A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media

JOURNAL OF COMPUTATIONAL PHYSICS

卷:228

Article

Baruch, G.²  Fibich, G.²  Tsynkov, S.¹ 

[1] N Carolina State Univ, Dept Math, Raleigh, NC 27695 USA

[2] Tel Aviv Univ, Sch Math Sci, Dept Appl Math, IL-69978 Tel Aviv, Israel

关键词: Nonlinear optics;    Kerr nonlinearity;    Inhomogeneous medium;    Material discontinuities;    Discontinuous coefficients;    Layered medium;    Nonparaxiality;    Forward scattering;    Backscattering;    (Narrow) solitons;    Paraxial approximation;    Nonlinear Schrodinger equation;    Arrest of collapse;    Finite-difference approximation;    Compact scheme;    High-order method;    Artificial boundary conditions (ABCs);    Two-way ABCs;    Traveling waves;    Complex valued solutions;    Frechet differentiability;    Newton's method;   

DOI  :  10.1016/j.jcp.2009.02.014

来源: Elsevier

【 摘 要 】

We present a novel computational methodology for solving the scalar nonlinear Helmholtz equation (NLH) that governs the propagation of laser light in Kerr dielectrics. The methodology addresses two well-known challenges in nonlinear optics: Singular behavior of solutions when the scattering in the medium is assumed predominantly forward (paraxial regime), and the presence of discontinuities in the optical properties of the medium. Specifically, we consider a slab of nonlinear material which may be grated in the direction of propagation and which is immersed in a linear medium as a whole. The key components of the methodology are a semi-compact high-order finite-difference scheme that maintains accuracy across the discontinuities and enables sub-wavelength resolution on large domains at a tolerable cost, a nonlocal two-way artificial boundary condition (ABC) that simultaneously facilitates the reflectionless propagation of the outgoing waves and forward propagation of the given incoming waves, and a nonlinear solver based on Newton's method. The proposed methodology combines and substantially extends the capabilities of our previous techniques built for I D and for multi-D. It facilitates a direct numerical study of nonparaxial propagation and goes well beyond the approaches in the literature based on the augmented paraxial models. In particular, it provides the first ever evidence that the singularity of the solution indeed disappears in the scalar NLH model that includes the nonparaxial effects. It also enables simulation of the wavelength-width spatial solitons, as well as of the counter-propagating solitons. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free