Xu Zhengfu, Bao Gang
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA.
J Opt Soc Am A Opt Image Sci Vis. 2010 Nov 1;27(11):2347-53. doi: 10.1364/JOSAA.27.002347.
A numerical scheme is presented to solve the nonlinear Helmholtz (NLH) equation modeling second-harmonic generation (SHG) in photonic bandgap material doped with a nonlinear χ((2)) effect and the NLH equation modeling wave propagation in Kerr type gratings with a nonlinear χ((3)) effect in the one-dimensional case. Both of these nonlinear phenomena arise as a result of the combination of high electromagnetic mode density and nonlinear reaction from the medium. When the mode intensity of the incident wave is significantly strong, which makes the nonlinear effect non-negligible, numerical methods based on the linearization of the essentially nonlinear problem will become inadequate. In this work, a robust, stable numerical scheme is designed to simulate the NLH equations with strong nonlinearity.
提出了一种数值格式，用于求解一维情况下模拟掺杂非线性χ(2)效应的光子带隙材料中的二次谐波产生（SHG）的非线性亥姆霍兹（NLH）方程，以及模拟具有非线性χ(3)效应的克尔型光栅中波传播的NLH方程。这两种非线性现象都是高电磁模密度与介质的非线性反应相结合的结果。当入射波的模强度足够强，使得非线性效应不可忽略时，基于本质非线性问题线性化的数值方法将变得不足。在这项工作中，设计了一种稳健、稳定的数值格式来模拟具有强非线性的NLH方程。
