Control of Spectral Extreme Events in Ultrafast Fiber Lasers by a Genetic Algorithm

Extreme wave events or rogue waves (RWs) are both statistically rare and of exceptionally large amplitude. They are observed in many complex systems ranging from oceanic and optical environments to financial models and Bose–Einstein condensates. As they appear from nowhere and disappear without a trace, their emergence is unpredictable and non‐repetitive, which makes them particularly challenging to control. Here, the use of genetic algorithms (GAs), which are exclusively designed for searching and optimizing stationary or repetitive processes in nonlinear optical systems, is extended to the active control of extreme events in a fiber laser cavity. Feeding real‐time spectral measurements into a GA controlling the electronics to optimize the cavity parameters, the wave events are able to be triggered in the cavity that have the typical statistics of RWs in the frequency domain. The intensity of the induced RWs can also be tailored. This accurate control enables the generation of optical RWs with a spectral peak intensity 32.8 times higher than the significant intensity threshold. A rationale is proposed and confirmed by numerical simulations of the laser model for the related frequency up‐ and downshifting of the optical spectrum that are experimentally observed.


Introduction
First introduced in the context of oceanic waves, [1] the concept of extreme events or rogue waves (RWs), i.e., statistically-rare DOI: 10.1002/lpor.202200470giant-amplitude waves, has been transferred to other natural environments such as the atmosphere, [1] as well as to the solid grounds of research laboratories.[7] In particular, fiber-optics systems provide ideal test beds for investigating the evolution dynamics of RWs [7] and the conditions for their emergence [8] owing to the possibility of recording a large number of events in a relatively short time and the relative simplicity and precision of fiber-optics experiments.Since the first observation of optical RWs on a platform of supercontinuum generation in a microstructured optical fiber [5] and thanks to the advances of ultrafast measurements, it has become possible to accurately measure the temporal and spectral characteristics of the nonlinear evolution of such waves and to control the amplitude and phase of the initial conditions fed to the system.Therefore, the occurrence of extreme events in cavity-free optical systems is now well understood and has been intimately associated with the modulation instability process and solitons over a finite background.As noise-driven dynamics are responsible for the appearance of RWs upon supercontinuum evolution, the ensuing dynamics are inherently unpredictable and uncontrollable.Yet, a wellcontrolled seed signal can help preserve the coherence of the continuum [6,9] or can be used to induce the collision of breather structures leading to the excitation of enhanced RWs. [10]However, such control methods cannot be implemented in more complex systems, such as those involving optical feedback, where the physical mechanisms underpinning the occurrence of RWs may differ from modulation-instability-related dynamics.In this context, ultrafast fiber lasers, which represent interesting realizations of dissipative nonlinear systems with dynamics driven by a complex interplay among the effects of nonlinearity, dispersion, and energy exchange, [11] provide excellent platforms to observe optical RWs.22] These markedly different scenarios call for a universal method to achieve on-demand generation of RWs.Machinelearning strategies, referring to the use of statistical techniques and numerical algorithms to carry out tasks without explicit www.lpr-journal.orgprogrammed and procedural instructions and being widely deployed in many areas of science and engineering, [23] appear to provide the appropriate solution.[26] Yet, these algorithms have been designed to target regimes of parameter-invariant, stationary pulse generation, [24,[27][28][29][30][31][32][33][34][35][36] while the latest advances relate to the generation of nonstationary repetitive patterns, such as breathing solitons and multi-breather complexes, [36] stationary soliton pairs, [37] or noise-like pulses. [38]By contrast, the on-demand generation of extreme waves, which are non-repetitive and rare events, represents a much harder task, and whether machine learning can be used to control such nonequilibrium dynamics widely present in many physical systems remains an open question.It is worthy to note that machine learning has been applied to compute the temporal intensity of optical RWs from spectral measurements only, [39] thereby relieving the experimental burden of direct timedomain observations.However, active control of extreme wave events by machine learning has not been demonstrated to date.
