection strength properly, the laser can be controlled to work at a given state. The effects of frequency detuning on the nonlinear behaviors are also investigated in terms of the bifurcation diagrams of photon density with the frequency detuning. For weak injection case, the nonlinear dynamics shown by the laser are quite different when the value of frequency detuning varies contrarily (positive and negative direction). If the optical injection is strong enough, the slave can be locked by the master even though the frequency detuning is relatively large. Ó 2005 Elsevier Ltd. All rights reserved. 1. Introduction Over the past few years, nonlinear dynamic behaviors of semiconductor lasers have attracted considerable attention due to its profound physics for understanding the fundamental characteristics of lasers and its potential applications [1 8]. In semiconductor lasers, various external perturbations such as optical injection from another laser [1,2], optical feedback from a distant reflector [3 6], and optoelectronic feedback from a photodetector that is added to the bias current [7,8], can induce the output of the laser to become unstable and yield the chaotic oscillation. Compared with the infinite dimensional system [9] composed of optical or optoelectronic feedbacks, the system subject to external optical injection is relatively simple, and is the most effective way to control and directly access to the fundamental mechanisms that lead to the instability. Unlike the external or optoelectronic feedback system where the perturbation strength is often limited by the laserõs own output, it allows one to perturb the laser with an independent optical field. The nonlinear dynamics of an external optical injection system are often governed by the competitions of the amplified spontaneous emission, the injected signal, and their beatings. For one hand, the injection signal can induce the instabilities such as the periodic and the chaotic behaviors. On the other hand, it can also be used as an effective mean to improve the frequency stability, enhance the bandwidth, and reduce the noise and distortion characteristics [10 13]. * Corresponding author. address: xfl79@163.com (X. Li) /$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi: /j.chaos

2 1388 X. Li et al. / Chaos, Solitons and Fractals 27 (2006) Recently, due to the unpredictability, the chaotic behavior of such system has also been brought forward for cryptographic applications [14 16]. However, most of the past researches about this issue are for the edge-emitting lasers (EELs), and there has little for the new vertical-cavity surface-emitting lasers (VCSELs), which have received starling development in recent twenty years and have been widely used in optical communication, optical interconnects, and other relevant fields [17 21]. As different semiconductor lasers generally have different intrinsic properties that lead to different dynamic characteristics, a deep understanding of the nonlinear dynamics of VCSEL with the external optical injection is necessary. In this paper, the nonlinear dynamics of an optically injected VCSEL are studied numerically and the corresponding Simulink model is established. For no frequency detuning case, the bifurcation diagrams of the photon density versus injection parameter are plotted, which show a period-doubling route to chaos in weak injection case and a period-halving route to the steady state in strong injection case. Then, in terms of temporal series, frequency spectrums and phase portraits, the influence of injection strength on the nonlinear behaviors of the considered system is investigated in detail. Then, the dependences of bifurcation diagram of photon density versus the injection parameter on the frequency detuning are also analyzed. At last, the bifurcation diagrams of photon density versus the frequency detuning are investigated. 2. Theory and model Fig. 1 shows the schematic setup of the master slave configuration under consideration. The laser surrounded by the broken line is the slave VCSEL (S-VCSEL), where the nonlinear behaviors are focused in this paper. The other laser providing the continuous wave (CW), which is injected into the slave, is called as the master VCSEL (M-VCSEL). The coupling of the master and the slave is unidirectional, which is guaranteed by an optical isolator set in the transmission line. Further, a variable attenuator is introduced to adjust the coupling strength of them. For simplicity, the same structure parameters are chosen for the master and the slave, which implies that the two VCSELs have the same eigenvalue of the bias current. The temporal changes of the complex optical fields in the cavity of the master and the slave lasers oscillating at the frequency x m and x s, respectively, can be expressed as [22 24] de m;s ðtþ ¼ 1 dt 2 ðg V gaþð1 ib c ÞE m;s ðtþþ½k inj E m ðtþ expði2pdmtþš s ð1þ where the subscripts m and s stand for the M-VCSEL and the S-VCSEL, respectively. V g is the group velocity, a is the equivalent cavity loss, b c is the linewidth broadening factor, and Dm =(x m x s )/(2p) is the frequency detuning between these two lasers. Considering the effect of the gain saturation as well as the multi-well structure of the active region, the equivalent gain can be expressed as [5] G ¼ CV g a N log½nðtþ=n 0 Š=½1 þ e NL PðtÞŠ ð2þ where C is the confinement factor, a N is the gain coefficient, e NL is the gain suppression factor, and N 0 is the carrier density at transparency. The parameter k inj is defined as [12]: k inj ¼ð1 RÞðR inj =RÞ 1=2 =s in ð3þ where R is the equivalent reflectivity of the injected DBR (distributed Bragg reflector), R inj is the injection parameter from M-VCSEL to S-VCSEL, and s in is the round-trip time inside the laser cavity. The internal complex optical field E(t) can usually be written as Aexp[ iu(t)], where A =(PV) 1/2 and U(t) are the complex slowly varying amplitude and the phase, respectively. P is the photon density, V = pw 2 d is the volume of the Fig. 1. Schematic diagram of external optically injected VCSEL system.

