
- Advanced Photonics
- Vol. 7, Issue 1, 016005 (2025)
Abstract
Keywords
1 Introduction
Nonlinearity is an intrinsic characteristic of dynamic systems, driving research across a broad range of disciplines, from physics and mechanical engineering to biology, and spanning scales from macroscopic to microscopic.1,2 Self-assembly and emergent many-body dynamics driven by nonlinear physics represent a frontier in modern science. These areas explore how complex systems—composed of multiple interacting components—exhibit behaviors due to intrinsic nonlinearity in their governing mechanisms.3
Passively mode-locked fiber lasers serve as ideal platforms for studying nonlinear physics, primarily due to two key factors. First, the ultrashort pulse sequences they generate possess extremely high peak powers, accentuating nonlinear effects, such as dissipative Talbot effects and modulation instability.19,20 Second, interactions between these ultrashort pulses can lead to the formation of compact soliton molecules, a phenomenon that, owing to its rich internal dynamics, has recently garnered significant attention for its profound parallels with related fields such as fluid dynamics and complex network systems.21
Existing studies on soliton molecule dynamics, such as the buildup process and spontaneous oscillations,38
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In this work, we venture into an unexplored regime of soliton molecule dynamics, where the concept of driven damped resonators is applied, for the first time to our knowledge, to intramolecular systems in ultrafast lasers. Through experimental observations and simulations, we reveal the susceptibility of Fano-type resonance in intrapulse separation, with resonance amplitudes confined to sub-100 fs scales, facilitated by the effective coupling between external modulation and laser gain. Here, we address a crucial challenge: attaining precise modulation of soliton molecule dynamics via external cavity injection to explore intramolecular nonlinear physics while preserving the original laser parameters and bound states structure under stable mode-locking conditions. We further confirm that resonance susceptibility exhibits a strong dependence on the amplitude of motion governed by nonlinear rather than linear dynamics. We uncover the Duffing-type nonlinearity in dissipative soliton interactions by modeling the backbone curve using a driven Duffing equation. This mechanism, tied to an anharmonic binding potential, enables direct probing of the overtones/subharmonic, even chaotic responses within intramolecular dynamics.
2 Results
2.1 Experimental Setup
Figure 1 illustrates the laser configuration and detection system employed in investigating the nonlinear response of soliton molecules. A typical dispersion-managed passively mode-locked fiber laser is used to generate multipulse states, utilizing a mode-locking mechanism based on nonlinear polarization rotation (NPR). A 980/1550 nm wavelength division multiplexer connects the laser diode and the gain fiber (Liekki Er 110-4/125), efficiently coupling the pump light into the laser cavity. The free-space components consist of three wave plates, an isolator, and a polarizing beam splitter, utilized, respectively, for polarization adjustment, ensuring unidirectional light transmission, and outputting the laser signals. The laser operates in a near-zero, slightly negative dispersion regime, with a net dispersion of approximately
Figure 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.
The mode-locked pulse train, optical spectrum, and radio-frequency (RF) spectrum of the laser repetition rate are characterized by an oscilloscope (Agilent Infinium), optical spectrum analyzer (Yokogawa AQ63700), and electrical spectrum analyzer (Rigol DSA815), respectively. A flippable mirror is utilized to direct the laser output into the BOC system [see Fig. 1(c)], enabling the detection of intramolecular dynamics. Scanning the movable mirror in one arm of the Michelson interferometer results in an S-shaped curve, with the linear region (marked in green) corresponding to the zero crossing. For details on the principle and adjustment of the BOC system, refer to our paper,48 in which the subfemtosecond resolution of the BOC technology was experimentally demonstrated. By implementing precise control techniques and high-resolution real-time observation methods, we have experimentally observed the resonant response of pulse separations within soliton molecules. The susceptibility of this response exhibits a pronounced shift with increasing resonance strength, providing deep insights into the anharmonic potential governing the interactions between optical solitons [Fig. 1(d)]. We achieved an excellent fit of the resonance frequency shift using the Duffing equation, indicating that Duffing-type nonlinearity plays a crucial role in the nonlinear resonant response within the molecule when the soliton molecule is analogized to a driven damped oscillator [Fig. 1(e)]. Furthermore, we successfully trigger a subharmonic response away from the fundamental resonance frequency. Under appropriate driving strengths, the subharmonic and chaotic response can dominate intramolecular dynamics, reflecting the multiple thresholds required for different oscillating modes in soliton interactions [Fig. 1(f)].
