Intermittent bursts in a directly modulated laser diode ... - UC3M

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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 36, NO. 11, NOVEMBER 2000

Intermittent Bursts in a Directly Modulated Laser Diode Guillermo Carpintero, Associate Member, IEEE, and Horacio Lamela, Member, IEEE

Abstract—Based on a rate equation description of a directly modulated laser diode, we investigate the origin of intermittent bursts that appear in the output power of the device. Sudden drops in the maximum amplitude of the laser’s output characterize this behavior, found in the period doubling sequence that appears when the modulation index varies. From the investigation of this behavior, this paper reveals the cause for the extreme sensitivity to noise of modulated laser diodes. Using bifurcation theory techniques, we explain how the unstable periodic solutions and the intrinsic noise of the device play a key role in the mechanism giving rise to these bursts. This paper shows that the mechanism is composed of two fundamental stages, which have been referred to as capture and reinjection. As we clarify the origin of the two stages, the different time scales involved in each one are also identified. From this analysis, we can conclude that the intermittent bursts are a characteristic signature of the process by which the laser stochastic dynamics progressively embeds one of the saddle cycles as the modulation index is varied. Index Terms—Nonlinear dynamics, numerical analysis, semiconductor lasers, stochastic differential equations.

I. INTRODUCTION

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HE NONLINEAR dynamics of laser diodes under sinusoidal direct current modulation have been the subject of numerous theoretical and experimental studies. Theoretical investigations using the standard rate equation model have commonly agreed that modulated laser diodes present a cascade of period doubling (PD) bifurcations to chaos as the strength of the modulation is increased. However, a clear difference has been established between the behavior in the absence of noise (deterministic dynamics) versus that resulting when random noise is introduced (stochastic dynamics). The role of random fluctuations on the three different routes to chaos described so far has always been a matter of concern in nonlinear systems theory [1]. In particular, it was discovered early on that noise could severely affect the PD sequence, destabilizing the large periodic orbits and enhancing the chaotic behavior due to the size scaling law discovered by Feigenbaum [2], [3]. Some theoretical studies that focused on the effect of noise in systems displaying periodic behavior when they are close to a dynamical instability led to the noise precursor theory. This theory states that the power spectrum of a system near a local bifurcation point presents characteristic marks that allow one to foresee the bifurcation before it actually takes place [4], which, in the case of a PD bifurcation, is known as the virtual Hopf phenomenon [5], [6]. Manuscript received July 6, 1999; revised July 17, 2000. This work was supported by the CAM Spanish Commission under Project 07T00021998. The authors are with Grupo de Optoelectrónica y Tecnología Láser, Universidad Carlos III de Madrid, Madrid 28911 Spain. Publisher Item Identifier S 0018-9197(00)09768-2.

On the other hand, experimental studies reported that only a few PD bifurcations are observed in modulated laser diodes [7], [8], as well as intermittent behavior [3]. Chaotic regimens have rarely been reported, and, to the best of our knowledge, a single experiment has been able to reach chaotic behavior in this particular setup [7]. The regimes that were found before the onset of chaos include two PD bifurcations, which were followed by a period tripling stage. Very recently, a numerical analysis of the rate equation model provided an explanation for this particular route to chaos, accounting for the different regimes that were experimentally observed. From this study, it turns out that random noise fluctuations play a major role in reaching an agreement between theory and experiments [9], [10]. The present analysis uncovers the reasons for the extreme sensitivity to noise of modulated laser diodes. In this paper, we show that, as a result of the laser’s sensitivity, sudden drops in the maximum output power appear in the laser when its behavior undergoes a PD sequence. We demonstrate that these drops are associated with a global bifurcation phenomena, in which the behavior is influenced by unstable solutions. This paper is organized as follows. Section II presents the rate equation model that we have used to describe the laser diode, including the noise sources that account for the random fluctuations. The next section provides a brief description of the different regimes that rule the deterministic dynamics of the laser diode as the modulation strength is increased. In Section IV, we report the bursting behavior that we obtain in the PD sequence, describing its origin using modern bifurcation theory. The conclusions of the present study are then summarized in Section V. II. RATE EQUATION DESCRIPTION The generation of laser light in a laser diode results from the interaction between carriers and photons occurring in the active layer. The dynamics that take place within the active layer are commonly described using two coupled ordinary differential equations, known as rate equations. However, this set of equations accounts for the deterministic dynamics of the laser. Following the Langevin approach, random noise fluctuations are included by the addition of a new term in each of the equations, called the Langevin source. The new set of equations defines a stochastic differential equation system (SDE), which in its normalized version is written as (1.1) (1.2) where and represent the carrier and photon densities by means of two dimensionless unit order variables. The different terms

