Junsong Peng

Zihan Zhao

Sonia Boscolo
email: s.a.boscolo@aston.ac.uk

Christophe Finot

Srikanth Sugavanam

Dmitry V Churkin

Heping Zeng
email: hpzeng@phy.ecnu.edu.cn

Breather molecular complexes in a passively mode-locked fibre laser

Breathing solitons are nonlinear waves in which the energy concentrates in a localized and oscillatory fashion. Similarly to stationary solitons, breathers in dissipative systems can form stable bound states displaying molecule-like dynamics, which are frequently called breather molecules. So far, the experimental observation of optical breather molecules and the real-time detection of their dynamics have been limited to diatomic molecules, that is, bound states of only two breathers. In this work, we report on the observation of different types of breather complexes in a mode-locked fibre laser: multi-breather molecules, and molecular complexes originating from the binding of two breather-pair molecules or a breather pair molecule and a single breather. The inter-molecular temporal separation of the molecular complexes attains several hundreds of picoseconds, which is more than an order of magnitude larger than that of their stationary soliton counterparts and is a signature of long-range interactions. Numerical simulations of the laser model support our experimental findings.

Moreover, non-equilibrium dynamics of breathing solitons are also observed, including breather collisions and annihilation. Our work opens the possibility of studying the dynamics of many-body systems in which breathers are the elementary constituents.

Introduction

The concept of soliton is universal and can be applied to a large class of solitary wave propagation phenomena that are observed in most branches of nonlinear science, ranging from fluid dynamics and biology to plasma physics and photonics [START_REF] Dauxois | Physics of solitons[END_REF]. The main feature of solitons is that they propagate for a long time without visible changes. In their original studies solitons were attributed to integrable Hamiltonian systems like the focusing one-dimensional nonlinear Schrödinger equation (NLSE) [START_REF] Zabusky | Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States[END_REF], but lately the initial concept has been extended to nonlinear dissipative systems, in which localized wave packets arise from the balance between dispersion or diffraction and nonlinearity (as the conventional solitons) and between gain and losses [START_REF] Akhmediev | Dissipative Solitons: From optics to biology and medicine[END_REF].

Besides their formation, their mutual interactions such as collisions and even the emergence of stable bound states are currently the subject of intense studies in laser physics by means of realtime ultrafast measurements [START_REF] Herink | Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules[END_REF][START_REF] Krupa | Real-Time Observation of Internal Motion within Ultrafast Dissipative Optical Soliton Molecules[END_REF][START_REF] Peng | Build-Up of Dissipative Optical Soliton Molecules via Diverse Soliton Interactions[END_REF][START_REF] Liu | Real-Time Observation of the Buildup of Soliton Molecules[END_REF]. Such dissipative multi-soliton complexes, also termed soliton molecules, result from self-organization phenomena in a laser cavity, and show dynamics similar to matter molecules, such as synthesis and vibration. In [START_REF] Kurtz | Resonant excitation and all-optical switching of femtosecond soliton molecules[END_REF], through transfer of general spectroscopy concepts to the case of dissipative solitons, resonant excitation and alloptical switching of soliton molecules in response to an external perturbation were demonstrated, thus extending the matter-like soliton molecule analogy. While the soliton-pair molecule, consisting of two bound solitons, has been the most studied multi-soliton structure, a recent study showed that two soliton-pair molecules can bind subsequently to form a stable molecular complex [START_REF] Wang | Optical soliton molecular complexes in a passively mode-locked fibre laser[END_REF].

Apart from stationary solitons, numerous nonlinear systems support breathing dissipative solitons exhibiting a periodic oscillatory behavior. Dissipative breathers were first demonstrated in passive Kerr cavities [START_REF] Leo | Dynamics of one-dimensional Kerr cavity solitons[END_REF], and subsequently reported in optical microresonators [START_REF] Yu | Breather soliton dynamics in microresonators[END_REF][START_REF] Lucas | Breathing dissipative solitons in optical microresonators[END_REF]. Dissipative systems whose averaged dynamics are governed by the complex Ginzburg-Landau equation (CGLE) also support breather solutions [START_REF] Deissler | Periodic, quasiperiodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation[END_REF]. In the context of passively mode-locked lasers, which constitute a suitable platform for the study of the properties and dynamics of nonlinear dissipative systems, the existence of regimes of operation in which the laser oscillator generates strongly breathing solitons was recently predicted theoretically [START_REF] Chang | Extreme soliton pulsations in dissipative systems[END_REF]. In [START_REF] Peng | Breathing dissipative solitons in mode-locked fiber lasers[END_REF], we have directly revealed the fast spectral and temporal dynamics of breathers in a mode-locked fibre laser by means of real-time detection techniques. Similarly to Akhmediev [START_REF] Akhmediev | Modulation instability and periodic solutions of the nonlinear Schrödinger equation[END_REF] and Kuznetsov-Ma [START_REF] Kuznetsov | Solitons in a parametrically unstable plasma[END_REF][START_REF] Ma | The perturbed plane-wave solutions of the cubic Schrödinger equation[END_REF] breathers in conservative nonlinear systems, breathing dissipative solitons are related to the Fermi-Pasta-Ulam recurrence [START_REF] Bao | Observation of Fermi-Pasta-Ulam Recurrence Induced by Breather Solitons in an Optical Microresonator[END_REF][START_REF] Mussot | Fibre multiwave mixing combs reveal the broken symmetry of Fermi-Pasta-Ulam recurrence[END_REF] a paradoxical evolution of nonlinearly coupled oscillators that periodically return to their original state. Breathing solitons are not only objects of fundamental interest in nonlinear science, but they are also attractive because of their potential for applications, such as in spectroscopy [START_REF] Liao | Chip-Scale Dual-Comb Source using a Breathing Soliton with an Increased Resolution[END_REF].

Contrary to the significant progress made in the study of the dynamics of stationary soliton molecules in ultrafast lasers, the existence of bound states of breathing solitons has been less explored. Patterns of breathers have been mostly studied in conservative systems described by the NLSE, and only breather collisions [START_REF] Frisquet | Collision of Akhmediev Breathers in Nonlinear Fiber Optics[END_REF] and periodic breather interactions [START_REF] Xu | Breather Wave Molecules[END_REF] have been reported. In fact, overlapping soliton pulses cannot form strong bonds in such systems as the effective interaction potential of the pulses is not a minimum [START_REF] Malomed | Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation[END_REF]. A potential minimum can be induced by dissipative effects only, thus enabling the formation of multi-pulse bound states. A number of recent works have reported on the observation of breathing soliton-pair molecules in mode-locked fibre lasers [START_REF] Peng | Breathing dissipative solitons in mode-locked fiber lasers[END_REF][START_REF] Chen | Dynamical diversity of pulsating solitons in a fiber laser[END_REF][START_REF] Wang | Real-time observation of dissociation dynamics within a pulsating soliton molecule[END_REF]. It is worth mentioning that besides the formation of breather molecules, various notable dynamics of dissipative localized structures have been experimentally studied in mode-locked lasers, including soliton and breather explosions [START_REF] Soto-Crespo | Pulsating, creeping, and erupting solitons in dissipative systems[END_REF][START_REF] Cundiff | Experimental evidence for soliton explosions[END_REF][START_REF] Runge | Observation of soliton explosions in a passively mode-locked fiber laser[END_REF][START_REF] Peng | Soliton collision induced explosions in a mode-locked fibre laser[END_REF][START_REF] Peng | Experimental Observations of Breathing Dissipative Soliton Explosions[END_REF] and rogue wave generation [START_REF] Lecaplain | Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser[END_REF].

In the present work, by employing an ultrafast fibre laser setup whose output is spectrally and temporally analyzed in real time, we demonstrate multi-breather molecules and two types of breather molecular complexes (BMCs). One originates from the stable binding of two basic molecules, each made up of a pair of breathing solitons ((2+2) BMC), and the other one arises from the binding of a breather-pair molecule and a single breather ((2+1) BMC). The intermolecular temporal separation of such complexes is more than an order of magnitude larger than that of their stationary soliton counterparts. A recent theoretical study has shown that the interaction of well-separated dissipative breathers by exchanging weakly decaying dispersive waves can lead to the formation of bound states as a result of harmonic synchronization [START_REF] Turaev | Long-range interaction and synchronization of oscillating dissipative solitons[END_REF].

Such dispersive waves emitted by breathers in the anomalous dispersion propagation regime is the driving mechanism for the formation of the breather complexes that are demonstrated in this work. We also explore the non-equilibrium dynamics of BMCs, including collisions of breathers and annihilation of an elementary breather within a BMC. Numerical simulations of the laser model described by the CGLE confirm our experimental observations.

