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We study a theoretical model describing a laser with a modulated parameter, concentrating on the appearance of extreme events, also called optical rogue pulses. It is shown that two conditions are required for the appearance of such events in this type of nonlinear system: the existence of generalized multi-stability and the collisions of chaotic attractors with unstable orbits in external crisis, expanding the attractor to visit new regions in phase space.
© 2014 Optical Society of America
1. Introduction
Jorge L. Borges in one of his literary pages wrote [1]: “... lo que llamamos azar es nuestra ignorancia de la compleja maquinaria de la causalidad.” (“ ... what we call chance is our ignorance of the complex mechanism of causality.”). This view point is rooted in more than two centuries of science [2]. Rogue extreme events in dynamical systems insert us into this subject: Do rare extreme and destructive events occur by chance? During the last two decades there is an increasing scientific interest on extreme events in general, and rogue waves in particular. Such interest is manifested by a fast increment in the number of publications on the subject in several disciplines: from oceanography [3–5] to astrophysics [6], and from geology [7] to optics [8–12]. Usually the main approach to study these events is to use the statistics properties of their occurrence. Looking at almost ancient literature a rogue wave is defined as an oscillation pulse whose amplitude is larger than a certain threshold. Such a threshold has been defined as twice the average value of the amplitude of the largest one-third maxima [13, 14]. Also, later on, it has been defined as an oscillation whose amplitude is four times larger than the standard deviation of the surface elevation [3]. In any case a definition based on particular statistics remains a definition based on an arbitrary limit. We prefer to adopt here that whatever pulse in the dynamics is much larger than the average it can be considered a rogue event, including in an optical system, where such extreme event of electromagnetic wave pulse has been called “optical rogue wave” [8–11,15]. In several natural phenomena rogue events can be destructive. Thus, it would be of extreme importance to study how far in advance we can predict and control them or, at least, to know our capacity of prediction. In order to answer questions about their control and predictability it is necessary to figure out how they are produced and which physical mechanism is at their origin.
Given their nonlinear nature, each different dynamical system may generate extreme events through specific mechanisms. The recent literature cited here contains studies of many experimental and theoretical systems that show extreme events. In this paper we analyze a theoretical simple model of a laser system and we show that it is possible to observe optical rogue light pulses in this system. We investigate and describe the dynamical process at the origin of such extreme events.
2. Extreme events in chaotic dynamics
As a matter of principle extreme events may appear on every statistical system and how much frequent an extreme event occurs depends on the distribution probability of the events in the dynamics. If we accept the above definition based on the standard deviation or the kurtosis [5,16] of the probability distribution, then it is clear that they are extremely rare for a Gaussian distribution while they become more frequent for a Levy’s type distribution possessing a long tail at high amplitude values. In classical mechanics, a statistical description is needed whenever the system is very complex and its description involves a very large number of variables or whenever the behavior is deterministically chaotic. Accordingly, determinism implies the existence of a unique evolution in phase space for each initial condition. A reduction of the phase space to a small number of variables and the inclusion of a statistical description often means a break of the determinism by allowing different possible evolutions in the reduced phase space for the same initial condition. We say then that there is noise in the system and stochastic methods are applied to describe it. If the amount of noise is small enough it will not disturb the trajectory in most of the phase portrait of the deterministic part of the system: a periodic or a quasi-periodic orbit will remain such and the behavior of the system is predictable. Thus, we do not need a statistical description; we do not even need to know the equations of motion and the initial conditions to figure out the past and the future evolution of the dynamical system. Within conditions where a chaotic deterministic trajectory exists, the predictability of the dynamical evolution is defined by the maximum positive Lyapunov exponent. If instead the noise plays a relevant role in modifying the attractor itself, then a statistical description is obligatory and our predicting capacity vanishes as fast as it vanish the autocorrelation function. As it is well known and we discuss in the following, changes of an attractor also occurs within fully deterministic models. Of course all the previous discussion losses significance for a quantum system, in which initial conditions cannot be determined with infinite precision, the only deterministic evolution is the evolution of the wave function. Thus there is an intrinsic role of the probability distribution and the only possible description is a statistical one. Therefore, having in mind the possibility of predicting the dynamical behavior of a system, we could define the following phenomenological categories, without claiming mathematical rigor:
Categories B and C systems require the knowledge of probability distributions and prediction can only be done on a statistical basis. The difference is that statistics is intrinsic to a system of category C while it is simply “misknowledge” in A and B. Here we consider a category A system. The system is deterministic. Then, its dynamical evolution in phase space is fixed by the equations of motion and the initial conditions. However, if the behavior is chaotic, it is not long term predictable by definition. Statistical treatments can be used, in principle, as much as it does for a system of category B and C. However, the power of recent computers to give direct numerical solutions of nonlinear equations permits studies like the one given here. Many previous papers already studied the probability distribution of deterministic chaotic behavior, mainly in maps [17–19]. In particular, [17] identifies some intermittent regime in which the probability distribution of the amplitude of the pulses becomes a Levy’s distribution. It then presents long tail at large amplitude as a consequence of the appearance of “extreme events” that do not become as rare as in a Gaussian distribution.