0018–9197/00$10.00 © 2000 IEEE

CARPINTERO AND LAMELA: INTERMITTENT BURSTS IN A DIRECTLY MODULATED LASER DIODE

in the rate equation model represent different processes within the active layer such as carrier injection, spontaneous carrier recombination, stimulated and spontaneous emission, and photon losses. In these normalized equations, is the quotient between the carrier lifetime and the photon lifetime, is the normalized gain compression factor, the normalized transparency carrier density, and the spontaneous emission coupling factor. The first term on the right-hand side of (1) models the current injection, which is assumed to introduce a sinusoidal time dependence, fol, in which is the lowing the expression is the modulation current amplitude, and is bias current, ). The behavior of the the modulation frequency (with period laserdiodeisanalyzedastheamplitudeofthemodulationisvaried at a fixed bias current and fixed modulation frequency. In this study, the variations in the modulation amplitude are expressed , in terms of the modulation index, defined as and the bias level is expressed through the bias index, defined , where is the normalized threshold current. as The random fluctuations are included by means of two Langevin and , delta-correlatedGaussian random varisources ables with zero mean. The auto- and cross-correlation of these sources can be found in [6], from which it becomes clear that the sources introduce multiplicative noise. Further details of this system can be found in [9]. In this system of coupled equations, it is always worth mentioning a few important facts. The first one is to notice that this model provides a description of the carrier and photon populations averaged over the entire active layer volume. Also, given the fact that a single photon rate equation is used, the present model can be applied to single-frequency laser diodes. This restriction agrees with the conditions under which chaotic behavior has been experimentally observed [7], where a distributed feedback (DFB) laser is used. From the point of view of nonlinear dynamics, it is important to notice that the laser diode is a dissipative system. Therefore, the time evolution of the system’s variables, known as trajectory, is largely independent on the initial conditions as it tends to a final state known as attractor. An attractor can be a fixed point, a periodic regime (known as limit cycle), or the more complex chaotic attractor, characterized by a bounded nonperiodic trajectory. Chaotic regimes are allowed in the rate equation system since the modulation of the current introduces the necessary third degree of freedom: time dependence. The laser diode can then be considered as a nonautonomous system in nonlinear dynamics jargon. Another important fact is that both equations run on different time scales. While the carrier rate equation evolves on the time scale , the photon rate equation runs on time scale, which is at least two orders of magnitude the above the carrier time scale. Among the direct consequences that the different time scales have is the appearance of the characteristic relaxation oscillations. The existence of fast and slow time scales, as will become clear in this study, is of great importance in the behavior of the device. III. THE FRAME FOR BURSTING Period doubling is one of the most ubiquitous scenarios for the appearance of chaos in deterministic dynamic systems. In