3 active region, w and d are the radius and thickness of the active layer, respectively. Substituting this expression into (1) and separating the real and image parts, one can obtain the following two equations: dp m;s ðtþ ¼ðG V g aþp m;s ðtþþb dt sp B½N m;s ðtþš 2 þf2k inj ½P m ðtþp s ðtþš 1=2 cos h inj ðtþg s ð4þ du m;s ðtþ ¼ 1 dt 2 b cðg V g aþ fk inj ½P m ðtþ=p s ðtþš 1=2 sin h inj ðtþg s ð5þ where the spontaneous emission is taken into account. h inj (t)=u s (t) U m (t)+2pdmt, b sp is the spontaneous emission factor, and B is the radiative recombination coefficient. In addition, the rate equation of carrier density is given by [24] dn m;s ðtþ dt ¼ X. Li et al. / Chaos, Solitons and Fractals 27 (2006) I m;s qpw 2 d N m;sðtþ GP m;s ðtþ ð6þ s e where I is the bias current, q is the electric charge, and s e is the carrier lifetime. w and d are the radius and thickness of the active region, respectively. The rate Eqs. (4) (6) are the basic of the numerical calculation of this paper. For a given bias current, the dynamics of the system shown in Fig. 1 is determined by the injection strength R inj and the frequency detuning Dm. In this paper, in terms of bifurcation diagrams, the effects of these two parameters on the dynamic behaviors of the S-VCSEL are investigated in detail. The Simulink provided by Matlab is used for the numerical calculation. Fig. 2 shows the corresponding schematic model established in this paper. The model is composed of the master and the slave modules. In each module, there have three smaller parts, which correspond to the rate equations of photon density (4), carrier density (6) and phase (5), respectively. This simulation tool is for the model-based and system-level design, and the established model can readily be masked and expanded, e.g. modifying the model shown in Fig. 2 properly, the mutually coupled system, the hybrid system including optical injection, optical feedback, and current modulation, etc., can all be studied. This method, to the best of our knowledge, has seldom been used for investigating the nonlinear dynamics of VCSEL. The typical device parameters of the considered VCSEL are listed in Table 1. Fig. 2. Simulink model for the master slave configuration.

4 1390 X. Li et al. / Chaos, Solitons and Fractals 27 (2006) Table 1 Values of VCSEL parameters Symbol Value Symbol Value k nm R w 5 lm s in 0.04 ps d 0.4 lm a 50 cm 1 C 0.07 e NL cm 3 V g cm/s s e 2.7 ns b c 4.8 a N cm 1 N cm 3 I th 4mA b sp I m 2.25I th B cm 3 /s I s 1.25I th 3. Numerical results It is worth pointing out that in the bifurcation diagrams considered in this paper, a single-point represents the CW operation, two points represent the single-periodic oscillation, and four points represent the double-periodic oscillation, and so on. Firstly, for no frequency detuning case, we begin our discussion with the bifurcation diagram of the photon density versus the injection parameter R inj. The simulation result is shown in Fig. 3. It can be seen that with increasing R inj the laser dynamics evolve from chaotic to multi-periodic state, then to single-periodic oscillation state, at last to the CW state (R inj > 0.018). We call this evolution as the period-halving route in following discussions. Similarly, the reverse process, from CW to chaotic state, is called as the period-doubling route, which will be discussed later. It is well known that most lasers themselves do not exhibit the chaotic behaviors, since the decay rates involved in the laser rate equations are much faster than the others [14]. However, if there has sufficient external perturbation, this steady state of free-running laser will be broken and the device will show a variety of nonlinear dynamics. From this point of view, when the optical injection from the master is relatively weak, the S-VCSEL should keep its intrinsic steady state, and with increasing this strength, it begins to exhibit nonlinear dynamics. In other words, there should have a period-doubling process, which has not been observed in Fig. 3. Further investigation shows that this process only happens in a relatively weak injection range. The simulation result is shown in Fig. 4, where it can be seen that the laser shows no apparent nonlinear phenomena when R inj is smaller than Beyond this value, the laser begins to exhibit periodic and chaotic behaviors. Especially, when R inj exceeds a critical value ( for our parameters), the laser enters the chaotic region. This value is so small that the period-doubling process cannot be found in Fig. 3. As shown in Figs. 3 and 4, the S-VCSEL can be controlled to work at a given nonlinear state, such as the chaotic, multi-periodic, single-periodic, or CW state, by choosing R inj properly. To further identify these nonlinear dynamic states, the time series, phase portraits, and frequency spectrums are needed to be investigated. Fig. 3. Bifurcation diagram of photon density versus the injection parameter, where Dm = 0.

