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Efficient Multiscale Algorithms for Simulating Nonlocal Optical Response of Metallic Nanostructure Arrays
SIAM Journal on Scientific Computing ( IF 3.0 ) Pub Date : 2021-07-15 , DOI: 10.1137/20m1324120 Yongwei Zhang , Chupeng Ma , Li-qun Cao , Dongyang Shi
SIAM Journal on Scientific Computing ( IF 3.0 ) Pub Date : 2021-07-15 , DOI: 10.1137/20m1324120 Yongwei Zhang , Chupeng Ma , Li-qun Cao , Dongyang Shi
SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page B907-B936, January 2021.
In this paper, we consider numerical simulations of the nonlocal optical response of metallic nanostructure arrays inside a dielectric host, which is of particular interest to the nanoplasmonics community due to many unusual properties and potential applications. Mathematically, it is described by Maxwell's equations with discontinuous coefficients coupled with a set of Helmholtz-type equations defined only on the domains of metallic nanostructures. To solve this challenging problem, we develop an efficient multiscale method consisting of three steps. First, we extend the system into the domain occupied by the dielectric medium in a novel way and result in a coupled system with rapidly oscillating coefficients. A rigorous analysis of the error between the solutions of the original system and the extended system is given. Second, we derive the homogenized system and define the multiscale approximate solution for the extended system by using the multiscale asymptotic method. Third, to fix the inaccuracy of the multiscale asymptotic method inside the metallic nanostructures, we solve the original system in each metallic nanostructure separately with boundary conditions given by the multiscale approximate solution. A fast algorithm based on the $LU$ decomposition is proposed for solving the resulting linear systems. By applying the multiscale method, we obtain results that are in good agreement with those obtained by solving the original system directly at a much lower computational cost. Numerical examples are provided to validate the efficiency and accuracy of the proposed method.
中文翻译：
用于模拟金属纳米结构阵列非局域光学响应的高效多尺度算法
SIAM 科学计算杂志，第 43 卷，第 4 期，第 B907-B936 页，2021 年 1 月。
在本文中，我们考虑了介电主体内金属纳米结构阵列的非局部光学响应的数值模拟，由于许多不寻常的特性和潜在应用，这对纳米等离子体学界特别感兴趣。在数学上，它由麦克斯韦方程组描述，该方程具有不连续系数，加上一组仅在金属纳米结构域上定义的亥姆霍兹型方程。为了解决这个具有挑战性的问题，我们开发了一种由三个步骤组成的高效多尺度方法。首先，我们以一种新颖的方式将系统扩展到介电介质占据的域中，并产生具有快速振荡系数的耦合系统。对原系统解与扩展系统解之间的误差进行了严格的分析。第二，我们通过使用多尺度渐近方法导出均质化系统并定义扩展系统的多尺度近似解。第三，为了解决金属纳米结构内部多尺度渐近法的不准确性，我们分别用多尺度近似解给出的边界条件求解每个金属纳米结构中的原始系统。提出了一种基于 $LU$ 分解的快速算法来求解由此产生的线性系统。通过应用多尺度方法，我们获得的结果与通过以低得多的计算成本直接求解原始系统所获得的结果非常吻合。提供了数值例子来验证所提出方法的效率和准确性。
更新日期：2021-07-16
In this paper, we consider numerical simulations of the nonlocal optical response of metallic nanostructure arrays inside a dielectric host, which is of particular interest to the nanoplasmonics community due to many unusual properties and potential applications. Mathematically, it is described by Maxwell's equations with discontinuous coefficients coupled with a set of Helmholtz-type equations defined only on the domains of metallic nanostructures. To solve this challenging problem, we develop an efficient multiscale method consisting of three steps. First, we extend the system into the domain occupied by the dielectric medium in a novel way and result in a coupled system with rapidly oscillating coefficients. A rigorous analysis of the error between the solutions of the original system and the extended system is given. Second, we derive the homogenized system and define the multiscale approximate solution for the extended system by using the multiscale asymptotic method. Third, to fix the inaccuracy of the multiscale asymptotic method inside the metallic nanostructures, we solve the original system in each metallic nanostructure separately with boundary conditions given by the multiscale approximate solution. A fast algorithm based on the $LU$ decomposition is proposed for solving the resulting linear systems. By applying the multiscale method, we obtain results that are in good agreement with those obtained by solving the original system directly at a much lower computational cost. Numerical examples are provided to validate the efficiency and accuracy of the proposed method.
中文翻译：
用于模拟金属纳米结构阵列非局域光学响应的高效多尺度算法
SIAM 科学计算杂志，第 43 卷，第 4 期，第 B907-B936 页，2021 年 1 月。
在本文中，我们考虑了介电主体内金属纳米结构阵列的非局部光学响应的数值模拟，由于许多不寻常的特性和潜在应用，这对纳米等离子体学界特别感兴趣。在数学上，它由麦克斯韦方程组描述，该方程具有不连续系数，加上一组仅在金属纳米结构域上定义的亥姆霍兹型方程。为了解决这个具有挑战性的问题，我们开发了一种由三个步骤组成的高效多尺度方法。首先，我们以一种新颖的方式将系统扩展到介电介质占据的域中，并产生具有快速振荡系数的耦合系统。对原系统解与扩展系统解之间的误差进行了严格的分析。第二，我们通过使用多尺度渐近方法导出均质化系统并定义扩展系统的多尺度近似解。第三，为了解决金属纳米结构内部多尺度渐近法的不准确性，我们分别用多尺度近似解给出的边界条件求解每个金属纳米结构中的原始系统。提出了一种基于 $LU$ 分解的快速算法来求解由此产生的线性系统。通过应用多尺度方法，我们获得的结果与通过以低得多的计算成本直接求解原始系统所获得的结果非常吻合。提供了数值例子来验证所提出方法的效率和准确性。
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