In this paper, we report on the intelligent control of optical RWs in a mode-locked fiber laser using GA-based optimization of the four-parameter intracavity nonlinear transfer function steered by electronically driven polarization control.We define merit functions relying on the statistical spectral properties of the laser output, which are capable to drive the formation of pulses in the laser displaying substructures with record-high intensities in the frequency domain.The intensity of the induced RWs can also be tailored by the GA.In particular, a super RW of intensity 32.8 times higher than the significant intensity threshold is synthetized.These extreme spectral events correlate with extreme variations of the pulse energy.Quite remarkably, significant frequency up-or downshifting of the optical spectrum is also associated with the emergence of these waves, which is confirmed by numerical simulations of the laser model based on the modified nonlinear Schrödinger equation.

Experimental Setup and Measurements
The experimental setup for generating optical RWs, sketched in Figure 1, is a typical fiber ring cavity in which a 1.3 mlong erbium-doped fiber (EDF) constitutes the gain medium, pumped by a laser diode operating at 980 nm through a wavelength-division multiplexer.Other fibers in the cavity are a section of dispersion-compensating fiber (DCF) and pieces of standard single-mode fiber (SMF) from the pigtails of the optical components used.The group-velocity dispersion (GVD) values ( 2 ) of the three fiber types are 65, 62.5, and −22.8 ps 2 km −1 , respectively, yielding a net normal dispersion of 0.028 ps 2 at the operating wavelength of ≈1.5 μm.The fundamental repetition frequency of the cavity is 16.765 MHz.The mode-locked laser operation is obtained thanks to an effective saturable absorber based on the nonlinear polarization evolution (NPE) effect. [40]he nonlinear transfer function of the NPE-based mode locking is controlled by an electronically driven polarization controller (EPC) working together with a polarization-dependent isolator.The EPC consists of four fiber squeezers oriented at 45°to each other and can generate all possible states of polarization over the Poincaré sphere, where each set of voltages applied on the squeezers corresponds to a specific state. [27,31,33,41]The overall envelope fluctuations of the laser output are directly measured via a fast photodiode (20 ps response time, 50 GHz bandwidth) connected to a real-time oscilloscope (33 GHz bandwidth, 80 GSa s −1 sampling rate).[47][48][49][50] In the latter case, the output of the laser generally contains both localized and non-localized waves.While the DFT data carefully reflects the spectral information for the localized waves, this is no longer the case for the non-localized waves.Therefore, simultaneous temporal intensity measurements are generally performed to assist the DFT data analysis and distinguish between localized and non-localized waves.][52] The DFT setup consists of a long segment of normally dispersive fiber that provides an accumulated dispersion of DL ≈−1200 ps nm −1 , thereby enabling the mapping of the spectrum of a short laser pulse in the time domain.From the photodetection of the DFT output signal on a fast photodiode, the optical spectrum for each pulse is obtained directly on the oscilloscope, with a resolution of  = 1∕(DL ⋅ BW) ≈0.025 nm, where BW is the bandwidth of the photodetection.The oscilloscope is connected to a computer that runs the GA and controls the EPC via a field programmable gate array (FPGA) and four digital-to-analog converters (DACs).During the searching process, the signals generated by the algorithm are delivered to the FPGA, which adjusts the control voltages of the EPC through the DACs.The DACs translate the instructions from the FPGA to the voltages and finally act on the EPC.

Genetic Algorithm Optimization
The intelligent search of RWs is realized via a GA whose principle, illustrated in Figure 1b,c, mimics mechanisms inspired by biological evolution: individuals composing a population progress through successive generations only if they are among the fittest. [53]In our case, an individual is a laser state, associated with the nonlinear transfer function defined by the four control voltages applied on the EPC; these voltages are therefore the genes of the individuals.The process begins with a randomly created population of individuals making up the initial generation.The output of the system is measured for each individual in the generation, evaluated by a user-defined merit function (also known as objective function or fitness), and assigned a score.The GA then creates the next generation by breeding individuals from the preceding generation, with the probability that an individual is selected to be a "parent" based on their score ("roulette wheel" selection).Two new individuals-children-are created from the crossover of two randomly selected parents, namely the interchange of their genes.A mutation probability is also specified, which can randomly alter the children's genes, thus allowing for the genetic sequence to be refreshed.The process repeats until the algorithm converges and an optimal individual is produced.
