Spatiotemporal Chaos Induces Extreme Events in an Extended Microcavity Laser

Cavity soliton inhibition of extreme events in lasers with injection

Vortex mediated turbulence can be the key element in the generation of extreme events in spatially extended lasers with optical injection. Here, we study the interplay of vortex mediated turbulence and cavity solitons on the onset of extreme events in semiconductor lasers with injection. We first analyze and characterize these two features separately, spatiotemporal chaotic optical vortices for low values of the injection intensity and cavity solitons above the locking regime. In regimes where vortex mediated turbulence and cavity solitons coexist, localized peaks of light inhibit instead of enhancing the generation of rogue waves by locally regularizing the otherwise chaotic phase of the optical field. Cavity solitons can then be used to manipulate and control extreme events in systems displaying vortex mediated turbulence.

Coexistence of gain-through-filtering and parametric instability in a fiber ring cavity

We experimentally and numerically investigate the dynamics of a fiber ring cavity in which two different instability can be excited: gain-through-filtering and parametric instability. We demonstrate that they can be triggered individually or collectively depending on the two main control parameters offered by the cavity, namely the pump power and the cavity detuning. The experimental observations are in good agreement with numerical simulations.

Extreme Events Prediction from Nonlocal Partial Information in a Spatiotemporally Chaotic Microcavity Laser

The forecasting of high-dimensional, spatiotemporal nonlinear systems has made tremendous progress with the advent of model-free machine learning techniques. However, in real systems it is not always possible to have all the information needed; only partial information is available for learning and forecasting. This can be due to insufficient temporal or spatial samplings, to inaccessible variables, or to noisy training data. Here, we show that it is nevertheless possible to forecast extreme event occurrences in incomplete experimental recordings from a spatiotemporally chaotic microcavity laser using reservoir computing. Selecting regions of maximum transfer entropy, we show that it is possible to get higher forecasting accuracy using nonlocal data vs local data, thus allowing greater warning times of at least twice the time horizon predicted from the nonlinear local Lyapunov exponent.

Chaos generated in a semiconductor microcavity

The dynamics of chaos in quantum systems has attracted much interest in connection with the fundamental aspects of quantum mechanics. We study the chaotic dynamics of both the excitonic mode and the cavity mode in a microcavity containing a quantum well driven by an external field. We investigate how the chaotic dynamics is influenced by the frequencies of the exciton and the cavity, the coupling constant between the exciton and cavity, the Coulomb interaction between excitons, and the response of the exciton to the cavity and the external field. We show that chaos can be generated synchronously in both the cavity and the excitonic mode by choosing appropriate parameters. Moreover, this kind of chaos can be controlled by the coupling constant, the strength of the interaction between excitons, the external field, the response of the excitons to the cavity, and the detuning between the cavity field and the excitonic field. The present study may have applications in chaos-based neural networks and extreme event statistics.

Spatiotemporal chaos induces extreme events in a three-element laterally coupled laser array

Extreme events are observed in the spatiotemporal chaos dynamics of a three-element laterally coupled laser array. With the help of statistical and dynamical analyses, we confirm that spatiotemporal chaos induces extreme pulses that are high enough to be identified as extreme events and cannot be found in synchronization chaos. Interestingly, our results show that extreme events always preferentially appear in the middle laser as the laser separation ratio is decreased (i.e., upon increasing the coupling strength), and then in the two outer lasers. This thus reveals the importance of the middle laser in the transition between synchronization chaos and spatiotemporal chaos states. Additionally, we show the evolution of extreme events in the plane of the pump level and laser separation ratio by calculating the corresponding proportion. Our results build a relation between extreme events and the spatiotemporal dynamics, which makes it easy to understand the formation mechanism of extreme events.