Results

Experimental setups

The experimental setup for generating breather complexes is an erbium-doped passively modelocked fibre laser with anomalous path-averaged cavity dispersion, sketched in Fig. 1 (see "Supporting Information" for details). Mode locking relies on the nonlinear polarization evolution (NPE) technique [START_REF] Noske | Subpicosecond soliton pulse formation from self-mode-locked erbium fibre laser using intensity dependent polarisation rotation[END_REF], which is realized through the inclusion of two fibre polarization controllers (PCs) and a polarization dependent isolator, and allows us to tune the nonlinear transfer function by simply rotating the loops, thereby modifying the linear cavity loss, hence the interactions among pulses. The output signal from the laser is analyzed in real time in both the time and frequency domains. We use the dispersive Fourier transform (DFT) method [START_REF] Goda | Dispersive Fourier transformation for fast continuous single-shot measurements[END_REF][START_REF] Mahjoubfar | Time stretch and its applications[END_REF] to acquire sub-nanometer resolution spectral measurements at the shot-to-shot level. This simple, yet powerful method, which is today a commonly used tool to obtain spectral dynamics on ultra-short time scales, maps the optical spectrum of the laser output onto a temporal waveform that is directly read out on a real-time oscilloscope. This is achieved by stretching the laser output pulses in a dispersive medium that cumulates a group-velocity dispersion large enough for the pulse propagation to satisfy the far-field condition, hence for the stretched waveform to represent the spectral intensity of the initial pulse waveform. The time-domain dynamics of the laser are characterized by measuring time traces of onedimensional intensity in real time, I(t), and then constructing from these traces the spatiotemporal intensity evolution I(t, z) [START_REF] Churkin | Stochasticity, periodicity and localized light structures in partially mode-locked fibre lasers[END_REF]. The latter reveals both the dynamics over the fast time t and the slow evolution propagation coordinate z, which is measured, in our case, as a number of cavity round trips.

Experimental results

The laser works in the stable single-pulse mode-locking regime, characterized by soliton-like pulse shaping, when the pump current is set to 44.9 mA. When the pump current is increased to 46 mA, self-starting mode locking is accompanied by the generation of multiple pulses per cavity round trip. The existence of the stable single-pulse regime in a narrow range of pump powers is due to the fact that the cavity length is longer than the typical length of most modelocked fibre lasers (less than 10 m). In our experiment, the pump current is fixed at 46 mA and we tune the cavity loss through small rotations of the PCs to generate various kinds of breather complexes. We note that the laser can also sustain regimes of emission of two, three and four bound stationary solitons. Figure 2 shows the dynamics of a breather-quartet molecule ('tetratomic molecule'). The spatio-temporal intensity evolution depicted in Fig. 2a reveals large periodic variations in the intensities of the four elementary breathers with a period of approximately 1000 round trips. The peak intensities within each period of pulsation change by nearly an order of magnitude (Fig. 2b). Concomitantly, the pulse temporal duration also varies, whilst the temporal resolution of our detection system (around 30 ps) does not allow us to capture such changes. The use of a time lens [START_REF] Ryczkowski | Real-time fullfield characterization of transient dissipative soliton dynamics in a mode-locked laser[END_REF] may potentially relax this constraint. However, this tecnhique may prove difficult to implement when the pulses experience large breathing. In spite of the periodic intensity variations, the pulse temporal separation remains nearly fixed at approximately 50 ps over consecutive cavity round trips, indicating a strong bond between the pulses. The fast evolutionary behavior of the pulse temporal intensity entails rapid variations of the optical spectrum, which are beyond the speed of traditional spectral measurement tools such as an optical spectrum analyzer. We employ real-time DFT spectral monitoring to capture the spectral evolution of the pulse quartet over cavity round trips. The DFT-based single-shot spectral measurements are shown in Fig. 2c. The spectrum features the typical interference pattern that is present in the spectrum of a soliton molecule [START_REF] Herink | Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules[END_REF], and the separation between the peaks of spectral intensity matches the pulse spacing in time. The evolution of the spectrum over cavity round trips is periodic, and the spectrum largely widens and narrows within each period, with the broadening (compression) naturally occurring in the vicinity of the position where the pulses reach the highest (lowest) peak intensity. The maximum width of the spectrum exceeds the minimum width by more than eight times (Fig. 2d). The energy of the pulse quartet (Fig. 2c, white curve), calculated by integration of the power spectral density over the whole wavelength band, evolves over round trips synchronously with the spectrum width and pulse temporal intensity. The calibration and accuracy of the DFT spectral measurements are checked by comparing, in the Supporting Information, the average of the measured consecutive singleshot spectra with the averaged spectrum recorded by an optical spectrum analyzer. To complete the quantitative dynamical picture of the breather-quartet molecule, we resolve in real-time the relative phases within the molecule using the DFT-based spectral interferometry method, which relies on computing the Fourier transform of each single-shot DFT spectrum and is now rather commonly used with stationary soliton states [START_REF] Wang | Optical soliton molecular complexes in a passively mode-locked fibre laser[END_REF], [START_REF] Xia | Experimental observation of shaking soliton molecules in a dispersion-managed fiber laser[END_REF]. It is worth to mention that the application of this method to the breather complexes generated in our laser cavity is quite challenging as the first-order single-shot autocorrelation traces are too weak when the breathers are at their intensity minima. The internal phase dynamics shown in the Supporting Information confirm the strong nature of the bond between the breather constituents of the molecule.

A notable phenomenon in the breathing process illustrated in Fig. 2 is the periodic appearance of wide Kelly sidebands in the shot-to-shot spectra (see, e.g., Fig. 2d). These Kelly sidebands are a manifestation of resonant dispersive waves in the temporal domain, which, in the usual perspective, radiate from solitons when they are perturbed by lumped nonlinear losses and various intra-cavity components in a round trip [START_REF] Komarov | Dispersive-wave mechanism of interaction between ultrashort pulses in passive mode-locked fiber lasers[END_REF]. The low-intensity background pattern synchronized to the breather pattern that is observable in Fig. 2a is formed out of the slowly decaying dispersive waves radiated by the breathers [START_REF] Du | Pulsating soliton with broadened Kelly sidebands in an ultrafast fiber laser[END_REF], which are the main agent of the breathers long-range interaction [START_REF] Turaev | Long-range interaction and synchronization of oscillating dissipative solitons[END_REF]. These dispersive waves are periodically emitted during hundreds of cavity periods and, as it can be seen from Fig. 2c (a magnified version of Fig. 2c is provided in the Supporting Information) and Fig. 2d, the central wavelength of the spectrum and the wavelengths of the Kelly sidebands oscillate synchronously with the pulse energy over cavity round trips [START_REF] Du | Pulsating soliton with broadened Kelly sidebands in an ultrafast fiber laser[END_REF][START_REF] Wang | Self-organized compound pattern and pulsation of dissipative solitons in a passively mode-locked fiber laser[END_REF]. This wavelength oscillation can be ascribed to cross-phase modulation of the breathers and the radiated dispersive waves [START_REF] Du | Pulsating soliton with broadened Kelly sidebands in an ultrafast fiber laser[END_REF]. As we can notice from Fig. 2a, this periodic wavelength shift is mirrored into a periodic shift of the pulse positions in the time domain. In the false color plot of Fig. 2a, the pulsations of the individual pulse intensities (round trip number from approximately 300 to 900, for example) are accompanied by a positive (to the right) time shift, while no time shift occurs over the duration of the intensity minima. As a result, the pulses at a given round trip number within a period of oscillation appear shifted in time with respect to the pulses at the same round trip number within the previous oscillation period while having the same group velocity. We note that this fact has no physical origin but, as explained in the Supporting Information, is a direct result of the spatio-temporal intensity representation. It is also noteworthy that the signal after temporal stretching has a rather long duration; for instance, the wavelength span in Fig. 2c corresponds to a time window of length 25 ns. Therefore, the impact of the artificial shift of the pulse temporal positions from one oscillation period to the next (approximately 100 ps in Fig. 2a) on the DFT measurements is negligible. The calculated physical time shift induced by the fibre dispersion over consecutive round trips shows good agreement with the artificial shift that is observed in the spatio-temporal intensity map. Although the Kerr effect can also induce a change in the pulse temporal positions, this effect is negligibly small (Supporting Information).

Another type of a breather complex is observed under different PC settings: a (2+2) BMC consisting of two bound breather pairs. The spatio-temporal intensity dynamics shown in Fig. 3a reveal the existence of two characteristic time scales, corresponding to the pulse separation within each breather-pair molecule and the separation between the two breather-pair molecules.

