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.

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 exhibit 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 test bed to explore ubiquitous synchronization dynamics of nonlinear systems. The locked breathing frequencies feature a high signal-to-noise ratio and can give rise to dense radio-frequency combs, which are attractive for applications.

Evolving Pulse Patterns Machine‐learning tools are mainly designed to target stationary pulses in mode‐locked lasers, while the intelligent excitation of evolving pulse patterns remains largely unexplored. In article number 2100191, Junsong Peng, Heping Zeng, and colleagues implement an evolutionary algorithm for the self‐optimization of the breather regime in a fiber laser. Depending on the specifications of the merit function used for the optimization procedure, various breathing‐soliton states are obtained.

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