Extreme events in two laterally-coupled semiconductor lasers

Rogue waves (RWs) are extreme and rare waves that emerge unexpectedly in many natural systems and their formation mechanism and prediction have been extensively studied. Here, we numerically demonstrate the appearance of extreme events (EEs) for the first time, to the best of our knowledge, in the chaotic regimes of a two-element coupled semiconductor laser array. Based on coupled-mode theory, we characterize the occurrence of EEs by calculating the probability distribution, which confirms the RW-type feature of the intensity pulses, i.e., non-Gaussian distribution. Combining with the results of the 0-1 test for chaos, we confirm that EEs originate from deterministic nonlinearities in coupled semiconductor laser systems. We show that EEs can be predicted with a long anticipation time. Furthermore, simulation results manifest that the occurrence probability of EEs can be flexibly tuned by tailoring the coupling parameter space. With the help of two-dimension maps, the effects of key parameters, i.e., the waveguide structure and the pump level, on the formation of EEs are discussed systematically. This work provides a new platform for the research of EEs in a highly integrated structure and opens up a novel investigation field for coupled semiconductor laser arrays.

Dynamics of soliton interaction solutions of the Davey-Stewartson I equation

In this paper, we first modify the binary Darboux transformation to derive three types of soliton interaction solutions of the Davey-Stewartson I equation, namely the higher-order lumps, the localized rogue wave on a solitonic background, and the line rogue wave on a solitonic background. The uniform expressions of these solutions contain an arbitrary complex constant, which plays a key role in obtaining diverse interaction scenarios. The second-order dark-lump solution contains two hollows that undergo anomalous scattering after a head-on collision, and the minimum values of the two hollows evolve in time and reach the same asymptotic constant value 0 as t→±∞. The localized rogue wave on a solitonic background describes the occurrence of a waveform from the solitonic background, quickly evolving to a doubly localized wave, and finally retreating to the solitonic background. The line rogue wave on the solitonic background does not create an extreme wave at any instant of time, unlike the one on a constant background, which has a large amplitude at the intermediate time of evolution. For large t, the solitonic background has multiple parallel solitons possessing the same asymptotic velocities and heights. The obtained results improve our understanding of the generation mechanisms of rogue waves.

Instabilities in quasiperiodic motion lead to intermittent large-intensity events in Zeeman laser

We report intermittent large-intensity pulses that originate in Zeeman laser due to instabilities in quasiperiodic motion, one route follows torus-doubling to chaos and another goes via quasiperiodic intermittency in response to variation in system parameters. The quasiperiodic breakdown route to chaos via torus-doubling is well known; however, the laser model shows intermittent large-intensity pulses for parameter variation beyond the chaotic regime. During quasiperiodic intermittency, the temporal evolution of the laser shows intermittent chaotic bursting episodes intermediate to the quasiperiodic motion instead of periodic motion as usually seen during the Pomeau-Manneville intermittency. The intermittent bursting appears as occasional large-intensity events. In particular, this quasiperiodic intermittency has not been given much attention so far from the dynamical system perspective, in general. In both cases, the infrequent and recurrent large events show non-Gaussian probability distribution of event height extended beyond a significant threshold with a decaying probability confirming rare occurrence of large-intensity pulses.

Evidence of Chaotic Dynamics in Three-Soliton Collisions

We observe chaotic optical wave dynamics characterized by erratic energy transfer and soliton annihilation and creation in the aftermath of a three-soliton collision in a photorefractive crystal. Irregular dynamics are found to be mediated by the nonlinear Raman effect, a coherent interaction that leads to nonreciprocal soliton energy exchange. Results extend the analogy between solitons and particles to the emergence of chaos in three-body physics and provide new insight into the origin of the irregular dynamics that accompany extreme and rogue waves.

Super rogue wave generation in the linear regime

Extreme or rogue waves are large and unexpected waves appearing with higher probability than predicted by Gaussian statistics. Although their formation is explained by both linear and nonlinear wave propagation, nonlinearity has been considered a necessary ingredient to generate super rogue waves, i.e., an enhanced wave amplification, where the wave amplitudes exceed by far those of ordinary rogue waves. Here we show, experimentally and theoretically, that optical super rogue waves emerge in the simple case of linear light diffraction in one transverse dimension. The underlying physics is a long-range correlation on the random initial phases of the light waves. When subgroups of random phases appear recurrently along the spatial phase distribution, a more ordered phase structure greatly increases the probability of constructive interference to generate super rogue events (non-Gaussian statistics with superlong tails). Our results consist in a significant advance in the understanding of extreme waves formation by linear superposition of random waves, with applications in a large variety of wave systems.