In our experiment, the initial population is set to 100 individuals, and the next generations have a smaller population composed of 30 individuals (10 parents and 20 children).Evaluation of the properties of an entire generation of individuals typically takes 2.5 min.
The merit function is the key ingredient of any GA, as it defines the optimization target.In our case, in order to connect the merit function to the emergence of spectral RWs, a longtime trace of the laser output is recorded by the oscilloscope, which enables through DFT measurements the accumulation of 12 × 10 3 successive single-shot spectra.From these measurements, the distribution of the intensity maxima I MAX for the spectra recorded at each round trip can be computed.We then design merit functions promoting the generation of ordinary and super spectral RWs based on two common statistical criteria for a wave event to be classified as an RW.One is that its height I MAX exceeds twice the significant wave height (SWH) I SWH , defined as the mean intensity of the upper third of intensity peaks.Therefore, the merit function for generating RWs is designed as , where this value is determined from the strongest RWs observed in the laser, and C RW is the target intensity relative to the significant intensity I SWH .It is straightforward to see that parameter C RW controls both the generation of RWs and their intensity: if it is set to 2, F RW returns a maximum value (C 1 ) when an RW is generated ( I MAX ∕I SWH = 2).Elseways, if it is set to a larger value, an RW with proportionately stronger intensity is generated as the GA tunes the laser parameters to make I MAX ∕I SWH always equals to C RW so as to maximize the fitness.
The other criterion qualifying a wave as an RW is that I MAX be greater than the mean intensity ⟨I⟩ of the intensity peaks by eight times the standard deviation  from the mean. [54]Super RWs refer to ultra-large deviations from the mean intensity [54,55] such that (I MAX − ⟨I⟩)∕ > 32.Therefore, the merit function for generating super RWs is designed as ,  and C 2 are constants set to 15 and 1.2, respectively ( is used to amplify the fitness), and C SRW is the parameter that controls the formation of super RWs.Based on the super RW defining criteria given above, C SRW is set to a value above 1 to guarantee that F SRW is maximum when a super RW is generated.To control the intensity of super RWs, we have merged the two merit functions F RW and F SRW into a new one: where the weights of the two components are determined empirically ( = 15,  = 1).
Examples of GA optimization curves for ordinary and super RW generation are presented in panels (a) and (b) of Figure 2, which show the best and average values of the respective ratios I MAX /I SWH and (I MAX − ⟨I⟩)∕, observed within the population for successive generations.We see that in the case of ordinary RWs, the best and average I MAX /I SWH values increase and converge to the optimization target of C RW = 8 ± 10% and an optimized value, respectively, after five generations.For the optimization time of the GA to be reasonably short (the average time is 25 min), the computer program stops once the output strength is within 10% of the target value.The best and average values of the fitness ratio used for generating super RWs display a similar evolution although their convergence is slower.We note that as RWs can be nonstationary processes, an important issue is whether they can be reproduced generation after generation.As we can see from Figure 2, for both types of RWs some generations after the optimal state has been first reached (blue diamonds between red ones) have smaller values than the target values.Nonetheless, the GA then modifies the genes of the individuals and the laser emits the target RWs again.These results evidence the ability of our GA to "lock" optical RWs: whilst some generations may deviate from the target RWs, the GA can successfully restore the laser to emitting RWs of the required intensity.S1 and S2 (Supporting Information).In this regard, optical super RWs resemble more their oceanic counterparts, which appear unexpectedly and disappear without a trace. [55] Discussion

Characteristics of Ordinary and Super Rogue Waves
Examples of statistical analysis for the observed ordinary and super RW operations of the laser are provided in Figure 3. Pan-els (a) and (b) show the associated distributions of spectral intensity maxima computed from the DFT recordings over 12000 round trips.Both histograms display a L-shaped profile, with extreme events occurring rarely, yet much more frequently than expected based on the relatively narrow distribution of typical events.However, the occurrence of extreme events with intensities ≈40 times the average results in a significantly extended tail of super RWs for the distribution of Figure 3b, which also displays a smaller standard deviation.Note that contrary to recent works using time-lens measurements, [17,56] in our experiment we do not have direct access to the real-time time-domain dynamics of the laser, hence we cannot measure the distribution of temporal intensity maxima for successive cavity round trips.Nonetheless, we have recorded the round-trip evolution of the energy of the output pulses calculated by integrating the DFT spectra and plotted it side by side with the corresponding evolution of the spectral intensity maxima (Figure 3c,d).