2.2 Experimental Resonant Excitation of Soliton Molecules
Stationary soliton molecules exhibiting constant intrapulse separation are realized by adjusting the pump strength within an appropriate range, specifically
Figure 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
Next, we proceed with the resonant excitation of the intrapulse separation in soliton molecules. The stationary state with constant separation depicted in Fig. 2(a) is employed as the initial state. We apply a logarithmic frequency-swept intensity-modulated sinusoidal signal with a central wavelength set within the gain spectrum of the Er:fiber at 1530 nm, and an average power of
The frequency-dependent susceptibility
2.3 Simulation Results
To verify the generality of the intramolecular resonance response and to uncover its deeper physical insights, we conduct numerical simulations using the scalar generalized nonlinear Schrödinger equation. 52 This equation incorporates fundamental physical effects within the laser cavity, including dispersion, self-phase modulation, finite gain bandwidth, and saturable gain. The key components of the model include an EDF as the gain fiber, two segments of single-mode fiber, and an output coupler. A fast saturable absorber model is employed to achieve the mode locking. The gain dynamics are controlled via round-trip—varying modulation of the small-signal gain coefficient
Figure 3(a) shows the resonant response of the internal dynamics of soliton molecules at a modulation depth of 0.1%. Both the intrapulse separation and pulse energy exhibit clear resonant responses, which are in high agreement with the experimental results. Additionally, we examine the consistency of energy fluctuations for each pulse within the molecule during the resonance process. The inset shows that pulses 1 and 2 always experience synchronized energy variations. This indicates that the resonant response of the intramolecular system is not directly but rather indirectly related to the gain dynamics, reflecting an intrinsic nonlinear dynamical behavior. Perturbations in gain simultaneously affect the energies of both pulses, leading to changes in the underlying interaction potential and ultimately driving the resonance in the intrapulse separation. Figure 3(b) shows the corresponding spectral evolution with the orange-highlighted spectral profile. Away from the resonance frequency, spectral changes are nearly imperceptible. However, the spectrum undergoes clear periodic broadening and narrowing at the resonance region, exhibiting a distinctive breathing pattern. Following the experimental procedure, we apply a short-time Fourier transform to the response of the intrapulse separation, revealing a distinct second-harmonic response in Fig. 3(c). Higher-order harmonics, such as third-harmonic observed in the experiment, do not manifest due to the minimal driving modulation depth of only 0.1%. In line with the experimental results, the susceptibility in Fig. 3(d) also follows a Fano line shape with a phase shift of
Figure 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
As a further note, it is inherently interesting to observe that the resonance frequency is significantly higher than the experimentally observed values. In our previous work, we applied semiclassical noise theory to analyze stationary soliton molecules under the influence of amplified spontaneous emission, focusing on their quantum noise limit.36 Although this was not explicitly highlighted, we indeed observe a characteristic eigenfrequency, which was designated as the cutoff frequency. Our prior experimental and simulation results demonstrated a pronounced correlation between this eigenfrequency and the intrapulse separation, which well supports the divergence observed between current experimental and simulation results: a longer separation, ranging from a few picoseconds to several tens of picoseconds, corresponds to a significantly lower cutoff frequency, which can drop to just a few kilohertz or even below 1 kHz. Conversely, a shorter separation (0.9 ps in the simulation) corresponds to a much higher cutoff frequency reaching several megahertz. Here, our simulation results are not intended for quantitative comparison with the experiment but are designed to qualitatively support the general resonance dynamics within soliton molecule dynamics.