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fact, a directly modulated laser diode without noise has been shown to go through a sequence of PD bifurcations as the amplitude of the modulation is increased. The modulation index constitutes what is known as the bifurcation parameter, which at certain values, known as bifurcation points, changes the qualitative behavior of the laser diode. In a PD sequence, at each PD bifurcation point, the attracting limit cycle doubles its period and an unstable saddle cycle with the same period of the original limit cycle appears. Therefore, this sequence can ideally produce periodic responses with an arbitrarily long period. Due to the Feigenbaum size scaling verified by this sequence, two conditions are imposed: 1) the periodic limit cycles are achieved up to a fixed value of the bifurcation parameter, called the accumulation point, and 2) as new period doubling bifurcations take place, the limit cycle becomes increasingly intertwined with the growing number of unstable saddle cycles. Beyond the accumulation point, the periodic regimes become irregular, and several disjointed segments are visited sequentially by the trajectory. The fact that, after a given number of iterations, we find ourselves in the same segment allows us to assign them periodicity; however, the behavior inside each segment is chaotic. As a general rule, the period of this chaotic attractor is given by the number of segments, which merge pairwise as the bifurcation parameter is increased and a period halving (PH) sequence is displayed beyond accumulation [11]. At this point, we would like to point out that the nonlinear behavior of modulated laser diodes is enhanced by biasing the device such that the resonance frequency is set at half the modulation frequency [7]. Since we are interested in the nonlinear behavior of the laser diode, we assume this condition from the start. The different regimes that appear as the bifurcation parameter is varied in the absence of noise (deterministic behavior) are presented in the bifurcation diagram in Fig. 1. This diagram is generated by stroboscopically sampling one of the system variables at a fixed phase of the modulation and to varying the modulation index from in steps , establishing a Poincaré section [11]. For each modulation index level, the thick dots represent the samples taken at time instants spaced by the modulation period. Therefore, a -periodic limit cycle becomes a single dot; two intersections -periodic limit cycle; and so on. When the indicate a modulation index increases beyond the accumulation point, the samples appear to be grouped in several segments, which provide an indication that chaotic regimes have been reached. In Fig. 1, the PD sequence has been highlighted by marking three PD bifurcation points with arrows. The PH sequence is also shown, and the merging process as the modulation index increases is displayed. Thisdiagramprovides apowerfultoolintheinvestigationofthe dynamic behavior of the system when all of the possible solutions, attracting (stable) and repelling (unstable), are represented. This has been achieved in the bifurcation diagram in Fig. 1, where the samples at the different modulation index levels corresponding to several saddle cycles (unstable limit cycles) are represented -periodic saddle by a full line, the -periodic by lines: the -periodic saddle using a saddle by a dashed line, and the dotted line. With this information on the diagram, one is now able to observe at the two first consecutive PD bifurcations

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Fig. 1. Bifurcation diagram of the carrier density versus the modulation index for the deterministic dynamics. The range of modulation index values where two regions are highlighted: the period doubling sequence using a solid trace and the period halving sequence using a waved trace. Arrows indicate three successive PD bifurcations. The unstable solutions that appear at each PD bifurcation are represented by a solid line (unstable T -periodic) and a dashed line (unstable 2T -periodic).

(at and ) how the limit cycle doubles its period, doubling the number of dots on the diagram, while the lines that appear at both levels indicate the birth of saddle , is marked , and cycles. The first one, for bifurcation, is marked the second one, for the . The lines indicate the evolution of the saddle cycles appearing at each PD bifurcation point. Further details about the bifurcation diagram represented in Fig. 1 can be found in [9]. It is in the PD sequence to chaos that we have found the existence of random bursts in the output. As we mentioned earlier and as can now be appreciated in Fig. 1, the PD sequence progressively intertwines the limit cycle with saddle ones. This fact was suspected to cause noise sensitivity in dynamic systems undergoing a PD sequence. In the next section, we will show that the origin of the bursts is not due to the approaching saddles to the limit cycle but due to the inset and outset associated with every saddle cycle. As we will shortly show, insets and outsets also play a fundamental role in the system dynamics, especially in the presence of noise.

