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博碩士論文 etd-0807116-141456 詳細資訊
Title page for etd-0807116-141456
論文名稱
Title
探討任意斜邊與邊角介質界面之二維亥姆霍茲方程緊湊係數
Analysis of Compact Stencils for Two-Dimensional Helmholtz Equation with Dielectric Corner and Inclined Interfaces
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
136
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2016-07-26
繳交日期
Date of Submission
2016-09-07
關鍵字
Keywords
赫茲向量、電磁場分量耦合方程式、材料平均、分段線性近似、相連局域場、有限差分緊緻模板、亥姆霍茲方程
transparent boundary condition, piece-wise linear approximation, local coordinate system, Connected Local Fields, material averaging, compact FD stencils, Helmholtz equation
統計
Statistics
本論文已被瀏覽 5694 次,被下載 578
The thesis/dissertation has been browsed 5694 times, has been downloaded 578 times.
中文摘要
計算電磁學在光電領域中乃是相當重要的一環,因如何精準快速地計算複雜的光波導元件是個非常關鍵的問題。一般而言,我們所著重的是光波導元件的穩態行為,因此模擬分析上,主要需要處理的對象便是描述電磁波在頻域下的亥姆霍茲方程式 (Helmholtz equation)。
傳統上來說,應用頻域有限差分法來處理波導問題時,最大的困難點發生在求取介質不連續區域附近的有限差分係數,此區域的處理亦常是主要的誤差來源。在本論文中,我們將藉由模擬計算二維矩形介電質結構與圓形介電質結構,以期進一步探討邊角介面或是介質介面附近的數值誤差。首先,我們會先推導出線性介面與邊角介面附近的解析形式的有限差分緊緻模板 (FD-CS),接著將所推得的FD-CS應用於模擬前述的兩種結構。在這兩種結構的模擬中,均勻區的係數都將採取先前實驗室所提出的相連局域場 (connected local fields, CLF) 方法的六階精密緊緻模板。由於均勻區具有高精密度,可以合理地推斷模擬計算的主要誤差來源乃是介面與邊角附近的差分係數不夠精確所造成的。矩形結構的不夠精準主要來自於方角錐附近的誤差,而圓形結構的不精確則是由於曲線的分段線性近似與介面誤差所致,利用高取樣密度下的模擬結果作為參考解,將低取樣密度下的數值解與之比較,可更進一步地了解介面係數與邊角係數的收斂特性。進一步探討在分段線性近似以及材料平均的準確性以與有效性,本論文實驗結果與解析正解比較在取樣密度單位波長為十點時,波核與波覆折射率小對比下幾乎沒有誤差而在大對比下(3.5:1.0),其場值誤差約在百分之二。
本篇論文的第五章節,簡單扼要地探討了不同形式的電磁場分量耦合方程式,並引用赫茲向量 (Hertz vector) 詳細地推導出介電質波導以軸向分量為主之電磁場分量耦合偏微分方程式。
Abstract
For passive dielectric waveguide devices, we are primary concerned with their steady-state behaviors or narrow-band characteristics. Hence the frequency-domain (FD) methods, such FD finite-difference (FD-FD) methods are often used for solving the Helmholtz equation. Traditional FD-FD methods cannot effectively process points near a dielectric interface with large index contrast. In recent years, we proposed the semi-analytical method of Connected Local Fields (CLF) to solve Helmholtz equation with complex dielectric structures.
In this thesis we will address several key mathematical techniques in order to demonstrate that CLF method can be a highly-accurate method for analysis of two-dimensional complex waveguide structures. First, we review several FD-FD methods in handling dielectric media with interfaces: These includes a straight forward implementation of the interface conditions (such as continuity of normal derivative), a “material averaging” approach for obtaining compact coefficients stencils (CS) of the Helmholtz equation near a dielectric interface and our CLF’s approach of obtaining CS for cells with an inclined linear interface. In chapter 4 we will develop CLF-worthy compact stencils for implementing the transparent boundary condition (TBC) based on local plane wave extrapolation. We have also developed a second-order accurate compact stencils for cell with dielectric corners. Finally we will apply piece-wise linear approximation (PWLA) for dielectric curved interfaces. Combining these techniques we study local corner errors in the 2D rectangular structure as well as PWLA errors in a 2D circular dielectric structure. We report here that compared with exact Green’s functions of the circular dielectric disk we found the PWLA errors are well under two percent for large core-cladding index contrast (3.5:1.0) and much less for small contrasts at low sampling density of 10 points per wavelength.
After two-D dielectric structures, the future work will be focused on the two and a half D dielectric waveguide problems where we encounter three-D wave fields inside a two-D structure. Here we exam a formulation based on coupled longitudinal EM components, namely and When combined with the CLF method, this formulation has the advantage of being able to choose an appropriate local coordinate system whose axes are parallel or perpendicular to the dielectric interface. Thus the complexity of coupled formulation may be greatly reduced.
目次 Table of Contents
目錄
頁次
論文審定書 i
誌謝 ii
中文摘要 iii
英文摘要 iv
目錄 vi
圖次 viii
第一章 緒論 1
1.1 研究背景 1
1.2 研究方法及回顧 2
第二章 數值方法簡介 4
2.1 數值計算方法 4
2.2 回顧傳統特殊介質界面處理方法 6
第三章 探討某些含介質界面之二階精密緊湊係數 25
3.1 二維頻域有限差分電介質水平與大對角界面緊湊係數 25
3.2 二維頻域有限差分方角電介質緊湊係數 28
3.3 改進邊角界面係數精密度 33
3.4 水平及大對角介面誤差分析 35
第四章 長方形以及圓形介質結構場場型分析與模擬 45
4.1 模擬架構與步驟 45
4.2 穿透邊界條件 51
4.3 二維數值格林函數模擬結果 54
4.4 長方形與圓形介質結構模擬與誤差分析 59
第五章 軸向場偏微分耦合方程式推導 102
5.1純量及位能函數以及赫茲向量 103
5.2橫向場偏微分耦合方程式推導 104
5.3 軸向場偏微分耦合方程式推導 106
第六章 結論與未來工作 116
參考文獻 119
參考文獻 References
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