六邊形共振腔具有特別的迴廊與準迴廊模態，靠的是電磁波在腔內的全反射將之侷限在內，固有相當高的品質因子。在本模擬室過去已經分別有人用整個六邊形來模擬氧化鋅與石墨烯，不過在這樣的情況下，會有兼併態而無法分辨(前人有些模態因為這樣只有找到兼併態的混合形式，無法分離)。由於六邊形具有特殊的對稱性，可以利用其特性將計算空間縮小，並將簡併態分開處理。本論文將此六邊形結構沿X,Y軸對切，只模擬四分之一的範圍，利用PEC/PMC(完美理想電/磁導體)處理軸對稱簡併模態，並進而重建整個模態。本次模擬所模擬的材料有氧化鋅晶柱，其共振腔可以將之模擬為一個二維介電物質，其原理較為簡單，我們分析其低階數模態到第20階，並且與推得的迴廊理論模態做比較。接下來是石墨烯六邊形奈米尺度片狀共振腔，其靠的是表面電漿的模態，因此其具有最短可允許的波長，其波長對應折射率的變化相當大，且波長遠大於共振腔的大小，我們成功地得出其低階數模態到第10階，然而因其成因為表面電漿無法用氧化鋅全反射的方式解析理論模態，必須用局域性表面電漿模態來分析。總結本模擬成功地利用六邊形軸對稱的四分之一結構重建六邊形的模態，並且去除掉同模態不同對稱性的簡併態，未來可以用此方法模擬其他情況節省大量計算時間與簡化簡併態的模態分析。

The hexagonal cavity has special resonate mode called whisper gallery mode(WGM) and quasi whisper gallery mode, they are formed by total reflection on the hexagonal resonator`s peripheral to trap the field and have high quality factor. These hexagonal cavity modes contain symmetry properties which can be utilized to reduce the computational time and separate degenerated modes. In this these, we analyze the hexagonal cavity mode of ZnO rod and Graphene flakes using the axis symmetry. ZnO is a dielectric material with similar refraction index over all wavelength regimes of our interests, so it is relativity easier to understand its property. On the other hand , graphene`s resonant mode is originated from surface plasmons due to the negative real part of its dielectric constant, and has the shortest cut-off wavelength around 1.5um at chemical doping at 0.6eV. Its effective refraction index differs from wavelength a lot. In the axis symmetry analysis, we cut one quarter of the hexagonal cavity along the X,Y axis, and use perfect electronic(magnetic) conductor PEC (PMC) to apply odd (even) symmetry respectively. This approach can separate degenerated mode with different symmetry, and reduce the computational time. We first analyze the dielectric ZnO case using 2D TE FDTD. Then we repeat the same analysis for graphene flakes using 3D FDTD. We successfully obtain the ZnO and graphene mode by this method. We further discuss these modes and compare them with the calculated WGM mode.

中文摘要……………………………………………………………………………I
Abstract…………………………………………………………………………… …II
致謝…………………………………………………………………………………XI
目錄………………………………………………………………………………XI
圖目錄……………………………………………………………………………XIII
表目錄………………………………………………………………………………XIV
第一章 序論…………………………………………………………………………1
1-1前言………………………………………………………………………………1
1-2研究動機…………………………………………………………………………2
1-3本文內容…………………………………………………………………………3
第二章 相關研究理論簡介…………………………………………………………4
2-1六角形迴廊模態對稱性…………………………………………………………4
2-2模態與共振波長關係式推導……………………………………………………5
2-3六角形共振腔的允許模態………………………………………………………8
2-4表面電漿…………………………………………………………………………9
第三章 有限差分時域法(FDTD) …………………………………………………10
3-1差分法之介紹……………………………………………………………………10
3-2 FDTD演算法……………………………………………………………………10
3-3同軸完美匹配層(UPML) ………………………………………………………13
3-4 PML之離散誤差…………………………………………………………………16
3-5完美理想導體……………………………………………………………………17
3-6 石墨烯能帶之FDTD演算………………………………………………………17
3-7 ghost point之使用原理…………………………………………………………21
第四章 模擬結果與討論……………………………………………………………23
4-1共振腔頻譜求法之介紹…………………………………………………………23
4-2氧化鋅之頻譜及模態結構………………………………………………………24
4-3氧化鋅模態之數據分析…………………………………………………………34
4-4石墨烯之共振腔頻譜及模態結構………………………………………………37
4-5石墨烯模態之數據分析…………………………………………………………45
第五章 結論與未來展望……………………………………………………………48
5-1結論………………………………………………………………………………48
5-2未來展望…………………………………………………………………………48
參考文獻………………………………………………………………………………52

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