三維光子晶體的計算軟體(FAME)開發以及說明｜國立清華大學博碩士論文庫

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研究生：	黃威智 Huang, Wei Zhi
論文名稱：	三維光子晶體的計算軟體(FAME)開發以及說明 User Guide for FAME(Fast Algorithm of Maxwell's Equations)
指導教授：	王偉成 Wang, Wei Cheng 林文偉 Lin, Wen Wei
口試委員:	林文偉 Lin, Wen Wei 黃聰明 Huang, Tsung Ming
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系 Department of Mathematics
論文出版年：	2015
畢業學年度：	103
語文別：	英文
論文頁數：	74
中文關鍵詞：	特徵向量、特徵值
外文關鍵詞：	eigenvalue, eigenvector, eigendecomposition, singular value decomposition
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查詢本校圖書館目錄查詢臺灣博碩士論文知識加值系統勘誤回報

為了做到的三維光子晶體的數值模擬。由於三維光子晶體是由麥克斯韋
方程組模型，我們使用議的有限差分法推導出明確的矩陣表示。我們得到
一個廣義特徵值問題（GEVP）。對應於所述光子晶體具有面心立方（FCC）
的晶格和簡單立方（SC）的晶格的GEVP 和我們將這些理論結果來推算
GEVP 到一個標準特徵值問題（SEVP）。既然我們想要做的數值模擬，我
們開發FAME 做到這一點，並提供了GUI 界面。本文是介紹了FAME。

In order to do numerical simulations of three dimensional photonic crystals.
Because three dimensional photonic crystals is modeled by the Maxwell
equations, we used Yee’s finite difference scheme to derive the explicit matrix
representation. We get a generalized eigenvalue problem (GEVP).
The GEVP corresponding to the photonic crystals with face centered cubic
(FCC) lattice and simple cubic (SC) lattice and we apply these theoretical
results to project the GEVP to a standard eigenvalue problem (SEVP).
Since we want to do numerical simulation, we developed FAME to do this
and provided a GUI interface. This thesis is a introduction to FAME.
1

Contents
Introduction 8
Mathematical model 9
1 Explicit Representations of matrices for Yee’s scheme . . . . . . . . 10
2 Explicit matrix representation of the curl operator in Simple Cubic 13
2.1 Matrix representation of ∇  E . . . . . . . . . . . . . . . . 13
2.2 Matrix representation of ∇  H . . . . . . . . . . . . . . . . 14
3 Eigendecomposition of Simple Cubic . . . . . . . . . . . . . . . . . 14
4 Explicit matrix representation of the curl operator in Face Centered
Cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1 Matrix representation of ∇  E . . . . . . . . . . . . . . . . 16
4.2 Matrix representation of ∇  H = ϵE . . . . . . . . . . . . 18
5 Eigendecomposition of Face Centered Cubic . . . . . . . . . . . . . 18
6 Eigenproblem of Photonic Crystal . . . . . . . . . . . . . . . . . . . 20
7 Eigenproblem of Chiral Medium . . . . . . . . . . . . . . . . . . . . 21
User Guide for FAME without GUI 21
1 User Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.1 Popt.model . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2 Popt.domain . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3 Popt.material . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Popt.graphic . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5 Popt.lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.6 Popt.boundary . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.7 Popt.hardwar . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.8 Popt.eigsolver.solver . . . . . . . . . . . . . . . . . . . . . . 28
2 User defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
User Guide for FAME with GUI 29
1 FAME Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Domain : Show 3D structure . . . . . . . . . . . . . . . . . 33
4 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6 Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4
7 Eigensolver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
9 Savedate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
10 Numerical result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Introduction to Functions and Parameters 41
1 User Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.1 Popt.model . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.2 Popt.domain . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.3 Popt.material . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.4 Popt.graphic . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.5 Popt.lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.6 Popt.boundary . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.7 Popt.hardwar . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.8 Popt.eigsolver.solver . . . . . . . . . . . . . . . . . . . . . . 48
2 FAME Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 FAME Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1 Construct Simple Cubic . . . . . . . . . . . . . . . . . . . . 51
3.2 Parameter for Simple Cubic . . . . . . . . . . . . . . . . . . 51
3.3 Locate Simple Cubic . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Locate Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5 Locate Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.6 User Defined . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 FAME Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 Lattice Parameter . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Lattice Construct . . . . . . . . . . . . . . . . . . . . . . . . 55
5 FAME Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1 Curl Parameter . . . . . . . . . . . . . . . . . . . . . . . . . 56
6 Multiplying matrix and vector . . . . . . . . . . . . . . . . . . . . . 59
6.1 Multiplying Photonic matrix Ar and vector . . . . . . . . . 60
6.2 Multiplying Photonic matrix Pr and vector . . . . . . . . . 61
6.3 Inverse Fast Fourier Transform for a vector field . . . . . . . 61
6.4 Multiplying Photonic matrix Qr and vector . . . . . . . . . 62
6.5 Multiplying Photonic matrix Qrs and vector . . . . . . . . . 63
6.6 Multiplying Photonic matrix Prs and vector . . . . . . . . . 63
6.7 Matrix Vector Multiple Invert Photon . . . . . . . . . . . . 64
7 FAME Eigen Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5
7.1 Eigen_Solver . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 Eigen Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.3 Eigenvector Restoration . . . . . . . . . . . . . . . . . . . . 66
Future Work 66
1 User Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2 Solver Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 Universal Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Curl Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5 Construct Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6 Compare parameter with the old User_Option . . . . . . . . . . . . 72
Appendices 73
Reference 73
List of Tables
Explanation the parameter of Pmesh . . . . . . . . . . . . . . . . . 49
Explanation the parameter of Pgrid . . . . . . . . . . . . . . . . . . 49
Explanation the parameter of Pgrid . . . . . . . . . . . . . . . . . . 49
Explanation the parameter of Pgrid . . . . . . . . . . . . . . . . . . 50
The meaning of the parameter . . . . . . . . . . . . . . . . . . . . . 53
The meaning of the parameter . . . . . . . . . . . . . . . . . . . . . 56
The meaning of the parameter . . . . . . . . . . . . . . . . . . . . . 56
The meaning of input parameter . . . . . . . . . . . . . . . . . . . 64
The meaning of input parameter . . . . . . . . . . . . . . . . . . . 66
parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
List of Figures
The translation vectors of simple cubic . . . . . . . . . . . . . . . . 11
The translation vectors of face centered cubic . . . . . . . . . . . . 11
The first Brillouin zone of a simple cubic lattice . . . . . . . . . . . 12
The first Brillouin zone of a fcc . . . . . . . . . . . . . . . . . . . . 12
A simple cubic lattice that consists of spheres and circular cylinders. 23
A simple cubic lattice that consists of semi-woodpile structure. . . . 23
Table for Popt.domain . . . . . . . . . . . . . . . . . . . . . . . . . 24
6
Table for Popt.material . . . . . . . . . . . . . . . . . . . . . . . . . 25
Table for Popt.graph . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Table for Popt.graph . . . . . . . . . . . . . . . . . . . . . . . . . . 27
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
FAME_introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 30
FAME_mode_photonic_introduction . . . . . . . . . . . . . . . . 31
FAME_domain_photonic . . . . . . . . . . . . . . . . . . . . . . . 32
FAME_domain_photonic show 3D structure . . . . . . . . . . . . . 33
FAME_domain_photonic show 3D structure . . . . . . . . . . . . . 33
FAME Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
FAME lattice change . . . . . . . . . . . . . . . . . . . . . . . . . . 35
FAME lattice setting . . . . . . . . . . . . . . . . . . . . . . . . . . 35
FAME_lattice_change . . . . . . . . . . . . . . . . . . . . . . . . . 36
FAME Boundary Condition . . . . . . . . . . . . . . . . . . . . . . 36
FAME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
FAME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Vector Fields will change over time . . . . . . . . . . . . . . . . . . 39
FAME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
A simple cubic lattice that consists of spheres and circular cylinders. 43
A simple cubic lattice that consists of semi-woodpile structure. . . . 43
Table for Popt.domain . . . . . . . . . . . . . . . . . . . . . . . . . 44
Table for Popt.material . . . . . . . . . . . . . . . . . . . . . . . . . 45
Table for Popt.graph . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Table for Popt.graph . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Describing the functions . . . . . . . . . . . . . . . . . . . . . . . . 50
Concept map functions for FAME Material . . . . . . . . . . . . . . 50
Concept map functions for FAME_Material_Simple_Cubic . . . . 51
Concept map functions for FAME_Lattice . . . . . . . . . . . . . . 55
FAME_Matrix_Vector_multiplication categories figure . . . . . . . 59
matrix vector multiplication for photon categories figure . . . . . . 60
The figure about eigensolver and functions to be used . . . . . . . . 65
eigensolver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7
                                