To obtain chaos in the dynamics of a lasers there are many techniques and we shall concentrate on modulation of one system parameter. In the following we focus in chaotic extreme pulses emitted by such lasers. The parameter modulation expands the phase space of simple model for the laser [22] and so allow for chaotic dynamics. As in other systems, the struct of the attractors have a natural role in the time evolution of any of its variables. The trajectories will change with the control parameters of the system, according to the bifurcations following the changes in the manifolds of the fixed points, the periodic orbits and the attractor borders. Details on these properties can be found in [20, 21]. The calculated and observed dynamical variable is the spiking optical pulses output, which is fairly accessible to experimental measurements, [22–28].
3. Laser with modulated losses: the model
The dynamical equations we use correspond to a common model of a solid state or some gas lasers, with a modulated control parameter. The so called rate equations for mono-mode laser with modulated losses [22]:
with In these equations, I(t) and N(t) are the normalized intensity and the population inversion, respectively; g is the coupling rate constant, N0 is the pump parameter, k0 is the constant loss rate, m and ω are the modulation amplitude and frequency, respectively, and γ‖ is the loss rate for the population inversion.We choose a modulation frequency ω close to the relaxation oscillation frequency Ω,
where A = N0g/k0.Setting S = (2g/γ‖)I, D = N(g/k0), and τ = k0 t, we get from Eqs. (1) and (2):
where β = ω/k0, and γ = γ‖/k0.Many properties of these equations have been studied theoretically and with experimental observation of this laser dynamics reported [22–28]. However none of them consider the occurrence of extreme events. Herein we focus on conditions for extreme pulses to appear.
4. Numerical results and analysis of rogue pulses
Numerical solutions were obtained using a standard Runge-Kutta algorithm. Taking the parameter m in Eqs. (3) and (5) as the changeable control parameter and collecting the maxima of the pulses the system shows a series of period doubling bifurcations until its dynamical behavior becomes chaotic. This is known to happen if the modulation frequency, ω, is close enough to the relaxation oscillation frequency, Ω. Figures 1(a) and 1(b) show the bifurcation diagram and the phase space portrait of a chaotic solution for a set of parameters where there is only one stable solution for each value of m. As the modulation amplitude m is further increased within the chaos, windows of periodic behavior occur along with the chaotic attractor expansion in phase space.
Fig. 1 (a) Bifurcation diagram graphing the maxima of intensity S as a function of the modulation amplitude m. The parameters used in the numerical integration are: A= 1.30, β = 0.09441, and γ = 0.027. (b) Phase space portrait, Intensity S as a function of population inversion D for m = 0.68. All other parameters values are the same as in Fig. 1(a).
Figure 1(b) shows the projection of a chaotic trajectory in phase space into the plane (S, D) (a phase space portrait). The area occupied by the trajectory appears densely visited. Essentially the phase space portrait shows that the chaotic attractor, once fully developed, cover almost homogeneously all possible values of the intensity and population inversion. Numerical verification of the actual density variation can be obtained by histograms of the pulse amplitude extracted directly from the time series as shown in Fig. 2.