The ratio between the two characteristic times being approximately one to seven indicates that the two times are plausibly associated with two different bond strengths. The intra-molecular pulse separation is nearly the same as that of the breather-quartet molecule in Fig. 2, but the inter-molecular separation comes to approximately 350 ps, an order of magnitude larger than that of the stationary soliton molecular complexes recently reported [START_REF] Wang | Optical soliton molecular complexes in a passively mode-locked fibre laser[END_REF]. Through rotation of the PCs, we can turn to a different (2+2) BMC (Fig. 3b). Compared to the previous case (Fig. 3a), the pulses of the two breather-pair molecules are bound at different time intervals, showing that a BMC can have varied intra-molecular separations.

By adjusting the linear cavity loss, we can also decrease the number of breathing solitons and prepare a robust breather triplet. An example is provided in Fig. 3c, which shows the spatiotemporal evolution of a BMC consisting of a breather-pair molecule bound to a single breather, where the latter can be regarded as a monoatomic molecule. Though the single breather is far apart from the breather pair by about 500 ps, its intensity evolves synchronously with the breather pair over cavity round trips, indicating a strong inter-molecular bond. We have also observed a single breather-pair molecule in the laser (see Supporting Information).

Soliton interaction is one of the most exciting areas of research in nonlinear dynamics. In nearly conservative systems, such as Bose-Einstein condensates of atoms confined to a quasidimensional waveguide, soliton collisions may result in annihilation depending on the relative phase between the solitons [START_REF] Nguyen | Collisions of matter-wave solitons[END_REF]. Soliton annihilation also occurs in dissipative systems [START_REF] Akhmediev | Dissipative soliton interactions inside a fiber laser cavity[END_REF]. In our experiment, we have observed breather collisions and destruction within unstable BMCs by rotating the PCs at fixed pump power. Examples are shown in Fig. 4. In Fig. 4a, the initial condition is a breather quartet. Breather fusions occur at round-trip numbers of approximately 2500 and 3500, followed by the splitting of each of the two resulting breathers into two pulses at a round-trip number of approximately 5000. Subsequently, one of the two breather pairs gradually disappears. This is the first time that splitting of breathing solitons is observed, following the observation of soliton splitting [START_REF] Peng | Build-Up of Dissipative Optical Soliton Molecules via Diverse Soliton Interactions[END_REF][START_REF] Peng | Real-time observation of dissipative soliton formation in nonlinear polarization rotation mode-locked fibre lasers[END_REF][START_REF] Liu | Revealing the Buildup Dynamics of Harmonic Mode-Locking States in Ultrafast Lasers[END_REF]. Figure 4b shows three collision events when the initial condition is a breather triplet. In all cases, the scenario resembles that of an elastic collision, in that the two colliding breathers just pass through each other and they are not affected by the collision: they do not merge and their traces do not change after the collision. This is different from the inelastic breather collisions that were recently observed in passive fibers, which represent an example of a conservative system [START_REF] Frisquet | Collision of Akhmediev Breathers in Nonlinear Fiber Optics[END_REF]. In that case, indeed, the two breathers merged into a single peak. It is also notable that different types of non-equilibrium dynamics of breather molecular complexes can be observed in our laser setup by fine adjustment of the PC settings. For instance, the time separation between the leading and trailing breather within a complex may increase or decrease monotonously over cavity round trips, a dynamical state that we could refer to as thermal expansion or contraction of a breather molecular complex.

Numerical simulations

In [START_REF] Du | Pulsating soliton with broadened Kelly sidebands in an ultrafast fiber laser[END_REF], the periodic shift of the central wavelength of the pulsating soliton spectrum as well as the appearance of wide Kelly sidebands on the spectrum were explained in the frame of a vectorial model for the laser operation, based on coupled extended NLSEs and including the effect of cross-phase modulation. Our focus is here on verifying the main features of the observed pulse complexes, that is, their periodic breathing behavior. To this end, we have bypassed a quite complex laser model and carried out numerical simulations of the laser based on the master-equation approach [START_REF] Haus | Theory of mode locking with a fast saturable absorber[END_REF], namely, we have used the cubic-quintic CGLE. For the context of a passively mode-locked fibre laser, the cubic-quintic CGLE has the form [START_REF] Grelu | Dissipative solitons for mode-locked lasers[END_REF] 𝑖𝜓 𝜁 + 𝐷 2 𝜓 𝜏𝜏 + |𝜓| 2 𝜓 + 𝜈|𝜓| 4 𝜓 = 𝑖𝛿𝜓 + 𝑖𝜀|𝜓| 2 𝜓 + 𝑖𝛽𝜓 𝜏𝜏 + 𝑖𝜇|𝜓| 4 𝜓 (1

)
where ζ is the normalized propagation distance traversed in the cavity, τ is the retarded time, ψ is the normalized envelope of the field, 𝐷 = -sgn(𝛽 ̅ 2 ), 𝛽 ̅ 2 is the path-averaged cavity dispersion, and ν corresponds to the saturation of the nonlinear refractive index. The dissipative terms are written on the right-hand side of Eq. ( 1), where δ > 0 (δ < 0) is the net linear gain (loss) coefficient, the term with ε represents the nonlinear gain (which arises, e.g., from saturable absorption), β > 0 accounts for spectral filtering, and μ < 0 represents the saturation of the nonlinear gain. This equation is one of the simplest models for a passively mode-locked laser with a fast saturable absorber, and can describe various complex nonlinear dynamics such as soliton explosions [START_REF] Soto-Crespo | Pulsating, creeping, and erupting solitons in dissipative systems[END_REF][START_REF] Cundiff | Experimental evidence for soliton explosions[END_REF][START_REF] Akhmediev | Strongly asymmetric soliton explosions[END_REF], rogue waves [START_REF] Lecaplain | Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser[END_REF], multiple pulsing [START_REF] Akhmediev | Multisoliton solutions of the complex Ginzburg-Landau equation[END_REF], switching dynamics [START_REF] Bao | Observation of Coexisting Dissipative Solitons in a Mode-Locked Fiber Laser[END_REF], and dissipative soliton resonances [START_REF] Chang | Dissipative soliton resonances[END_REF] among the others [START_REF] Grelu | Dissipative solitons for mode-locked lasers[END_REF][START_REF] Renninger | Dissipative solitons in normal-dispersion fiber lasers[END_REF].

By solving the CGLE numerically for the set of parameters given in the caption of Fig. 5, we have found periodic breathing dynamics of a bound pulse quartet, which qualitatively account for the behaviors observed in Fig. 2. We have also found numerically the different types of BMCs generated in the experiment (Fig. 3), as shown in Fig. 6.

Conclusion

One of the remarkable properties of dissipative solitons, which are mostly absent in integrable systems, is the ability to form robust multi-pulse bound states (soliton molecules). An optical cavity constitutes an ideal propagation medium to study multiple soliton interactions, since even very weak interactions can be revealed through the virtually unlimited propagation time [START_REF] Akhmediev | Dissipative soliton interactions inside a fiber laser cavity[END_REF][START_REF] Kutz | Stabilized pulse spacing in soliton lasers due to gain depletion and recovery[END_REF][START_REF] Jang | Ultraweak long-range interactions of solitons observed over astronomical distances[END_REF]. While soliton pairs constitute the central soliton molecule case, soliton molecules can exist in various isomers [START_REF] Leblond | Cis' bound states of three localized pulses of the cubic-quintic CGL equation[END_REF], and a large population of optical solitons can self-assembly into macromolecules and soliton crystals [START_REF] Haboucha | Analysis of soliton pattern formation in passively mode-locked fiber lasers[END_REF], and even into highly-ordered supra-molecular structures through the tailoring of their long-range interactions [START_REF] He | Formation of optical supramolecular structures in a fibre laser by tailoring long-range soliton interactions[END_REF]. Although breathers are fundamentally different from stationary solitons, they appear to exhibit similar collective dynamics. The experimental generation of breather-pair molecules in a fibre laser cavity has been reported on a few recent occasions [START_REF] Peng | Breathing dissipative solitons in mode-locked fiber lasers[END_REF][START_REF] Chen | Dynamical diversity of pulsating solitons in a fiber laser[END_REF][START_REF] Wang | Real-time observation of dissociation dynamics within a pulsating soliton molecule[END_REF]. The emphasis of the present article is on pushing the similarity in collective behavior between breathing and stationary solitons further.

We have reported on the experimental observation and real-time dynamic characterization of different types of breather complexes in a passively mode-locked fibre laser, including tetratomic molecules, and molecular complexes formed by the binding of two diatomic molecules or a diatomic and a monoatomic molecule. We have also observed breather annihilation within an unstable molecular complex when the complex is in the phase of intensity drop. Since breathing dissipative solitons are fundamental modes of many nonlinear physical systems, it is reasonable to assume that the breather dynamics observed in this work will incentivize the investigation of BMCs in various other systems.

