Farey tree and devil's staircase of frequency-locked breathers in ultrafast lasers

Nonlinear systems with two competing frequencies show locking or resonances. In lasers, the two interacting frequencies can be the cavity repetition rate and a frequency externally applied to the system. Conversely, the excitation of breather oscillations in lasers naturally triggers a second characteristic frequency in the system, therefore showing competition between the cavity repetition rate and the breathing frequency. Yet, the link between breathing solitons and frequency locking is missing. Here we demonstrate frequency locking at Farey fractions of a breather laser. The winding numbers show the hierarchy of the Farey tree and the structure of a devil's staircase. Numerical simulations of a discrete laser model confirm the experimental findings. The breather laser may therefore serve as a simple model system to explore universal synchronization dynamics of nonlinear systems. The locked breathing frequencies feature high signal-to-noise ratio and can give rise to dense radio-frequency combs, which are attractive for applications.


Introduction
Nonlinear systems with two competing frequencies show locking or resonances, in which the system locks into a resonant periodic response featuring a rational frequency ratio 1 . The locking increases with nonlinearity, and at subcritical values of the nonlinearity, the system has quasiperiodic responses between locked states, whilst the supercritical system may exhibit chaotic as well as periodic or quasi-periodic responses. A general feature of frequency locking is the robustness of the locked states to variations of system parameters, namely, the constancy of the frequency ratio (or winding number) over a range of parameters. Resonances have been investigated theoretically and experimentally in many physical systems including coupled oscillators 2 , charge-density waves 3 , Josephson junctions 4 , 5 and the Van der Pol oscillator 6 amongst others 7 , and their distribution in parameter space in the form of a devil's staircase 8 is well understood from the number theory concept of Farey trees 9,10,11,12,13,14 . In optics, frequency-locking phenomena have been extensively studied in modulated semiconductor lasers, where an external frequency can be readily coupled to the nonlinear system by using a radio-frequency (RF) source 11,15,16,17,18 , and the hierarchy of the Farey tree and structure of a devil's staircase can be rather easily observed when tuning the external frequency 11 . Frequency locking has also been demonstrated in other laser structures, such as fibre lasers with external loss modulation 19 or solid-state lasers operating in a two-mode regime 20 . Furthermore, although not explicitly mentioned by the authors, the subharmonic, harmonic and rational harmonic operation regimes of Kerr micro-resonators that were reported in Refs. 21,22 imply a frequencylocking process. The generation of soliton molecules (i.e., stable bound states of two solitons) in a titanium-sapphire laser that was reported in Ref. 23 also evidences the occurrence of frequency locking: a subharmonic response of the soliton molecule was observed when the strength of the external driving force exceeded a certain threshold.
All the frequency-locking examples mentioned above relate to nonlinear systems where an external, accurately controllable modulation adds a new characteristic frequency to the system.
Far less is experimentally known, by comparison, when the second frequency is not externally controlled and is intrinsic to the nonlinear system. This is particularly relevant to breathing solitons that have recently emerged as a ubiquitous mode-locked regime of ultrafast fibre lasers 24,25,26,27,28 . Breathing solitons, manifesting themselves as localised temporal / spatial structures that exhibit periodic oscillatory behavior, are found in various subfields of natural science, such as solid-state physics, fluid dynamics, plasma physics, chemistry, molecular biology and nonlinear optics 29 . Optical breathers were first studied experimentally in Kerr fibre cavities 30 and subsequently reported in optical micro-resonators 21,31,32 . They are currently attracting significant research interest in virtue of their connection with a range of important nonlinear dynamics, such as rogue wave formation 33,34 , the Fermi-Pasta-Ulam recurrence 35,36,37 , turbulence 38 , chimera states 39,40 , chaos 41 and modulation instability phenomena 42 . From a practical application perspective, breathers can increase the resolution of dual-comb spectroscopy 43 as the breathing frequency comes along additional tones in a frequency comb, and the breather regime in a laser oscillator can be used to generate high-amplitude ultrashort pulses without additional compressors 44,45 .
In this paper, we present the first in-depth study of the locking of breather oscillations to the cavity repetition frequency in a fibre laser. Besides the hurdle represented by the absence of an external driver to realize frequency locking, the excitation of breathing solitons in a fibre laser requires fine tuning of the laser parameters, where the breather mode-locking regime exists in a narrower parameter space than stationary mode locking 44 . Therefore, targeting frequencylocked breather states in the laser via trial and error is a laborious task. Here we show that such a difficulty can be circumvented by using an evolutionary algorithm (EA) based on the optimal parameter tuning of the intracavity nonlinear transfer function through computer-controlled polarisation control. Machine-learning strategies, referring to the use of statistical techniques and numerical algorithms to carry out tasks without explicit programmed and procedural instructions, are widely deployed in many areas of engineering and science 46 . In the field of ultrafast photonics, machine-learning approaches and the use of genetic and evolutionary algorithms have recently led to several dramatic improvements in dealing with the multivariable optimisation problem associated with reaching desired operating regimes in fibre lasers. In the present study, the merit function used in the EA optimisation procedure can distinguish between frequency-locked and unlocked breather states, thereby enabling fast and precise tuning of the laser to the target frequency-locked breather operation. The locked breather states show two unambiguous features: persistence under pump power and polarisation perturbations, and narrow linewidth and high signal-to-noise ratio (SNR) of the oscillation frequency in the electrical spectrum of the laser emission. Importantly, frequency-locked states occur in the sequence they appear in the Farey tree and within a pump-power interval given by the width of the corresponding step in the devil's staircase. This demonstrates that breather mode-locked fibre lasers exhibit the universal properties characteristic of nonlinear systems driven by two competing frequencies.