5 X. Li et al. / Chaos, Solitons and Fractals 27 (2006) Fig. 4. Detail bifurcation diagram of photon density versus the injection parameter when injection strength is relatively weak, where Dm =0. Fig. 5(a) (g) display the simulation results for different values of R inj, where the typical temporal series (first column), phase portraits (second column) and frequency spectrums (third column) corresponding to the different nonlinear regions are plotted, among which Fig. 5(a) (d) and (d) (g) correspond to the period-doubling and period-halving process, respectively. Fig. 5(a) and (g) represent the steady state, where a constant output in the time-domain, a single dot in the phase portrait, and a close-zero intensity in the frequency-domain can be observed easily. The single-period behavior is shown in Fig. 5(b) and (f). It is seen that the laser outputs periodically and the phase portrait, which is called as the attractor in dynamical system terminology, shows a closed curve. Further, there has a prominent frequency component in the frequency-domain corresponding to the fluctuation frequency of the laser output. In addition, there has Fig. 5. The temporal series, phase portraits, and frequency spectra for different injection parameter: (a) , (b) , (c) , (d) , (e) , (f) 0.005, and (g) 0.02.

6 1392 X. Li et al. / Chaos, Solitons and Fractals 27 (2006) an apparent difference between Fig. 5(b) and (f): in the former figure, the laser oscillates approximately at the free-running relaxation oscillation frequency; nevertheless, the one shown in the posterior figure is much higher, which verifies that the external optical injection surely can be beneficial to enhance the modulation bandwidth of semiconductor lasers, which is a hot topic that many researchers are studying now. Increasing R inj in period-doubling route or decreasing R inj in period-halving one, double-period behavior takes place, and the results are shown in Fig. 5(c) and (e), where R inj = and are chosen separately. At this time, the laser outputs with two different peak values alternately. Further, two closed curves can be seen in the phase portrait, and a subharmonic frequency is observed in the frequency-domain. At last, the chaotic state [25], which has received much attention for secure communication in recent years, is considered. There has a relatively wide range for chaotic operation in optical injected system. Here, we chose R inj = as an example, and the result is plotted in Fig. 5(d) in which the signatures of chaos, such as the noise-like oscillations in the time-domain, the broadened spectrum, and the random distribution in the phase portrait, are observed. In previous discussions, we have neglected the frequency detuning, which is often present in many practical applications. Next, we will further investigate the effects of frequency detuning on the nonlinear behaviors of the laser. Fig. 6 shows the bifurcation diagram of the photon density versus the injection parameter for different values of frequency detuning, where Dm = 6, 3, 0.5, 0.5, 3, and 6 GHz correspond to Fig. 6(a) (f), respectively. It is easily observed that the variations are asymmetric when the value of Dm changes contrarily (positive and negative direction), moreover, with the increasing of jdmj, the chaotic region is shortened apparently. For example, for Dm = 3 GHz (Fig. 6(b)), the laser only shows the periodic and CW behaviors within the plotted area, where there has a critical value for R inj, beyond which the laser transforms from double-periodic fluctuations to steady state suddenly. However, it is much different for the negative Dm case. Take Dm = 3 GHz an example (Fig. 6(e)). With increasing R inj, the laser undergoes the single-periodic, double-periodic, single-periodic, and steady region orderly. Especially, for the range of R inj 2 [0.029,0.039], the effects of external perturbation and optical field in the laser cavity on the laser behaviors are almost equivalent, which lead the laser to operate in the regions of single-period and steady state alternately. For 6 and 6 GHz cases, the double-periodic region vanishes, but the other characteristics are similar to those shown in Fig. 6(b) and (e). To further investigate the influence of frequency detuning on the laser dynamic behaviors, the bifurcation diagrams of photon density versus the frequency detuning for different injection parameters are plotted in Fig. 7, where R inj =1 10 6,1 10 5,1 10 4, 0.01, 0.05, and 1 for Fig. 7(a) (f), respectively. For relatively weak injection case, as shown in Fig. 7(a), the bifurcation diagram