Similarly to the stationary soliton scenario [START_REF] He | Formation of optical supramolecular structures in a fibre laser by tailoring long-range soliton interactions[END_REF], we believe it will be possible to assemble supra-molecular structures of optical breathers in a fibre laser by using intense opto-acoustic effects in photonic crystal fibres. So far, the studies of localized wave structures in ultrafast fibre optics are limited to the time and frequency domains, as these structures are generated in single-mode fibres. The generation and propagation of pulses in multimode fibre systems have recently drawn great attention [START_REF] Wright | Spatiotemporal mode-locking in multimode fiber lasers[END_REF]. A multimode fibre laser provides a new degree of freedom in the control of coherent light fields: the space domain. We can therefore anticipate that the spatio-temporal engineering of optical pulses will enable the generation of even more complex breather structures.

Supporting Information

Figure S1 :

 S1 Figure S1: Validation of the dispersive Fourier transform (DFT) implementation.

Figure S2 :

 S2 Figure S2: Characterisation of a breather-quartet molecule.

Figure S3 :

 S3 FigureS3: Evidence for the nonlinear interaction between breathing solitons and coherent

Figure S4 :

 S4 Figure S4: Graphical explanation of the artefact of the spatio-temporal intensity

Figure S5 :

 S5 Figure S5: Relationship between physical and artificial time shifts in the spatio-temporal

Figure S6 :

 S6 Figure S6: Dispersion-induced time shift.

Figure S7 :

 S7 Figure S7: Dynamics of a breathing soliton-pair molecule.

Figure 1 .

 1 Figure 1. Schematic of the laser and real-time detection system. WDM, wavelength division multiplexer; PC, polarization controller; PDI, polarization dependent isolator; SMF, single mode fibre; EDF, erbium doped fibre; DCF, dispersion compensating fibre.

Figure 2 .

 2 Figure 2. Dynamics of a breathing soliton quartet molecule. (a) Temporal evolution of the intensity relative to the average round-trip time over consecutive round trips. (b) Temporal intensity profiles at representative round-trip numbers of maximal and minimal energies within a period of oscillation. (c) DFT recording of single-shot spectra. The evolution of the energy is also shown (solid white line). (d) Single-shot spectra at representative round-trip numbers of maximal and minimal spectrum extents within a period.

Figure 3 .

 3 Figure 3. Dynamics of various breather molecular complexes. Temporal evolution of the intensity relative to the average round-trip time over 4000 consecutive round trips for: (a) a (2+2) breather molecular with equal intra-molecular pulse separations; (b) a (2+2) breather molecular complex with different intra-molecular separations; and (c) a (2+1) breather molecular complex.

Figure 4 .

 4 Figure 4. Examples of non-equilibrium dynamics of breather molecular complexes. (a) Breather splitting and annihilation of elementary breathers around a round-trip number of 5000. (b) Three collision events; the weak breather ultimately vanishes after the collisions.

Figure 5 .

 5 Figure 5. Simulation of a breather-quartet molecule. Evolutions of the (a) temporal intensity and (c) spectrum found for the set of CGLE parameters: (ν = 2, δ = -0.01, β = 0.3, ε = 5, μ =-0.02). The evolution of the energy 𝑄(𝜁) = ∫ 𝑑𝜏|𝜓(𝜁, 𝜏)| 2 is also shown. (b) Temporal intensity profiles at representative round-trip numbers of maximal and minimal energies within a period of oscillation. (d) Corresponding spectra of maximal and minimal extents.

Figure 6 .

 6 Figure 6. Simulation of various breather molecular complexes. Evolution of the temporal intensity for: (a) a (2+2) breather molecular complex with equal intra-molecular pulse separations; (b) a (2+2) breather molecular complex with different intramolecular separations; and (c) a (2+1) breather molecular complex. The CCQGLE parameters are (ν = 2, δ = -0.23, β = 0.4, ε = 6, μ = -0.02) for the solution shown in panel (a), while the same set of parameters as that of Fig. 5 is used for the solutions shown in panels (b) and (c).

Acknowledgements (We acknowledge the support from the National Natural Science Fund of China (11621404, 11561121003, 11727812, 61775059, and 11704123), National Key Research and Development Program (2018YFB0407100), Key Project of Shanghai Education. Commission (2017-01-07-00-05-E00021), and Science and Technology Innovation Program of Basic Science Foundation of Shanghai (18JC1412000)).

Competing interests: There are no financial competing interests.

Keywords: (ultrafast fibre lasers, breathers, mode locking)


Xiuqi Wu

Junsong Peng
email: jspeng@lps.ecnu.edu.cn

Sonia Boscolo

Ying Zhang

Christophe Finot

Heping Zeng
email: hpzeng@phy.ecnu.edu.cn

Title (Intelligent breathing soliton generation in ultrafast fibre lasers)

Keywords: (ultrafast fibre lasers, breathers, mode locking)

Harnessing pulse generation from an ultrafast laser is a challenging task as reaching a specific mode-locked regime generally involves adjusting multiple control parameters, in connection with a wide range of accessible pulse dynamics. Machine-learning tools have recently shown promising for the design of smart lasers that can tune themselves to desired operating states. Yet, machine-learning algorithms are mainly designed to target regimes of parameter-invariant, stationary pulse generation, while the intelligent excitation of evolving pulse patterns in a laser remains largely unexplored. Breathing solitons exhibiting periodic oscillatory behavior, emerging as ubiquitous mode-locked regime of ultrafast fibre lasers, are attracting considerable interest by virtue of their connection with a range of important nonlinear dynamics, such as exceptional points, and the Fermi-Pasta-Ulam paradox. Here, we implement an evolutionary algorithm for the self-optimisation of the breather regime in a fibre laser modelocked through a four-parameter nonlinear polarisation evolution. Depending on the specifications of the merit function used for the optimisation procedure, various breathingsoliton states are obtained, including single breathers with controllable oscillation period and breathing ratio, and breather molecular complexes with a controllable number of elementary constituents. Our work opens up a novel avenue for exploration and optimisation of complex dynamics in nonlinear systems.

Introduction

Breathing solitons form an important part of many different classes of nonlinear wave systems, manifesting themselves as localised temporal/spatial structures that exhibit periodic oscillatory behavior. Breathers were first studied experimentally in Kerr fibre cavities [START_REF] Leo | Dynamics of one-dimensional Kerr cavity solitons[END_REF] and subsequently reported in optical micro-resonators, [START_REF] Yu | Breather soliton dynamics in microresonators[END_REF][START_REF] Lucas | Breathing dissipative solitons in optical microresonators[END_REF][START_REF] Cole | Subharmonic Entrainment of Kerr Breather Solitons[END_REF] where the dynamics are governed by the Lugiato-Lefever equation. Recently, they have also emerged as an ubiquitous mode-locked regime of ultrafast fibre lasers. [START_REF] Peng | Breathing dissipative solitons in mode-locked fiber lasers[END_REF][START_REF] Xian | Subharmonic Entrainment Breather Solitons in Ultrafast Lasers[END_REF][START_REF] Du | Spatio-spectral dynamics of the pulsating dissipative solitons in a normal-dispersion fiber laser[END_REF] These nonlinear waves are attracting significant research interest in optics by virtue of their strong connection with the Fermi-Pasta-Ulam recurrence [START_REF] Bao | Observation of Fermi-Pasta-Ulam Recurrence Induced by Breather Solitons in an Optical Microresonator[END_REF][START_REF] Mussot | Fibre multi-wave mixing combs reveal the broken symmetry of Fermi-Pasta-Ulam recurrence[END_REF] a paradoxical evolution of nonlinearly coupled oscillators that periodically return to their initial state, the formation of rogue waves, [START_REF] Suret | Singleshot observation of optical rogue waves in integrable turbulence using time microscopy[END_REF][START_REF] Dudley | Instabilities, breathers and rogue waves in optics[END_REF] turbulence, and modulation-instability phenomena.

