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研究生:許永昌
研究生(外文):Yong-Chang Hsu
論文名稱:二維與非平面光子晶體帶隙分析及光學元件應用
論文名稱(外文):Two-Dimensional and Out-Of Plane Photonic Crystal Band Gap Analysis and Its Application to Optical Devices
指導教授:楊照彥
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:應用力學研究所
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:中文
論文頁數:108
中文關鍵詞:光子晶體能帶平面波展開法藕合器共振腔光波導管
外文關鍵詞:photonic crystalbandgapplane wave methodcouplercavityoptical waveguide
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本論文中所採用之平面波展開法(PWM)會探討到頻率域問題。本論文也探討不同倒晶格空間格點數對頻率能帶準確度之影響。在平面波展開法中Maxwell 方程將以特徵值-特徵函數型式出現。我們以平面波展開法(PWM)分析不同晶格週期排列所會造成之頻率能帶。我們分析光學共振腔以及光學波導管耦合器,找出這些光學元件的模態性質。在光學共振腔中元件我們可發現多種不同共振頻率及共振模態。在光波導管耦合器元件中我們可發現波導管間之距離將會影響著能量耦合的效能。
此外,光子晶體光纖的另ㄧ發展重點是光子晶體之非平面入射。不同的角度入射於晶體中也將會產生不同的頻率能帶,藉由此分析可找出在何種的角度內可找出造成全反射的頻率能帶。隨著不同晶格排列會有不同的模態出現,在本論文中將只探討六角晶格以及正方晶格的排列形狀。本論文之模擬數值可和實驗或期刊上的數據作對照,在所有的實例中幾乎都可得到正確的結果。因此我們可驗證本論文之程式準確性。
In this thesis we investigate the frequency domain problem of 2-D photonic crystals using the Plane Wave Method (PWM). The influence on the accuracy due to different grid numbers in reciprocal lattice space is also examined. In plane wave method , the Maxwell’s equations are represented in eigenvalue-eigenfunction form and the band gaps caused by different periodic lattice arrangement are calculated. We analyze optical cavity and directional coupled waveguide to find the properties of mode. In optical cavities we can find many different resonant frequencies and resonant modes. In optical coupled waveguides we can find that the energy coupling efficiency is affected by distance between waveguides.
Furthermore, another developing emphasis of photonic crystal fiber is the out-of plane propagation of electromagnetic wave on photonic crystal. Different incident angle to crystal leads to different frequency band gap. By this analysis, we can find the total reflection frequency band gap in any incident angle. Different lattice shapes and arrangements also cause different modes, we only examine the arrangements by hexagonal lattice and square lattice in this work. Comparing the simulation results with those available in journals or experiments, we almost get correct results with them and excellent agreement has been obtained in all cases. We can prove the accuracy of program codes within the thesis.
摘要 …………………………………………………………… II
目錄 …………………………………………………………… IV
第一章 概 論
1-1 光子晶體理論介紹 …………………………………… 1
1-2 光子晶體理論應用 …………………………………… 1
1-3 文獻回顧 …………………………………… 3
1-3.1 光子晶體能帶 …………………………………… 3
1-3.2 能帶計算方法 …………………………………… 3
第二章 理論分析
2-1 數值方法介紹 ………………………………………… 5
2-2 固態物理基本理論 ……………………………………… 6
2-3 光子晶體理論推倒 ……………………………………… 8
2-3.1 晶體材料分佈 …………………………………… 9
2-3.2 平面波方程式 …………………………………… 10
2-3.3 TE-TM mode …………………………………… 15
第三章 數值模擬結果
3-1 介電質分佈 ………………………………………… 17
3-2 光子晶體能帶分析 ………………………………… 20
3-2.1 何謂能帶 ……………………………………… 20
3-2.2 六角晶格收斂性分析 …………………………… 21
3-2.3 六角晶格疊代誤差分析 ………………………… 23
3-2.4 六角晶格能帶分析 ………………………… 27
3-2.5 正方晶格收斂性分析 …………………………… 28
3-2.6 六角晶格疊代誤差分析 ………………………… 30
3-2.7 正方晶格能帶分析 ………………………… 33
3-3 波傳之表面模態 …………………………………… 35
3-3.1 正方晶格 ……………………………………… 37
3-3.2 六角晶格 ……………………………………… 39
3-4 波傳之能態密度 …………………………………… 41
3-4.1 六角晶格 ……………………………………… 41
3-4.2 正方晶格 ……………………………………… 43
3-5 能帶之純量化性質 …………………………………… 45
3-5.1 六角晶格 ……………………………………… 46
3-5.2 正方晶格 ……………………………………… 49
第四章 光學元件之應用
4-1 概論 ………………………………………………… 52
4-2 共振腔 ……………………………………………… 53
4-2.1 正方晶格之共振腔 ……………………………… 55
4-2.2 六角晶格之共振腔 ……………………………… 62
4-2.3 超晶胞(supercell 5×5)與缺陷模態之收斂性關係 70
4-2.4 多重supercell對缺陷模態影響 ……………… 72
4-3 正方晶格光波導耦合器 ……………………………… 74
4-3.1 光學耦合器簡介 ……………………………… 75
4-3.2 耦合器效能計算 ……………………………… 76
4-3.3 耦合器長度計算 ……………………………… 79
4-3.4 不同耦合長度計算 ……………………………… 80
第五章 非平面入射
5-1 概論 ……………………………………………… 82
5-2 理論分析 ……………………………………………… 82
5-2.1 入射角之定義 …………………………………… 84
5-3 六角晶格之非平面入射 …………………………… 86
5-3.1 入射角度計算 ………………………………… 86
5-3.2 入射角度 ………………………………… 90
5-3.3 帶隙比 ……………………………… 91
5-3.4 完全能帶(Complete Band-Gap) ……………… 93
5-4 正方晶格之非平面入射 ……………………………… 94
5-4.1 入射角度計算 ………………………………… 94
5-4.2 入射角度 ………………………………… 98
5-4.3 帶隙比 ………………………………… 99
5-4.4 完全能帶(Complete Band-Gap) ……………… 100
5-5 討論 ………………………………………………… 101
第六章 結論與展望
6-1 結論 ……………………………………………… 102
6-2 展望 ……………………………………………… 103
參考文獻 104
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