IV. BURSTING BEHAVIOR IN A MODULATED LASER DIODE In this section, we present the mechanism that allows intermittent bursts in modulated laser diode dynamics. In the previous section, we stated that it is found in the PD sequence, which we described thoroughly in the absence of noise. The only

difference that we are about to introduce now is the presence of noise by turning on the Langevin sources in the rate equation model. For the sake of comparison with the deterministic results, Fig. 2 presents, along with the results of Fig. 1, the bifurcation diagram that is a result of stroboscopically sampling a single realization of the stochastic process at the same modulation index levels. From this figure, two known facts of stochastic dynamics can now be appreciated. The first one is that the stochastic trajectory wanders around the deterministic limit cycles, showing a small dispersion due to noise fluctuations. This can be appreciated in Fig. 2 by the thin lines that appear on each of the dots of the deterministic dynamics. The second fact is that the chaotic behavior is enhanced, appearing at lower levels of the bifurcation parameter. The continuous segments that appear for the modulation levels above the second PD bifurcation point, which is characteristic of the chaoticity of the regime, appear at lower levels of the modulation index. The increase in the dispersion of the stochastic trajectory means that chaotic dynamics are due to the fact that saddle cycles are embedded in the trajec-periodic saddle, tory. Initially, the segments embed the -periodic behavior, and finally masking the deterministic -periodic saddle, producing a single end by embedding the continuous line in the bifurcation diagram. The embedding of the saddles has been shown to produce pedestal components in the power spectra [10], which have been observed experimentally [7]. To examine the process by which noise enhances the chaotic behavior embedding saddle cycles, we turn to the time


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Fig. 2. Bifurcation diagram of the carrier density versus the modulation index for the stochastic dynamics, represented with light dots over the deterministic bifurcation diagram in Fig. 1.

evolution of the stochastic process within this range of modulation index levels. For this purpose, Fig. 3 represents the time evolution of the normalized photon density for four levels of the modulation index in a single realization of the stochastic process at each level. At the lowest modulation level, where the modulation , the time evolution of the system apindex is set to pears as a continuous dark trace. The influence of noise appears in the small fluctuations on the maximum amplitude [Fig. 3(a)]. If we were to find the periodicity of this behavior, one could perform the fast Fourier transform of the time sequence as reported in [10] and observe the frequencies where the peaks are located. At this particular level, noise pedestals are a result of the embed-periodic saddle cycle. However, as the modulading of the tion index is increased, the time evolution presents intermittent bursts that appear at apparently random time intervals, causing a sudden drop in the maximum amplitude of the output. As is observed in the figure, the frequency of the appearance of these bursts increases as the modulation index grows [Fig. 3(b)–(d)]. This behavior is what we have defined as intermittent bursting in the modulated laser diode. The decrease that the bursts produce in the maximum output power is best appreciated in Fig. 4, where we plot a detailed time evolution of the photon density along with its envelope, calculated by low-pass when filtering the photon time evolution. In this figure, one observes the difference between regular behavior (also known as “laminar” phases) and the intermittent bursts at random moments. In order to gain insight into the dynamics of the bursts in a directly modulated laser diode, let us now examine in detail the time evolution inside the burst. The blowup of the time axis at a

single burst is presented in Fig. 5. This figure shows the sudden drop in the maximum output photon density. When the time series appearing in this figure is sampled at the same rate used to build the bifurcation diagrams, the resulting samples are those points that appear highlighted by thick dots. The time evolution of this collection of points presents a double exponential divergence toward “laminar” behavior, where the intensity of one component grows while the intensity of the other component decreases. Any explanation of the intermittent bursts will have to provide an indication of the origin of this particular temporal evolution, which reminds us of the intermittence route to chaos. It is worth mentioning that intermittent bursts have been experimentally observed in laser diodes subject to delayed feedback [12], where part of the emitted light is reinjected back into the laser using an external mirror. Although many papers have been published on their origin in this configuration [13], [14], it is commonly agreed that saddle type solutions play a key role in the appearance of the bursts [15]. This fact provides us with a clue about where to start our investigation on the origin of the bursts in modulated laser diodes, the saddle cycles that appear in the PD sequence. V. ORIGIN OF THE INTERMITTENT BURSTS So far, we have presented different aspects of the temporal evolution of the intermittent bursts of the modulated laser diode. Beginning with an explanation of their origin, we would like to first point out that the bursts appear within the same range of values of the modulation index where the enhancement of the chaotic behavior occurs. It is, therefore, natural to raise the

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=

Fig. 3. Time evolution of the normalized photon density at different modulation index levels (from bottom to top): (a) m 4:2, (b) m = 4:3, (c) m = 4:4, and (d) m = 4:5. The system parameters are T = 500, k = 0:004, n = 0:6429, A = 2:8, and = 5E -5 and the time spans over 2000 modulation periods.