Breathers are also closely related to exceptional points, [START_REF] Xian | Subharmonic Entrainment Breather Solitons in Ultrafast Lasers[END_REF] thereby opening a novel avenue for the exploration of exceptional-point physics. Besides their significance in nonlinear science, breathers also bear interesting possibilities for practical applications. For instance, they can increase the resolution of dual-comb spectroscopy, [START_REF] Liao | Chip-Scale Dual-Comb Source using a Breathing Soliton with an Increased Resolution[END_REF] and the breather regime in a laser oscillator can be used to generate high-amplitude ultrashort pulses without additional compressors. [START_REF] Chang | Extreme soliton pulsations in dissipative systems[END_REF][START_REF] Chang | Extreme amplitude spikes in a laser model described by the complex Ginzburg-Landau equation[END_REF] In addition to their growing use as sources of ultrashort pulses for many applications, modelocked fibre lasers constitute an ideal platform for the fundamental exploration of complex nonlinear wave dynamics. Indeed, the high levels of linear and nonlinear effects accumulated during a single round trip (RT) in such lasers, along with the nonlinear polarisation evolution (NPE) effect -which is often exploited as the saturable absorption mechanism to drive the mode-locking process [START_REF] Noske | Subpicosecond soliton pulse formation from self-modelocked erbium fibre laser using intensity dependent polarisation rotation[END_REF] -entail a wealth of complex short-pulse dynamics [START_REF] Grelu | Dissipative solitons for mode-locked lasers[END_REF] that can be accessed through tuning of the control cavity parameters. These include pulsating regimes, [START_REF] Soto-Crespo | Pulsating, creeping, and erupting solitons in dissipative systems[END_REF] soliton explosions, [START_REF] Soto-Crespo | Pulsating, creeping, and erupting solitons in dissipative systems[END_REF][START_REF] Cundiff | Experimental evidence for soliton explosions[END_REF][START_REF] Runge | Observation of soliton explosions in a passively mode-locked fiber laser[END_REF][START_REF] Peng | Soliton collision induced explosions in a mode-locked fibre laser[END_REF] and various multi-pulsing regimes, where the interactions between the solitons lead to harmonic mode locking or self-organised patterns such as soliton bunches and molecules, [START_REF] Grelu | Phase-locked soliton pairs in a stretchedpulse fiber laser[END_REF][START_REF] Krupa | Real-Time Observation of Internal Motion within Ultrafast Dissipative Optical Soliton Molecules[END_REF][START_REF] Herink | Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules[END_REF][START_REF] Peng | Real-time observation of dissipative soliton formation in nonlinear polarization rotation modelocked fibre lasers[END_REF] soliton molecular complexes [START_REF] Wang | Optical soliton molecular complexes in a passively mode-locked fibre laser[END_REF] and supramolecular structures. [START_REF] He | Formation of optical supramolecular structures in a fibre laser by tailoring long-range soliton interactions[END_REF] Chaotic pulsed states also abound, including noise-like pulse emission [START_REF] Horowitz | Noiselike pulses with a broadband spectrum generated from an erbium-doped fiber laser[END_REF] and rogue-wave formation. [START_REF] Lecaplain | Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser[END_REF] Operating regimes in which a laser oscillator generates strongly breathing solitons were predicted theoretically in Ref [13]. In Ref [5], we have directly resolved the fast temporal and spectral dynamics of breathers in a mode-locked fibre laser using real-time detection techniques.

More recently, we have also reported on the observation of different types of breather molecular complexes (BMCs) in a fibre laser, whose formation is driven by the dispersive waves that are emitted by breathers in the anomalous-dispersion propagation regime. [START_REF] Peng | Breather molecular complexes in a passively mode-locked fibre laser[END_REF] However, reaching a desired operating regime in a fibre laser generally depends on precisely adjusting multiple parameters in a high-dimensional space, which is usually performed through an often lengthy, trial-and-error experimental procedure, due to the lack of analytic relationship between the cavity parameters and the pulse features. The practical difficulties associated with such a procedureincluding the problem of repeatability -, along with the inadequacy of systematic numerical propagation modelling, hamper the possibilities of obtaining an augmented subspace of useful ultrafast dynamics through intracavity parameter adjustment.

Nevertheless, such a difficulty can be circumvented by machine-learning strategies and the use of evolutionary and genetic algorithms, which are well-suited to the global optimisation problem of complex functions. [START_REF] Brunton | Self-tuning fiber lasers[END_REF][START_REF] Baumeister | Deep learning and model predictive control for self-tuning mode-locked lasers[END_REF] The application of advanced algorithmic tools and adaptive feedback and control has recently greatly boosted the progress in the search for a truly selfoptimising laser, and a number of groups have reported on different approaches to automate optimisation of one or more parameters of the laser cavity to reach and maintain a desired operating state. [START_REF] Andral | Fiber laser mode locked through an evolutionary algorithm[END_REF][START_REF] Andral | Toward an autosetting mode-locked fiber laser cavity[END_REF][START_REF] Woodward | Towards 'smart lasers': self-optimisation of an ultrafast pulse source using a genetic algorithm[END_REF][START_REF] Winters | Electronic initiation and optimization of nonlinear polarization evolution mode-locking in a fiber laser[END_REF][START_REF] Woodward | Genetic algorithm-based control of birefringent filtering for selftuning, self-pulsing fiber lasers[END_REF][START_REF] Kokhanovskiy | Machine learning methods for control of fibre lasers with double gain nonlinear loop mirror[END_REF][START_REF] Pu | Intelligent programmable mode-locked fiber laser with a human-like algorithm[END_REF][START_REF] Wei | Harnessing a multi-dimensional fibre laser using genetic wavefront shaping[END_REF][START_REF] Girardot | Autosetting mode-locked laser using an evolutionary algorithm and time-stretch spectral characterization[END_REF][START_REF] Genty | Machine learning and applications in ultrafast photonics[END_REF] A recent work [START_REF] Pu | Intelligent control of mode-locked femtosecond pulses by time-stretch-assisted real-time spectral analysis[END_REF] introduced extra novelty by incorporating fast spectral measurements into the feedback loop of the laser setting which, along with an intelligent polarisation search algorithm, enabled real-time control of the spectral width and shape of ultrashort mode-locked pulses. Despite these significant advances, the intelligent generation of breathing solitons in a fibre laser remains challenging because breathers refer to a highly dynamical state in which the pulse spectral and temporal characteristics change drastically within a period of oscillation, while existing machine-learning strategies are mostly designed to target laser generation regimes of parameter-invariant, stationary pulses.

In the present experimental work, we implement an evolutionary algorithm (EA) for the selfoptimisation of the breather regime in a mode-locked fibre laser, based on the optimal fourparameter tuning of the intracavity nonlinear transfer function through electronically driven polarisation control. We define compound merit functions relying on the characteristic features of the radiofrequency (RF) spectrum of the laser output, which are capable to locate various self-starting breather regimes in the laser, including single breathers with controllable breathing ratio and period, and BMCs with a controllable number of elementary constituents.

Results

Experimental setups and principle

The experimental setup is sketched in Figure 1(a). The laser is a fibre ring cavity in which a 1.3-m-long erbium-doped fibre constitutes the gain medium, pumped by a laser diode operating at 980 nm through a wavelength-division multiplexer. Other fibres in the cavity are a section of dispersion-compensating fibre and pieces of standard single-mode fibre from the pigtails of the optical components used. The group-velocity dispersion values of the three fibre types are 65, 62.5, and -22.8 ps 2 /km, respectively, yielding a normal cavity dispersion of 0.028 ps 2 at the operating wavelength of ~1.5 μm. The repetition rate of the laser is 16.765 MHz. By using a normally dispersive laser cavity, the generated breathing solitons appear to be more robust than in the anomalous-dispersion case. Indeed, with anomalous dispersion, the Kelly sidebands radiating from the solitons may cause energy variations that do not refer to a breather state, so that the design of an efficient EA for optimisation of the breather laser regime may become more challenging. More importantly, since pulses generated in the anomalous-dispersion regime are typically much shorter, the time resolution of the detection system used may not enable capturing the changes experienced by the temporal duration of breathers within an oscillation period [START_REF] Bao | Observation of Fermi-Pasta-Ulam Recurrence Induced by Breather Solitons in an Optical Microresonator[END_REF] . The mode-locked laser operation is obtained thanks to an effective saturable absorber based on the NPE effect. [START_REF] Noske | Subpicosecond soliton pulse formation from self-modelocked erbium fibre laser using intensity dependent polarisation rotation[END_REF] The nonlinear transfer function of the NPE-based mode locking is controlled by an electronically driven polarisation controller (EPC) working together with a polarisation-dependent isolator. The EPC consists of four fibre squeezers oriented at 45° to each other and each controlled by an applied voltage signal and can generate all possible states of polarisation over the Poincaré sphere, where each set of voltages corresponds to a specific state. Therefore, only one EPC is required in the laser setup to achieve the manipulation of the intracavity polarisation states enabling a complete control of the nonlinear transfer function, as in [START_REF] Andral | Fiber laser mode locked through an evolutionary algorithm[END_REF] [START_REF] Woodward | Genetic algorithm-based control of birefringent filtering for selftuning, self-pulsing fiber lasers[END_REF][START_REF] Pu | Intelligent programmable mode-locked fiber laser with a human-like algorithm[END_REF][START_REF] Shen | Electronic control of nonlinear-polarization-rotation mode locking in Yb-doped fiber lasers[END_REF] . Furthermore, with a response time of 0.4ms, the EPC can change the laser operation regime quickly, thus enabling the implementation of an EA to find a userdefined optimal laser regime within a realistic time scale. The laser output is split into two ports: a fraction is directly detected by a fast photodiode (Finisar XPDV2320R; 20-ps response time, 50-GHz bandwidth) plugged to a real-time oscilloscope (Agilent; 33-GHz bandwidth, 80-GSa/s sampling rate). The so measured time traces of one dimensional intensity, I(t), in real time are then used to construct the spatio-temporal intensity evolution I(t,z) [START_REF] Churkin | Stochasticity, periodicity and localized light structures in partially mode-locked fibre lasers[END_REF] to characterise the timedomain dynamics of the laser. The remaining laser output is sent through a time-stretch dispersive Fourier transform (DFT) setup consisting of a long segment of normally dispersive fibre that provides a total accumulated dispersion of DL≈-1200 ps/nm. After propagation through such a long dispersive link, a short laser pulse gets significantly stretched so that its spectrum is mapped into the time domain. [START_REF] Goda | Dispersive Fourier transformation for fast continuous single-shot measurements[END_REF] 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 Δλ=1/(DL*BW)≈0.025 nm, where BW is the bandwidth of the photodetection. The oscilloscope is connected to a computer that runs the EA and controls the EPC via a field programmable gate array (FPGA) and four digital-to-analogue converters (DACs; 125-MHz sampling rate, 14-bit resolution). 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.

