光子晶體結構是由週期性介電材料所組成，因為具有特殊的光子能帶，所以可以有效控制光波的傳播。本論文使用平面波展開法分析正方晶格的光子能隙，藉由時域耦合模態理論來設計光子晶體波導光能量分配器，並使用有限時域差分法模擬光波在分配器內的傳播情況，以探討分配器內波導的圓柱尺寸與傳遞係數的關係。此外，本論文亦對耦合共振光學波導的幾何結構進行建模與數值計算，由結果得知兩倍密四方晶格的排列方式在中心波長具有寬頻且高傳輸的特性。延伸此波導架構，本論文提出一個多模態的光子晶體光能量分光器，設計出同時兼具波導、分波與濾波能力的微米級光學元件，此研究成果應可用在日後的積體光子迴路設計中。

Photonic crystals are composed of periodic dielectric that has photonic band gap. It can effectively control the light of propagation. In this thesis, we analysis the band gaps of the square lattice photonic by plane wave expansion, and design of optical power divider by using time-domain coupled-mode theory. We simulated the light wave propagation in power divider by using finite-difference time-domain method. We also discussed the relationship of the cylindrical size and the transmission coefficient in the divider. Besides, We have simulated the structure of coupled-resonator optical waveguides. Simulation results show that the double dense cubic photonic crystals have wide high-transmission bandwidths near the center of wavelength. Further, we proposed the power splitter of multimode interference in photonic crystal waveguides. It is a nanometer scale of optical component and it has the ability to guide, split and filter. The results of this study should be applied for photonic integrated circuit in the future.

目錄
中文摘要 Ⅰ
英文摘要 Ⅱ
目錄 III
圖目錄 V
表目錄 IX
第一章 前言 1
1.1 研究動機與目的 1
1.2 光學的歷史背景 2
1.3 光子晶體的研究與應用 3
1.4 論文架構 4
第二章 光子晶體的基本理論 6
2.1 晶格、倒晶格向量、週期函數與光子能隙 6
2.2 光子晶體波導 9
2.3 研究方法 10
第三章 數值模擬的計算方法 12
3.1 波動方程式 12
3.2 平面波展開法 15
3.3 有限時域差分法 17
第四章 光子晶體波導光能量分配器 19
4.1 設計架構 19
4.2 元件設計與分析 21
第五章 耦合共振光學波導之研究 29
5.1 單一區域的結構分析 29
5.1.1 改變CROW區域介電圓柱的數量 29
5.1.2 改變Taper區域介電圓柱的數量 31
5.2 單一區域不同圓柱數量的差異 32
5.2.1 改變CROW區域介電圓柱的數量 32
5.2.2 改變Taper區域介電圓柱的數量 32
5.3 改變共振腔間距的影響 33
5.4 耦合共振波導尺寸的改進 34
5.5 耦合共振波導與晶格排列方式之分析 35
第六章 多模態干涉結構 39
6.1 正方晶格多模態結構的最佳化 39
6.2 圓柱半徑與能量流的週期變化關係 48
6.3 兩倍密四方晶格耦合共振波導加多模態結構 50
6.4 分光比例的改變 54
第七章 結論及未來展望 57
參考文獻 59