We can clearly see that for both the ordinary and super RW regimes, the peak intensity fluctuations in the frequency domain are tightly connected with changes in the pulse energy, thereby entailing a L-shaped profile of the associated distribution of energy (Figure 3e  Without active control, the intensities of the RWs generated in many optical systems are generally just above the threshold of 2 times the SWH.For example, RWs of intensities ≈10 times the SWH have been reported in optical multiple filamentations; [57] Raman scattering in fiber lasers generates RWs with intensities ≈7 times the SWH. [16]Yet, the intensity of RWs has never been actively controlled in any physical systems to date.Here, the merit functions used by the GA permit to achieve RWs with tailored intensity and to find operating conditions of the laser that enable record-high intensity values.The results obtained with the merit function F RW (see Section 2.2) based on the optimization of the ratio I MAX /I SWH are summarised in Figure 4a.The highest RW so-produced has an intensity exceeding the SWH by a factor of 32.8.We also note that whilst the RWs generated by the use of the merit function F RW can be very strong, only some of them fulfil the defining criteria of super RWs from which the merit function F SRW is derived.These super RWs are indicated by red circles in Figure 4a.In other words, a high I MAX /I SWH ratio is not always a synonym of super RWs.Yet, by using the composite merit function F TSRW , the GA not only is able to tune the intensity of the RWs, but also to guarantee that they are super RWs, as shown in Figure 4b.Quite importantly, Figure 4a,b together highlight that ordinary and super RWs cannot be distinguished by their intensity only, since the former can have large intensities, e.g., 25 times the SWH (Figure 4a) while the latter can feature relatively small intensities, e.g., only 10 times the SWH (Figure 4b).A second degree of freedom is therefore required to discriminate between ordinary and super RWs, and the standard deviation of the distribution of intensity maxima is fit for purpose.Figure 4c shows the distribution of the ordinary and super RWs emitted by the laser in the plane of peak intensity I MAX and standard deviation .A dashed black line separating the regions of existence of ordinary and super RWs is drawn to ease visualization.We can clearly see that relatively low-intensity events still correspond to super RWs when the standard deviation is quite small.By contrast, high-intensity events do not qualify as super RWs if the standard deviation is too large.

Dynamics of the Rogue Events
In our laser setup, the automated and high-resolution control of the NPE transfer function enables access to the nonlinear cavity dynamics, and the precise measurements enabled by the DFT technique allow us to track the spectral changes experienced by the circulating pulses in the cavity.This provides a precious tool for controlling the emergence of RWs in the cavity.We are then able to reveal four different generation mechanisms of RWs: Q-switching instability with frequency down or up-shifting, multiple pulsing, transient noise-like pulsing, and temporal spiking preceding mode-locking.We focus here on the frequency shifting process occurring under Q-switching instability operation of the laser, while the other three processes are discussed in the Figures S5, S6, and S10 (Supporting Information).Figure 5a,d shows typical DFT-recorded evolutions of the intensity maxima I MAX of the laser spectrum over successive cavity roundtrips, and magnified versions of the areas supporting the largest I MAX values are shown in the insets to highlight the development and disappearance of RWs.Since this usual 1D representation of the laser intensity evolution precludes a detailed investigation of the underlying dynamics, a 2D-spatiospectral-representation [49,51,[58][59][60] is used in Figure 5b,e to capture the transient dynamics of RWs.We can see that the laser spectrum has different structures before and after the occurrence of the RW peak.Indeed, as revealed by the single-shot spectra at the roundtrips that proceed the emergence of the RW peak (Figure 5c,f), the spectrum features well-defined oscillations, which are quite typical of the spectral broadening of a coherent wave induced by self-phase modulation (SPM) in the fiber.
