Langevin model for soliton molecules in ultrafast fiber ring laser cavity: investigating experimentally the interplay between noise and inertia

TL;DR

This study addresses the noisy, inertia-influenced dynamics of soliton molecules in ultrafast fiber lasers by constructing a data-driven Langevin framework via Sparse Identification of Nonlinear Dynamics (SINDy). A normalized inter-soliton distance $u = ( au - ig angle au ig angle)/\sigma_{ au}$ is modeled with a seven-parameter latent equation $ rac{d^2u}{dt^2} = oldsymbol{\xi} ext{ terms}$, while a Langevin force $\sigma_L$ captures stochastic fluctuations, forming a full Langevin description $ rac{d}{dt}[ au abla u] = [ abla u F_{SINDy}] + [ heta imes heta + \sigma_L]$. The Gaussian, state-independent noise is shown to be small yet dynamically informative, and the full model reproduces experimental dynamics while enabling non-invasive perturbation analysis through a numerical twin. The work demonstrates that noise can enhance model robustness and reveals how increasing noise amplitude or introducing correlated noise can destabilize the molecule, offering a path toward interpretable, controllable stochastic models for complex laser systems. Overall, the methodology provides a general framework to disentangle latent dynamics from fluctuations in trackable systems and can be extended to other domains where constituent particle tracking is possible.

Abstract

The dynamics of soliton molecules in ultrafast fiber ring laser cavity is strongly influenced by noise. We show how a parsimonious Langevin model can be constructed from experimental data, resulting in a mathematical description that encompasses both the deterministic and stochastic properties of the evolution of the soliton molecules. In particular, we were able to probe the response dynamics of the soliton molecule to an external kick in a sub-critical approach, namely without the need to actually disturb the systems under investigation. Moreover, the noise experienced by the dissipative solitonic system, including its distribution and correlation, can now be also analyzed in details. Our strategy can be applied to any systems where the individual motion of its constitutive particles can be traced; the case of optical solitonic-system laser presented here serving as a proof-of-principle demonstration.

Figures (5)

• Figure 1: (a) Experimental setup. PBS: Polarization Beam Splitter. PC: Polarization Controller. PD: Photodiode. HDF: Highly Dispersive Fiber (total dispersion of 50ps/nm). (b) Evolution of time separation $\tau$ between the two dissipative solitons. Its exhibits a characteristic oscillation with a period $T_0 = 117.6 ~ \text{RTs}$. $<\tau> = 3.48 ~ \text{ps}$. (c) Evolution of the optical phase difference $\Delta\Phi$ between the two dissipative solitons. (d) Phase portrait in the plane $\{\tau,d{\tau}\}$ corresponding to (b).
• Figure 2: (a) $<\tau> = 3.48 ~ \text{ps}$. Blue-circles: Phase portrait of the SM oscillation in the $\{\tau; \dot{\tau}\}$ plane ($d{\tau}/dt = \dot{\tau}$). Black line: limit-cycle defined by Eq. (\ref{['eq2']}). Red-dashed line: recovery dynamics for a noiseless model when the SM has been initially kicked from the limit-cycle, to the yellow dot, which serves as starting point. (b) Details of the recovery of the red-dashed trajectory presented in (a). The geometrical distance from the limit cycle evolves as an exponential decaying law with a characteristic thermalization time $\tau_{th}=98.2 \text{RTs}$. (c) Comparison between the experimental oscillations (solid blue) and the result of Eq. (\ref{['eq2']}) (dashed red). (d) Evolution of the deviation between the SINDy prediction and the actual experimental evolution. The deviation is measured as the euclidean distance in the phase portrait.
• Figure 3: (a) Evolution of the $\xi_i$ coefficients as function of $\lambda$, the free parameter controlling the LARS regression. (b) Number of non-zero coefficients. (c) Corresponding reconstruction error. $\lambda=3.7$ is chosen for the present work, corresponding to 7 non-zero parameters.
• Figure 4: ($a_1$,$b_1$) Results of Eq. (\ref{['eq:LangevinSINDY']}). Cyan line indicates the average drift force, attributed to $\theta_{\tau}$ (resp. $\theta_{\dot{\tau}}$). ($a_2$-$b_2$) Resulting distribution of the Langevin's Force $\sigma_L$, according to Eq. (\ref{['eq:Langevin']}). It is obtained after substraction of the drift $\theta$ in ($a_1$,$b_1$). By definition $F(\tau,\cdot{\tau}) = F_{SINDy}(\tau,\cdot{\tau}) + \theta(\tau,\cdot{\tau})$. According to the Langevin's formalism, ($a_2$) must be null, and therefore is ascribed to numerical artifact. (c) Blue : Histogram of the Langevin's force shown in ($b_2$). Gray : histogram of the distribution of the numerical errors, construted from ($a_2$). (d) Correlation $<\sigma_L(t+t_0)\sigma_L(t)>_t/<\sigma_L(t)^2>_t$ as a function of the time offset $t_0$.
• Figure 5: (a) Observed dynamics.(b) Nominal Langevin Model as defined by Eq. (\ref{['eq:LangevinSINDY0']}). (c) red-dash : Result of Langevin simulation where the noise amplitude (i.e. $std(\sigma_L)$) has been multiplied by two. The SM breaks appart quickly. (d) red-dash : Result of Langevin simulation where the frequency of the noise correlation has been set to the oscillation period. The SM breaks appart eventally, after a rather long evolution.