Fig. 2 Histogram of the calculated maxima of intensity corresponding to the time signal of Fig. 1(b). The vertical dashed line indicates the threshold corresponding to the AI calculated as the mean value plus four times the standard deviation of the probability distribution. Clearly no “optical rogue events” occur.
By simple observation of Fig. 2 we can see that in this case extraordinary high intensity pulses do not occur. The histogram shows a distribution of peaks that is fairly uniform. It also evidences that there is a sharp cut on the maximum of the variable. In analogy with the logistic map, larger density of peak values appear somewhere within and on the extremes of the attractor. This can be better observed in Fig. 3, which contains a blow-up of Fig. 1(a) and shows the concentration of maxima at very low values near zero power. Similar blow up gives evidence to the higher density on the shap top extreme of the attractor. The relevant point is that a quantitative exam of the probability distribution of the pulses confirms that there is no long tail at high values of the variable. No pulse is obtained whose maximum overcomes the average value plus four times the standard deviation. Thus, during the integration time where we have over 106 maxima, no optical rogue pulses have been generated and most probably they do not exist for such parameter values. The dynamical behavior is chaotic but the trajectory evolves in a sharply bounded region of phase space.
Fig. 3 Blow-up of lower portion of the bifurcation diagram of Fig. 1(a). This figure shows how the density of pulse maxima increases, accumulating near zero value.
Decreasing the dissipation of the system, i. e. decreasing (Aγ‖/Ω) [28], the bifurcation diagram undergoes a significant changes as seen in Fig. 4. The same diagram was calculated increasing and decreasing the value of the m parameter an using at each step of the new time series calculation the precedent values of the dynamical variables as initial conditions. This procedure was applied to search for regions of parameter giving bi-stability and multi-stability. They are not shown if Fig. 4 but do appear for low values of m (near 0.005) when we calculate with different sets of initial conditions. Through the range where there is chaos, the diagrams indicate a single attractor.
Fig. 4 Bifurcation diagram as in Fig. 1(a) but with the following parameter values: A=1.3; β = 0.0135; and γ = 0.00027.
Two main observation are then in order. First there exist several coexisting periodic attractors for the same parameter values implying existence of generalized multi-stability [20, 27]. Secondly an increase of the modulation amplitude, after chaos is reached, results in a series of abrupt expansions of the chaotic attractor yielding a kind of staircase in the bifurcation diagram. Each step marks the occurrence of a bifurcation due to the collision of a chaotic attractor with an unstable orbit in what was named an external crisis of the chaotic attractor in reference [20].
Related types of bifurcations have been studied in the modulated laser, proposing mechanisms to the origin of mixed-mode laser operation [21]. These bifurcations lead to bursts of pulses between two chaotic conditions in the laser.
The generalized multi-stability is not a surprise if we just consider that, in the conservative limit, infinite number of periodic orbits coexist for all values of the control parameters. As dissipation increases the number of coexisting orbits decreases. For realistic parameter values corresponding to a CO2 laser or solid state lasers several stable periodic orbits may coexist.
As we mention before, the abrupt changes in the attractors correspond to bifurcations called “external crisis” [29] because they are produced when there is a collision of the strange attractor with an unstable orbit of a different bifurcation branch. Before the collision, the chaotic attractor is confined in a region of phase space limited by the stable manifold of the unstable orbit, but after the collision, trajectories can explore a much larger region in phase space “propelled” by the unstable manifold of the unstable orbit. This process generates pulses whose maximum intensity are much larger than those previous to the collision and which remain rare compare to the density of pulses with lower maximum intensity. In order to emphasize the occurrence of the rare rogue pulses in our system a very long time is given in Fig. 5. The segment in this figure is densely packed and hides the individual pulses. To visualize some individual pulses a short segment of Fig. 5 is presented in Fig. 6. The shapes of the pulses, be smaller or large and rare, are not very different.