The spectrum extent is up to 10 nm.By contrast, at the round trips succeeding the RW peak emergence, the spectrum structure becomes much more chaotic, seemingly noise-like, with significant changes from round trip to round trip and spanning over 40 nm.Another noticeable feature revealed by the DFT measurements is the progressive and continuous frequency down-or upshifting of the spectrum that occurs during the rising stage of RWs.Frequency downshifting due to Raman scattering has been identified as the main wave mechanism of RW formation in supercontinuum generation [5,6] and one of the possible RW formation mechanisms in mode-locked lasers. [16]However, no significant Raman effect occurs in our laser due to the short length of the cavity (12.26 m).We can see in Figure 5c that in the beginning, the frequency components on the long-wavelength edge of the spectrum are only slightly stronger than the frequencies on the short-wavelength edge, but then the long-wavelength end of the spectrum grows with increasing pulse energy over the following roundtrips to finally evolve into an RW.As revealed by Figure 5f, frequency up-shifting may also be a signature of the development of RWs in the cavity, which confirms our assumption that Raman scattering is not the key physical process driving the observed intra-pulse energy transfer.Both types of frequency shift phenomena shown in Figure 5 feature a "winner takes all" scenario: the stronger spectrum end at the start grows even stronger and develops into a spectral RW.To confirm that the DFT measurements are valid in these cases, the corresponding temporal intensity measurements are also provided (Figure S7, Supporting Information).
To illustrate further these intriguing dynamics, we have performed numerical simulations of the laser using a scalar-field, lumped model that includes the dominant physical effects of the system on the evolution of a pulse over one round trip inside the cavity, namely, GVD and SPM for all the fibers, gain saturation and bandwidth-limited gain for the active fiber, [61] and the discrete effects of a saturable absorber element.The pulse propagation in the optical fibers is modeled by a generalized nonlinear Schrödinger equation, which takes the following form: [62] where  = (z, t) is the slowly varying electric field moving at the group velocity along the propagation coordinate z, and  is the Kerr nonlinearity coefficient.We used the nominal experimental GVD values and the estimated nonlinear coefficients of  = 5 (W km) −1 ,  = 1.1 (W km) −1 and  = 5 (W km) −1 for the EDF, SMF, and DCF, respectively.The dissipative terms in Equation (1) represent linear gain as well a parabolic approximation to the gain profile with the bandwidth Ω.The gain is saturated according to g (z) = g 0 ∕(1 + E∕E s ), where g 0 is the small-signal gain, which is non-zero only for the gain fiber, E (z) = ∫ dt|| 2 is the pulse energy, and E s is the gain saturation energy determined by the pump power.The effective nonlinear saturation involved in the NPE mode-locking technique is modeled by an instantaneous and monotonous nonlinear transfer function for the field amplitude: , where P (z, t) = |(z, t)| 2 the instantaneous pulse power.We took as typical values q 0 = 0.8 for the unsaturated loss due to the saturable absorber and P s = 0.3 W for the saturation power.Linear losses are imposed after the passive fiber segments, which summarise intrinsic losses and output ).e) Intracavity evolution of the spectral intensity was recorded over two consecutive traversals of the cavity (round-trip numbers 5 and 6), showing that the gain fiber is responsible for magnifying the energy difference between the long and short wavelength components of the spectrum.f) Corresponding intracavity evolution of the intensity of the spectral peak.SA: saturable absorber, OC: output coupler.
coupling.The numerical model is solved with a standard symmetric split-step propagation algorithm, and inspired by the experimentally observed asymmetry of the laser spectrum, we have used a picosecond pulsed initial condition with an asymmetric temporal waveform for our model.