Fig. 4. Detail of the normalized photon density’s time evolution at the modulation index level m = 4:4 with its calculated envelope function.

question of the relation between both. Our first step has been to represent the dynamics of the system in another type of diagram: the state space. At every moment, the state of the laser is completely specified when the carrier and photon densities are given. Therefore, using the Poincaré section method, and plotting the photon density versus the carrier density at fixed time intervals spaced by the modulation period (which constitute the intersections of the trajectory with the Poincaré section) for a given modulation index, provides a picture of the dynamic state of the , a level laser. In Fig. 6, we depict such state space for

at which the stochastic dynamics shows bursting and the deterministic dynamics a -periodic limit cycle. In this figure, each of the thick dots represents a carrier density, photon density pair of values. By looking at this figure, one observes that the samples from the stochastic dynamics of the system accumulate mainly on two separate segments. These segments constitute the attractor of the system, to which all the trajectories of the system tend toward as the time increases. Since each segment is visited alternatively by the stochastic trajectory, the be-periodic. However, as the samples havior can be said to be


Fig. 5. Time evolution of the normalized photon density inside a single burst at m of the forcing period.

Fig. 6. State space diagram of the dynamics when the modulation index is set to m = 4:4. The dots represent the samples of the stochastic trajectory and the four white dots represent the deterministic 4T -periodic limit cycle. The two white squares represent the 2T -periodic saddle that appeared in the second PD bifurcation and the T -periodic saddle cycle is represented by . Arrows (a) and (b) represent the capture and slow evolution processes, respectively.

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from the -periodic saddle cycle and the -periodic limit cycle of the deterministic dynamics lie within this attractor (that is, within the segments), we can state that the segments correspond to a chaotic attractor. The fact that two segments are observed is related to the fact that it is the period of the unstable saddle embedded in the stochastic dynamics that also provides the stretching mechanism needed for chaos [11]. In addition to the two segments in Fig. 6, some of the dots lie in the area between both, in the vicinity of the intersection of -periodic saddle cycle with the Poincaré section. What the is really interesting about these points is that they correspond to samples taken within the bursts. To demonstrate this fact,

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= 4 4. Consecutive dots represent time intervals separated by integer multiples :

in Fig. 6 we highlighted numerically according to the order of appearance, the sequence of dots that come from samples within the burst in Fig. 5. As can be appreciated on the diagram, while the first of these points is located at one segment’s extreme, the next one is located in the region surrounding the intersection of the -periodic saddle. This fact already allows -periodic saddle us to relate the embedding of the unstable with the bursting behavior. The bursts are shown to take place in two steps. First, the stochastic trajectory is captured by the -periodic saddle cycle [arrow (a) in Fig. 6], and secondly, after the capture, the trajectory shows a slow evolution toward the segments [arrow (b)], where the stochastic dynamics spends most of the time. If we were to follow the evolution of the sequence of points in Fig. 6, starting at the point labeled 2, it would show that the evolution toward the segments has the double exponential characteristic than that observed within the burst. Since the evolution of the trajectory relaxing back toward the attractor shows a -periodic characteristic. The right–left alternation, it has a fact that the capture and the reinjection process have very different time scales (each takes a different number of intercepts) accounts for the bursting behavior in the time domain, while the fact that noise is responsible for the capture process accounts for the random appearance of these bursts. However, in this process, the mechanism for the capture of the trajectory remains to be clarified. From the comparison of the stochastic versus the deterministic dynamics, it becomes apparent that the role of noise is -periodic mainly to allow the trajectory to be captured by the saddle cycle, which otherwise remains isolated from the system dynamics. However, in order to understand the capture mechanism of the trajectories, it is important to notice that saddle cycles play a key role in organizing the behavior of the system [11]. This is due to two special sets of points, formed by the