Fig. 5 Long series of pulse Intensity S as a function of time measured in units of the modulation period. The parameter values are the same as in Fig. 3 with m = 0.01236. This time series contains rare abnormally large amplitude spikes.
Fig. 6 Pulse Intensity S as a function of time measured in units of the modulation period. This is a time zoom of Fig. 5 (with displaced time scale).
In Figs. 7(a) and 7(b) we present a pair of histograms of the maxima intensity corresponding to different values of the control parameter before and after an external crisis is produced at m = 0.01219 (the second step in Fig. 4).
Fig. 7 (a) Probability distribution of the maxima intensity of the pulses for a modulation amplitude m = 0.01215; all other parameter values are the same as in Fig. 4, (b) same as Fig. 7(a) for a modulation amplitude m=0.01225.
It is clear that before the crisis there are no pulses satisfying the usual criteria to define it as a rogue pulse (RP). Even if the distribution is not homogeneous, there is a sharp limit in the intensity of the maxima and such limit is below the threshold to define an extreme event. Just after the crisis instead, it will appear a longer tail in the probability distribution indicating the appearance of extreme events. Furthermore we observe a growing tendency towards the end of the distribution. In other words there is an increase in the number of events showing an absolute maximum value for the laser intensity. This is so because there is always a maximum value that the laser intensity can not overcome.
The occurrence of extreme events has a highly nonlinear dependence with the control parameters in the system. In Fig. 8 is plotted the number of events satisfying the condition (based on the AI defined as the average value plus four times the standard deviation) of being an extreme event as a function of the control parameter m for values close to an external crisis. For the parameters we explored, the subsequent steps of expansion in the chaotic attractor shown in the bifurcations of Fig. 3 have the rare larger spikes as the first one and a tail appears on the histograms after the bifurcations. However, because the size of the steps do not grow while the attractor increase their average size, the criteria of a tails of pulses matching rogue pulse condition is only verified if one lowers the threshold from average value plus four times the standard deviation, used in the first bifurcation at m = 0.01219. Immediately after the crisis the number of extreme events increases, reaches a maximum and then decreases to vanish before a new external crisis is produced. Thus, extreme events happen immediately after an external crisis. As long as the extreme events are very rare, the average value of the maxima intensity and its variance remain similar to the values previous to the crisis. However as higher intensity pulses become more frequent the probability distribution becomes larger and it is not possible to reach the threshold for pulses to be considered rogue events. Each time a new crisis happens, the strange attractor can suffer a rapid expansion in phase space and generate pulses whose maxima are much higher than the average and extreme events appear.
Fig. 8 Number of events (RP) overcoming the AI as a function of the modulation amplitude m. Before the crisis at m = 0.0122 extreme events have not been observed. After the crisis the number of extreme events grows, reaches a maximum, and decreases back to zero at m = 0.0125.
A similar dynamical mechanism based on the existence of a crisis to generate extreme events was apparently experimentally observed in a laser with injected signal [12]. It is worthwhile to notice that the probability distribution showed in Fig. 7(b) is similar to those obtained experimentally in reference [30] using a complex electronic circuit.
5. Conclusion
In conclusion we identified here a very simple laser system that would experimentally be able to generate deterministic optical rogue pulses. Such extreme events are generated by a collision of a chaotic attractor with an unstable orbit of a different bifurcation branch in what is usually called an external crisis. Then we have create enough knowledge to say that such extreme events are not produced just by chance but following a precise causal mechanism which imposes also a well defined trajectory in phase space. Therefore we are in possess of all the tools to study the predictability of extreme events by a crisis, and maybe it is possible to manipulate them in order to avoid or to generate them at our convenience.
Acknowledgments
Work partially supported by Brazilian Agencies Conselho Nacional de Pesquisa e Desenvolvimento (CNPq) and Fundação de Ciência de Pernambuco (FACEPE) Pronex APQ 0630-1.05/06, by a Brazil-France Capes-Cofecub project 456/04 and project OPTIROC of the French Research Agency ANR.
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