多维层状介质中非线性Helmholtz方程的高阶数值方法
A High-Order Numerical Method for the Nonlinear Helmholtz Equation in Multidimensional Layered Media
摘要：
本文提出了一种新的计算方法来求解控制激光在克尔介质中传输的标量非线性亥姆霍兹方程。该方法解决了非线性光学中的两个众所周知的挑战：当介质中的散射假设主要是向前（近轴制度）时，解决方案的奇异行为，以及介质的%线性和非线性光学特性中存在的不连续性。具体地说，我们考虑一个板的非线性材料，它可以在传播方向光栅，这是沉浸在线性介质作为一个整体。该方法的关键组成部分是一个半紧凑的高阶有限差分格式，保持精度的不连续性，并使子波长分辨率在大的域在可容忍的成本，一个非本地的双向人工边界条件（ABC），同时促进无反射传播的传出波和向前传播的给定传入波，和一个非线性求解器基于牛顿的方法。所提出的方法相结合，并大大扩展了我们以前的技术为1Dand多维的能力。它有利于直接的数值研究nonparaxial传播，远远超出了文献中的方法的基础上的“增强”傍轴模型。特别是，它提供了有史以来第一个证据表明，奇异性的解决方案确实消失在标量NLH模型，包括nonparaxial效果。它也使模拟的波长宽度的空间孤子，以及反向传播的孤子。
英文摘要：
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 % linear and nonlinear 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 1Dand 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.
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