[Paper Review] Broad Band Solitons in a Periodic and Nonlinear Maxwell System

Dmitry E. Pelinovsky, Gideon Simpson+1 more|arXiv (Cornell University)|Jun 18, 2011

Advanced Fiber Laser Technologies16 references3 citations

TL;DR

This paper studies broad band solitons in a one-dimensional periodic and nonlinear Maxwell system with a refractive index composed of Dirac delta functions. Using asymptotic, analytical, and numerical methods, it establishes the existence of small-amplitude, spatially localized gap solitons near band edges via bifurcation theory and numerical continuation, showing robustness in truncated systems and validating the NLS approximation for long-time dynamics.

ABSTRACT

We consider the one-dimensional Maxwell equations with low contrast periodic linear refractive index and weak Kerr nonlinearity. In this context, wave packet initial conditions with a single carrier frequency excite infinitely many resonances. On large but finite time-scales, the coupled evolution of backward and forward waves is governed by nonlocal equations of resonant nonlinear geometrical optics. For the special class of solutions which are periodic in the fast phase, these equations are equivalent to an infinite system of nonlinear coupled mode equations, the so called it extended nonlinear coupled mode equations, xNLCME. Numerical studies support the existence of long-lived spatially localized coherent structures, featuring a slowly varying envelope and a train of carrier shocks. In this paper we explore, by analytical, asymptotic and numerical methods, the existence and properties of spatially localized structures of the xNLCME system, which arises for a refractive index profile consisting of periodic array of Dirac delta functions. We consider the limit of small amplitude solutions with frequencies near a band-edge. In this case, stationary xNLCME is well-approximated by an infinite system of coupled, stationary, nonlinear Schrödinger equations, the extended nonlinear Schrödinger system, xNLS. We embed xNLS in a one-parameter family of equations, xNLS$^ε$, which interpolates between infinitely many decoupled NLS equations ($ε=0$) and xNLS ($ε=1$). Using bifurcation methods we show existence of solutions for a range of $ε\in(-ε_0,ε_0)$ and, by a numerical continuation method, establish the continuation of certain branches all the way to $ε=1$. Finally, we perform time-dependent simulations of truncated xNLCME and find the small-amplitude solitons to be robust to both numerical errors and the NLS approximation.

Motivation & Objective

• Investigate the existence of spatially localized coherent structures in the extended nonlinear coupled mode equations (xNLCME) arising from a periodic, nonlinear Maxwell system.
• Analyze the system in the limit of small-amplitude solutions near a band-edge, where xNLCME is approximated by an infinite system of coupled nonlinear Schrödinger equations (xNLS).
• Establish the existence of non-trivial, localized solutions for a range of parameters in a one-parameter family interpolating between decoupled and fully coupled NLS equations.
• Demonstrate the robustness of near-band-edge gap solitons under time-dependent simulations and numerical errors, validating the NLS approximation.
• Identify open problems regarding existence and function space regularity for localized solutions in xNLCME and xNLS with general refractive index profiles.

Proposed method

• Derive the extended nonlinear coupled mode equations (xNLCME) from the Maxwell equations with low-contrast, periodic refractive index and weak Kerr nonlinearity using multiple-scale asymptotic analysis.
• Approximate xNLCME by an infinite system of stationary nonlinear Schrödinger equations (xNLS) in the small-amplitude, band-edge limit.
• Introduce a one-parameter family of equations, xNLSϵ, interpolating between decoupled NLS (ϵ=0) and xNLS (ϵ=1), to study solution bifurcations.
• Apply bifurcation theory to prove existence of non-trivial solutions for ϵ∈(−ϵ₀,ϵ₀), and use numerical continuation to extend solution branches to ϵ=1.
• Perform time-dependent simulations of truncated xNLCME systems with pseudo-spectral spatial discretization and RK4 time stepping to assess soliton stability.
• Use Rayleigh-Ritz variational approximations (single parameter and Gaussian) to probe existence of critical points and validate solution structure.

Experimental results

Research questions

• RQ1 Do non-trivial, spatially localized solutions exist for the extended nonlinear coupled mode equations (xNLCME) with a periodic delta-function refractive index?
• RQ2 Can the existence of localized solutions in the xNLS system be established via bifurcation methods for a range of coupling parameters?
• RQ3 How robust are small-amplitude, near-band-edge gap solitons under time evolution in truncated xNLCME systems?
• RQ4 To what extent does the nonlinear Schrödinger (NLS) approximation accurately describe the dynamics of these localized structures?
• RQ5 What are the appropriate function spaces for the existence of solutions, and can existence be proven for arbitrarily large finite truncations of xNLCME?

Key findings

• Non-trivial, localized solutions exist for the xNLCME system in the small-amplitude, band-edge regime, confirmed via bifurcation analysis for ϵ∈(−ϵ₀,ϵ₀).
• Numerical continuation successfully extends solution branches from ϵ≈0 to ϵ=1, indicating the persistence of localized structures in the fully coupled limit.
• Time-dependent simulations of truncated xNLCME systems show that small-amplitude, near-band-edge gap solitons remain localized and robust to numerical errors and truncation effects.
• The NLS approximation accurately captures the dynamics of these solitons, with better agreement observed as μ→0 (smaller nonlinearity).
• The Gaussian and single-parameter Rayleigh-Ritz approximations suggest the existence of critical points, though rigorous existence proofs remain open.
• Robustness is observed across different truncations (2-mode, 4-mode) and domain sizes, with minimal distortion at low nonlinearity (μ=0.1) and increased oscillations at higher μ (μ=0.4).