is symmetric with the center (Dm = 0 GHz), and the laser solely displays single-periodic oscillation. Further, we can see that, when Dm = ±1.6 GHz, the value of the peak and the value photon density achieve the maximum and minimum, respectively. With increasing R inj, the nonlinear effects become stronger. As seen from Fig. 7(b) and (c) that the chaotic behavior is excited around the center. However, when R inj exceeding some critical degree, larger injection strength cannot induce stronger nonlinear dynamics, but drives the system towards the steady state. Fig. 7(d) (f) demonstrate the results, where , 0.01, and 0.05 are chosen for R inj. We can easily Fig. 6. Bifurcation diagram of photon density versus the injection parameter for different frequency detuning: (a) 6, (b) 3, (c) 0.5, (d) 0.5, (e) 3, and (f) 6 GHz.

7 X. Li et al. / Chaos, Solitons and Fractals 27 (2006) Fig. 7. Bifurcation diagram of photon density versus the frequency detuning for different injection parameter: (a) , (b) , (c) , (d) 0.01, (e) 0.05, and (f) 1. get that the nonlinear effects become weaker with the further increase of R inj. Especially, as can be seen in Fig. 7(f) that the laser can operate at the steady state for a large variation of Dm as long as R inj is large enough. Moreover, it can be observed that the bifurcation diagram is not symmetric any more as the increasing of injection parameter, which has also been mentioned in the past discussions. Above simulation results indicate that there have various nonlinear dynamic behaviors in an optically injected VCSEL system. From the physics point of view, these nonlinear phenomena can be considered as the result of the interactions between the external perturbation and the optical field in the laser cavity; moreover, the interaction includes the following two aspects: the amplitude determined by the injection parameter, and the phase determined by their phases as well as the frequency detuning. When the strength of in-cavity one is much stronger than the injected one, the laser is hardly affected by the perturbation. However, when the optical injection dominates the other, the laser can be locked by the master. At this time, the effect of the phase can be neglected, and the system solely shows the steady behavior. However, when the strength of the intrinsic field and the perturbation are equivalent each other, the dynamics of the system will mainly be determined by the phase. At this time, if the frequency detuning is relatively small, as shown in the rate equation for S-VCSEL, the variation of the phase is much complex, which induce the nonlinear dynamic behaviors mentioned above. However, if the frequency detuning is so large that the phase of the salve can be neglected, the variation of the slaveõs phase is not so complex any longer, since the phase of the master and the frequency detuning is time-constant; therefore, the nonlinear behaviors cannot be stimulated accordingly. It is worth pointing out that, for any particular injection parameter (R inj ), there always has a detuning between the master and the slave to produce a frequency-locked output state. 4. Conclusion In conclusion, the nonlinear dynamics of a VCSEL with external optical injection are studied numerically. Under different injection conditions, the bifurcation diagrams of photon density versus both the injection parameter and the frequency detuning, the time series, the frequency spectrums, and the phase portraits are investigated numerically. With increasing the injection strength to some degree, the laser undergoes the CW, periodic, and chaotic region orderly along period-doubling bifurcation route. However, further increasing injection level, along period-halving route, the laser reenters into the steady region, where the laser is locked by the master. The typical time series, frequency spectrums, and phase portraits for different nonlinear states are observed. We further notice that, when the injection strength is relatively weak, the frequency detuning of the master and the slave affects the nonlinear dynamics of the system dramatically; however, this effect is much different when the value of frequency detuning varies contrarily (positive and negative direction). It is also shown that the nonlinear behaviors often take place at the round of zero frequency detuning. With increasing injection parameter from zero, the nonlinear dynamics get more complex firstly; however, the further increasing of the injection strength dose not increase the complexity of the system, but drive the laser to the steady state region contrarily.

