On the Predictability of Optical Rogue Waves in a Semiconductor Laser with Optical Injection

Extreme events and rogue waves are ubiquitous in complex systems [1] and a lot of efforts are nowadays focused in developing reliable analysis techniques for their detection and prediction [2]. Laser systems displaying extreme pulses in their output intensity are ideal candidates for performing laboratory controlled experiments that allow testing novel diagnostic tools to detect warning signals of upcoming extreme events. Here we study numerically the predictability of extreme intensity pulses emitted by an optically injected semiconductor laser [3, 4]. We show that symbolic ordinal time-series analysis [5] allows identifying the patterns of intensity oscillations that are likely to occur before an extreme pulse [6]. Within the ordinal approach, a time series y(t) is divided into non-overlapping segments of length L, and each segment is assigned a symbol, s, (known as ordinal pattern, OP) according to the ranking of the values inside the segment. For example, with L = 3, if y(t) < y(t+1) < y(t+2), s(t) is '012', if y(t) > y(t + 1) > y(t + 2), s(t) is '210', and so forth. In this way, the symbols take into account the relative temporal ordering of the values in the serie. We apply the ordinal method to the sequence of intensity peak heights {... I-i, Ii+1, Ii+2.} (indicated with dots in the time series shown in Fig. 1). For each intensity peak that is above a given threshold, the L previous peaks are used to define the ordinal patterns. The threshold is measured in units of the standard deviation, s, because the intensity time series is first normalized to zero mean and sigma = 1. Because extremely high intensity pulses are rare events, very long simulations are needed in order to compute the probabilities of the patterns that anticipate them with good statistics. The number of possible patterns increases with L as L! Therefore, we limit the analysis to the patterns defined by L = 3 peaks, which gives 6 possible patterns. As shown in Fig. 1, this approach gives not only information about the patterns that are likely to occur before an extreme pulse, but also, provides information about patterns which are unlikely to occur before extreme pulses. The specific patterns identified capture the topology of the underlying attractor and depend on the model parameters. The method proposed here can be useful for analysing the output of other laser systems that generate extreme intensity fluctuations.