目前光波導積體元件有許多部分是連續彎曲(continuously bending)
形式的波導，所以彎曲形式的波導場型分析顯得格外重要。光在傳統上
彎曲波導的計算大部分是光束傳播法(BPM)，雖然光束傳播法計算快
速，但使用單向傳播與近光軸的近似，在大角度彎曲波導元件的分析算
出來的場型不夠精確。一般目前最常用時域有限差分法(FD-TD)來分析
光波導元件光場的傳播，雖然時域有限差分法是個有效的方法，但是分
析尺寸較大的元件時，需要大量記憶體和運算時間。假如光在連續彎曲
波導元件傳播沒有什麼反射情形產生，為求加快運算速度，保持精確，
我們提出以圓柱波導模態(circular modes)配合全特徵模態展開法(Full
eigen-mode expansion technique, FEMET), 圓柱坐標聯立橫模耦合積分
方程(cylindrical couple transverse-mode integral-equation, C-CTMIE)來對
不同彎曲程度的彎曲波導做分析與精算。
利用FEMET及CTMIE來分析彎曲波導需要用到圓柱座標基底模態
(circular mode basis)，但圓柱座標基底模態涉及Bessel functions，其模態
並不好找，所以我們提出一個利用直角座標平行層的特徵模態來展解圓
柱座標彎曲波導特徵態的方法。透過圓柱座標重新歸化(renormalization)
直角座標平行層的特徵模態、得到一個矩陣特徵值、特徵向量的方程式。
此方法可以較快速、簡單地找出圓柱座標基底模態。最後我們提出以
whisper gallery modes 的設計，可以減少彎曲波導的輻射耗損(radiation
loss) 。

Present day optical integrated circuits contain many continuously
bending waveguides making it important to study EM field profiles of
bending waveguides. The mostly widely used numerical method for
analyzing bending waveguides is the beam propagation method (BPM).
Although it can calculate very fast, BPM results are not accurate enough in
many wide-angle applications due to BPM’s intrinsic paraxial approximation.
Recently, full-wave based finite-difference time-domain technique has
become quite popular and has been used to study many optical devices.
Unfortunately it can not be used to study smoothly bending waveguides due
to huge computational resource requirements needed for these large optical
devices. In the absence of reflection in a bending waveguide, other one
way, wide-angle methods can be applied. In this thesis we propose two
such methods to analyze different kinds of bending waveguides. We use
full eigen-mode expansion technique (FEMET) when reflection is negligible.
In cases where reflection is strong, we propose a cylindrical couple
transverse-mode integral-equation (C-CTMIE) to do the job.
Both FEMET and CTMIE methods are built on complete sets of circular
layered modes of the underlying structure. These modes are not easy to
solve because the standard cylindrical mode solver requires extensive
references to Bessel functions of complex arguments and orders. Here in
this thesis, we proposed to expand cylindrical layered modes from planar
layered modes of an equivalent structure. In essence, we renormalize the
existing planar layered waveguide modes and turn them into desired circular
layered modes. We show that using a matrix eigenvector formulation, this
relatively simple technique is not only quite fast but also produces very
accurate results. Finally using these circular modes various S-bend
waveguides are analyzed. We also present a design to minimize the
radiation loss of a circular waveguide using whisper gallery modes.

第一章 導論 1
1.1 簡介.............................................................................1
1.2 彎曲波導特徵模態.....................................................4
1.3 保角轉換.....................................................................8
第二章 平行層波導模態展解彎曲波導特徵模態 12
2.1 理論推導...................................................................12
2.2 彎曲波導模態數值模擬結果...................................19
2.3 分析與討論...............................................................43
第三章 全特徵模態展開法分析彎曲波導 46
3.1 TE 入射FEMET 理論架構推導..............................46
3.2 FEMET 模擬分析結果............................................ 52
3.3 分析與討論...............................................................76
第四章 CTMIE 理論分析彎曲波導 77
4.1-1 CTMIE(TE wave)方程式推導.................................77
4.1-2 PQRS 聯立積分方程式............................................84
4.1-3 重疊積分...................................................................88
4.2 CTMIE 模擬分析結果.............................................96
第五章 結論與未來研究 104
參考文獻 106
中英文對照表 109

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