﻿ 铝基微结构光栅几何参数反演<sup>*</sup>
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1. 哈尔滨工业大学 能源科学与工程学院, 哈尔滨 150001;
2. 上海卫星工程研究所, 上海 201100

Inverse estimation of geometric parameters of aluminum matrix microscale structure grating
SUN Shuangcheng1, QI Hong1, SUN Jianping1, RUAN Shiting2, LYU Zhongyuan1, RUAN Liming1
1. School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China;
2. Shanghai Satellite Engineering Research Institute, Shanghai 201100, China
Received: 2017-02-13; Accepted: 2017-06-07; Published online: 2017-08-04 16:24
Foundation item: National Natural Science Foundation of China (51476043)
Corresponding author. QI Hong, E-mail: qihong@hit.edu.cn
Abstract: Micro structure grating is a widely used electronic component. The geometric parameters of one-dimensional rectangular aluminum matrix grating are inversely estimated using stochastic particle swarm optimization (SPSO) algorithm. The theoretical overview of rigorous coupled wave analysis (RCWA) algorithm and particle swarm optimization algorithm is introduced, and RCWA algorithm is employed to solve the electromagnetic field problem within grating. The objective function is formulated based on the spectral reflectance obtained by the direct problem, and then the SPSO algorithm is used to optimize the objective function. The geometric parameters such as the grating period, ridge width and groove depth are retrieved simultaneously. The effects of population size and searching space on the inverse estimation results are also investigated. The retrieval results show that SPSO algorithm is effective and robust for estimating geometric parameters of grating and the population size is suggested as 30.
Key words: inverse estimation of grating structure     stochastic particle swarm optimization (SPSO) algorithm     rigorous coupled wave analysis (RCWA) algorithm     spectral reflectance     radiative property

1 正问题模型

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RCWA法求解电磁场问题主要可分为3步[14]：①求解Maxwell方程组得到反射区和透射区的电磁场解的形式；②对光栅区电磁场进行Fourier展开得到耦合波方程；③对反射区、光栅区和透射区边界面上施加边界条件，并与耦合波方程联立求得各阶衍射波。下面以横电波(TE波)为例介绍RCWA法的求解过程。

 图 1 TE波入射矩形光栅示意图 Fig. 1 Schematic of rectangular grating with a TE wave irradiation

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z=0和z=d处，电磁场边界条件可分别表示为[14]

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2 反问题模型 2.1 微粒群优化算法

PSO算法中，每只觅食的鸟用一个没有质量和体积的微粒代替，每个微粒代表空间内的一个潜在解，个体之间相互协作寻找搜索区域内的最优解。基本PSO算法采用简单的速度-位移模型，在D维搜索空间内，每个微粒的速度和位置更新公式表示为[10]

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PSO算法计算流程图如图 2所示。

 图 2 PSO算法计算流程图 Fig. 2 Calculation flowchart of PSO algorithm
2.2 随机微粒群优化算法

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1) 如果Pg=Pj，表示随机产生的微粒即为当前全局最优位置，此时保存最优位置并在搜索空间内再重新产生一个微粒，其他微粒在更新PgPj后继续演化。

2) 如果PgPj，且当代没有新的全局最优位置产生，那么所有微粒均按式(24)更新位置。

3) 如果PgPj，且当代有新的全局最优位置产生，即存在微粒k(kj)使得Xk(t+1)=Pg，此时微粒k在搜索空间内停止更新和再生，其他微粒在更新PgPj后继续演化。

3 结果与讨论 3.1 单槽矩形光栅结构反演

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TM波以θ=30°入射到几何结构为aest=[1.0, 0.9, 0.6] μm的矩形光栅时，光谱反射率如图 3中实点所示。采用SPSO算法对此结构光栅进行反演计算，参数设置如表 1所示，MVmaxε1分别为种群数、最大速度、收敛精度。

 图 3 单槽光栅光谱反射率 Fig. 3 Spectral reflectance of single-slot grating

 参数 数值 M 50 Vmax 3.0 C1 1.2 C2 0.8 tmax 100 ε1 10-6 λ1/μm 0.1 λ2/μm 5.0 n 100

 图 4 单槽光栅几何参数反演结果 Fig. 4 Inverse estimation results of single-slot grating geometry parameters

 图 5 4种PSO算法目标函数值变化曲线 Fig. 5 Changing curves of objective function values of four PSO algorithms

 入射角度/(°) aest/μm 平均相对误差/% 最大相对误差/% 0 [1.000 6±1.26×10－3, 0.899 2±3.61×10－3, 0.600 4±2.51×10－3] 0.380 5 2.852 6 30 [0.999 3±3.09×10－3, 0.899 3±4.59×10－3, 0.599 2±2.99×10－3] 0.368 9 2.863 1 60 [0.999 4±2.90×10－3, 0.900 5±3.23×10－3, 0.599 3±3.02×10－3] 0.376 6 2.799 9

 aexa/μm aest/μm 平均相对误差/% 最大相对误差/% [1.0, 0.9, 0.6] [0.999 3±3.09×10－3, 0.899 3±4.59×10－3, 0.599 2±2.99×10－3] 0.368 9 2.863 1 [3.0, 2.5, 1.5] [2.999 1±3.11×10－3, 2.500 5±1.19×10－3, 1.499 4±2.95×10－3] 0.359 6 2.904 0 [5.0, 4.0, 2.0] [5.000 6±3.33×10－3, 3.999 2±4.14×10－3, 2.000 6±3.20×10－3] 0.331 5 2.882 3

3.2 SPSO算法参数影响分析

 算例 Λ /μm l /μm d/μm 迭代次数 目标函数值 平均相对误差/% 算例1 [0.9, 1.1] [0.8, 1.0] [0.5, 0.7] 59 3.25×10-7 0.020 9 算例2 [0.8, 1.2] [0.7, 1.1] [0.4, 0.8] 71 9.19×10-7 0.115 5 算例3 [0.5, 1.5] [0.4, 1.4] [0.1, 1.1] 100 8.11×10-6 0.206 4

SPSO算法种群大小直接影响每代计算时间和每次所能处理的候选解个数，因而对反演结果有很大影响。分别设置为种群数M=10、30和50，反演过程中目标函数值变化曲线如图 6所示。

 图 6 不同种群大小下目标函数值变化曲线 Fig. 6 Changing curves of objective function values under different population sizes

3.3 双槽矩形光栅结构反演

 图 7 双槽矩形光栅示意图 Fig. 7 Schematic of rectangular double-slot grating

 图 8 双槽光栅光谱反射率 Fig. 8 Spectral reflectance of double-slot grating
 图 9 双槽光栅几何参数反演结果 Fig. 9 Inverse estimation results of double-slot grating geometry parameters
4 结论

1) SPSO算法可以准确地反演得到光栅的几何形状参数，包括单槽和双槽矩形光栅，平均相对误差分别为0.38%和0.42%，计算精度较高。

2) SPSO算法在计算效率方面要明显优于基本PSO算法、标准PSO算法及PSO-GA算法。

3) 分析了SPSO算法中搜索区间及种群大小对光栅结构参数反演的影响。结果表明，随着搜索区间增大，所需迭代次数增加，相对误差也随之增大；综合考虑计算时间和收敛速度，推荐种群数取30。

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#### 文章信息

SUN Shuangcheng, QI Hong, SUN Jianping, RUAN Shiting, LYU Zhongyuan, RUAN Liming

Inverse estimation of geometric parameters of aluminum matrix microscale structure grating

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(11): 2199-2206
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0055