在計算光電磁學的領域中，涵蓋兩大類型的數值模擬方法，其一為積分方程式方法，其二為有限差分法。前者具有極高的計算精密度，但只要元件改變即必須重新建構模擬的演算規則，應用上缺乏彈性。而後者，有限差分法，其原理簡單能廣泛地應用於各種問題的分析。在光電被動元件的分析上，最重要的是元件的穩態表現或是窄頻特性，因此我們的模擬分析著重在頻域，而相應的電磁波控制方程式便是Helmholtz equation。使用標準的有限差分公式將原始偏微分方程式轉換為離散化方程式，為了得到足夠的精準度，必須有相當高的取樣密度（以數值色散誤差而言，大約每個波長要有15個以上的取樣點，誤差才能低於1%），這造成我們必須進行龐大的稀疏矩陣的反演運算。有鑒於此，本論文提出「相連局域場方法」對Helmholtz equation進行離散化處理，簡稱「CLF」。
相連局域場方法係利用Helmholtz equation在均勻區的通解 Fourier-Bessel series 對緊緻網格 (compact stencil) 上的取樣位置作局域場展開，進而獲得方程式本身的離散化形式 (LFE-9) 以及能夠表示取樣位置以外的區域的場函數重構公式。藉著數值色散誤差分析可知，對於LFE-9而言，當每個波長涵蓋2.1個取樣點以上時，相對誤差在1%以下，且此離散化方程相對於原始方程式具有六階精確，達到橢圓系偏微分方程式在緊緻網格下離散化的最高精確階數，因而得以大幅度地降低數值模擬的儲存量與計算量。更進一步地，我們使用一階近似法對LFE-9的色散相對誤差進行嚴格分析，得到了誤差作為取樣密度與平面波行進方向的封閉形式函數關係，確實地掌握這兩個因素對於色散誤差的影響。

In the thesis, we propose a newly-developed method called the method of Connected Local Fields (CLF) to analyze opto-electromagnetic passive devices. The method of CLF somewhat resembles a hybrid between the finite difference and pseudo-spectral methods. For opto-electromagnetic passive devices, our primary concern is their steady state behavior, or narrow-band characteristics, so we use a frequency-domain method, in which the system is governed by the Helmholtz equation. The essence of CLF is to use the intrinsic general solution of the Helmholtz equation to expand the local fields on the compact stencil. The original equation can then be transformed into the discretized form called LFE-9 (in 2-D case), and the intrinsic reconstruction formulae describing each overlapping local region can be obtained.
Further, we present rigorous analysis of the numerical dispersion equation of LFE-9, by means of first-order approximation, and acquire the closed-form formula of the relative numerical dispersion error. We are thereby able to grasp the tangible influences brought both by the sampling density as well as the propagation direction of plane wave on dispersion error. In our dispersion analysis, we find that the LFE-9 formulation achieves the sixth-order accuracy: the theoretical highest order for discretizing elliptic partial differential equations on a compact nine-point stencil. Additionally, the relative dispersion error of LFE-9 is less than 1%, given that sampling density greater than 2.1 points per wavelength. At this point, the sampling density is nearing that of the Nyquist-Shannon sampling limit, and therefore computational efforts can be significantly reduced.

Acknowledgements i
Chinese Abstract and Keywords ii
English Abstract and Keywords iii
Contents iv
List of Figures vii
Chapter 1: Introduction 1
1.1 Research Motivation 1
1.2 Research Scope 4
1.3 Research Method 6
1.4 Overview of the Thesis 9
Chapter 2: Concepts and Literature Review 10
2.1 Discretization of Computation Domain 10
2.2 Standard FD-Form Helmholtz Equation 17
2.3.1 1-D Case: FD2-3 17
2.3.2 2-D Case: FD2-5 and FD2-9 20
2.3 Improved FD-like Schemes of Helmholtz Equation 25
2.3.1 Jo-Shin-Suh’s Optimal Formula 25
2.3.2 Nehrbass-Jevtic-Lee’s Formula (RD-FD) 26
2.3.3 Singer-Turkel’s 4th-Order Formula (FD4-9) 27
2.3.4 Singer-Turkel’s 6th-Order Formula (FD6-9) 27
2.3.5 Hadley’s Formula and Chang-Mu’s Work 28
Chapter 3: Numerical Dispersion 31
3.1 The Meaning of Numerical Dispersion 31
3.2 General Techniques of Numerical Dispersion Analysis 33
3.2.1 Derivation of Dispersion Equation 33
3.2.2 Dispersion Analysis at Low Frequency: Taylor Expansion 36
3.3 Dispersion Characteristics of Classical FD Schemes 38
3.3.1 Numerical Dispersion Characteristic: FD2-5 38
3.3.2 Numerical Dispersion Characteristic: FD2-9 38
3.3.3 Dispersion Equations under Low Frequency 39
3.4 A Short Conclusion 40
Chapter 4: The Theory of Connected Local Fields 42
4.1 The 1-D Exact CLF Formula: LFE-3 42
4.2. The 2-D Five-Point CLF Formula: LFE-5 48
4.2.1 The 2-D Five-Point CLF Formula: LFE-5 48
4.2.2 The 2-D Nine-Point CLF Formula: LFE-9 51
4.2.3 The 2-D Reconstruction Formulae for LFE-9 53
4.3 Investigation of Numerical Dispersion 56
4.3.1 V versus B Curves 56
4.3.2 Relative Dispersion Error versus Sampling Density 59
4.3.3 Numerical Study of LFE-9 Spatial Dispersion 63
4.4 Principles of CLF 67
Chapter 5: Thorough Investigation on CLF Formulae 70
5.1 Local Truncation Error 70
5.1.1 Local Truncation Error of LFE-5 70
5.1.2 An Alternate Derivation for the LFE-9 Formula 72
5.1.3 Local Truncation Error of LFE-9 73
5.2 Rigorous Analysis of Numerical Dispersion 74
5.2.1 Summary of Dispersion Characteristics 74
5.2.2 First-Order Analysis 75
5.2.3 First-Order Dispersion Error for LFE-5 76
5.2.4 First-Order Dispersion Error for LFE-9 77
5.2.5 Numerical Verification of First-Order Error Analysis 79
5.3 Summary 80
Chapter 6: Conclusions 83
6.1 Summary: The Theory of CLF 83
6.2 Future Works 84
References 86

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