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(a)

(b)

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Fig. 7. (a) State space diagram where the inset [W (T )] and outset [W (T )] corresponding the T -periodic saddle cycle (represented by ) of a modulated laser diode when the modulation index is set to m = 4:4 are represented. (b) Detail of the squared area in (a), where the folded structure of the outset can be observed.

trajectories that either head directly to or directly away from the attractor, receiving the name of inset (or stable manifold) and outset (or unstable manifold) respectively. The important fact is that the manifolds form barriers in the state space that the trajectories cannot cross. With these considerations, any trajectory that approaches the inset of any saddle will be dragged toward it along this manifold, only to be pushed away from it along the outset. We have computed the inset and outset of -periodic saddle when the modulation index is the unstable using the deterministic rate equation model. fixed at Both sets are represented in the state space diagram in Fig. 7(a), -periodic saddle where the triangle shows the location of the ] cycle. Those points that belong to their inset [labeled are represented with thick gray dots, while the outset [labeled ] is in thin black dots. It can be observed in this figure that the extremes of the outset very much resemble the area where the segments were found to lie in the stochastic dynamics.

This is due to the fact that, in a PD bifurcation point, the outset of a -periodic saddle cycle becomes the inset for the period doubled (higher order) limit cycle that is born. However, it is important to notice in the figure that the two manifolds are very close to one another. In Fig. 7(b), an enlarged picture of the window area marked in Fig. 7(a) is shown, and it can be observed that both manifolds are, in fact, close to each other but there is no intersection between them. From our study of the manifold structure, several statements can be made about the system. -periodic saddle pushes the trajecto1) The outset of the ries of the system toward the higher order periodic cycles of the PD sequence, where the attractor of the system is to be found. 2) Around the higher order periodic cycles, the outset folds over itself, having a very complicated structure. This causes the system to be highly sensitive to noise fluctuations.


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TABLE I

Fig. 8. State space diagram where the inset [W (T )] and outsets [W (T )] are represented along with the stochastic trajectory when m is set to 4.4. The stochastic trajectory, represented with gray squares, occupies a large region as it embeds the part of outset of the unstable T -periodic saddle close to the deterministic regime.

3) As the modulation index is increased, the inset and outset -periodic saddle cycle approach each other, of the making us foresee an intersection between them both (which will be shown later). One of the main effects of noise in a dynamic system is that it allows the trajectories to “sense” the surrounding state space. In our system, this statement is translated into the broad segments in Fig. 6 over which the stochastic trajectory spent most of the time. Due to the folded nature of the manifolds, the system presents an extreme sensitivity to the random fluctuations, and the stochastic trajectory evolves over the area occupied by these manifolds. This fact is presented in Fig. 8, where the unstable -periodic saddle is plotted using thin dots. manifold of the The stochastic trajectory, which is also represented in Fig. 8 using gray squares, shows a spread around the intercepts on the Poincaré plane corresponding to the deterministic solution: -periodic cycle. The area over which the stochastic traa jectory spreads occupies the deterministic solution as well as part of the unstable manifold. This is due to the fact that the random fluctuations continually push the stochastic trajectory away from the deterministic solution. The push locates the trajectory on a given point close to the unstable manifold. The trajectory then rapidly relaxes toward the unstable manifold, over which it now relaxes back toward the deterministic solution. As is appreciated in Fig. 8, this mechanism allows the trajectory to develop over a large area in the state space. The “sensing” characteristic of the stochastic trajectory be-periodic saddle comes most important when the inset of the approaches its outset, and therefore, approaches the region on which the stochastic trajectory evolves. The noise fluctuations can then place the trajectory at random intervals of time close -periodic saddle to be dragged to it. enough to the inset of the This accounts for the capture mechanism, which takes place on a very short time scale. Once the trajectory has been captured by the saddle, the time evolution continues along the saddle’s outset toward the higher periodic orbit. This process, which

