different biasing levels. We studied the route to chaos followed by the deterministic system when the modulation frequency is two times the resonance frequency.
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 3, MARCH 1998
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Truncation of the Feigenbaum Sequence in Directly Modulated Semiconductor Lasers Horacio Lamela, Guillermo Carpintero, Associate Member, IEEE, and Pablo Acedo
Abstract—In this paper, we analyze the dynamic behavior of directly modulated semiconductor lasers as the modulation index is varied, with an emphasis on the influence of noise at two different biasing levels. We studied the route to chaos followed by the deterministic system when the modulation frequency is two times the resonance frequency. We found that the behavior is more complex than that of a Feigenbaum sequence. In addition, a period tripling stable solution, which appears due to a fold bifurcation, coexists with the Feigenbaum sequence for certain values of the modulation index. Their coexistence gives rise to hysteresis loops and chaotic bifurcations, namely boundary crisis. Due to the coexistence of solutions in the deterministic system, the role of noise can be expected to be of great importance. When noise fluctuations are introduced in the model, the behavior evolves from the single periodic response through period doubling, period quadrupling, and period tripling in accordance with recent experimental studies. We have also found agreement in the behavior at different conditions of the analysis by varying the biasing point and the modulation frequency. Our results show that in the route to chaos, the period-doubling sequence is effectively truncated due to random noise. The reason for the truncation is found in the nearby coexisting period-three solution. Index Terms— Chaos, diode laser dynamics, period doubling, stochastic analysis.
I. INTRODUCTION
T
HE dynamic behavior of laser diodes has been extensively studied using a set of coupled ordinary differential equations, broadly known as rate equations. Many theoretical studies using this model have shown that directly modulated semiconductor lasers should present a route to chaos through period-doubling bifurcations as the modulation index is increased [1]. The occurrence of a period-doubling bifurcation has been confirmed experimentally [2] and only recently, a route to chaos via period doubling, period quadrupling, and period tripling has been found [3], [4]. This experimental route to chaos differs from the one expected theoretically, via successive period-doubling bifurcations following a Feigenbaum sequence. Since noise was ignored in early theoretical studies relying on deterministic models of the laser, the experimental absence of stable high period bifurcations has been attributed to the laser noise fluctuations [2]. The reason is that due to the Feigenbaum size scaling, as successive bifurcations take place
and the period of the solution increases, the deterministic amplitude variations narrow. Therefore, one would expect the sequence of bifurcations to stop when the amplitude of the noise fluctuations becomes comparable to these deterministic variations. Nevertheless, to the best of our knowledge, the only study in detail of the effect of noise on the dynamic behavior of semiconductor lasers analyzed its behavior close to the first period doubling bifurcation [5]. This study was based on the noise precursor theory developed by Wiesenfeld [6], demonstrating that noise can act as an important precursor for the period doubling bifurcations allowing us to forecast the occurrence of a period doubling bifurcation by observing the output spectrum. In this paper, we analyze the period-doubling route to chaos obtained in directly modulated laser diodes as the modulation index is increased. Using standard methods in nonlinear dynamics, as well as sophisticated continuation theory methods, the deterministic model is investigated thoroughly. We have found the well-known period-doubling route to chaos which coexists, as we shall present, with a periodthree solution introduced in the system by a fold bifurcation. As a result of this coexistence, hysteresis effects appear in the dynamic behavior of the laser diode [7]. Once the deterministic solutions are known, Langevin noise sources are incorporated to the rate-equation description of the laser. When the noisy model is used, the laser shows a jump to the period-three solution as soon as a positive Lyapunov exponent appears in the period-doubling branch. Therefore, the coexistence of solutions, as well as the noise fluctuations allowing the transition, are responsible for the truncation of the Feigenbaum sequence. II. THE STOCHASTIC LASER DIODE MODEL The stochastic model that we have used for the intrinsic laser diode is based on the rate equation system in which Langevin noise sources have been introduced [5]–[8]. The present analysis employs normalized unit-order dimensionless variables for the photon and carrier densities as well as time. The normalization is such that and . The system of equations is given by
