Nonlinear dynamics in breathing-soliton lasers

TL;DR

Breathing solitons in mode-locked fibre lasers provide a powerful testbed for nonlinear dissipative dynamics, enabling detailed study of two-frequency interactions and complex synchronization. The authors outline measurement-to-model workflow, contrasting the CQGLE master-equation framework with a lumped NLSE-based model and showcasing advanced diagnostics such as time-stretched dispersive Fourier transform. Key contributions include the discovery of frequency-locked breathers organized by Farey-tree fractal hierarchies, unconventional Arnold tongues with holes, and the formation of breather molecular complexes, all controllable via genetic-algorithm optimization. The work highlights practical implications for dense RF-comb generation and ultrafast laser control, while pointing to broader relevance for multi-frequency nonlinear dynamics and potential extensions to spatiotemporal mode-locked systems and hyperchaotic regimes.

Abstract

We review recent advances in the study of nonlinear dynamics in mode-locked fibre lasers operating in the breathing (pulsating) soliton regime. Leveraging advanced diagnostics and control strategies -- including genetic algorithms -- we uncover a rich spectrum of dynamical behaviours, including frequency-locked breathers, fractal Farey hierarchies, Arnold tongues with anomalous features, and breather molecular complexes. We also identify a novel route to chaos via modulated subharmonic states. These findings underscore the utility of fibre lasers as model systems for exploring complex dissipative dynamics, offering new opportunities for ultrafast laser control and fundamental studies in nonlinear science.

Figures (9)

• Figure 1: Experimental setup. Schematic of a typical fibre laser cavity used to generate and characterise breathing solitons. The setup includes a set of diagnostic tools for detailed observation of the pulsating structures, as well as passive and active components enabling mode locking via nonlinear polarisation rotation. FPC, fibre polarisation controller; POL, polariser; COL, collimator; QWP/HWP, quarter-/half-wave plates; PBS, polarisation beam splitter; LC, liquid crystal phase retarder; EPC, electronically driven polarisation controller; DAC, digital-to-analog converter.
• Figure 2: Typical properties of a breathing soliton with a long pulsation period, observed in a laser cavity operating at normal average dispersion. The cavity repetition rate is 16.765$\,$MHz. (a) Temporal evolution of the intensity relative to the average over successive cavity round-trips, recorded using a 50-GHz photodiode with a 20-ps response time and a 33-GHz bandwidth oscilloscope operating at an 80-GSa/s sampling rate. (b) DFT measurement of single-shot spectra over consecutive round-trips; the white curve indicates the pulse energy evolution. The accumulated dispersion is $-1200,\mathrm{ps/nm}$, yielding a spectral resolution of 0.025$\,$nm. (c) RF spectrum obtained by Fourier transformation of the photodiode signal. Data adapted from LPR_Wu_2022, acquired following laser optimisation via a GA.
• Figure 3: Genetic algorithm principles. (a) Flow chart of the algorithm; (b) "Roulette wheel" selection diagram. Results adapted from LPR_Wu_2022.
• Figure 4: (a, b) Experimental characterisation of synchronised and unsynchronised breathing-soliton states. (a1, b1) Photodetected dispersive Fourier transform (DFT) signals captured over consecutive cavity roundtrips ($T_{\mathrm{r}}$ denotes the roundtrip time). (a2, b2) Corresponding single-shot DFT spectra; white curves trace the energy evolution. (a3-a4, b3–b4) Associated RF spectral measurements. The synchronised state (a3–a4) shows a single-mode oscillation at the subharmonic breathing frequency over spans of 50$\,$kHz and 100$\,$Hz. In contrast, the unsynchronised state (b3–b4) exhibits unstable multimode oscillation of a non-subharmonic breathing frequency over 50-kHz and 10-kHz spans. The reference frequency corresponds to one-fifth of the fundamental repetition rate. (c) Experimental observation of frequency locking: RF spectrum of the laser output versus pump power, showing the sequential emergence of rational winding numbers. Adapted from NC_Wu_2022.
• Figure 5: Farey tree and devil's staircase. (a) Measured breathing frequency (winding number) plotted as a function of pump power. The inset shows the relevant portion of the Farey tree, with the observed winding numbers highlighted in blue. (b) RF spectra corresponding to frequency-locked states with winding numbers 1/5, 2/9, and 9/41, respectively. In each case, a set of equidistant spectral lines emerges within the frequency span defined by the cavity repetition rate $f_{\mathrm{r}}=34.2\,$MHz. (c) Simulated breathing frequency (winding number) as a function of the gain saturation energy, varied with step sizes of 10$\,$pJ and 1$\,$pJ, respectively. The finer step reveals additional plateaux, indicating a fractal structure in the frequency-locking behaviour. Insets display the relevant sections of the Farey tree with the observed Farey fractions. Adapted from NC_Wu_2022.