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博碩士論文 etd-0706105-172702 詳細資訊
Title page for etd-0706105-172702
論文名稱
Title
向量磁場正交基底分析任意圓柱座標介電質波導
Analysis of arbitrarily profiled cylindrical dielectric waveguides using vectored magnetic orthogonal bases
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
77
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
2005-06-24
繳交日期
Date of Submission
2005-07-06
關鍵字
Keywords
向量基底、貝索函數、馬克斯威爾、正交
Bessel function, photonic crystal, orthogonal
統計
Statistics
本論文已被瀏覽 5627 次,被下載 15
The thesis/dissertation has been browsed 5627 times, has been downloaded 15 times.
中文摘要
光波導在光通訊的領域中扮演者很重要的角色,且隨著對頻寬需求愈高,光波導的形式就愈複雜。目前對複雜光波導的解法大約有:向量橫模積分方程式、頻域有限差分、有限元素法、向量邊界積分方程式、直角座標的簡單基底法等。但眾多的波導解法,有些只適合解一維結構,並不適合解二維複雜波導。
在本論文中,我們發展了一套新方法,以嚴謹的向量正交基底,針對圓柱座標磁場,以一維的結構展圓柱座標波導的模態及場形分佈情形。本方法是考量在邊界上切面和垂直面連續的條件下,使用磁場的 和 作基底,可全區定義,經線性組合亦滿足Helmholtz方程式,由於基底與結構解非常近似,可降低計算項數而大大減少計算量。因為場量本身沒有奇異點,但耦合方程式的運算子因含 項會產生奇異點問題,此部分會由基底的貝索函數本身函數的定義,與原耦合方程式產生的 互相消去,解決發散問題,而且是用函數本身的定義而不是用一些特殊數值運算的方式蓋過原點。
推導的過程雖較為複雜,但本身基底的貝索函數(Bessel’s function)即是特徵值,因此使得最後結果的形式較為簡單,從而大幅減少模擬時的計算量,又由於本身是結構解,所以解的精確度很高,可靠度也很高。
Abstract
The dielectric waveguide component has become a mature industry in these days in making it and understanding how it works. There are many theoretical and numerical methods to solve these waveguide modes. For example, the rigorous vectorial coupled transverse mode integral equation formulation (VCTMIE), finite-difference frequency-domain method (FDFD), vectorial beam propagation method (VBPM) and modal expansion method with simple bases (MEMSB)…etc. With the exception of the MEMSB method and the alike, all these methods work very hard to handle interface boundaries and many require many terms meet the convergence requirements.
In this thesis, we propose a rigorous modal expansion method based on a set of orthogonal transverse magnetic bases to analyze arbitrarily profiled 2-D cylindrical dielectric waveguides. First, we expand the mode field solution of a general multi-layered waveguide by linear combination of 1D homogeneous solutions. Magnetic and components are chosen for its continuity property across the material interface. The choice of Magnetic field over electric field also reduces the number of terms and minimizes the Gibb’s phenomenon. Our new vector bases eliminate the numerical difficulty of working with the singular term of the cylindrical differential operators. When compared with the results using simple bases, we further reduce one quarter of terms without loosing any accuracy.
Although the process of deriving the formulation of this vector cylindrical basis expansion technique is complex because Bessel functions and their derivatives are involved, the resulting matrix eigenvalue-eigenvector equation is much simpler than that of the simple bases and the new the result is also more accurate. We also extended the analysis to study the 2-D cylindrical dielectric waveguide problem.
目次 Table of Contents
誌謝 I
中文摘要 II
英文摘要 III
目錄 IV
圖表目錄 VI
第一章 導論 1
1-1 研究動機 1
1-2 簡介 3
1-3 馬克斯威爾方程式 5
第二章 磁場圓柱座標向量基底 7
2-1 邊界條件 7
2-2 圓柱基底 9
2-3 不同邊界下的圓柱基底 13
2-4 磁場圓柱座標向量基底 16
第三章 橫向磁場耦合方程式及正交特性 23
3-1 橫向磁場耦合方程式 23
3-2 向量基底的正交特性 26

第四章 特徵函數推導證明 32
第五章 全橫向磁場公式架構 41
5-1 磁場對磁場正交特性 41
第六章 二維結構圓柱光纖波導架構 44
第七章 數值模擬結果 51
7-1 特徵值特徵向量矩陣化 51
7-2 磁場對磁場特徵函數矩陣化 55
7-3 數值結果 57
第八章 結論 65
參考文獻 67
中英對照表 68
參考文獻 References
[1] Chin-ping Yu, Hung-chun Chang, “Compact finite-difference frequency-domain method for the analysis of two dimensional photonic crystal”, OSA,Vol.12, No.7 5, April 2004
[2] Tzong-Lin Wu, Chia-Hsin Chao, “Photonic crystal fiber analysis through the Vector Boundary-Element Method: effect of elliptical air hole”, IEEE, Vol.16, No.1, Jan. 2004
[3] Tzong-Lin Wu, Jung-Sheng Chiang, Chia-Hsin Chao, “A Novel Approach Calculating the Dispersions of Photonic Crystal Fibers”, IEEE, Vol.16, No.6, June 2004
[4] 曹碩芳, “簡單正交基底之二維向量介質波導模態解”, 國立中山大學光電工程研究所碩士畢業論文, June 2004
[5] 張世明, “精簡正交基底之二維向量介電質波導模態解”, 國立中山大學光電工程研究所碩士畢業論文, June 2004
[6] Akira Ishimaru, “Electromagnetic Wave Propagation, Radiation, and Scattering”, Englewood Cliffs, N.J: Prentice-Hall, 1991
[7] Amnon Yariv, “Optical Electronics in Modern Communications”, Fifth editon, 1997
[8]房景威, “任意軸對稱光纖波導理論與數值分析”, 國立中山大學光電工程研究所碩士畢業論文, June 2002
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