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研究生:江柏叡
研究生(外文):Po-Jui Chiang
論文名稱:適於光波導與光子晶體之多區塊類頻譜法模態解析模型的發展
論文名稱(外文):Development of Multidomain Pseudospectral Mode Solvers for Optical Waveguides and Photonic Crystals
指導教授:張宏鈞
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:光電工程學研究所
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2007
畢業學年度:95
語文別:英文
論文頁數:174
中文關鍵詞:類頻譜
外文關鍵詞:PSMS
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本論文發展一項採用多區塊類頻譜法之全向量模態解析模型以分析任意階射率分布之光波導問題。藉由柴比雪夫和雷建德多項式類頻譜法,我們將介質分割成多個區塊,再利用曲線變形技巧使得每個區塊組合成元件截面結構的形狀,最後再將區塊與區塊介面間之適當的邊界條件引入,如此可大大增加分析的精準程度。除此之外,為了處理洩漏波導問題,我們應用完美匹配層吸收邊界於模型中,得以計算複數傳播常數。本研究利用此新模型分析傳統的光波導結構以及新穎的光子晶體光纖結構,並與正解和其他數值分析方法做比較,以證明新模型分析結果的可靠性及準確性。
再者,本研究首次利用類頻譜模態解析法計算二維光子晶體的頻帶圖,並且展示此方法對於如三角晶格與四方晶格等不同週期性結構,不論是橫向電場或橫向磁場光波都能獲得高精準度收斂的結果。本研究亦展示對於極微小頻帶寬度間的能隙的分析亦可獲得極佳的精準度,證明所建立新方法的價值與重要性。
  本論文在附錄中另提出一項利用柴比雪夫類頻譜法做內差的方式計算各種光纖的色散係數,相較於傳統的直接差分計算須利用相當多波長取樣點的方式,本方法僅須用到數個波長上的有效折射率,因此有極佳的計算效率。
A new full-vectorial pseudospectral mode solver based on multidomain pseudospectral
methods for optical waveguides with arbitrary step-index profile is presented. Both Legendre and Chebyshev collocation methods are employed. The multidomain advantage helps in proper fulfillment of dielectric interface conditions, which is essential in achieving high numerical accuracy. Suitable multidomain division of the computational domain is performed to deal with general curved interfaces of the permittivity profile and field continuity conditions are carefully imposed across the dielectric interfaces. Therefore, a curvilinear coordinate mapping technique is introduced to perfectly deal with curved boundaries. Each contiguous subdomain is joined by intensionally imposing different types of boundary conditions to enhance the accuracy. Moreover, perfectly matched layer (PML) absorbing boundary conditions are incorporated into the model so that leaky
modes with complex propagation constants can be analyzed. The solver is applied to the calculation of guided modes on optical fibers, fused fiber couplers, D-shaped fibers, channel waveguides, rib waveguides, and photonic crystal fibers, and comparison with analytical results or reported ones based on other methods is made. It is demonstrated that numerical accuracy in the effective index up to the remarkable 10􀀀10 order can be easily achieved.
The multidomain pseudospectral scheme is for the first time applied to the calculation of the band diagrams of two-dimensional photonic crystals with the inclusion of the required periodic boundary conditions, and is again shown to possess excellent numerical convergence behavior and accuracy. The proposed method shows uniformly excellent convergence characteristics for both the transverse-electric and transverse-magnetic waves in the analysis of di_erent structures. The analysis of a mini band gap with the normalized frequency gap width as small as on the order of 107 is also shown to demonstrate the extremely high accuracy of the proposed method.
A novel numerical calculation of chromatic dispersion coefficients of optical fibers including holy fibers is also proposed in this research using a procedure involving Chebyshev-Lagrange interpolation polynomials. Only numerically determined effective indices at several wavelengths are needed for obtaining the dispersion curve and no direct numerical differentiation of the effective refractive index is involved.
1 Introduction 1
1.1 History of the Spectral Methods . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Application of Spectral Methods to Electromagnetics Problems . . . . . . 3
1.3 Modal Analysis of Optical Waveguides and Photonic Crystals . . . . . . . 4
1.4 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Contributions of the Present Work . . . . . . . . . . . . . . . . . . . . . . 8
2 Fundamentals of Spectral Methods 14
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Spectral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Galerkin-Type Method . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Pseudospectral Method . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Pseudospectral Fourier Method . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Pseudospectral Legendre Method . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Pseudospectral Chebyshev Method . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Mathematical Formulation and Boundary Conditions for OpticalWaveg-
uides 31
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 The Physical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 The Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 The Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
i
3.3.1 2-D Pseudospectral Formulae in Curvilinear Form . . . . . . . . . . 35
3.3.2 Formulation of the PSMS . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Setting Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.1 Dielectric Boundary without Corners . . . . . . . . . . . . . . . . . 42
3.5 The Perfectly Matched Layer . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5.1 The Berenger''s Perfectly Matched Layer . . . . . . . . . . . . . . . 44
3.5.2 The Anisotropic Perfectly Matched Layer . . . . . . . . . . . . . . . 47
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Numerical Results for Analysis of Various Optical Waveguide Struc-
tures 58
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Circular Waveguide: Comparison of Pseudospectral Chebyshev and Legendre
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 The D-Shaped Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Symmetrical 2 2 Strongly Fused Fiber-Optical Couplers . . . . . . . . . . 61
4.4.1 Fused Coupler with No Cores . . . . . . . . . . . . . . . . . . . . . 63
4.4.2 Fiber-Core E ects in Fused Fiber-Optical Coupler . . . . . . . . . . 64
4.5 Analysis of Waveguides with Corners . . . . . . . . . . . . . . . . . . . . . 64
4.5.1 Dielectric Boundary with Corners . . . . . . . . . . . . . . . . . . . 65
4.5.2 Rectangular Channel Waveguide . . . . . . . . . . . . . . . . . . . 66
4.5.3 Rib Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.6 Photonic Crystal Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Analysis of Band Structures in Two-Dimensional Photonic Crystals 110
5.1 The Physical Picture for 2-D Photonic Crystals . . . . . . . . . . . . . . . 110
5.2 Mathematical Formulation of the PSMS . . . . . . . . . . . . . . . . . . . 111
5.3 Imposing Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3.1 Boundary Conditions Between Adjacent Sub-domains . . . . . . . . 113
ii
5.3.2 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 117
5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.1 PC with Square Lattice . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.2 PC with Triangular Lattice . . . . . . . . . . . . . . . . . . . . . . 123
5.4.3 PC with Large Air Holes . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4.4 PC with Large Dielectric Pixels . . . . . . . . . . . . . . . . . . . . 124
5.4.5 Analysis of a Mini Band Gap . . . . . . . . . . . . . . . . . . . . . 125
6 Summary and Future Works 145
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A Robust Calculation of Chromatic Dispersion Coe cients of Optical Fibers
from Numerically Determined E ective Indices Using Chebyshev-Lagrange
Interpolation Polynomials 1 148
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A.2 The Chebyshev Collocation Method . . . . . . . . . . . . . . . . . . . . . . 150
A.3 An Idealistic Waveguide Having Analytical Chromatic Dispersion Characteristics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.4 Dispersion in Holey Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
A.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
B List of Acronyms 167
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