The evolution of the spectral intensity over successive cavity round trips illustrated in Figure 6a,b indicates that the initial asymmetry in the time domain is converted into a progressively developing asymmetry in the spectrum, with an intensity peak growing on one edge of the spectrum.These results qualitatively agree with the experimental observations.A red or blue shift of the spectrum occurs depending on the side of the initial temporal asymmetry, thereby showing that the initial condition dictates which spectral components will grow into an RW.The dynamics of RWs induced by frequency upshifting are presented in Figure S8 (Supporting Information).Details of the intracavity spectral dynamics over two roundtrips are provided in Figure 6e,f.We can see that most of the SPM-induced spectral broadening occurs in the EDF, and the temporal asymmetry of the initial pulse results in an energy difference between the short and long wavelength components of the spectrum. [61]or completeness, the corresponding round-trip evolution of the laser temporal intensity is also plotted in Figure 6c,d.Whereas in the first stages of the evolution, the temporal profile of the pulse remains rather unchanged, pronounced oscillations eventually develop on the pulse edge with a larger intensity gradient, which is reminiscent of an intra-pulse shock wave.These oscillations would ultimately lead to pulse collapse and the emergence of noise-like broadband structures similar to those observed in Figure 5. Therefore, the dynamics of our laser operating at normal net dispersion appear to be rather different from those arising in laser cavities with net anomalous dispersion where the pulse propagation can be destabilized by modulation-instability processes or soliton fission.We should note that while our laser model is able to capture the evolution dynamics relating to the formation of RWs, it cannot reproduce the dynamics connected with their disappearance.A full description of the transient nature of RWs is a particularly challenging task, mainly due to the fact that a Q-switching-type phenomenon is involved in the dynamics as illustrated by Figure S9 (Supporting Information).
Identifying the key parameters of the laser cavity that govern the dynamics of RWs is an important question.Our work shows that these are the parameters of the saturable absorber (NPE) and the pump power.Indeed, when the pump power is around the threshold for stationary mode locking and the parameters of the saturable absorber do not fulfill the stability condition, [63] Q-switching instability disrupts mode locking and leads to the generation of RWs.By contrast, RWs generated by multiple pulsing are observed when the pump power is increased such that it is far beyond the mode-locking threshold, as illustrated by Figure S10 (Supporting Information).Intensity distributions of the RWs generated by Q-switching and multiple pulsing are provided in Figure S11 (Supporting Information), which shows that Q-switching can promote the generation of RWs with a wide range of intensities (from 5 to 32 I SWH ) while multiple pulsing induces RWs with smaller intensities within a narrower range ≈7 I SWH ).

Conclusion
We have demonstrated, for the first time, the possibility of using GAs to promote the emergence and control the intensity of RWs in nonlinear optical systems.Both ordinary and super RWs with tailored intensity can be generated on demand using merit functions that are based on their statistical defining characteristics.This control strategy is general and independent of the physical model considered.In the present work dealing with a fiber laser cavity, we have experimentally demonstrated the handling of extreme spectral events which also correlate with extreme fluctuations of the pulse energy.The real-time capture of a large number of spectral data through DFT measurements along with the guidance provided by numerical simulations of the laser model have suggested a new physical scenario for the emergence of these extreme waves: an initially coherent but asymmetric pulse circulating in the cavity causes one edge of the spectrum to grow and eventually evolve into a RW through the effect of SPM, while a concomitant intra-pulse shock wave develops in the time domain.This shock wave ultimately leads to pulse collapse and the emergence of noise-like broadband structures.By tuning the cavity parameters, the GA can trigger the formation of the initially asymmetric waveform sowing the RW dynamics.
