Fig. 1. Concept and demonstration of resonant excitation of nonlinear dynamics of soliton molecules. (a) Experimental setup of the mode-locked fiber laser: LD, laser diode; WDM, wavelength division multiplexer; EDF, Er-doped fiber; OC, output coupler; Col., collimator; PBS, polarization beam splitter; QWP, quarter-wave plate; HWP, half-wave plate; ISO, isolator. Driving module: CW, continuous wave; EDFA, erbium-doped fiber amplifier; F.G., function generator; and EOM, electro-optic modulator. (b) Mode-locking status monitoring module: PD, photodiode; OSC, oscilloscope; ESA, electrical spectrum analyzer; OSA, optical spectrum analyzer. (c) BOC system used to detect variations in intrapulse separation. (d) The resonant excitation induces forced oscillations in the intrapulse separation, implying that the system periodically traverses different positions on the anharmonic potential curve under external driving. (e) Schematic of resonance frequency shifts induced by Duffing-type nonlinearity. (f) Schematic of harmonic/subharmonic responses and chaotic dynamics. ωd: driving frequency; ωr: eigenfrequency.

Fig. 2. Experimental measurements of the mode-locked state and resonant excitation of soliton molecules. (a) Optical spectrum and (b) pulse sequence. (c) Response of intrapulse separation under external driving detected by the BOC system. Resonant pulse energy for a driving period is also shown as the brown curve. (d) Resonant susceptibility χ(f) with Fano line shape fitting, where the red curve represents phase evolution. The driving frequency is also presented as the gray curve. (e) The two-dimensional spectra extracted from the short-time Fourier transform of (c) display fundamental response and overtones. (f) RF spectrum of the laser repetition rate under external driving.

Fig. 3. Simulation of resonant excitation of soliton molecules under 0.1% external driving. (a) Resonance response of intrapulse separation, with the pulse energy response shown by the brown curve. The inset illustrates the energy variation of each pulse within the molecule, highlighting a high consistency. (b) Spectral evolution under frequency sweeping, with the white box highlighting an enlarged view at the resonance region. The orange curve represents the single-shot spectra. (c) Two-dimensional spectra extracted from the short-time Fourier transform of intrapulse separation in (a). (d) Resonant susceptibility χ(f) with Fano line shape fitting, where the red curve represents phase evolution. Inset: the corresponding Nyquist plot in the complex plane. (e) Transient behavior of intrapulse separation transitioning from no driving to stable forced oscillations, with the driving frequency fixed at 5.8 MHz.

Fig. 4. Simulation of resonant excitation of soliton molecules under driving strengths ranging from 0.5% to 1.2%. (a) Resonant susceptibility χ(f). (b) Trend of maximum response amplitude and its linear fit. (c) Backbone curve with linear and Duffing equation fitting. (d) Fano fittings of the resonant susceptibility. Inset: phase evolution. (e) Nyquist plots in the complex plane.

Fig. 5. Subharmonic response and deterministic route to intramolecular chaos obtained by sweeping driving strength far from the fundamental resonance region. (a) Intrapulse separation response. Inset: an enlarged view of the boxed region shows a distinct amplitude threshold, above which the subharmonic response dominates. The transition from the fundamental to subharmonic modes manifests period-doubling bifurcation characteristics. (b) Two-dimensional spectra extracted from the short-time Fourier transform of intrapulse separation in (a). The dashed lines spanning (a) and (b) indicate the system’s operation across different modes. (c) Chaotic separation response at a fixed driving strength of 11%. (d) Lyapunov exponent analysis of intrapulse separation in (c). Inset: enlarged view of the linear region. (e) Chaotic spectral evolution.