Fig. 9. State space diagram where the inset (dash line) and outset (gray dots) of the modulated laser diode are represented along with the deterministic solution (black dots) for the modulation index level m = 4:8. The appearance of an homoclinic trajectory is demonstrated as the inset and outsets of the T -periodic saddle intersect.

takes place on a slower time scale, evolves at the frequency of -pethe solution to which the trajectory tends toward, the riodic saddle, embedded in the segment. In order to give an estimation of the time scales involved in the evolution along each -perimanifold, we computed the Floquet multipliers of the odic saddle [11]. The modulus of these multipliers at different and , is provided in Table I. modulation index levels, As it corresponds to an externally excited system, the absolute value of the multipliers of the saddle solution is one lower than unity, and the other greater than one. Each of these multipliers can be associated, respectively, with the rate of contraction and expansion of trajectories in state space due to this saddle cycle. As the modulation index is increased beyond this level, the outset and the inset continue to approach each other until in fact, both sets intersect as depicted in Fig. 9 (where the mod). At this level, a homoclinic ulation index is set at intersection occurs, giving rise to a homoclinic connection, a trajectory that connects the saddle point to itself. The appearance of the homoclinic connection does not affect the deterministic dynamics of the system, which is also depicted in Fig. 9, where the deterministic attractor is represented by dark dots that . The deterministic dynamics at this lie over the outset -perilevel, shown in the bifurcation diagram in Fig. 1, is a odic chaotic attractor. In this figure, it can be observed that the -periodic saddle drives the trajectories toward outset of the

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the chaotic attractor. Also, this figure reveals that, in the absence of noise, the system is not allowed to evolve using the outset -periodic saddle. The deterministic dynamics are unof the able to detect the homoclinic connection. Under these condi-petions, the random bursts implying excursions along the riodic saddle do not appear. Only when the modulation index is -periodic chaotic attractor collides with increased, and the the inset of the saddle, is the deterministic trajectory embedded in this solution. This corresponds to the level at which the deterministic bifurcation diagram shows the merging of the two segments into a single one. VI. CONCLUSION To summarize the results that were presented in the previous sections, we confirm that, under certain conditions, the output of a directly modulated laser diode shows bursting behavior. These bursts constitute a sudden decrease in the maximum amplitude of the photon and carrier densities. The conditions under which this behavior has been described are within the period doubling sequence that appears when the modulation index is varied and random noise fluctuations are taken into account. The origin of bursting behavior can be explained in terms of a global bifurcation phenomena taking place in the system. In fact, this paper demonstrates that bursting behavior acts as an indicator of the process by which the trajectory embeds a saddle cycle. In the modulated laser diode, two major conditions have been identified as playing a key role to allow this process to take place: 1) the complexity of the manifolds associated with the saddle being embedded by the system’s trajectory and 2) the formation of an homoclinic tangle, which implies the approximation of the inset of the saddle to its outset. Together with these two conditions, the introduction of noise fluctuations gives rise to the described phenomenon. This study provides, to the best of our knowledge, the first explanation of the origin of the intermittent patterns that appeared in experimental observations of directly modulated laser diodes. The analysis of this behavior, which required the use of bifurcation theory, has shown the complexity of the system. An investigation to apply present knowledge to other laser configurations where intermittent bursts have been reported is under way. REFERENCES [1] J. P. Eckmann, “Roads to turbulence in dissipative dynamical systems,” Rev. Mod. Phys., vol. 53, no. 4, pp. 643–654, 1981. [2] J. P. Crutchfield, F. D. Farmer, and B. A. Huberman, “Fluctuations and simple chaotic dynamics,” Phys. Rep., vol. 92, no. 2, pp. 45–82, 1982. [3] 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. [4] K. Wiesenfeld, “Noisy precursors of nonlinear instabilities,” J. Stat. Phys., vol. 38, no. 5/6, pp. 1071–1097, 1985. [5] , “Virtual Hopf phenomenon: A new precursor of period-doubling bifurcations,” Phys. Rev. A., vol. 32, no. 3, pp. 1744–1751, 1985.