To verify the ability of our control method to promote the emergence of RWs with user-defined intensity in different laser cavities, we have applied the method to a laser with a quite different net cavity dispersion (−0.09 ps 2 ).The results are summarised in Figure S12 (Supporting Information), showing that the intelligent system can still control RWs.
The occurrence of RWs is a very general phenomenon in fiber lasers.RWs have been observed in various fiber laser systems, including ring, linear [56,64] and figure-eight cavities, [16] normal-, anomalous-, [18] and near-zero dispersion [65] cavities, long [15,66] and short cavities.As mentioned in Section 1, various physical processes have been identified as the main drivers of RW forma-tion in mode-locked fiber lasers.Generally, these processes can be readily accessed by increasing of the laser pump power.Besides, RWs may also appear when turning on a laser.We argue that RWs do not represent a severe threat to fiber lasers modelocked through the NPE technique.Possibly, the inclusion of a physical saturable absorber, such as a semiconductor saturable absorber mirror, in the fiber cavity makes the laser more susceptible to damage.RWs could also be a problem for the detection devices, such as photodiodes, used to monitor the laser emission or for the sample under investigation.Moreover, after the occurrence of an RW event, the laser may switch to very different modes of operation (continuous-wave or nonstationary modes), which is not usually welcomed in the context of applications.To control the nonlinear transfer function of the NPE-based laser mode-locking, electronic polarization controllers have been largely used, [27,29,33] featuring a faster control speed than that of liquid-crystal phase retarders. [67,68]However, RWs have been observed also in fiber laser layouts where polarization control is not used. [16]It is reasonable to assume that, compatibly with the types of optical elements incorporated in the laser cavity and their offered number of degrees of freedom, it is possible to achieve intelligent control over the generation and intensity of RWs also in fiber lasers not deploying polarization controllers.
RWs are ubiquitous in nature, and understanding their generation mechanisms in different physical contexts is an important problem. [8,69]Mode-locked lasers represent ideal testbed systems for studying optical RWs.In this respect, we believe that the conceptually different generation mechanisms of RWs that our machine learning-based control method enables revealing will be of great interest to the laser photonics community.
From the perspective of practical applications, a mode-locked laser must remain in the stable mode-locking regime without switching to nonstationary operation states such as RWs.Therefore, knowing the conditions for the emergence of RWs is important to avoid such states.Our work indicates that RWs can be stimulated by Q-switching or multiple pulsing in mode-locked fiber lasers, thus one can suppress these two effects to avoid RWs.Q-switching instability can be suppressed by operating the laser at pump powers well above the mode-locking threshold or, alternatively, by deploying a saturable absorber with low modulation depth and small saturation energy. [63]To suppress multiple pulsing, a short fiber cavity can be used.
It is reasonable to expect the machine-learning method used in this work to be applicable to the control of RWs in other nonlinear systems.In particular, it would be of great interest to be able to control RWs in water wave tanks, where the dynamics are closely related to those in the ocean. [70]Besides, RWs refer to unstable states of complex systems.As demonstrated here in the case of a laser system, the use of control algorithms can make these instabilities accessible in a "repetitive" manner so that laborious manual tuning of the system's parameters is no longer required, and this facilitates the exploration of the rich underlying physics.Therefore, our work may open the way to the control and study of instabilities in a wide range of complex systems.

Figure 1 .
Figure 1.a) Schematic of the laser system.WDM: wavelength-division multiplexer; EDF: erbium-doped fiber; EPC: electronic polarization controller; PDI: polarization-dependent isolator; DCF: dispersion-compensating fiber; FPGA: field programmable gate array; DAC: digital-to-analog converter.b,c) Illustration of the GA principle for generating RWs with controllable intensity and controlling the intensity of super RWs, respectively.

Figure 2 .