Manuscript received March 3, 1997; revised November 13, 1997. This work was supported by the CICYT Spanish Commission under Project TIC96-1415E. The authors are with Grupo de Optoelectr´onica y Tecnolog`ia L`aser, Departamento de Ingenier`ia El´ectrica, Electr´onica y Autom´atica, Universidad Carlos III de Madrid, Legan`es, Madrid 28911, Spain. Publisher Item Identifier S 0018-9197(98)01768-0. 0018–9197/98$10.00 1998 IEEE
(1) (2) (3)
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TABLE I LASER-DIODE PARAMETERS
where and . In these expressions, is the linear gain coefficient, and are the carrier and photon lifetimes, is the confinement factor, the spontaneous emission factor and is the threshold carrier density of the laser diode. In (1) and (2), and are the normalized carrier transparency population density and nonlinear gain compression factor, respectively. and are the Langevin noise sources arising from the discrete nature of carrier generation and recombination processes and the spontaneous emission, respectively [8]. On the right-hand side of (1), the first term is the injection of current, composed of a constant bias term and of a sinusoidally varying modulation, with normalized angular frequency ( is the applied modulation frequency). The bias and modulation levels are defined, respectively, by the bias index and modulation index . The modulation index is the control parameter of the present study. The values of the laser parameters, obtained from recent publications, are presented in Table I. In our normalized notation, we obtain that 500, 0.004, 0.6429, 2.8, and . III. NUMERICAL ANALYSIS OF THE NONLINEAR BEHAVIOR The numerical data that we are going to present in this section corresponds to the deterministic analysis of the rateequation system. This implies that the Langevin noise sources in (1) and (2) have not been taken into account. For the sake of comparison between the deterministic and the stochastic results (presented in the next section), and due to the particularities of the numerical solution of stochastic systems, we have used an Euler method to solve both the deterministic and the stochastic systems. The data are presented in a collection of bifurcation diagrams. For their construction, the variables of interest are sampled at a fixed phase point of the sinusoidal input current modulation. When this stroboscopic recording of the system variables has been obtained for several levels of the modulation index, the samples at each level are then plotted versus the modulation index. In Fig. 1, we present the bifurcation diagram obtained with the deterministic laser model when the modulation index is swept forward and backward. The bias index is 1.25 and the modulation frequency is chosen to be twice the resonance
Fig. 1. Bifurcation diagram of the Normalized Carrier Population versus the 1.25 and f =fres 2. Thick traces mark the modulation index with p unstable branch of the period-three solution.
=
m
=
Fig. 2. Lyapunov exponents for the solutions found in the upward sweep. The continuous traces in the bifurcation diagram, with positive Lyapunov exponents, show chaotic behavior.
frequency. The resonance frequency is then a subharmonic of the modulation frequency, and the device is therefore subject to a parametric forcing, showing the most interesting types of behavior [3]–[5]. As we mentioned earlier, we have taken the modulation index as bifurcation parameter. At the lowest indices of modulation, the laser follows the periodic variation imposed by the modulation current. This behavior corresponds to a single spot in the bifurcation diagram. As the modulation index is increased, three successive perioddoubling bifurcations (to periods 2, 4, and 8 behavior) appear consecutively as the modulation index reaches 1, 3.9, and 4.5, respectively. At the modulation index 4.6, the bifurcation diagram shows continuous traces indicating that chaotic behavior has been reached, as was predicted in many previous studies [1]. The chaoticity of the behavior has been tested calculating the Lyapunov exponents, which measure the sensitivity to initial conditions. The Lyapunov exponents are calculated by integrating the equivalent linear system of (1)–(3) over several periods of the solution ( periodic) with a unit modulus random initial condition. In order to keep its growth limited, at every period of the solution
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(a)
493
(b)
(c) Fig. 3. Bifurcation diagrams of the Normalized Carrier Population versus the modulation index with: (a) p 375 and f =fres 1.411. (c) p 1.375 and f =fres 1.209.
