臺灣博碩士論文加值系統

在本篇論文中,我們發展了新的電磁解析模型來分析一維波導、二維波導和光子晶體結構。同時,我們也推導了新的同步近似項的邊界條件來配合新的方程式。另外,由於樂勤得(Legendre)內差方程式準確的近似特性,我們也使用了譜方法中的樂勤得法來對空間分割。
不同於大部分是以赫姆霍茲(Helmholtz)方程式為基礎來解波導模態,我們提出了一種綜合了馬克斯威爾旋度以及散度方程式的新模型來產生特徵值(eigenvalue)問題的方法。對於在介面的邊界條件,我們則是使用了同步近似項的邊界條件,這種邊界條件可以在數學上證明具有穩定數值的效果。雖然在時域上,使用同步近似項邊界條件的譜方法是一個被證實具有不錯收斂性質的數值方法,但其使用的方程式跟本篇研究所發展的、在頻域使用的方程式有本質上的不同。在時域上我們所考慮的電磁場一律是實數,但在頻域上因為使用相位法的因素,所有的電磁場必須使用複數來表示。所以我們重新推導了分別給一維波導、二維波導和光子晶體使用的同步近似項邊界條件。
而為了確認新提出演算法的效率程度和數值收歛性,我們做了一些數值實例分析。這些光實例包括平板波導、部份填滿波導、圓柱形波導、具有尖角的埋入式波導、肋型波導以及四角晶格和三角晶格這兩種光子晶體結構。利用新的演算法來解這些結構時,我們可以得到高精度的傳播常數和特徵頻率以及觀察到指數收斂的特性。另外要特別說明的是,對於有尖角的介質波導,我們在沒有對邊界作任何特殊處理的情況下,得到了比以往更準確的數值解,這代表了在光波導分析領域的一項重大突破。

In this thesis, mode solvers for one-dimensional (1D) and 2D waveguides, and photonic crystals (PCs) with new electromagnetic formulations are developed. The new penalty-type boundary conditions are derived to work with new formulations, and the pseudospectral Legendre method is adopted to perform spatial discretization for its accurate approximation property.
Unlike many waveguide mode solvers which are based on Helmholtz equations, we propose new formulations which combine Maxwell’s curl and divergence equations to derive the eigenvalue problem. For the interface boundary condition treatment, penalty-type boundary conditions are employed and mathematically proved that they can stablize the scheme. Athlough pseudospectral time-domain (PSTD) methods with penalty-type boundary conditions have been known to offer good convergence property, the related frequency-domain formulations developed in this work prossess intrinsic difference. In time-domain simulations, the electromagnetic fields considered are all real quantities, while in frequency-domain analysis, the fields are complex ones with the phasor technique. And new penalty-type boundary conditions in the frequency-domain mode analysis of 1D and 2D waveguides, and PCs, are respectively derived.
Numerical examples are considered to examine the efficieney and numerical convergency property of the proposed algorithms. Optical structures in these examples include slab waveguides, partial-filled waveguide, fiber waveguides, channel waveguides with sharp corners, rib waveguides, PCs with square lattice, and PCs with triangular lattice. Spectral convergence property with very high-accuracy modal effective index and igenfrequency calculation is achieved. In particular, for the dielectric waveguide with corners, higher numerical accuracy than reported results is obtained without doing field singularity treatment at the corners as in the latter. This represents significant advancement in the numerical analysis of optical waveguide problems.

1 Introduction 1
1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . 3

2 Mathematical Formulation for 1D Waveguide analysis 5
2.1 Equations Used for 1D Waveguide Structure . . . . . . . . . . 5
2.2 Well-posedness Analysis . . . . . . . . . . . . . . . . . . . 6
2.3 The Energy Method . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Characteristic Representations of Physical Boundary Condtions 10
2.5 Design of the Scheme . . . . . . . . . . . . . . . . . . . . 12
2.6 Final Form of the Formulation . . . . . . . . . . . . . . . . 14

3 Mathematical Formulation for 2D Waveguide Analysis 18
3.1 Equations Used for 2D Waveguide Structure . . . . . . . . . . 18
3.2 Well-posedness Analysis . . . . . . . . . . . . . . . . . . . 20
3.3 The Energy Method . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Characteristic Representations of Physical Boundary Condtions 25
3.5 Design of the Scheme . . . . . . . . . . . . . . . . . . . . 27
3.6 The Final Form of The Six-Equation Version. . . . . . . . . . 29
3.7 The Final Form of Three-Equation Version. . . . . . . . . . . 31

4 Mathematical Formulation for 2D Photonic Crystal Analysis 35
4.1 Equations Used for PC Structure. . . . . . . . . . . . . . . . 35
4.2 Well-posedness Analysis. . . . . . . . . . . . . . . . . . . . 36
4.3 The Energy Method. . . . . . . . . . . . . . . . . . . . . . . 37
4.4 Characteristic Representations of Physical Boundary Condtions. 38
4.5 Design of the Scheme . . . . . . . . . . . . . . . . . . . . . 40
4.6 The Final Form of The Three-Equation Version . . . . . . . . . 42
4.7 The Final Form of the One-Equation Version . . . . . . . . . . 43

5 Pseudospectral Method and Shifted Inverse Power Method 45
5.1 The Pseudospectral Method. . . . . . . . . . . . . . . . . . . 45
5.1.1 Overview of the Pseudospectral Method. . . . . . . . . 45
5.1.2 The Pseudospectral Legendre Method . . . . . . . . . . 46
5.1.3 Curvilinear Representation of The Pseudospectral Method 48
5.2 The Shifted Inverse Power Method . . . . . . . . . . . . . . . 51
5.2.1 The Algorithm of SIPM. . . . . . . . . . . . . . . . . 51
5.2.2 The Iterative Method . . . . . . . . . . . . . . . . . 52
5.2.3 Guessing the Initial Eigenvector Using Former Data . . 54

6 Numerical Results For Waveguide Problems 57
6.1 Symmetric Slab Waveguides. . . . . . . . . . . . . . . . . . . 57
6.2 Asymmetric Slab Waveguides . . . . . . . . . . . . . . . . . . 58
6.3 Partially Filled Metallic Waveguides . . . . . . . . . . . . . 58
6.4 Circular Metallic Waveguides . . . . . . . . . . . . . . . . . 59
6.5 Fiber Waveguides . . . . . . . . . . . . . . . . . . . . . . . 60
6.6 Channel Waveguides with Sharp Corners. . . . . . . . . . . . . 61
6.7 Rib Waveguides . . . . . . . . . . . . . . . . . . . . . . . . 62

7 Numerical Results For Photonic Crystal Problems 86
7.1 Square-Lattice Photonic Crystals . . . . . . . . . . . . . . . 86
7.2 Triangular-Lattice Photonic Crystals . . . . . . . . . . . . . 87

8 Conclusion 96

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