The tipping times in an Arctic sea ice system under influence of extreme events

In light of the rapid recent retreat of Arctic sea ice, the extreme weather events triggering the variability in Arctic ice cover has drawn increasing attention. A non-Gaussian α-stable Lévy process is thought to be an appropriate model to describe such extreme events. The maximal likely trajectory, based on the nonlocal Fokker-Planck equation, is applied to a nonautonomous Arctic sea ice system under α-stable Lévy noise. Two types of tipping times, the early-warning tipping time and the disaster-happening tipping time, are used to predict the critical time for the maximal likely transition from a perennially ice-covered state to a seasonally ice-free one and from a seasonally ice-free state to a perennially ice-free one, respectively. We find that the increased intensity of extreme events results in shorter warning time for sea ice melting and that an enhanced greenhouse effect will intensify this influence, making the arrival of warning time significantly earlier. Meanwhile, for the enhanced greenhouse effect, we discover that increased intensity and frequency of extreme events will advance the disaster-happening tipping time, in which an ice-free state is maintained throughout the year in the Arctic Ocean. Finally, we identify values of the Lévy index α and the noise intensity ϵ in the αϵ-space that can trigger a transition between the Arctic sea ice state. These results provide an effective theoretical framework for studying Arctic sea ice variations under the influence of extreme events.

Control of dissipative rogue waves in nonlinear cavity optics: Optical injection and time-delayed feedback

We investigate and review the formation of two-dimensional dissipative rogue waves in cavity nonlinear optics with transverse effects. Two spatially extended systems are considered for this purpose: the driven Kerr optical cavities subjected to optical injection and the broad-area surface-emitting lasers with a saturable absorber. We also consider a quasi-two-dimensional system (the two dimensions being space and time) of a fiber laser describing the complex cubic-quintic Ginzburg-Landau equation. We show that rogue waves are controllable by means of time-delayed feedback and optical injection. We show that without delayed feedback, transverse structures are stationary or oscillating. However, when the strength of the delayed feedback is increased, all the systems generate giant two-dimensional pulses that appear with low probability and suddenly appear and disappear. We characterize their formation by computing the probability distribution, which shows a long tail. Besides, we have computed the significant wave height, which measures the mean wave height of the highest third of the waves. We show that for all systems, the distribution tails expand beyond two times the significant wave height. Furthermore, we also show that optical injection may suppress the rogue wave formation in a semiconductor laser with a saturable absorber.

Soliton Maxwell demons and long-tailed statistics in fluctuating optical fields

We demonstrate experimentally in biased photorefractive crystals that collisions between random-amplitude optical spatial solitons produce long-tailed statistics from input Gaussian fluctuations. The effect is mediated by Raman nonlocal corrections to Kerr self-focusing that turn soliton-soliton interaction into a Maxwell demon for the output wave amplitude.

Statistical Properties and Predictability of Extreme Epileptic Events

The use of extreme events theory for the analysis of spontaneous epileptic brain activity is a relevant multidisciplinary problem. It allows deeper understanding of pathological brain functioning and unraveling mechanisms underlying the epileptic seizure emergence along with its predictability. The latter is a desired goal in epileptology which might open the way for new therapies to control and prevent epileptic attacks. With this goal in mind, we applied the extreme event theory for studying statistical properties of electroencephalographic (EEG) recordings of WAG/Rij rats with genetic predisposition to absence epilepsy. Our approach allowed us to reveal extreme events inherent in this pathological spiking activity, highly pronounced in a particular frequency range. The return interval analysis showed that the epileptic seizures exhibit a highly-structural behavior during the active phase of the spiking activity. Obtained results evidenced a possibility for early (up to 7 s) prediction of epileptic seizures based on consideration of EEG statistical properties.

