光波導的製作在積體光學中是極為重要的一環，因此如何分析各種
形式的光波導是當前的重要課題。彎曲光波導在以往的分析中是一個困難的問題，傳統上波導計算方式是光束傳播法(Beam PropagationMethod，BPM），但是因為BPM 使用大量近似，會因為在彎曲所造成的斜面左右兩側的座標系統不完全匹配，而造成計算上的困難。
此論文中引入解析連續法去處理因為彎曲所導致的斜面邊界問
題，可以處理原本斜面兩側座標系統的不匹配，計算量與誤差可以大幅降低，再配合上連續條件與微分連續的邊界條件，推得兩組積分方程矩陣。但推導出的積分方程矩陣，計算上較複雜，所以利用波導具有對稱結構的特性，簡化計算過程與矩陣繁衍，可以有效地降低計算量。為了克服彎曲的角度過大時，推衍出來的矩陣會奇異(singular)的問題，而造成計算的誤差會加大，所以推導矩陣元素均以積分正解公式來計算，再配合上特殊的數值方法，將所能計算的彎曲角度範圍擴大。利用我們理論，微波波導可以計算到彎曲30 度的結構，介電質波導可以計算到彎曲15 度的結構，且可算出相當小的反射係數，大約為-60dB，這是其他方法無法做到的。透過場型的計算分析後，可以發現到波導中能量傳播的情況也與我們理論預期的結果相符。

Dielectric waveguides are crucial devices in the making of
integrated-optical circuits. It is very important to analyze this type of waveguides so we can optimize the design for better performance.
Analysis of bent waveguides has been a difficult problem in the past. In a bent waveguide, two coordinate systems are needed to fully describe the ongoing complex scattering process in the transition region of the
waveguide. It is extremely hard to analyze such problems for methods built on a single coordinate system such as the finite-difference,finite-element methods and the beam propagation method (BPM).
In this thesis, we adopt dual mode-field representations (for all the low and higher-order modes), one for the incident and reflected waves and the other for the transmitted waves, to study bending effects. To calculate
the wave fields, we apply the analytic continuity principle to allow the waves to analytically extend and join smoothly on the bordering line. By matching the two continuity conditions of both the fields and their normal
derivatives we get two matrix equations for the reflection and transmission coefficients. For symmetrical bending waveguide, the task can be further reduced to solving two smaller problems each with even or odd symmetry on the bordering line.
As the bent angle increases the governing matrix equation becomes more singular. As a result, all the elements in the matrix are calculated with closed-form formulae to minimize the stability problem. In addition, special numerical methods are used to extend the range of the bending angles that this method can handle. In conclusion, our theory can calculate microwave bending waveguides up to 30 degrees and for dielectric slab waveguide with 15 degree bent angle. With this method we are able to compute small reflection coefficients of about -60dB and less.

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
第二章 彎曲微波波導的理論推導 4
2-1：解析連續法的理論模型 4
2-2：對稱結構的理論模型 5
2-3：彎曲對稱結構的分析 8
第三章 彎曲微波波導數值計算結果 18
3-1：數值積分的結果 18
3-2：特殊數值分析的結果 21
第四章 彎曲三層介電質波導的理論推導 24
4-1：三層結構的模態的分析 24
4-1-1：二層結構的模態的分析與歸一化係數的推導 26
4-1-2：二層結構轉對稱三層結構的分析 43
4-2：三層彎曲對稱結構的分析 48
第五章 彎曲三層介電質波導數值計算結果 66
第六章 結論與未來研究 76
參考文獻 78
中英對照表 80

[1] M. Rivera, “A finite difference BPM analysis of bent dielectric waveguides", J.Lightwave Technology., vol. 13, pp. 233-238, Feb. 1995.
[2] J. Ya Yan Lu and Pui Lin Ho;” Beam Propagation Modeling of Arbitrarily Bent Waveguides”, Photonics Technology Letters, IEEE , Volume: 14 , Issue: 12 , Dec. 2002
[3] ”Calculation of Bending Losses in Dielectric Waveguides Using Eigenmode Expansion and Perfectly Matched Layers”, Peter Bienstman, E. Six, M. Roelens, M. Vanwolleghem, and Roel Baets, 2001
[4] “Determination of Bend Mode Characteristics in Dielectric Waveguides”Andrea Melloni, Federico Carniel, Raffaella Costa, and Mario Martinelli JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 19, NO. 4, APRIL 2001
[5] Sangin Kim & Adand Gopinath ;” vector analysis bend dielectric waveguides using finite-difference method”, J.Lightwave Technology., vol. 14,1996
[6] Ishimaru,A.,”Electromagnetic Wave Propagation ,Radiation ,and Scattering ,”Englewood Cliffs,N.J.:Prentice-Hall,1991
[7] 施有彰You-Jang Shih ;”Analysis and Numerical Study of Rectangular Waveguide with Large Bending Angles”,國立中山大學光電工程研究所碩士畢業論文,June 2001
[8] 黃志文Chi-Wen Huang ;”Modal Analysis of Roof-top Ridged Waveguides”, 國立中山大學光電工程研究所碩士畢業論文,July 1997
[9] 游能忠Neng-Jung You ;”Application of coupled E/H field formulation to the design of multiple layer AR coating for large incident angles”,國立中山大學光電工程研究所碩士畢業論文,June 2001
[10] Scarmozzino,R ;”Numerical Techniques for Modeling Guided-Wave Photonic Device”, IEEE Journal of Selected topic in Quantum electronic,
January 2000
[11] Tso-Lun Wu and Hung-Wen Chang, “Guiding mode expansion of a TE and TM transverse-mode integral equation for dielectric slab waveguides with an abrupt termination”J. Opt. Soc. Am. A, vol. 18, no. 11, pp.2823-2832,
November 2001.
[12] Tso-Lun Wu and Hung-Wen Chang,“Analysis of TE to x and TM to x Mode for Dielectric Slab Waveguides ”Optics and Photonics Taiwan ’01 Proceeding in Optical wavegudide Modeling, TB1-2 pp. 146-148,December 13-14, 2001.
[13] 吳宙秦Chou-Chin Wu ;”具有小角度斜切角終端波導之特性分析”,國立中山大學光電工程研究所碩士畢業論文,June 2004
[14] L. M. Johnson and F. J. Leonberger ;” Low-loss LiNbO3 waveguide bends with coherent coupling” OPTICS LETTERS, February 1983 / Vol. 8, No. 2