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

Abstract

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.

Fig. 1. The experimental configuration.

a The breather fibre laser setup. WDM wavelength-division multiplexer, EDF erbium-doped fibre, ISO isolator, LC liquid crystal phase retarder, PBS polarisation beam splitter, Col collimator, OC optical coupler, NDF normally dispersive single-mode fibre involved in the dispersive Fourier transform measurements, PD photodetector, OSC oscilloscope, CYM cymometer, ESA electronic spectrum analyser, OSA optical spectrum analyser. b The flowchart of the evolutionary algorithm.

Fig. 2. Two different breather operations of the laser observed over 50 cavity roundtrips.

a, b Frequency-locked breather state showing a well-defined periodicity, and c, d frequency-unlocked breather state featuring degraded periodicity. Panels a, c show the photo-detected DFT (dispersive Fourier transform) output signals (T_r is the round-trip time), and panels b, d are the corresponding DFT recordings of single-shot spectra. The white curves in b, d represent the energy evolutions.

Fig. 3. RF spectral measurements of the breather states shown in Fig. 2.

The reference frequency is one-fifth of the fundamental repetition frequency. a, b Single-mode oscillation of the breathing frequency when frequency locking occurs measured over spans of 50 kHz and 100 Hz, respectively. c, d Unstable multimode oscillation of the breathing frequency measured over 50-kHz and 10-kHz spans. e Change in breathing frequency over time for the locked (red) and unlocked (blue) breather states, as measured with a cymometer. The standard deviation (SD) of the breathing frequency values is 2.05 Hz for the frequency-locked state and 7175.78 Hz for the unlocked state.

Fig. 4. The machine-learning results.

a Evolution of the mean (black diamonds) and best (blue squares) merit scores of the individuals for each generation. Also shown is the corresponding evolution of the SNR (signal-to-noise ratio) of the breathing frequency (red circles). Persistence of the optimal state (frequency-locked breathers) with a variation of the b pump power and c polarisation (varied by changing the voltage on the liquid crystal LC2).

Fig. 5. Farey tree and devil’s staircase measured in experiments.

a Measured breathing frequency (winding number) as a function of the pump power. In the inset is shown the part of the Farey tree containing the observed winding numbers (blue). b–d RF spectra measured with the ESA showing dense frequency combs for the frequency-locked states corresponding to the winding numbers 1/5, 2/9 and 9/41, respectively. A new set of equidistant spectral lines fills in the frequency interval corresponding to the cavity repetition rate $f_{\mathrm{r}}$ (34.2 MHz). e Map of spectral intensity in the space of radio-frequency and pump power, showing the build-up of rational winding numbers.

Fig. 6. Farey tree and devil’s staircase observed in the numerical simulations.

a, b Breathing frequency (winding number) as a function of the gain saturation energy E_sat (related to the pump power in the experiment) varied with a step of 10 and 1 pJ, respectively. With the smaller step, more plateaux are observed, evidencing a fractal pattern. In the insets are shown the parts of the Farey tree containing the observed Farey fractions. Since the plateau representing winding number 4/19 is very narrow, it is magnified in the inset in panel (b). c Map of spectral intensity in the space of radiofrequency and gain saturation energy, showing the build-up of rational winding numbers.