Quantum-dot micropillar lasers subject to coherent time-delayed optical feedback from a short external cavity

We investigate the mode-switching dynamics of an electrically driven bimodal quantum-dot micropillar laser when subject to delayed coherent optical feedback from a short external cavity. We experimentally characterize how the external cavity length, being on the same order than the microlaser's coherence length, influences the spectral and dynamical properties of the micropillar laser. Moreover, we determine the relaxation oscillation frequency of the micropillar by superimposing optical pulse injection to a dc current. It is found that the optical pulse can be used to disturb the feedback-coupled laser within one roundtrip time in such a way that it reaches the same output power as if no feedback was present. Our results do not only expand the understanding of microlasers when subject to optical feedback from short external cavities, but pave the way towards tailoring the properties of this key nanophotonic system for studies in the quantum regime of self-feedback and its implementation to integrated photonic circuits.

Alternation of Defects and Phase Turbulence Induces Extreme Events in an Extended Microcavity Laser

Out-of-equilibrium systems exhibit complex spatiotemporal behaviors when they present a secondary bifurcation to an oscillatory instability. Here, we investigate the complex dynamics shown by a pulsing regime in an extended, one-dimensional semiconductor microcavity laser whose cavity is composed by integrated gain and saturable absorber media. This system is known to give rise experimentally and theoretically to extreme events characterized by rare and high amplitude optical pulses following the onset of spatiotemporal chaos. Based on a theoretical model, we reveal a dynamical behavior characterized by the chaotic alternation of phase and amplitude turbulence. The highest amplitude pulses, i.e., the extreme events, are observed in the phase turbulence zones. This chaotic alternation behavior between different turbulent regimes is at contrast to what is usually observed in a generic amplitude equation model such as the Ginzburg-Landau model. Hence, these regimes provide some insight into the poorly known properties of the complex spatiotemporal dynamics exhibited by secondary instabilities of an Andronov-Hopf bifurcation.

Dynamics of a light beam suffering the influence of a dispersing mechanism with tunable refraction index

The dynamics of a monochromatic light beam is studied inside the oval billiard with an inner scatter circle, which can be interpreted as a cross section of a long optical fiber. The outer oval boundary acts as a perfect reflector for the light beam while the scatter circle encloses a medium with changeable refraction index. The light beam refracts when it enters inside this circle and some drastic changes in the phase space are observed. The increase of the refractive index destroys the center of stability in the phase space leading to a spread of scattered light and the islands of traps for the light propagation become more pulverized. Numerical results are presented and discussed.

Extreme events and single-pulse spatial patterns observed in a self-pulsing all-solid-state laser

The passively Q-switched, self-pulsing all-solid-state laser is a device of widespread use in many applications. Depending on the condition of saturation of the absorber, which is easy to adjust, different dynamical regimes are observed: continuous-wave emission, stable oscillations, period doubling bifurcations, chaos, and, within some chaotic regimes, extreme events (EEs) in the form of pulses of extraordinary intensity. These pulses are sometimes called "dissipative optical rogue waves." The mechanism of their formation in this laser is unknown. Previous observations suggest they are caused by the interaction of a few transverse modes. Here we report a direct observation of the pulse-to-pulse evolution of the transverse pattern. In the periodical regimes, sequences of intensities are correlated with sequences of patterns. In the chaotic ones, a few different patterns alternate, and the EEs are related with even fewer ones. In addition, the series of patterns and the pulse intensities before and after an EE are markedly repetitive. These observations demonstrate that EEs follow a deterministic evolution, and that they can appear even in a system with few interacting modes. This information plays a crucial role for the development of a mathematical description of EEs in this laser. This would allow managing the formation of EE through control of chaos, which is of both academic and practical interest (laser rangefinder).