Frequency-locked and unlocked breathers in the laser
To investigate the dynamics of breathers, we have built the fibre ring cavity that is sketched in

Evolutionary algorithm optimisation of frequency-locked breathers
Reaching a frequency-locked breather state in our laser depends on precisely adjusting four parameters: the pump strength and three polarisation controllers, which is quite difficult to do manually. In Ref. 49 , we have introduced an approach based on an EA for the search and optimisation of the breather mode-locking regime in ultrafast fibre lasers, which relies on specific features of the RF spectrum of the breather laser output. In the self-tuning regime, the operation state of the laser is characterised in real time with the oscilloscope, which is connected to a computer running the EA and controlling the polarisation state through the voltages applied on the LCs via a driver to lock the system to the desired breather regime (Figs.

1(a) and (b)
). Yet, the merit function of the breather mode locking used in Ref. 49 is unable to distinguish between frequency-locked and unlocked breather states, where it usually breeds unlocked (unstable) states which have a wider parameter space. Here, we further develop our approach to directly pinpoint frequency-locked breathers so that the EA tunes the laser to these states only. To this end, we define a new merit function which takes into account the distinguishing trait of frequency-locked breather states, namely, a high SNR of the breathing frequency as shown in Fig. 3(a, b). The new merit function is given in Eq. (2) in the "Methods" section. An example of an optimisation curve (referring to a breather state with a winding number of 1/5) is presented in Fig. 4(a), which shows the evolution of the best and average merit scores of the population, as defined by Eq. (2)