8 1394 X. Li et al. / Chaos, Solitons and Fractals 27 (2006) Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No and ), the Key Project of Chinese Ministry of Education (Grant No ), and the Doctor Innovation Fund of Southwest Jiaotong University (Grant No. 2005). References [1] Hwang SK, Liu JM. Dynamical characteristics of an optically injected semiconductor laser. Opt Commun 2000;183: [2] Sukow DW, Gauthier DJ. Entraining power-dropout events in an external-cavity semiconductor laser using weak modulation of the injection current. IEEE J Quantum Electron 2000;36: [3] Yu SF. Nonlinear dynamics of vertical-cavity surface-emitting lasers. IEEE J Quantum Electron 1999;35: [4] Law JY, Agrawal GP. Effects of optical feedback on static and dynamic characteristics of vertical-cavity surface-emitting lasers. IEEE J Sel Top Quantum Electron 1997;3: [5] Masoller C, Sicardi Schifino AC, Cabeza C. The nonlinear gain and the onset of chaos in a semiconductor laser with optical feedback. Chaos, Solitons & Fractals 1995;6: [6] Uchida A, Takahashi T, Kinugawa S, Yoshimori S. Dynamics of chaotic oscillations in mutually coupled microchip lasers. Chaos, Solitons & Fractals 2003;17: [7] Lin FY, Liu JM. Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback. Opt Commun 2003;221: [8] Lin FY, Liu JM. Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback. IEEE J Quantum Electron 2003;39: [9] Mork J, Tromborg B, Mark J. Chaos in semiconductor lasers with optical feedback: theory and experiment. IEEE J Quantum Electron 1992;28: [10] Simpson TB, Liu JM, Gavrielides A. Bandwidth enhancement and broadband noise reduction in injection-locked semiconductor lasers. IEEE Photonic Technol Lett 1995;7: [11] Wang J, Haldar MK, Li L, Mendis FVC. Enhancement of modulation bandwidth of laser diodes by injection locking. IEEE Photonic Technol Lett 1996;8:34 6. [12] Murakami A, Kawashima K, Atsuki K. Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection. IEEE J Quantum Electron 2003;39: [13] Yabre G. Effect of relatively strong light injection on the chirp-to-power ratio and 3 db bandwidth of directly modulated semiconductor lasers. IEEE J Lightwave Technol 1996;14: [14] Ohtsubo J. Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback. IEEE J Quantum Electron 2002;38: [15] Murakami A. Phase locking and chaos synchronization in injection-locked semiconductor lasers. IEEE J Quantum Electron 2003;39: [16] Fujiwara N, Takiguchi Y, Ohtsubo J. Observation of the synchronization of chaos in mutually injected vertical-cavity surfaceemitting semiconductor lasers. J Opt Lett 2003;28: [17] Takaoka K, Ishikawa M, Hatakoshi G. Low-threshold and high-temperature operation of InGaAlP-based proton-implanted red VCSELs. IEEE J Sel Top Quantum Electron 2001;7: [18] Chang-Hasnain CJ, Harbison JP, Hasnain G, Von Lehmen AC, Florez LT, Stoffel NG. Dynamic, polarization and transverse mode characteristics of VCSELÕs. IEEE J Quantum Electron 1991;27: [19] Chang-Hasnain CJ. Progress and prospects of long-wavelength VCSELs. IEEE Commun Mag 2003;41: [20] Li XF, Pan W, Luo B, Ma D, Deng G. Theoretical analysis of multi-transverse-mode characteristics of vertical-cavity surfaceemitting lasers. Semicond Sci Technol 2005;20: [21] Li XF, Pan W, Luo B, Ma D, Li NH. Suppressing nonlinear dynamics induced by external optical feedback in vertical-cavity surface-emitting lasers. Opt Laser Technol 2005;37: [22] Yu SF, Wong WN, Shum P, Li EH. Theoretical analysis of modulation response and second harmonic distortion in vertical cavity surface emitting lasers. IEEE J Quantum Electron 1996;35: [23] Lang R. Injection locking properties of a semiconductor laser. IEEE J Quantum Electron 1982;18: [24] Agrawal GP, Dutta NK. Long-wavelength semiconductor lasers. 2nd ed. New York: Van Nostran Reinhold; [25] Saha P, Banerjee S, Chowdhury AR. Some aspects of synchronization and chaos in a coupled laser system. Chaos, Solitons & Fractals 2002;14: