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Existence and stability of soliton-based frequency combs in the Lugiato-Lefever equation

Lukas Bengel, Björn de Rijk

TL;DR

The paper proves that the Lugiato-Lefever equation supports robust, soliton-based frequency combs by constructing periodic multipulse solutions composed of N well-separated bright solitons on a single period and proving their diffusive spectral stability and nonlinear stability. The approach reformulates the LLE as a reversible spatial dynamical system, uses Lin's method to concatenate 1-pulse homoclinics into N-pulse trains, and applies Evans-function and Floquet-Bloch techniques to control the spectrum and establish stability against localized and subharmonic perturbations. By exploiting an exact parameter regime with small damping and forcing, the authors show that both the amplitude and the spectral bandwidth of these combs can be made arbitrarily large, and the pulse spacing and period can be tuned over wide ranges. These results provide a rigorous foundation for the experimentally observed, highly stable, soliton-based frequency combs and offer a principled pathway to chip-scale, broadband comb implementations with strong stability guarantees. The work thus bridges rigorous nonlinear wave theory with practical frequency-comb technology, confirming that LLE-based soliton trains can serve as stable, tunable broadband frequency combs in microresonators.

Abstract

Kerr frequency combs are optical signals consisting of a multitude of equally spaced excited modes in frequency space. They are generated by converting a continuous-wave pump laser within an optical microresonator. It has been observed that the interplay of Kerr nonlinearity and dispersion in the microresonator can lead to a stable optical signal consisting of a periodic sequence of highly localized ultra-short pulses, resulting in broad frequency spectrum. The discovery that stable broadband frequency combs can be generated in microresonators has unlocked a wide range of promising applications, particularly in optical communications, spectroscopy and frequency metrology. In its simplest form, the physics in the microresonator is modeled by the Lugiato-Lefever equation, a damped nonlinear Schrödinger equation with forcing. In this paper we demonstrate that the Lugiato-Lefever equation indeed supports arbitrarily broad Kerr frequency combs by proving the existence and stability of periodic solutions consisting of any number of well-separated, strongly localized and highly nonlinear pulses on a single periodicity interval. We realize these periodic multiple pulse solutions as concatenations of individual bright cavity solitons by phrasing the problem as a reversible dynamical system and employing results from homoclinic bifurcation theory. The spatial dynamics formulation enables us to harness general results, based on Evans-function techniques and Lin's method, to rigorously establish diffusive spectral stability. This, in turn, yields nonlinear stability of the periodic multipulse solutions against localized and subharmonic perturbations.

Existence and stability of soliton-based frequency combs in the Lugiato-Lefever equation

TL;DR

The paper proves that the Lugiato-Lefever equation supports robust, soliton-based frequency combs by constructing periodic multipulse solutions composed of N well-separated bright solitons on a single period and proving their diffusive spectral stability and nonlinear stability. The approach reformulates the LLE as a reversible spatial dynamical system, uses Lin's method to concatenate 1-pulse homoclinics into N-pulse trains, and applies Evans-function and Floquet-Bloch techniques to control the spectrum and establish stability against localized and subharmonic perturbations. By exploiting an exact parameter regime with small damping and forcing, the authors show that both the amplitude and the spectral bandwidth of these combs can be made arbitrarily large, and the pulse spacing and period can be tuned over wide ranges. These results provide a rigorous foundation for the experimentally observed, highly stable, soliton-based frequency combs and offer a principled pathway to chip-scale, broadband comb implementations with strong stability guarantees. The work thus bridges rigorous nonlinear wave theory with practical frequency-comb technology, confirming that LLE-based soliton trains can serve as stable, tunable broadband frequency combs in microresonators.

Abstract

Kerr frequency combs are optical signals consisting of a multitude of equally spaced excited modes in frequency space. They are generated by converting a continuous-wave pump laser within an optical microresonator. It has been observed that the interplay of Kerr nonlinearity and dispersion in the microresonator can lead to a stable optical signal consisting of a periodic sequence of highly localized ultra-short pulses, resulting in broad frequency spectrum. The discovery that stable broadband frequency combs can be generated in microresonators has unlocked a wide range of promising applications, particularly in optical communications, spectroscopy and frequency metrology. In its simplest form, the physics in the microresonator is modeled by the Lugiato-Lefever equation, a damped nonlinear Schrödinger equation with forcing. In this paper we demonstrate that the Lugiato-Lefever equation indeed supports arbitrarily broad Kerr frequency combs by proving the existence and stability of periodic solutions consisting of any number of well-separated, strongly localized and highly nonlinear pulses on a single periodicity interval. We realize these periodic multiple pulse solutions as concatenations of individual bright cavity solitons by phrasing the problem as a reversible dynamical system and employing results from homoclinic bifurcation theory. The spatial dynamics formulation enables us to harness general results, based on Evans-function techniques and Lin's method, to rigorously establish diffusive spectral stability. This, in turn, yields nonlinear stability of the periodic multipulse solutions against localized and subharmonic perturbations.

Paper Structure

This paper contains 14 sections, 10 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Embedding in the mathematical literature
  4. Dynamical systems approach
  5. Outline of paper
  6. Soliton-based pulse solutions
  7. Spatial dynamics formulation
  8. Primary 1-pulse solutions
  9. N-pulse solutions
  10. Periodic N-pulse solutions
  11. Stability analysis
  12. Spectral a-priori bounds
  13. Stability of N-pulse solutions
  14. Stability of periodic N-pulse solutions

Key Result

Theorem 1

Fix $N \in \mathbb{N}$. Set $n = {\lfloor \frac{N}{2} \rfloor}$ and $\alpha_0 = N\! \mod 2 \in \{0,1\}$. Assume that $\zeta, f > 0$ and $\theta_0 \in \mathbb{R}$ obey $8 \zeta < \pi^2 f^2$, $\pi f \cos \theta_0 = 2 \sqrt{2 \zeta}$ and $\sin \theta_0 > 0$. Then, there exist constants $C_0,\varepsilon such that equation LLE admits a stationary smooth solution $u_{m,\mathbf{k},\varepsilon} \colon \ma

Figures (2)

  • Figure 1: Periodic approximations of the primary 1-pulse solutions established in Theorem \ref{['thm:1-pulse']} (see also Theorem \ref{['thm:ex_periodic']}) with parameters $\zeta =1, f=2$. The solutions were computed with the MATLAB package pde2pathUecker2014pde2path. The top row shows a stable 1-pulse bifurcating with $\sin\theta_0>0$, its spectrum against co-periodic perturbations, and the corresponding frequency comb obtained by plotting $\log|\hat{u}(k)|$ against the Fourier frequency variable $k$. The bottom row depicts an unstable 1-pulse bifurcating with $\sin\theta_0<0$, the associated spectrum, and its frequency comb.
  • Figure 2: The figure shows periodic approximations of 2- and 3-pulses, which were established in Theorem \ref{['thm:ex_N-pulse']}, see also Theorem \ref{['thm:ex_periodic']}, computed with pde2pathUecker2014pde2path. The parameters are $\zeta =1$ and $f=2$. Continuation was performed in the small parameter $\varepsilon$ starting from superpositions of bright NLS-solitons.

Theorems & Definitions (29)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Definition 1
  • Definition 2
  • Theorem 2: Bengel2024Stability
  • Definition 3
  • Lemma 1
  • proof
  • Proposition 1
  • ...and 19 more