Dragon-king-like extreme events in coupled bursting neurons

We present evidence of extreme events in two Hindmarsh-Rose (HR) bursting neurons mutually interacting via two different coupling configurations: chemical synaptic- and gap junctional-type diffusive coupling. A dragon-king-like probability distribution of the extreme events is seen for combinations of synaptic coupling where small- to medium-size events obey a power law and the larger events that cross an extreme limit are outliers. The extreme events originate due to instability in antiphase synchronization of the coupled systems via two different routes, intermittency and quasiperiodicity routes to complex dynamics for purely excitatory and inhibitory chemical synaptic coupling, respectively. For a mixed type of inhibitory and excitatory chemical synaptic interactions, the intermittency route to extreme events is only seen. Extreme events with our suggested distribution is also seen for gap junctional-type diffusive, but repulsive, coupling where the intermittency route to complexity is found. A simple electronic experiment using two diffusively coupled analog circuits of the HR neuron model, but interacting in a repulsive way, confirms occurrence of the dragon-king-like extreme events.

Drifting cavity solitons and dissipative rogue waves induced by time-delayed feedback in Kerr optical frequency comb and in all fiber cavities

Time-delayed feedback plays an important role in the dynamics of spatially extended systems. In this contribution, we consider the generic Lugiato-Lefever model with delay feedback that describes Kerr optical frequency comb in all fiber cavities. We show that the delay feedback strongly impacts the spatiotemporal dynamical behavior resulting from modulational instability by (i) reducing the threshold associated with modulational instability and by (ii) decreasing the critical frequency at the onset of this instability. We show that for moderate input intensities it is possible to generate drifting cavity solitons with an asymmetric radiation emitted from the soliton tails. Finally, we characterize the formation of rogue waves induced by the delay feedback.

Abnormal chiral events in a semiconductor laser with coherent injection

We study experimentally and theoretically the dynamics of a spatially extended (along the propagation direction) oscillatory medium with coherent forcing. We observe abnormally high events, responsible for a different statistics of intensity and pulse height, in a regime where solitons and roll patterns are unstable. We focus on the formation of these high-peak events and their connection to the phase dynamics. Each abnormal event can be associated with a change in the slope of the phase time trace. Furthermore, the coexistence of ±2π phase rotations inside the cavity can be associated to the observation of abnormal events, similarly to recent predictions in bidimensional vortex turbulence.

Extreme events induced by collisions in a forced semiconductor laser

We report on the experimental study of an optically driven multimode semiconductor laser with a 1 m cavity length. We observed a spatiotemporal regime where real-time measurements reveal very high-intensity peaks in the laser field. Such a regime, which coexists with the locked state and with stable phase solitons, is characterized by the emergence of extreme events that produce heavy tail statistics in the probability density function. We interpret the extreme events as collisions of spatiotemporal structures with opposite chirality. Numerical simulations of the semiconductor laser model, showing very similar dynamical behavior, substantiate our evidences and corroborate the description of interactions such as collisions between phase solitons and transient structures with different phase rotations.

Generation of Rogue Waves in Gyrotrons Operating in the Regime of Developed Turbulence

Within the framework of the average approach and direct 3D PIC (particle-in-cell) simulations, we demonstrate that the gyrotrons operating in the regime of developed turbulence can sporadically emit "giant" spikes with intensities a factor of 100-150 greater than the average radiation power and a factor of 6-9 exceeding the power of the driving electron beams. Together with the statistical features such as a long-tail probability distribution, this allows the interpretation of generated spikes as microwave rogue waves. The mechanism of spikes formation is related to the simultaneous cyclotron interaction of a gyrating electron beam with forward and backward waves near the waveguide cutoff frequency as well as with the longitudinal deceleration of electrons.

Extreme and superextreme events in a loss-modulated CO_{2} laser: Nonlinear resonance route and precursors

We investigate the occurrence of extreme and rare events, i.e., giant and rare light pulses, in a periodically modulated CO_{2} laser model. Due to nonlinear resonant processes, we show a scenario of interaction between chaotic bands of different orders, which may lead to the formation of extreme and rare events. We identify a crisis line in the modulation parameter space, and we show that, when the modulation amplitude increases, remaining in the vicinity of the crisis, some statistical properties of the laser pulses, such as the average and dispersion of amplitudes, do not change much, whereas the amplitude of extreme events grows enormously, giving rise to extreme events with much larger deviations than usually reported, with a significant probability of occurrence, i.e., with a long-tailed non-Gaussian distribution. We identify recurrent regular patterns, i.e., precursors, that anticipate the emergence of extreme and rare events, and we associate these regular patterns with unstable periodic orbits embedded in a chaotic attractor. We show that the precursors may or may not lead to the emergence of extreme events. Thus, we compute the probability of success or failure (false alarm) in the prediction of the extreme events, once a precursor is identified in the deterministic time series. We show that this probability depends on the accuracy with which the precursor is identified in the laser intensity time series.