Figure 2. a,b) Evolutions of the average (squares) and maximum (diamonds) values of the ratios I MAX /I SWH and (I MAX − ⟨I⟩)∕ for ordinary and super RW optimization, respectively, over successive generations.The dashed lines denote the respective target fitness ratios C RW and C SRW .The blue diamonds (between the red diamonds) depict the generations that no longer meet the target fitness ratio; nevertheless, the target RW is later restored by the GA.c, d) DFT recordings of the intensity maxima I MAX over 12000 successive cavity round trips for the optimised ordinary and super RW laser operations.The blue and red dashed lines denote the peak intensity thresholds for ordinary and super RWs, respectively.

Figure 3 .
Figure 3. a,b)Histograms showing distributions of the spectral intensity maxima for 12000 successive cavity round trips, for ordinary and super RW laser operations, respectively.The black lines denote the associated significant wave heights (I SWH ).c,d) Evolutions of the spectral intensity maxima (blue and red curves) and pulse energies (green and black curves) over 12000 successive roundtrips for ordinary and super RW laser operations, respectively.e,f) Corresponding histograms of the energy of the pulses for ordinary and super RW laser operations, respectively.The red lines denote the associated significant wave heights (E SWH ).In panels (c)-(f), <E> represents the average energy within a roundtrip.

Figure 4 .
Figure 4. a) GA-based tuning of the strength of spectral RWs relying on the optimization of the ratio I MAX /I SWH (merit function F RW ): Regression between GA output and target I MAX /I SWH values.Although the generated RWs can have large intensities, only three of them fulfill the definition of super RWs (red circles).b) GA-based tuning of the strength of spectral super RWs relying on the joint optimization of the ratios I MAX /I SWH and (I MAX − Ī)∕ (composite merit function F TSRW ): Regression between GA output and target I MAX /I SWH values.c) Distribution of ordinary and super RWs in the plane of peak intensity I MAX and standard deviation .The blue and red arrows point at the strongest ordinary RW and the weakest super RW, respectively.

Figure 5 .
Figure 5. Experimentally observed RW dynamics induced by frequency downshifting and upshifting.a,d) Evolutions of the intensity maxima of the photo-detected signals after time stretching over 12 × 10 3 successive cavity round trips.The insets show magnified versions of the dashed rectangular regions.b,e) Spatio-spectral representations of the laser intensity evolutions are shown in the insets of panels a,d), revealing the RW evolution dynamics.c,f) Single-shot spectra at the round-trip numbers indicated by dashed lines in panels (b,e), highlighting the intensity growth of the downshifted and upshifted frequency components.

Figure 6 .
Figure 6.Dynamics of frequency downshifting induced RWs as obtained from numerical simulations of the laser model.a) Evolution of the spectral intensity over successive cavity round trips.b) Spectral intensity profiles at the round-trip numbers indicated by dashed lines in (a).c) Corresponding round-trip evolution of the temporal intensity.d) Temporal intensity profiles at the same round-trip numbers as in (b).e) Intracavity evolution of the spectral intensity was recorded over two consecutive traversals of the cavity (round-trip numbers 5 and 6), showing that the gain fiber is responsible for magnifying the energy difference between the long and short wavelength components of the spectrum.f) Corresponding intracavity evolution of the intensity of the spectral peak.SA: saturable absorber, OC: output coupler.
MAX of the photo-detected signals after time stretching (DFT data) over 12000 successive cavity round trips when the laser operates in the optimal ordinary and super RW modes are depicted in panels (c) and (d) of Figure2, respectively.We see that while the emergence of ordinary RWs in the laser cavity is accompanied by a pronounced background, which is induced by relaxation oscillations, super RWs feature low fluctuations in the background.Such a difference has been evidenced by several experiments, two examples of which are shown in Figures Such locking ability of the GA has been confirmed by numerous experiments, where additional examples are given in the FiguresS1 and S2(Supporting Information).Representative evolutions of the intensity maxima I