After successfully validating the experimentally observed resonance dynamics of intrapulse separation, we further investigate the evolution of the resonance response under strong driving conditions in the simulation model. This scenario is particularly challenging to achieve in experimental settings due to the detrimental effect of experimental noise, which diminishes the robustness of soliton molecules to resist intense external perturbations. As illustrated in Fig. 4(a), the waveforms of the susceptibilities remain stable across varying driving strengths. The peak amplitude increases with driving strengths, with its linear growth trend depicted in Fig. 4(b). Notably, a significant shift in the resonance frequency is observed in Fig. 4(a), which indicates a strong dependence of the resonance characteristics on the driving strength. Such behavior is interpreted within the framework of nonlinear physics and is associated explicitly with the softening effect induced by Duffing-type nonlinearity.3
Figure 4.Simulation of resonant excitation of soliton molecules under driving strengths ranging from 0.5% to 1.2%. (a) Resonant susceptibility
We extract the resonance frequencies and plot them on the vertical axis against the maximum amplitudes on the horizontal axis to construct the backbone curve. The result highlights a distinct critical amplitude, beyond which the system, driven with an amplitude significantly exceeding typical operational conditions, manifests pronounced nonlinear effects, marking the dominance of high amplitude-driven operations. As depicted in Fig. 4(c), the resonance frequency decreases with increasing response amplitude, exhibiting a behavior that significantly deviates from a linear fitting (dashed black line). We model this by solving the amplitude for a driven Duffing oscillator,
Equation (2) is used to fit the backbone curve of the resonant soliton molecules, achieving a high degree of consistency, as indicated by a fitting
Next, we shift our focus to the frequency range far from the fundamental resonance mode. This approach allows for significantly stronger driving without destabilizing the binding structure of molecules due to excessive resonance effects. We set the driving frequency to 10 MHz, approximately twice the eigenfrequency, and increase the driving strength from 5% to 13%. The resulting separation response is shown in Fig. 5(a), with its short-time Fourier transforms presented in Fig. 5(b). Within the driving strength range of 5% to 8%, the response amplitude increases smoothly with increasing driving strength. However, when the driving strength surpasses 8%, a distinct critical behavior is observed, characterized by a rather sharply defined order-of-magnitude increase in the amplitude and the emergence of a subharmonic frequency at 5 MHz. This behavior indicates that the system has crossed a critical subharmonic threshold, where second-order nonlinear interactions become sufficiently strong to initiate and sustain oscillations at half the driving frequency (5 MHz). Below this threshold, the driving force at 10 MHz primarily excites the fundamental mode, with energy transfer to other modes, including the subharmonic, being too weak to overcome critical damping. However, as the driving amplitude surpasses the threshold, the nonlinear coupling between the fundamental and subharmonic modes intensifies, leading to more efficient energy transfer. This results in a bifurcation in the separation response, where the subharmonic mode not only emerges but stabilizes, increasing in amplitude until it can coexist with or even dominate the oscillation.54
Figure 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.
We observe extreme sensitivity at the bifurcation point in the full-record spectrogram, where pronounced transient instabilities emerge and dissipate. These transient processes are frequently accompanied by spontaneous self-modulation and occur during transitions between various oscillation modes.55 As we further increase the driving strength to 9%, beyond the subharmonic threshold, an additional frequency component
To provide quantitative evidence of chaos, we perform a Lyapunov exponent analysis on the response signal of intrapulse separation obtained with a fixed driving strength of 11%. Lyapunov exponents quantify the rate of exponential divergence between nearby trajectories in phase space, which is essential for identifying chaotic dynamics. As shown in Fig. 5(d), the average initial logarithmic divergence of neighboring trajectories follows a linear trend, confirming the presence of chaotic dynamics through positive maximal Lyapunov exponents.57 The enlarged view of the linear region is shown in the inset. Finally, the spectral evolution corresponding to Fig. 5(c) is presented in Fig. 5(e), revealing pronounced spectral fluctuations accompanied by oscillations in the interference fringes. These oscillations deviate from the regular breathing periodicity characteristic of breather molecules,39 instead exhibiting erratic and unpredictable oscillatory behavior. We note that the emergence of chaos here does not follow the classic, well-defined routes, but rather involves a new modulated subharmonic route characterized by a “subharmonic–quasi-periodic–chaotic” sequence, highlighting an intrinsic complexity analogous to that of low-dimensional nonlinear systems, as observed in dissipative solitons.58,59
3 Discussion