The intelligent search of breathers is realised via an EA whose principle, as illustrated in Fig. 1(b), mimics mechanisms inspired by Darwin's theory of evolution: individuals composing a population progress through successive generations only if they are among the fittest. [START_REF] Mitchell | An introduction to genetic algorithms[END_REF] In our case, an individual is a laser regime, associated with the nonlinear transfer function defined by the four control voltages applied to the EPC; these voltages are therefore the genes of the individuals. The process begins with a collection of individuals or 'population' (making up the first generation), each comprising a set of randomly assigned genes. The system output is measured for each individual in the generation, evaluated by a user-defined merit function (also known as fitness or objective function) and assigned a score. The EA 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, [START_REF] Mitchell | An introduction to genetic algorithms[END_REF] Fig.

1(c))

. While elitist selection -in which the best individuals are cloned to the next generation to ensure their high-quality genes are preserved -, works well to search for stationary modelocking states, [START_REF] Girardot | Autosetting mode-locked laser using an evolutionary algorithm and time-stretch spectral characterization[END_REF] roulette selection is found more efficient for the breather mode locking here.

Two new individualschildren -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. This process repeats until the algorithm converges and an optimal individual is produced.

The algorithm is initialised with a population of 100 individuals, and the population size of the next generations is kept constant to 50 individuals. This relatively large population size is crucial to targeting the breather mode-locking regime, which exists in a narrower parameter space than stationary mode locking. [START_REF] Chang | Extreme amplitude spikes in a laser model described by the complex Ginzburg-Landau equation[END_REF] Crossover is realised by generating random numbers from 0 to 1 and correlating them to each pair of individuals. If a pair has a correlated number smaller than 0.6, crossover within the pair occurs; otherwise, no crossover happens. Mutation is realised in a similar manner, with a smaller probability: an individual is subject to mutation if its correlated number is smaller than 0.02. The mutation implementation is described in the Supporting Information. Evaluation of the properties of an entire generation of individuals typically takes 3.3 minutes.

A critical factor to the success of a self-optimising laser implementation is the merit function, which must return a higher value when the laser is operating closer to the target regime. Merit functions based on the peak height of the cavity repetition frequency or pulse count work well for searching stationary mode locking regimes [START_REF] Pu | Intelligent programmable mode-locked fiber laser with a human-like algorithm[END_REF][START_REF] Pu | Intelligent control of mode-locked femtosecond pulses by time-stretch-assisted real-time spectral analysis[END_REF] but cannot be used as standalone to target breather modes of operation. Therefore, we formulate a compound merit function based on the following observations. To select a breather regime, we need a merit function that discriminates between breather and stationary pulsed operations. The oscillation frequency of the breathers manifests itself as sidebands in the RF spectrum of the laser output, as illustrated in Fig. 1(d), where |f±1 -fr| represents the breathing frequency, and fr and f±1 are the cavity repetition frequency and sideband frequencies, respectively. There are no sidebands located at f±1 when the laser works in a stationary mode locking regime. Therefore, we can design a merit function that exploits the intensity ratio of the central band located at fr to the sidebands at f±1 . In practice, however, it is easier to operate with the intensity ratio of fr to the frequency interval from f-1 to f+1. Hence, the merit function for the breather operation derived from this feature is However, equation (1) alone cannot guarantee the generation of breathers in the laser because other laser modes, such as relaxation oscillations or noise-like pulse emission, may also feature sidebands in the RF spectrum (see Supplementary Figs. 1 and2). Therefore, to exclude these possibilities, we use the merit function relating to the mode-locked laser operation [START_REF] Pu | Intelligent programmable mode-locked fiber laser with a human-like algorithm[END_REF] , which is derived from the feature that mode-locked pulses have a significantly higher intensity than free-running states,

1 1 ( ) ( ) () r r f f f f f f f f b ff ff I f I f C If                (1) where
    1 , =, 0, iL i i i th i ml i i th I I I I CI II L            (2)
where L is the number of laser output intensity points recorded by the oscilloscope (L=2 [START_REF] Peng | Real-time observation of dissipative soliton formation in nonlinear polarization rotation modelocked fibre lasers[END_REF] , corresponding to a time trace of ~2700 cavity RTs), Ii is the intensity at point i and Ith is a threshold intensity that noise should not exceed. Therefore, Cml represents the average of pulses' intensities. We can then define the total merit function of the breather mode-locking regime as the weighted sum of Cml and Cb,

merit ml b F C C      (3)
where the weights of the two components are determined empirically (𝛼=40000, 𝛽=10).

The search for single-breathing soliton mode locking is implemented here as a three-stage optimisation procedure. The first stage involves testing and ranking each individual according to its fitness with respect to the merit function Cml which enables the exclusion of those individuals that correspond to relaxation oscillation regimes whose merit score is low.

Meanwhile, to exclude noise-like pulse mode locking, which returns moderate values of Cml similar to stationary mode locking, the maximum peak intensity of the pulses is checked: if it is extremely high, a new search begins. The second stage involves pulse counting to select mode locking at the fundamental repetition frequency (the details of pulse count are provided in the Supporting Information). Finally, the individuals passing through the first two stages, are scored against the compound merit function given in equation (3) to exclude stationary pulse states.

The inclusion of additional components in the definition of Fmerit is the key to achieving advanced control of the characteristics of the breather state, such as tuning of the oscillation period or breathing ratio. The corresponding merit functions are given in the Supporting Information. Further, the use of equation ( 3) followed by pulse count at a different pump-power level enables the generation of BMCs with an optimised number of elementary constituents (see Supporting Information).

Experimental results and discussion

We start with the objective of obtaining a single-breather laser regime. To this end, we fix the pump power applied to the gain fibre to 70 mW. In a first series of experiments, starting from a unique, randomised set of polarisation parameters, we generate breathers without imposing any additional constraint on the features of the breather solution that is targeted. An example of an optimisation curve is presented in Fig. 2(a), which shows the evolution of the best and average merit scores of the population, as defined by equation ( 3), for each generation. We see that the best merit score quickly increases and converges to an optimised value after only 2 generations (ie., after 6 minutes). This fast convergence to the optimal state is facilitated by the relatively large population size of the generations. The average score of the population gradually increases to converge to almost the same value. The fluctuations of the average score, between the 6 th and 9 th generations, are ascribed, in some extent, to fluctuations of the measured features of established laser states, and predominantly to the mutation process. The spectral and temporal characteristics of the optimal state are summarised in panels (b-d) of Fig. 2. The various measurements confirm the operation of the laser in the targeted mode [START_REF] Peng | Breathing dissipative solitons in mode-locked fiber lasers[END_REF] : a RF spectrum exhibiting two symmetrical sidebands around the cavity repetition frequency (Fig. 2 It is even more appealing to achieve intelligent control of the parameters of the breathers formed in the laser. By using our EA, we are able to generate breathers with tunable breathing ratio and period of oscillation. The breathing ratio is defined as the ratio of the largest to the narrowest width of the pulse spectrum within a period, Δλmax/Δλmin. Since the strength of the frequency sidebands in the RF spectrum is proportional to the breathing ratio, we can optimise the latter by taking into account the sidebands' strength in the definition of the merit function (see Supporting Information). Figure 3 shows the spectral and temporal dynamics of three examples of breathers with different breathing ratios that can be generated in the laser cavity by setting corresponding values in the merit function. They refer to the weakest breathing regime found in the cavity, which features a breathing ratio of 1.076 (Fig. 3(a)), the strongest breathing regime with a breathing ratio of 1.816 (Fig. 3(g)), and a moderate breathing regime (Fig. 3(d), breathing ratio of 1.471). By comparing the pulse spectra at the round-trip numbers of maximal and minimal spectrum extent within a period in panels (c, f, i) of Fig. 3, we can see that a larger breathing ratio is not achieved by an increased width of the widest spectrum but by a decrease of the narrowest spectrum extent. This is due to the fact that the maximum width of the spectrum is saturated by the gain bandwidth of the gain fibre. We have confirmed that 1.816 is the maximum breathing ratio achievable in the laser by manually tuning the pump power and the EPC under the single-breather mode-locked laser operation, with the results

showing that the laser mode locking is destroyed when trying to increase the breathing ratio above 1.816.