References
[1] T.-M. Huang, W.-W. Lin, and W. Wang. ”Matrix Representations of Discrete
Differential Operators and Operations in Electromagnetis”.
73
[2] R.-L. Cherny, H.-E. Hsieh, T.-M. Huang, W.-W. Lin, W. Wang. ”Singular
Value Decompositions for Single-Curl Operators in Three-Dimensional
Maxwell’s Equations for Complex Media”. SIAM. J. Matrix Anal. Appl.,
36(1), 203–224.
[3] T.-M. Huang, H.-E. Hsieh, W.-W. Lin, and W. Wang. ”Matrix representation
of the double-curl operator for simulating three dimensional photonic
crystals”.
[4] K. Yee. Numerical solution of initial boundary value problems involving
Maxwell’s equations in isotropic media. IEEE Trans. Antennas and Propagation,
14:302-307, 1966.
[5] T.-M. Huang, H.-E. Hsieh, W.-W. Lin, and W. Wang. ”Eigendecomposition
of the Discrete Double-Curl Operator with Application to Fast Eigensolver for
Three-Dimensional Photonic Crystals”. SIAM. J. Matrix Anal. Appl., 34(2),
369–391
[6] T.-M. Huang, W.-J. Chang, Y.-L. Huang, W.-W.Lin, and W. Wang. ”Preconditioning
bandgap eigenvalue problems in three-dimensional photonic crystals
simulations”. Journal of Computational Physics 229 (2010) 8684–8703
[7] Z. Jia and C. Li. ”The shift-invert residual arnoldi method and the jacobidavison
method:Theory and algorithms,” arXiv preprint arXiv:1109.5455,
2011.
[8] Z. Jia and C. Li. ”Inner iterations in the shift-invert residual arnoldi method
and the jacobidavison method,” Science China Mathematics, vol. 57, no. 8,
pp. 1733-1752, 2014.
[9] G. L. Sleijpen and H. A. Van der Vorst. ”A jacobi-davidson iteration method
for linera eigenvalue problems,” SIAM Review, vol. 42, no. 2, pp. 267-293,
2000.
[10] M. Luo and Q. H. Liu. ”Three-dimensional dispersive metallic photonic crystals
with a bandgap and a high cutoff frequency, ” Vol. 27, No. 8/August
2010/J. Opt. Soc. Am. A 1879.
74

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