Our work casts light on a series of unresolved questions about soliton molecule dynamics. First, what are the eigenfrequencies involved in intramolecular dynamics? Our combined experimental and simulation efforts reveal key aspects of the intramolecular resonance mechanism within the framework of nonlinear dynamics. Stationary soliton molecules exhibit strong internal damping that exceeds the critical threshold, thereby suppressing spontaneous oscillatory behavior such as limit-cycle dynamics. However, supercritical bifurcations occur when driven externally with sufficient strength, leading to sustained forced oscillations. Upon cessation of the driving force, the system naturally returns to a fixed-point attractor, characterized by stable intrapulse separation, a behavior consistent with overdamped systems. This reversion is confirmed by gradually reducing the driving signal to zero. These eigenfrequencies are intrinsic to the solitonic bound state and are solely determined by the characteristics of the underlying binding potential, such as the damping properties. Although they can be excited through modulation of the laser gain, they do not directly depend on gain dynamics. Similar observations have been confirmed in composite dissipative Kerr solitons within Kerr resonators.47
Second, how can soliton molecules be driven to achieve resonance in fiber lasers? Previous works have developed pump current modulation strategies that have been successfully applied to switching between discrete soliton doublet states and phase-tailored assembly.44,45 However, considering that the resonance response of the intrapulse separation is minimal (
Last, what are the manifestations of soliton nonlinear interactions? Here, we observe a range of intriguing nonlinear phenomena near and far from the fundamental resonance region, including resonance frequency shifts, which exhibit a universal nature that could be characterized using the Duffing model. Additionally, we explore the critical subharmonic response dynamics across varying drive strengths, underscoring the intricate nature of intramolecular dynamics. By leveraging multifrequency nonlinear mode coupling, we induce a distinct chaotic response through strong external driving.
4 Conclusion
Although soliton molecules have been discovered in ultrafast lasers for over 30 years, the dynamical processes that govern them remain poorly understood, primarily due to the absence of a universal physical model that could describe their intramolecular motion. In this study, we first introduce the concept of a driven damped harmonic oscillator to the realm of soliton molecules dynamics, revealing a well-defined resonant response in intrapulse separation characterized by the universal Duffing model. This approach offers a critical framework for investigating the nonlinear mechanisms underlying soliton interactions within the laser system.
Our findings contribute to the fundamental understanding of multisoliton physics within a broad context and offer new insights into the buildup mechanisms and interactions of multisoliton states across a wide range of physical systems, including Mamyshev oscillators and passive optical resonators. Additionally, our work is expected to inspire further research into questions, such as the symmetry-breaking dynamics of internal motion in vector soliton molecules and marginal stability within the context of symmetry-breaking dynamics of vector dissipative solitons,60 the nature of eigenfrequencies in trisoliton systems,61 the potential existence of multiple eigenfrequencies, and whether their nonlinear resonance responses reveal a more intricate framework. Notably, resonant excitation provides a general approach to generating breathers, laying the groundwork for investigating the fractal dynamics within intramolecular dynamics.62 For practical applications, ultrafine scanning of temporal separations of ultrashort laser pulses, ranging from a few femtoseconds to hundreds of femtoseconds, shows great potential in areas requiring high detection sensitivity.63
Defeng Zou received his PhD from Tianjin University (TJU) in 2023 and is currently working as a postdoctoral fellow in the Department of Electronic and Electrical Engineering at the Southern University of Science and Technology. His research interests include ultrafast lasers and soliton dynamics. He has received awards such as the Wang Daheng Optical Award and the Guo Guangcan Optical Award for Scientific and Technological Innovation from the Chinese Optical Society.
Minglie Hu received the BSc degree in electrical engineering from the School of Precision Instruments and Optoelectronics Engineering, Tianjin University, Tianjin, China, in 2000. He received the PhD degree in optical engineering from Tianjin University in 2005. He is now a professor at Tianjin University. His current research interests include mode-locking laser oscillators and amplifiers, fiber lasers, nonlinear and linear propagation in the photonic crystal fibers, and microstructure optical devices.
Perry Ping Shum is an SPIE Fellow. He received his BEng and PhD degrees in Electronic and Electrical Engineering from the University of Birmingham, UK, in 1991 and 1995, respectively. In 1999, he joined Nanyang Technological University. In 2020, he joined Southern University of Science and Technology. He has served as the Director of the Network Technology Research Centre and has also held the positions of founding Director of the OPTIMUS-Photonics Centre and founding Director of the Centre for Optical Fibre Technology. Prof. Shum has published over 400 papers and has an H-index of 70. He has served as chair, committee member, and international advisor for numerous international conferences. His research interests encompass laser technology, biophotonics, silicon photonics, and fiber-based devices.
Biographies of the other authors are not available.
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