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Numerical simulation of Lugiato-Lefever equation for Kerr combs generation in Fabry-Perot resonators

Mouhamad Al Sayed Ali, Stéphane Balac, Germain Bourcier, Gabriel Caloz, Monique Dauge, Arnaud Fernandez, Olivier Llopis, Fabrice Mahé

TL;DR

This work analyzes Kerr frequency comb generation in Fabry-Perot resonators modeled by the FP-LLE, a nonlocal variant of the Lugiato–Lefever equation with periodic boundary conditions and an integral nonlinearity. It develops and compares three numerical frameworks—Split-Step dynamics, Collocation for steady states, and pseudo-arclength continuation—to capture both dynamic evolution and bifurcation structures of steady solutions. The study reveals rich multistability and intricate bifurcation landscapes, including multiple flat-state Branches and numerous bifurcations from flat solutions, under representative parameter sets; it also provides practical guidelines for identifying steady states and interpreting Kerr comb spectra. The results, complemented by open-source Matlab tools, offer a robust computational toolkit for predicting and analyzing Kerr combs in FP resonators across time-dependent and stationary regimes.

Abstract

Lugiato-Lefever equation (LLE) is a nonlinear Schrödinger equation with damping, detuning and driving terms, introduced as a model for Kerr combs generation in ring-shape resonators and more recently, in the form of a variant, in Fabry-Perot (FP) resonators. The aim of this paper is to present some numerical methods that complement each other to solve the LLE in its general form both in the dynamic and in the steady state regimes. We also provide some mathematical properties of the LLE likely to help the understanding and interpretation of the numerical simulation results.

Numerical simulation of Lugiato-Lefever equation for Kerr combs generation in Fabry-Perot resonators

TL;DR

This work analyzes Kerr frequency comb generation in Fabry-Perot resonators modeled by the FP-LLE, a nonlocal variant of the Lugiato–Lefever equation with periodic boundary conditions and an integral nonlinearity. It develops and compares three numerical frameworks—Split-Step dynamics, Collocation for steady states, and pseudo-arclength continuation—to capture both dynamic evolution and bifurcation structures of steady solutions. The study reveals rich multistability and intricate bifurcation landscapes, including multiple flat-state Branches and numerous bifurcations from flat solutions, under representative parameter sets; it also provides practical guidelines for identifying steady states and interpreting Kerr comb spectra. The results, complemented by open-source Matlab tools, offer a robust computational toolkit for predicting and analyzing Kerr combs in FP resonators across time-dependent and stationary regimes.

Abstract

Lugiato-Lefever equation (LLE) is a nonlinear Schrödinger equation with damping, detuning and driving terms, introduced as a model for Kerr combs generation in ring-shape resonators and more recently, in the form of a variant, in Fabry-Perot (FP) resonators. The aim of this paper is to present some numerical methods that complement each other to solve the LLE in its general form both in the dynamic and in the steady state regimes. We also provide some mathematical properties of the LLE likely to help the understanding and interpretation of the numerical simulation results.

Paper Structure

This paper contains 17 sections, 3 theorems, 73 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview of mathematical properties of the LLE model
  3. Properties of the time-dynamic FP-LLE problem
  4. Steady state solutions
  5. Flat solutions
  6. Properties of FP-LLE steady state solutions
  7. Some numerical methods for solving FP-LLE
  8. Symmetric Split-Step method
  9. A Collocation method
  10. Pseudo-arclength continuation method
  11. Numerical illustrations
  12. Bifurcation diagram for $\beta=-0.2$ and $F=1.6$
  13. Solutions to the dynamic LLE for $\beta=-0.2$, $F=1.6$ and $\alpha=6$
  14. Stationary solutions computed by the Collocation method
  15. Conclusion
  16. ...and 2 more sections

Key Result

Proposition 1

Any solution $\psi = u_1+\mathrm{i}\xspace u_2$ to the steady state FP-LLE eq:mods satisfies the energy estimate Moreover, $\int_{-\pi}^{\pi} u_1( \theta)\mathrm{\ d}\theta >0$.

Figures (13)

  • Figure 1: Number of flat solutions to the LLE depending on the values of the parameters $(\alpha, F^2)$ for $\sigma=1$.
  • Figure 2: Variation of $\rho_\bullet = |\psi_\bullet|^2$ as a function of $F$ for $\alpha\in\{1,5,10,15\}$ and $\sigma=1$.
  • Figure 3: Variation of $\rho_\bullet = |\psi_\bullet|^2$ as a function of $\alpha$ for $F\in\{1,2,3,5\}$ and $\sigma=1$.
  • Figure 4: Bifurcation diagram. Integers refer to the label $\ell$ of the BP as given in Table \ref{['table:1659']}. Top: BP (black circle) on the curve of flat solutions (black curve) for $\beta=-0.2$ and $F=1.6$ and branches of solutions bifurcating from the BP (color curves). Middle: Zoom on the bifurcation diagram of the larger rectangle area where the bifurcation curves cross the line $\alpha=6$. Bottom: Zoom on the bifurcation diagram corresponding to the smaller rectangle area.
  • Figure 5: Solutions to the FP-LLE for $\alpha=6, \beta = -0.2$ and $F=1.6$ (left) and corresponding Kerr frequency combs (right) obtained on the bifurcation branches starting from BP 9, 10, 11, 12 and 13 (from top to bottom). On the left, the real part of the solution is drawn in blue dashed line, its imaginary part in magenta dashed line and it modulus in red solid line. The frequency comb on the right is a representation of the sequence $10\,\log_{10}(| c_n|^2)$ where $(c_n)_{n\in\mathbb{Z}\xspace}$ are the Fourier coefficients of the solution.
  • ...and 8 more figures

Theorems & Definitions (7)

  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof