﻿ 基于COMSOL弱形式方程求解色散光子晶体能带 Band Diagram Calculations of Dispersive Photonic Crystals Based on COMSOL Weak Form Equation

Applied Physics
Vol.07 No.05(2017), Article ID:20726,10 pages
10.12677/APP.2017.75021

Band Diagram Calculations of Dispersive Photonic Crystals Based on COMSOL Weak Form Equation

Yunfei Xu, Zhihong Hang

Collaborative Innovation Center of Suzhou Nano Science and Technology, College of Physics Optoelectronics and Energy, Soochow University, Suzhou Jiangsu

Received: May 6th, 2017; accepted: May 24th, 2017; published: May 27th, 2017

ABSTRACT

In order to study dispersive photonic crystals (PCs), we introduce a numerical finite element method based on weak form equation in COMSOL, which transforms the complex band diagram problem into a simple eigenvalue problem by solving the eigenvalue with respective to wave vector k by frequency. The advantages of the method are illustrated by two examples. The equi-fre- quency contours close to the Dirac-like cone dispersion of a square-lattice PC can be calculated with much less time than traditional methods. We also succeeded in obtained the edge states on the bearded edge and the zigzag edge of a honeycomb PC with a dispersive negative background medium which is numerical unstable for traditional methods.

Keywords:Numerical Methods, Weak Form, Photonic Crystals, Band Diagram

1. 引言

1987年由Yablonovitch [1] 和John [2] 分别独立提出光子晶体的概念，将不同折射率的材料在空间做周期性排列，构成光子晶体。其具有和半导体类似的导带、禁带等光子晶体能带结构使得人们可以对光的传播进行任意调控。光子晶体的初期主要工作就是寻找光子禁带结构，将光局域正在很小局域能够大大提高光子的使用效率，具有很高的应用价值。光子晶体的发展是伴随着凝聚态理论发展的。拓扑绝缘体的研究刚刚获得了2016年的诺贝尔物理学奖。随着具有狄拉克色散关系光子晶体能带的发现 [3] ，如何构建拓扑绝缘体的光学类比，研究光子晶体能带的体拓扑性质，构建具有拓扑性质的光子界面态/边缘态是光子晶体研究的最新发展。

2. 偏微分方程的弱形式介绍

(1)

(2)

(3)

(4)

(5)

COMSOL中未知函数(因变量)u及试函数v的函数表达如下：未知函数u,表达为u，写为(),在二维电磁波方程弱形式求解中，未知函数为Ez为()；类似的试函数v及分量写为，，将弱形式方程输入COMSOL求解域的若干项中，便会自动进行求解运算，其每一个子域可以有不同定义，COMSOL将自动整合不同子域方程。

(a) (b)

Figure 1. (a) Parameter setting of weak form module in COMSOL; (b) Periodic boundary condition setting

Table 1. Parameter setting

kx, ky是布洛赫波矢k的x，y分量, epsilon，mu分别为介电常数和磁导率，不同子域中单独设定，所有子域中初始值设置为Ez。原胞的周期性边界条件的设置非常重要，其需要满足奎特周期性边界，即目标端和源项场值上相差一个相位因子，相位因子的值由边界和波矢的相对距离确定。以图1(b)中二维正方晶格为例，以Ez为约束条件分别设置两对周期性边界，即1,3为一对，2,4为一对分别将1和2边界Ez的值映射到3和4，同时源项和目标项的顶点也需要一一对应，如1,3这对周期边条中，1的源项顶点5,8要分别映射到3边界的顶点6，7。

3. 利用弱形式方程数值求解复杂光子晶体能带

3.1. 等频率曲线计算中的应用

(a) (b)

Figure 2. (a) The band structure of square lattice calculated by traditional methods; (b) The band structure calculated by Weak Form

3.2. 色散负背景材料能带计算中的应用

Figure 3. Equi-frequency diagrams of close to Dirac cone dispersion

(6)

ωp为有效等离子频率，可以通过计算金属柱阵列的S参数得出。也就是说，如果在光子晶体柱的空气背景中，插入大量细金属柱，就可以在ω < ωp的频率范围内，用负的有效介电常数材料取代原本的空气背景。根据引文 [17] 的理论，我们设计使用了晶格常数为4.9205mm，半径为0.00375 mm的金属柱阵列，其等效的等离子频率为17.03 GHz (结构见图4(a))。通过调控金属柱占空比，我们可以轻松的改变等效等离子频率，即调控背景材料的有效介电常数，即光子晶体柱之间的倏矢波耦合系数，从而调控光子晶体能带。

Figure 4. (a) The unit cell of a honeycomb lattice PC; (b) The first Brillouin zone of the system; (c) Comparison of the calculated bulk photonic band diagram of the metal wire background and the effective medium background

(a) (b)

Figure 5. (a) The calculated band diagrams with edge states; (b) The corresponding eigen field distribution of edge mode with bearded edge.

4. 结论

Band Diagram Calculations of Dispersive Photonic Crystals Based on COMSOL Weak Form Equation[J]. 应用物理, 2017, 07(05): 149-158. http://dx.doi.org/10.12677/APP.2017.75021

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