m
=
=
m
=
the linear system solution is normalized to unity dividing by its modulus at that time instant . After periods, the Lyapunov exponent is calculated as [9] (4) The maximum Lyapunov exponent for the solutions on the upward sweep presented in Fig. 1 is plotted in Fig. 2. The region where positive exponents are obtained corresponds to the appearance of continuous traces in the bifurcation diagram, clearly indicating the chaotic behavior. Also, the period-doubling bifurcation points already mentioned appear, effectively showing null exponents. When the modulation index is increased to 5.6, the chaotic behavior disappears and the laser is left in a periodthree stage in which it remains up to a modulation index level at which chaotic behavior appears again until the end of the forward sweep. Starting at this maximum value and sweeping backward, we start with the period-three solution. The behavior changes to period four when reaches 4.5. From then on down to 0.1, the same bifurcation
= 1.25 and fm =fres = 1.478. (b) p = 1.
diagram as that of the upward sweep is obtained. We have therefore identified an interval of the modulation index in which two solutions coexist, clearly identified in Fig. 1. Within the interval ranging from 4.6 to 5.5, a hysteresis loop is formed between the period-doubling cascade and the period-three solution as the modulation index is varied [7]. At this point, it is interesting to show the bifurcation diagram at a different conditions of the parameters: the bias index and the modulation frequency . First, we obtained the bifurcation diagram for the same biasing level as in Fig. 1, 1.25, but with a different modulation frequency. For the presentation of the following bifurcation diagrams, we found it useful to introduce a new parameter into the analysis, the ratio between the modulation frequency and the resonance 2. We frequency. The frequency ratio for Fig. 1 was 1.478, with the modulanow decrease this ratio to tion frequency less than half that of the modulation frequency. Under these conditions, the bifurcation diagram presented in Fig. 3(a) shows two period-doubling bifurcations to a perioddouble and period-quadruple behavior, followed by a period halving which leaves the system with a period two solution. This confirms the fact that modulation frequencies close to the harmonics of the resonance frequency unstabilize the system.
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(a)
(b)
(c)
(d)
Fig. 4. Stochastic bifurcation diagrams of the Normalized Carrier Population versus the modulation index: (a) p and f =fres 1.478, (c) p 1.375 and f =fres 1.411, (d) p 1.375 and f =fres 1.209.
m
=
=
m
=
=
A similar result can be obtained if instead of decreasing the modulation frequency, we increase the resonance frequency. This is achieved by increasing the bias index. In this study, we present the bifurcation diagrams that were obtained for the 1.375. In Fig. 3(b), the frequency ratio is set bias index 1.411, and the results are similar to those in at Fig. 3(a), as we expected. The fact that decreasing modulation frequencies lead to an increasingly stable system is also proven 1.209 at when the frequency ratio is decreased to 1.375 bias level. The behavior of the system under the this last set of conditions is presented in Fig. 3(c), showing a suppression of the period-quadrupling behavior. This behavior, where the nonlinear regimens are changed with variations values is in good agreement to that obtained of and experimentally by Ngai and Liu [3], [4]. In order to understand the hysteresis phenomena, we have used a continuation method to obtain the unstable branch of the period three solution in Fig. 1 [7]. The return points of the newly calculated unstable solution were also plotted in Fig. 1. The spots corresponding to this solution have been drawn with a thicker line, providing a clear picture of what is 4.6, a tangent bifurcation introduces really happening. At simultaneously stable and unstable period three solutions. This
m
=
= 1. 25 and fm =fres = 2, (b) p = 1.25