[6] Y. H. Kao and H. T. Lin, “Virtual Hopf precursor of period-doubling route in directly modulated semiconductor lasers,” IEEE J. Quantum Electron., vol. 29, no. 6, pp. 1617–1623, 1993. [7] H. F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 m InGaAsP distributed feedback laser,” IEEE J. Quantum Electron., vol. 29, no. 6, pp. 1668–1675, 1993. [8] S. Bennet, M. Snowden, and S. Iezekiel, “Nonlinear dynamics in directly modulated multiple-quantum-well laser diodes,” IEEE J. Quantum Electron., vol. 33, no. 11, pp. 2076–2083, 1997. [9] H. Lamela, G. Carpintero, and F. Mancebo, “Period tripling and chaos in the dynamic behavior of directly modulated diode lasers,” IEEE J. Quantum Electron., vol. 34, pp. 1797–1801, 1998. [10] G. Carpintero and H. Lamela, “Underlying dynamics of the pedestal components in a modulated laser diode with noise,” Opt. Lett., vol. 24, no. 23, pp. 1711–1713, 1999. [11] C. O. Weiss and R. Vilaseca, Dynamics of Lasers. Weinheim, Germany: VCH, 1991. [12] I. Fischer, G. H. M. Tartwijk, A. M. Levine, W. Elsässer, E. Göbel, and D. Lenstra, “Fast pulsing and chaotic itineracy with a drift in the coherence collapse of semiconductor lasers,” Phys. Rev. Lett., vol. 76, no. 2, pp. 220–223, 1996. [13] G. H. M. Tartwijk, A. M. Levine, and D. Lenstra, “Sisyphus effect in semiconductor lasers with optical feedback,” IEEE J. Select. Topics Quantum Electron 1, pp. 466–472, 1995. [14] M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E, vol. 55, no. 6, pp. 6414–6418, 1997. [15] T. Sano, “Antimode dynamics and itineracy in the coherence collapse of semiconductor lasers with optical feedback,” Phys. Rev. A, vol. 50, no. 3, pp. 2719–2726, 1997.

Guillermo Carpintero graduated from the Universidad Politécnica de Madrid in the Telecommunications engineering degree in 1993, and received the doctor degree from the Universidad Carlos III de Madrid in 1999. His doctoral thesis analyzed the nonlinear behavior of modulated laser diodes including the effect of noise. He is currently an Assistant Professor and member of the Optoelectronics and Laser Technology research group at the Universidad Carlos III de Madrid. He is involved with different national and European research projects, being his major research interests semiconductor lasers, device modeling and applied bifurcation theory.

Horacio Lamela received the Industrial engineering degree from the Universidad Politécnica 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 inteferometry from the Conservatoire d’Arts et Metiers of Paris, 1985, 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 Post-Doctoral Fellow at the Electrical Engineering and Computer Sciences Department, working on LiNbO spatial light modulators and as a Visiting Scholar through Fullbright-MEC Commission at the Research Laboratory of Electronics in semiconductor laser frequency stabilization for cesium atomic clocks. He is presently an Associate Professor of the Departamento de Ingeniería Eléctrica, Electrónica y Automática and leader of the Optoelectronics and Laser Technology Group at Universidad Carlos III de Madrid, Spain. He is currently involved in various Spanish and European Projects in the fields of high sensitive optical sensor measurements and applications of diode lasers and optoelectronics. His research interests are high-speed semiconductor laser dynamics, laser-diode fiber-optic communications and sensor applications, laser interferometry and fiber-optic interferometric sensors.