Farey tree and devil's staircase of the breather laser
Benefiting from a reliable and efficient EA-based optimisation approach, we have explored the transitions between the different breather states of the laser that can be accessed by varying the pump power starting from the range corresponding to a 1/5 frequency-locked state.   states. Therefore, even though the winding numbers 7/32 and 9/41 display only one point in Fig. 5(a), they can clearly be identified in this map, which reveals a richness of detail that has been largely overlooked in previous studies due to lack of high-quality RF spectral measurements. It is also noteworthy that changing the pump power by only 10% is enough to find seven frequency-locked states for the laser, whose power-stability properties are dictated by a devil's staircase. As a further note, we would like to emphasize that the frequency-locked states observed are reproducible but not self-starting, meaning that if the pump power is turned off when the laser operates in a locked state and then it is turned back on again, the laser does not return to that state instantaneously. To restore the frequency-locked operation, one can run the EA controlling the polarisation states again, which will quickly reset the laser to the desired state. Many such experimental tests have confirmed the reproducibility of the locked states.  systems. The EA approach used in this paper could benefit the control of the frequency-locking process in such systems as well as in others. We also believe that our EA-based approach for the control of frequency locking in fibre lasers is not restricted to NPE-based configurations and can be extended to other laser mode-locking schemes that entail period multiplication, such as the Mamyshev oscillator 51,52 .
Optical breathing solitons have been extensively studied in open-loop nonlinear systems such as, for example, single-pass fibre systems 36,53 . However, in the absence of a frequencylocking mechanism, these breathers may suffer from instabilities originating from the noise of the input light. By contrast, we have studied the dynamics of breathers in a closed-loop system -a laser resonator. In this system, the universal frequency-locking process is tailored through the nonlinear interaction between the cavity repetition frequency provided by the laser resonator and the breathing frequency. Ergo, frequency-locked breathers can be generated, showing excellent stability against cavity parameter perturbations.
Frequency-locked breathers give rise to wide and dense RF combs which are not constrained by the length of the laser cavity. We have shown an example of a comb where the density of the spectral lines is increased by a factor of 41 compared with stationary single-pulse mode locking, thus enabling a line spacing in the sub-MHz range. Another example of a dense comb (where the density increase factor is 35) is given in Supplementary Fig. 4. Therefore, representing an alternative to fibre cavities of hundreds of meters which are regarded as being highly unstable, controlled frequency-locked breather lasers are attractive for many applications, for instance, in high-resolution spectroscopy.
We note that subharmonic entrainment of breather oscillations to the cavity repetition rate in a fibre laser was recently reported and explained as arising between the exceptional points of a non-Hermitian system involving two coupled modes with different detunings 26  Dispersion plays an important role in determining the pulse dynamics in ultrafast fibre lasers 54,55,56 . The laser cavity used in this work has a nearly zero net dispersion. We have observed that frequency locking of breathers does not occur when the laser is operated at moderate or large normal dispersion 24,49 . Thus, a very small net cavity dispersion seems to be crucial to the emergence of frequency-locked breathers in a fibre laser. It is worth noting in this regard that breathing solitons at nearly zero net dispersion and at large normal dispersion differ quite significantly in respect to their period of oscillation. Indeed, the former oscillate with a period ranging from several to dozens of round trips while the latter generally feature a much longer period of the order of hundreds of round trips 24 , indicating that the underlying formation mechanism could be different. Future work will thoroughly investigate the connection between the frequency locking mechanism and the cavity dispersion.

Methods
Farey tree. The Farey tree represents a particular ordering of the rational numbers by applying the Farey-sum or median operation ⨁ to two neighboring fractions, m/n and p/q, which gives a new fraction in the next lower level of the tree by adding the numerators and the denominators separately: ⨁ = + + . The physically motivated hypothesis invoked to explain the local ordering of the hierarchy of (two-frequency) resonances is that the larger the denominator, the smaller the plateau. The Farey fraction or Farey mediant is the fraction with smallest denominator between m/n and p/q, if they are sufficiently close that | − | = 1 -when they are called adjacents -hence it is the most important resonance in the interval. The Farey tree provides a qualitative local ordering of two-frequency resonances and gives rise to a curve with an infinite number of steps showing self-similarity, which is known as the devil's staircase.
For a detailed review see, e.g., Ref. 10 . 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. In the present work, we have defined and tested the following merit function for the auto-setting of an optimised self-starting frequency-locked breather regime: where N is the number of laser output intensity points recorded by the oscilloscope (N=2 24 , corresponding to a time trace of 7174 cavity round-trips), Ii is the intensity at point i and Ith is a threshold intensity that noise should not exceed. Thus, Fml represents the average of pulses' intensities, and is used to exclude laser modes, such as relaxation oscillations and noise-like pulse emission, which may display similar RF spectral features to the breather regime. The second term Fb is a merit function that discriminates between breather and stationary pulsed operations, derived from the feature that the breathing frequency b manifests itself as two symmetrical sidebands ±1 around the cavity repetition frequency r in the RF spectrum of the laser output ( b = | ±1 − r |).There are no sidebands when the laser works in a stationary mode-locking regime. Therefore, Fb is designed to exploit the intensity ratio of the central band , where 0 is the unsaturated loss due to the absorber, m is the saturable loss (modulation depth), ( , ) = | ( , )| 2 is the instantaneous pulse power, and sat is the saturation power. Linear losses are imposed after the passive fibre segments, which summarise intrinsic losses and output coupling. The numerical model is solved with a standard symmetric split-step propagation algorithm and using similar parameters to the nominal or estimated experimental values (see Supplementary Table 1).

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability
The code that support the findings of this study are available from the corresponding author on request.