Another important parameter of the breathing solitons that it is desirable to control is their period of oscillation. Indeed, it was recently found that the breathing period is a key parameter for the characterisation of subharmonic entrainment [START_REF] Cole | Subharmonic Entrainment of Kerr Breather Solitons[END_REF][START_REF] Xian | Subharmonic Entrainment Breather Solitons in Ultrafast Lasers[END_REF] . Although this parameter is somehow related to breathing ratio, it is relevant to adapt the merit function to the specific optimisation of this feature (see Supporting Information), thus avoiding the need for an empirical manual adjustment of the gain/loss properties of the laser cavity. The results are summarised in Fig. 4, which shows the spectral and temporal dynamics of three examples of breathers with different oscillation periods. These results confirm that the designed merit function (cf. Supporting Information) can indeed be reliably used to tune the breathing period automatically. The maximum and minimum breathing periods found in the laser equal 251 cavity RTs (Fig. 4(a,b))

and 103 RTs (Fig. 4(e-f)), respectively.

In like manner to their stationary counterparts, [START_REF] Krupa | Real-Time Observation of Internal Motion within Ultrafast Dissipative Optical Soliton Molecules[END_REF][START_REF] Herink | Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules[END_REF][START_REF] Wang | Optical soliton molecular complexes in a passively mode-locked fibre laser[END_REF][START_REF] Peng | Build-Up of Dissipative Optical Soliton Molecules via Diverse Soliton Interactions[END_REF] breathing solitons in a laser cavity can interact and, within specific cavity parameter ranges, form robust multi-breather bound states, [START_REF] Peng | Breathing dissipative solitons in mode-locked fiber lasers[END_REF][START_REF] Peng | Breather molecular complexes in a passively mode-locked fibre laser[END_REF] also termed breather molecules. Soliton molecules can be generated in fibre lasers by solely increasing the pump power above the fundamental mode locking regime, with the number of solitons within a molecule scaling with the pump power. [START_REF] Krupa | Real-Time Observation of Internal Motion within Ultrafast Dissipative Optical Soliton Molecules[END_REF][START_REF] Wang | Optical soliton molecular complexes in a passively mode-locked fibre laser[END_REF] However, the procedure for the excitation of breather molecules is not so straightforward. Further to this, it is difficult to generate multi (>2)-breather complexes in a normally dispersive fibre laser cavity because in the normal-dispersion propagation regime breathers do not emit dispersive waves. [START_REF] Peng | Breathing dissipative solitons in mode-locked fiber lasers[END_REF] It is therefore of great interest to implement an EA for the generation of BMCs. To this end, we apply an optimisation procedure that involves using the merit function given in equation (3) when the pump power is set to a level that favours multi-pulse self-starting of the laser, and subsequently applying pulse count (cf. Support Information) to control the number of breathers in the established multi-breather bound states. As an example, we show here the generation of BMCs composed of two, three and four elementary breathers. The optimisation time varies from a minimum of 2 min to a maximum of ~40 min.

We first perform an EA-based search for the laser polarisation parameters leading to the formation of breather-pair molecules ("diatomic" molecules). In these experiments, the pump power is increased to 120 mW. Considering that the individuals of a population are scored only against the number of breathers constituting the formed multi-breather states, several types of breather-pair molecules with very different dynamics can be accessed. Two such examples are presented in Fig. 5, which gives the round-trip evolutions of the DFT spectra, the first-order single-shot optical autocorrelation traces computed by Fourier transform of the DFT spectra, and the relative phases within the molecules retrieved from the autocorrelation traces. [START_REF] Wang | Optical soliton molecular complexes in a passively mode-locked fibre laser[END_REF] The recording of the real-time spectral interferogram for successive roundtrips shown in Fig. 5(a) displays a very dense pattern of spectral fringes (a magnified version is given in Fig.

5(b))

. The extremely small spectral-fringe separation corresponds to a large pulse separation of 268 ps within the molecule (Fig. 5(c)). The relative phase ϕ21 between the trailing and leading breathers features an approximately linear evolution with time (number of cavity roundtrips) (Fig. 5(d), red curve). Since the slope of the phase evolution function is proportional to the intensity difference between the two bound pulses, [START_REF] Krupa | Real-Time Observation of Internal Motion within Ultrafast Dissipative Optical Soliton Molecules[END_REF][START_REF] Herink | Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules[END_REF][START_REF] Soto-Crespo | Multisoliton regime of pulse generation by lasers passively mode locked with a slow saturable absorber[END_REF] the quasi-linear phase evolution shown in Fig. 5(c) indicates an almost fixed intensity difference between the two synchronously evolving breathers, with the trailing breather being more intense than the leading one. A second type of a breather diatomic molecule found by the EA is presented in Fig. 5(e-h). In this case, we observe a significantly larger breathing of the optical spectrum, and an almost threefold reduction of the intra-molecular pulse separation (the pulse separation being 98 ps). In sharp contrast to the first molecule (Fig. 5(d)), the dynamics of the relative phase feature a pronounced oscillating behavior (Fig. 5(h), red). This indicates that the two breathers continuously exchange energy with each other. At the roundtrip numbers where the phase evolution function has extrema, the two breathers feature equal intensities, and the total energy is highest (Fig. 5(h), black).

Bound states of three breathers are also found in the laser using the EA when the pump power is increased to 150 mW. In a similar manner to the breather-pair case, the EA allows us to find different bound breather triplets, which are representative of three breather complex different from each other, and this also impacts the dynamics of the relative phases and the energy exchanges within the complexes (Supplementary Fig. 3).

Our EA can also be employed to search for bound states of four breathers in the laser under a pump power of 170 mW. complexes (Supplementary Fig. 4). It is noteworthy that other types of quadri-breather complexes, such as (3+1) and (2+2) BMCs, were also found in the laser, but the characterisation of their internal dynamics was limited by the electronic-based spectrum resolution of our system, hence only partial information was obtained (Supplementary Figs. 5 and6).

Conclusion

We have demonstrated, for the first time, the possibility of using EAs to perform search and optimisation of the breathing soliton regime in a fibre laser cavity. Through exploration of the nonlinear cavity dynamics, which can be accessed by automated control of the NPE transfer function, we have shown that composite merit functions, derived from specific features of the RF spectrum of the breather laser output, permit to achieve single-breather states with tailored parameters such as the breathing ratio and period of oscillation. At laser pump powers that favour multi-pulse emission, different types of BMCs with a controllable number of elementary constituents have also been generated automatically in the laser cavity. Our experimental setup combines a computer-interfaced single measurement device (an oscilloscope) with the quick setting of intracavity parameters through electronic polarisation control, thus allowing the EA to optimise the breather regime within a realistic time scale. While the EA concept presented here is ideally suited to mode-locked lasers, the breather-tailored merit functions designed in this work could also benefit the exploration of breather waves and related nonlinear dynamics in other systems, such as microresonators, [START_REF] Yu | Breather soliton dynamics in microresonators[END_REF][START_REF] Lucas | Breathing dissipative solitons in optical microresonators[END_REF] fibre Kerr resonators, [START_REF] Leo | Dynamics of one-dimensional Kerr cavity solitons[END_REF] and single-pass fibre systems. [START_REF] Mussot | Fibre multi-wave mixing combs reveal the broken symmetry of Fermi-Pasta-Ulam recurrence[END_REF][START_REF] Kibler | The Peregrine soliton in nonlinear fibre optics[END_REF] Contrary to the generation regimes of stationary pulses that have been mainly addressed by previous works using EAs, breathing solitons exhibit a fast evolutionary behaviour. In this respect, our work opens novel opportunities for the exploration of highly dynamic, non-stationary operating regimes of ultrafast lasers, such as soliton explosions, nonrepetitive rare events and intermittent nonlinear regimes. [START_REF] Lapre | Real-time characterization of spectral instabilities in a mode-locked fibre laser exhibiting soliton-similariton dynamics[END_REF] The generation and propagation of pulses in multimode fibre systems have recently drawn great attention. [START_REF] Wright | Spatiotemporal mode-locking in multimode fiber lasers[END_REF][START_REF] Wright | Mechanisms of spatiotemporal mode-locking[END_REF][START_REF] Jing | Spatio-temporal-spectral imaging of non-repeatable dissipative soliton dynamics[END_REF][START_REF] Guo | Real-time multispeckle spectraltemporal measurement unveils the complexity of spatiotemporal solitons[END_REF] The nonlinear multimodal interference in multimode fibres has intensity discrimination properties that can be applied in mode-locked fibre lasers to generate a variety of different types of ultrashort pulses. [START_REF] Wang | Graded index multimode fibre as saturable absorber induced by nonlinear multimodal interference for ultrafast photonics[END_REF] In these emerging laser designs, the vast parameter space makes systematic exploration impracticable, yet ideally suited to optimisation by an EA.