Characterization of spatiotemporal chaos in a Kerr optical frequency comb and in all fiber cavities

Complex spatiotemporal dynamics have been a subject of recent experimental investigations in optical frequency comb microresonators and in driven fiber cavities with Kerr-type media. We show that this complex behavior has a spatiotemporal chaotic nature. We determine numerically the Lyapunov spectra, allowing us to characterize different dynamical behavior occurring in these simple devices. The Yorke-Kaplan dimension is used as an order parameter to characterize the bifurcation diagram. We identify a wide regime of parameters where the system exhibits a coexistence between the spatiotemporal chaos, the oscillatory localized structure, and the homogeneous steady state. The destabilization of an oscillatory localized state through radiation of counter-propagating fronts between the homogeneous and the spatiotemporal chaotic states is analyzed. To characterize better the spatiotemporal chaos, we estimate the front speed as a function of the pump intensity.

Chaos and extreme events in an azimuthally polarized Nd:GdVO4 laser with pump modulation

Nonlinear dynamics and extreme events were explored in a spatially distributed polarization laser based on an azimuthally polarized Nd:GdVO₄ laser with pump modulation. When the modulation frequency approached the relaxation oscillation frequency, a period-doubling route to chaos was observed with an increasing modulation depth, and spatiotemporal extreme events occurred with the generation of the chaos. An efficient azimuthal polarization was kept, even though the intensity fluctuated in the chaotic region.

Two-dimensional dissipative rogue waves due to time-delayed feedback in cavity nonlinear optics

We demonstrate a way to generate two-dimensional rogue waves in two types of broad area nonlinear optical systems subject to time-delayed feedback: in the generic Lugiato-Lefever model and in the model of a broad-area surface-emitting laser with saturable absorber. The delayed feedback is found to induce a spontaneous formation of rogue waves. In the absence of delayed feedback, spatial pulses are stationary. The rogue waves are exited and controlled by the delay feedback. We characterize their formation by computing the probability distribution of the pulse height. The long-tailed statistical contribution, which is often considered as a signature of the presence of rogue waves, appears for sufficiently strong feedback. The generality of our analysis suggests that the feedback induced instability leading to the spontaneous formation of two-dimensional rogue waves is a universal phenomenon.

Alignment of Lyapunov Vectors: A Quantitative Criterion to Predict Catastrophes?

We argue that the alignment of Lyapunov vectors provides a quantitative criterion to predict catastrophes, i.e. the imminence of large-amplitude events in chaotic time-series of observables generated by sets of ordinary differential equations. Explicit predictions are reported for a Rössler oscillator and for a semiconductor laser with optoelectronic feedback.

Extreme events induced by spatiotemporal chaos in experimental optical patterns

Extreme events such as rogue waves are often associated with the merging of coherent structures. We report on experimental results in the physics of extreme events emerging in a liquid-crystal light valve subjected to optical feedback, and we establish the relation of this phenomenon with the appearance of spatiotemporal chaos. This system, under particular conditions, exhibits stationary roll patterns that can be destabilized into quasi-periodic and chaotic textures when control parameters are properly modified. We have identified the parameter regions where extreme fluctuations of the amplitude can emerge and established their origin through its direct relation with the experimental largest Lyapunov exponents, the proportion of extreme events, and the normed kurtosis.

Complementary optical rogue waves in parametric three-wave mixing

We investigate the resonant interaction of two optical pulses of the same group velocity with a pump pulse of different velocity in a weakly dispersive quadratic medium and report on the complementary rogue wave dynamics which are unique to such a parametric three-wave mixing. Analytic rogue wave solutions up to the second order are explicitly presented and their robustness is confirmed by numerical simulations, in spite of the onset of modulation instability activated by quantum noise.