newly created pair of solutions coexists with the perioddoubling cascade. One can also observe in Fig. 1 that as the modulation index is increased, the basin of attraction of the period-doubling sequence is gradually reduced. This reduction finally ends when the chaotic attractor collides with the unstable branch of the period-three solution. As a result of this phenomenon, called boundary crisis [10], the chaotic solution disappears. However, what we have just described is the behavior of the deterministic model. IV. RESULTS
OF THE
STOCHASTIC ANALYSIS
The question that we want to answer now is whether the noise introduces significant changes in the behavior of the laser diode presented above. We expect changes only in the case where we find a coexistence of different solutions. For this reason, we have obtained the bifurcation diagrams when noise is introduced into the system. In this paper, we present single realizations of the stochastic process under the different sets of conditions presented in the deterministic study. The stochastic bifurcation in Fig. 4(a) corresponds to that presented in Fig. 1. Comparing both diagrams, the crucial role of noise is clearly displayed. First of all, it can be observed that the lines in the bifurcation diagram have
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transition from this branch of solutions to the coexisting period-three solution. The transition to chaos occurs via period doubling followed by period quadrupling and period tripling in accordance with experimental data rather than through a Feigenbaum sequence. This new route to chaos, where the Feigenbaum sequence is truncated, could be related to other types of laser experiments, like frequency-modulated CO lasers [10], [11] or -switched CO lasers [12] where similar observations have been found. REFERENCES
Fig. 5. FFT spectra of the time evolution of the photon density when Langevin noise sources are included (modulation index at m 4.2).
=
become thicker. Being a single realization, the small deviations introduced by noise over the stable deterministic solutions appear in the plot. Whereas the first period-doubling bifurcation can be clearly appreciated, at the period-quadrupling behavior the amplitude difference is blurred by the noise fluctuations making it look like a noisy period-two behavior. Fourier analysis of the time evolution shows a peak at the frequency corresponding to the period quadrupling bifurcation (Fig. 5). The main difference between both diagrams is the truncation of the period doubling route to chaos. The system no longer reaches the chaotic behavior at the end of the period-doubling cascade. As soon as the deterministic system reaches to a solution with positive Lyapunov exponents, the introduction of noise allows it to jump to a coexisting stable period-three solution. It is worth mentioning that the route to chaos found using the stochastic model coincides qualitatively with that found in the experimental work by Ngai and Liu in laser diodes [3], [4]. The stochastic bifurcation diagrams presented in Fig. 4(b)–(d) correspond to the deterministic ones presented in Fig. 3(a)–(c). It can be observed that the behavior does not change qualitatively from the observed in the deterministic case. As we expected, since the initial branch of solutions remains stable throughout the entire span of the modulation index, noise does not introduce significant changes of behavior. V. CONCLUSION In our study of the deterministic laser model we have found that the period-doubling route to chaos coexists with a periodthree solution. This coexistence gives rise to a hysteresis loop in which both transitions from one attractor to the other occur via catastrophic bifurcations, which are a kind of global bifurcation. It is worth mentioning that this type of bifurcation has no associated precursors and therefore their occurrence cannot be forecast as they were with the perioddoubling bifurcations. Using a stochastic rate-equation model, introducing the noise effects into the model, we have shown for the first time that the period-doubling sequence to chaotic behavior is truncated in favor of the coexisting stable periodthree solution. As soon as a positive Lyapunov exponent appears on the period-doubling sequence, noise induces the