Another promising area of research will be the expansion of the EA approach to use a wider range of tools in the general field of machine learning. Neural networks, for example, have previously been applied to the control of pulse shaping [START_REF] Boscolo | Artificial neural networks for nonlinear pulse shaping in optical fibers[END_REF] and the classification of different regimes of nonlinear propagation [START_REF] Närhi | Machine learning analysis of extreme events in optical fibre modulation instability[END_REF][START_REF] Salmela | Predicting ultrafast nonlinear dynamics in fibre optics with a recurrent neural network[END_REF] in single pass fibre geometries, and the optimisation of white-light continuum generation in bulk media. [START_REF] Valensise | Deep reinforcement learning control of white-light continuum generation[END_REF] Their extension to active control of mode locking has already been studied theoretically. [START_REF] Baumeister | Deep learning and model predictive control for self-tuning mode-locked lasers[END_REF] This extension appears a natural next step in the field, and with these techniques it may even be possible to control a broader range of processes within the plethora of complex nonlinear dynamics of mode-locked lasers.

Supporting Information

Text: 1, Details of the mutation.

2, Merit functions for:

breathing ratio tuning; breathing period tuning;

breather molecular complex generation.

  measured across the frequency interval from f-to f+ and the width 2Δ of the frequency band centred on fr, respectively. Therefore, the difference between these two quantities relates to the intensities of the sidebands at f±1 , hence Cb represents the relative strength of the sidebands with respect to the frequency interval [f-1, f+1]. Accordingly, if Cb approaches zero, it means that fr prevails and there are no sidebands, indicating a stationary mode locking state. On the contrary, a Cb value far from zero evidences the presence of strong sidebands in the RF spectrum, indicating possible breather formation in the laser cavity. In our experiments, the RF spectrum is obtained directly from the oscilloscope that processes the fast Fourier transform of the laser output intensity recording, and a span of 381.44 KHz (fr ±190.72 KHz) is chosen for the measurements.

  (b)), and a periodic compression and stretching of the optical spectrum over cavity roundtrips (Fig.2(c)), accompanied by synchronous periodic changes of the pulse energy (Fig.2(c), white curve), peak intensity and duration of the pulse in the time domain (Fig.2(d)).

  categories: (2+1) and (1+2) BMCs, and breather-triplet molecules. A (2+1) or (1+2) BMC originates from the stable binding of a breather-pair molecule and a single breather, with the breather-pair molecule being at the leading or trailing edge of the complex, while a breather-triplet molecule ('triatomic' molecule) comprises three nearly equally spaced breathers. The results are summarised in Fig.6. In all breather complex cases, the DFT-based single-shot spectral measurements and the spatio-temporal intensity evolution clearly show a periodic breathing of the optical spectrum accompanied the synchronous variations of the pulse intensities. The pulse separations within a complex can be readily resolved from the spatiotemporal intensity map. For example, Fig.6(d)shows that the intra-molecular separation in the leading breathing-pair molecule is 86 ps, while the trailing breather is 212-ps apart from the molecule. The spectral and temporal dynamics of the three breather complexes are markedly

Figure 7

 7 depicts the dynamics of two examples of (1+3) BMCs, arising from the binding of a single breather and a breather-triplet molecule. The spatiotemporal intensity evolutions shown in panels (d) and (h) reveal very different pulse separations within the two complexes. Again, from more dedicated measurements, it is possible to gain insight into the BMCs' internal motion and appreciate the various behaviours of the different breather complexes, which differ from those usually exhibited by stationary soliton molecular

Fig. S1 :

 S1 Fig. S1: Relaxation oscillations give rise to RF sidebands.

Fig. S2 :

 S2 Fig. S2: Noise-like pulse emission also shows frequency sidebands in the RF spectrum.

Fig. S3 :

 S3 Fig. S3: Internal phase dynamics of BMCs formed of three breathers.

Fig. 1 (

 1 Fig. 1 (a) Experiment setup of the self-optimising breather mode-locked laser. WDM: wavelength-division multiplexer; EDF: erbium-doped fibre; EPC: electronic polarisation controller; PDI: polarisation dependent isolator; DCF: dispersioncompensating fibre; FPGA: field programmable gate array; DAC: digital-to-analogue converter. (b) Illustration of the EA principle. (c) Schematic of the 'roulette wheel' selection. (d) Sketch of the RF signal under breather mode locking, where fr is the cavity repetition frequency, and the sideband frequencies f±1 are a manifestation of breathers with the oscillation frequency | f±1 -fr|.

Fig. 2 (

 2 Fig. 2 (a) Evolution of the average (red circles) and maximum (blue squares) merit scores over successive generations, for the merit function given in Eq. (3). (b-d) Characteristics of the optimised state: (b) RF spectrum obtained by Fourier transform of the signal from the photodiode. (c) DFT recording of single-shot spectra over consecutive cavity round trips; the white curve represents the energy evolution. (d) Temporal evolution of the intensity relative to the average round-trip time over consecutive round trips.

Fig. 3

 3 Fig. 3 EA optimisation results for breathing solitons with a tunable breathing ratio: dynamics of breathers with (a-c) small, (df) moderate and (g-i) large breathing ratios. (a,d,g): DFT recording of single-shot spectra over consecutive cavity round trips. (b,e,h): Temporal evolution of the intensity relative to the average round-trip time over consecutive round trips. (c,f,i): Singleshot spectra at the round-trip (RT) numbers of maximal and minimal spectrum extents within a period.

Fig. 4

 4 Fig. 4 EA optimisation results for breathing solitons with a tunable oscillation period: Dynamics of breathers with (a-b) large, (c-d) moderate and (e-f) small oscillation periods. (a,c,e): DFT recording of single-shot spectra over consecutive round trips. The white curve represents the energy evolution. (b,d,f): Temporal evolution of the intensity relative to the average round-trip time over consecutive round trips.

Fig. 5

 5 Fig. 5 Typical EA optimisation results for breather-pair molecules: Dynamics of (a-d) "increasing-phase" and (e-h) "oscillatingphase" breather molecules. (a,e) DFT recording of single-shot spectra over consecutive cavity round trips. A Moiré interference pattern is visible in panel (a). (b,f) Close-up view of the DFT recording. (c,g) Evolution of the first-order single-shot autocorrelation trace over consecutive round trips. (d,h) Evolution of the phase difference between the two breathers (red curve) and the energy of the molecule (black curve) as a function of the RT number.

Fig. 6

 6 Fig. 6 Typical EA optimisation results for BMCs formed of three breathers: Dynamics of (a-d) a (2+1) BMC, (e-h) a (1+2) BMC, and (i-l) a breather-triplet molecule. (a,e,i) DFT recording of single-shot spectra over consecutive cavity round trips. The white curve represents the energy evolution. (b,f,j) Close-up view of the DFT recording. (c,g,k) Corresponding temporal intensity. (d,h,l) Temporal evolution of the intensity relative to the average round-trip time over consecutive round trips.

Fig. 7

 7 Fig. 7 Typical EA optimisation results for BMCs formed of four breathers: Dynamics of (1+3) BMCs with (a-d) large and (eh) small internal pulse separations. (a,e) DFT recording of single-shot spectra over consecutive cavity round trips. The white curve represents the energy evolution. (b,f) Close-up view of the DFT recording. (c,g) Corresponding temporal intensity. (d,h) Temporal evolution of the intensity relative to the average round-trip time over consecutive round trips.

Acknowledgements

We acknowledge the support from National Key Research and Development Program (2018YFB0407100), the National Natural Science Fund of China (11621404, 11561121003, 11727812, 61775059, and 11704123), Key Project of Shanghai Education Commission (2017-01-07-00-05-E00021), Science and Technology Innovation Program of Basic Science Foundation of Shanghai (18JC1412000) and National Key Laboratory Foundation of China (6142411196307).

Competing interests: There are no financial competing interests.