[1] C. H. Lee, T. H. Yoon, and S. Y. Shin, “Period doubling and chaos in a directly modulated laser diode,” Appl. Phys. Lett., vol. 46, no. 1, pp. 95–97, 1985. [2] Y. C. Chen, H. G. Winful, and J. M. Liu, “Subharmonic bifurcations and irregular pulsing behavior of modulated semiconductor lasers,” Appl. Phys. Lett., vol. 47, no. 3, pp. 208–210, 1985. [3] W. F. Ngai and H. F. Liu, “Observation of period doubling, period tripling, and period quadrupling in a directly modulated 1.55 m InGaAsP distributed feedback laser diode,” Appl. Phys. Lett., vol. 62, no. 21, pp. 2611–2613, 1993. [4] H. F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 m InGaAsP distributed feedback lsemiconductor laser,” IEEE J. Quantum Electron., vol. 29, pp. 1668–1675, 1993. [5] Y. H. Kao and H. T. Lin, “Persistent properties of period doubling in directly modulated semiconductor lasers,” Phys. Rev. A, vol. 48, no. 3, pp. 2292–2298, 1993. [6] K. Wiesenfeld, “Noisy precursors of nonlinear instabilities,” J. Stat. Phys., vol. 38, no. 56, pp. 1071–1097, 1985. [7] G. Carpintero and H. Lamela, “Hysteresis in directly modulated semiconductor lasers,” in Physics and Simulation of Optoelectronic Devices V, M. Osinski and W. W. Chow, Eds., Proc. SPIE, vol. 2994, pp. 242–249, 1997. [8] D. Marcuse, “Computer simulation of laser photon fluctuations: Theory of single-cavity laser,” IEEE J. Quantum Electron., vol. QE-20, pp. 1139–1148, 1984. [9] R. C. Hilborn, Chaos and Nonlinear Dynamics. Oxford, U.K.: Oxford, 1994, pp. 205–206. [10] D. Dangoisse, P. Glorieux, and D. Hennequin, “Laser chaotic attractors in crisis,” Phys. Rev. Lett., vol. 57, no. 21, pp. 2657–2660, 1986. [11] T. Midavaine, D. Dangoisse, and P. Glorieux, “Observation of chaos in a frequency-modulated CO2 laser,” Phys. Rev. Lett., vol. 35, no. 19. pp. 1989–1992, 1985. [12] F. T. Arecchi, R. Meucci, G. Puccioni, and J. Tredicce, “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser,” Phys. Rev. Lett., vol. 49, no. 17, pp. 1217–1220, 1982.
Horacio Lamela received the Industrial engineering Degree from the Universidad Polit´ecnica de Madrid (UPM) in 1980, the “Diplome d’Etudes Approfondies” (DEA) from the University of Paris XI in 1981, and the “Docteur-Ingenieur” degree in optical interferometry from the Conservatoire National d’Arts et Metiers of Paris for his work in wavelength measurements of He-Ne stabilized lasers by saturation absorption at the Institut National de Metrologie, France. From May 1985 to November 1987, he worked at the Massachusetts Institute of Technology, Cambridge, as a Post-Doctoral Fellow at the Electrical Engineering and Computer Science Department, working on LiNbO3 spatial light modulators, and as a Visiting Scholar through a Fullbright-MEC Comission at the Research Laboratory of Electronics with semiconductor laser frequency stabilization for cesium atomic clocks. He is presently working as Associate Professor with the Optoelectronics and Laser Technology Group, Departamento de Ingenieria El´ectrica, El´ectronica y Autom´atica of Universidad Carlos III de Madrid, Madrid, Spain. His current research interests are semiconductor laser modeling and nonlinear dynamics, laser interferometry, optical sensors, and optoelectronics.
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Guillermo Carpintero (M’95–A’96) received the Telecommunication Engineering Degree from the Universidad Polit´ecnica de Madrid in 1993. He is currently working towards the Ph.D. degree in electrical engineering at the Universidad Carlos III de Madrid, Madrid, Spain. His doctoral work focuses on nonlinear dynamics of directly modulated semiconductor lasers. Since 1993, he has worked as a Doctoral Researcher with the Optoelectronics and Laser Technology Group at the Universidad Carlos III de Madrid. His research interests are semiconductor laser modeling and nonlinear dynamics.
Pablo Acedo received the Telecommunication Engineering Degree from the Universidad Polit´ecnica de Madrid in 1993. He is currently working towards the Ph.D. degree in electrical engineering at the Universidad Carlos III de Madrid, Madrid, Spain. His doctoral work concerns laser interferometry for plasma fusion diagnostics. Since 1993, he has worked as a Doctoral Researcher with the Optoelectronics and Laser Technology Group at the Universidad Carlos III de Madrid. His research interests are laser interferometry, semiconductor laser modeling, and optoelectronics.