圖目錄
圖1.1 空間中一維、二維、三維光子晶體示意圖 4
圖2.1 二維正方晶格之晶格向量與倒晶格向量示意圖 6
圖2.2 TE模態之光子能隙圖 7
圖2.3 TM模態之光子能隙圖 8
圖2.4 流程圖 11
圖3.1 二維光子晶體偏振方向示意圖 14
圖3.2 正方晶格之單位晶胞結構示意圖 15
圖3.3 FDTD電場與磁場以交叉位置所組成的晶格點 18
圖4.1 分配器的架構圖 19
圖4.2 分光器之細部結構 22
圖4.3 當d = 3a、ri = 0, 0.07a, 0.09a、rw = 0、rc = 0.2a時的傳輸頻譜 22
圖4.4 入射波長1550 nm、d = 3a、ri = 0.09a、rw = 0、rc = 0.2a時的電場分佈圖 23
圖4.5 當d = 3a、ri = 0.07a、rw = 0, 0.07a, 0.09a、rc = 0.2a時的傳輸頻譜 24
圖4.6 入射波長1550 nm、d = 3a、ri = 0.07a、rw = 0.07a、rc = 0.2a時的電場分佈圖 24
圖4.7 當d = 3a、ri = 0.07a、rw = 0、rc = 0, 0.2a, 0.235a時的傳輸頻譜 25
圖4.8 入射波長1550 nm、當d = 3a、ri = 0.07a、rw = 0、rc = 0時的電場分佈圖 25
圖4.9 上下方輸出波導中圓柱不對稱時的傳輸頻譜 26
圖4.10 入射波長1550 nm上下方輸出波導中圓柱不對稱時的電場分佈圖 27
圖4.11 當d = 4a、ri = 0.08a、rw = 0、rc = 0.23a時的傳輸頻譜 27
圖5.1 耦合共振光學波導 29
圖5.2 在CROW區域改變介電圓柱的數量 30
圖5.3 (a)當C分別為2和4的傳輸頻譜 (b)當C分別為6和8的傳輸頻譜 30
圖5.4 在錐形區域改變介電圓柱的數量 31
圖5.5 當C固定為2，L分別為3和9時的傳輸頻譜 31
圖5.6 當L固定為9，C分別為5和9時的傳輸頻譜 32
圖5.7 當C固定為7，L分別為0、3、9時的傳輸頻譜 33
圖5.8 改變共振腔間距 34
圖5.9 相鄰的介電圓柱彼此圓心的距離為2a、2.5a、3a時的傳輸頻譜 34
圖5.10 晶格常數a分別為558nm和600nm的傳輸頻譜 35
圖5.11 平移耦合共振波導朝向線缺陷波導方向 35
圖5.12 比較平移0.5a前後差異的傳輸頻譜 36
圖5.13 耦合共振波導平移0.5a後之結構 36
圖5.14 兩倍密四方晶格排列 36
圖5.15 進行傳輸頻譜的比較 37
圖5.16 改變縱軸晶格的長度圖 37
圖5.17 (a)縱軸長度分別為√3 a、1.8a、1.9a的傳輸頻譜 (b)縱軸長度分別為2a、2.1a、2.2a的傳輸頻譜 38
圖6.1 縱向長度為4a的多模態結構 39
圖6.2 縱向長度4a，入射波長1550nm時的能量流 39
圖6.3 縱向長度為4a、橫向長度為5.5a之電場分佈圖 40
圖6.4 縱向長度為4a，改變橫軸長度的傳輸頻譜 41
圖6.5 橫軸長度分別為5.5a、5.6a及5.7a的傳輸頻譜 41
圖6.6 縱向長度6a，入射波長1550nm時的能量流 42
圖6.7 縱向長度為6a，改變橫軸長度的傳輸頻譜 43
圖6.8 橫軸長度為11.3a與11.4a的傳輸頻譜 43
圖6.9 縱向長度8a，入射波長1550nm時的能量流 44
圖6.10 縱向長度為8a，改變橫軸長度的傳輸頻譜 45
圖6.11 橫軸長度為6.3a與6.4a的傳輸頻譜 45
圖6.12 橫軸長度為7.6a與7.7a的傳輸頻譜 46
圖6.13 縱向長度4a、6a及8a最佳結構的傳輸頻譜 46
圖6.14 縱向長度為4a、橫向長度為5.5a多模態結構與d = 3a、ri = 0.07a、rw = 0、rc = 0.235a單一模態複合共振腔結構的傳輸頻譜 47
圖6.15 導入圓柱半徑為0.08a的結構 48
圖6.16 圓柱半徑為0.08a時的電場分佈圖 49
圖6.17 圓柱半徑為0.08a時的能量流 49
圖6.18 圓柱半徑為0.09a時的電場分佈圖 49
圖6.19 圓柱半徑為0.09a時的能量流 49
圖6.20 圓柱半徑與能量流的週期變化關係 50
圖6.21 正方排列的耦合共振波導加多模態結構 50
圖6.22 (a)正方晶格排列的能量流 (b)兩倍密四方晶格排列的能量流 51
圖6.23 耦合共振波導加多模態結構 (a)正方晶格 (b)兩倍密四方晶格 52
圖6.24 兩倍密四方晶格排列下 (a)橫軸長度為5a的傳輸頻譜 (b)橫軸長度為5.5a的傳輸頻譜 (c)橫軸長度為6a的傳輸頻譜 53
圖6.25 (a)多模態結構前無耦合共振波導的傳輸頻譜 (b)多模態結構前有耦合共振波導的傳輸頻譜 53
圖6.26 左邊輸出波導加入圓柱半徑0.07a示意圖 54
圖6.27 (a)輸出波導端加入圓柱半徑為0.06a的傳輸頻譜圖 (b)輸出波導無圓柱半徑的傳輸頻譜圖 55
圖6.28 (a)輸出波導端加入圓柱半徑為0.07a的傳輸頻譜圖 (b)輸出波導無圓柱半徑的傳輸頻譜圖 55
圖6.29 (a)輸出波導端加入圓柱半徑為0.08a的傳輸頻譜圖 (b)輸出波導無圓柱半徑的傳輸頻譜圖 56
圖6.30 (a)輸出波導端加入圓柱半徑為0.09a的傳輸頻譜圖 (b)輸出波導無圓柱半徑的傳輸頻譜圖 56
 
表目錄
表4.1 當d = 3a、ri = 0, 0.07a, 0.09a、rw = 0、rc = 0.2a時的傳輸頻譜比較表 23
表4.2當d = 3a、ri = 0.07a、rw = 0、rc = 0, 0.2a, 0.235a時的傳輸頻譜比較表 26
表4.3當d = 4a、ri = 0.08a、rw = 0、rc = 0.23a時的傳輸頻譜比較表 28
表6.1 縱向長度為4a，各個橫軸長度的傳輸頻譜比較表 40
表6.2 縱向長度為6a，各個橫軸長度的傳輸頻譜比較表 42
表6.3 縱向長度為8a，各個橫軸長度的傳輸頻譜比較表 44
表6.4 縱向長度4a、6a及8a最佳結構的傳輸頻譜比較表 47
表6.5 縱向長度為4a、橫向長度為5.5a多模態結構與d = 3a、ri = 0.07a、rw = 0、rc = 0.235a單一模態複合共振腔結構的傳輸頻譜比較表 48

[1] K. Sakoda, Optical Properties of Photonic Crystals, Springer, Berlin (2001).
[2] S. G. Johnson, and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice, Springer, Berlin (2002).
[3] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals - Molding the Flow of Light, Princeton University Press (1995).
[4] 欒丕綱、陳啟昌,光子晶體:從蝴蝶翅膀到奈米光子學.五南,台北 (2010).
[5] J. C. Maxwell, “A dynamical theory of the electromagnetic field,” Philosophical Transactions of the Royal Society of London 155, 459 (1865).
[6] C. Kittle, Introduction to solid state physics, John Wiley & Sons, New York (1996).
[7] E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[8] S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 (1987).
[9] K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structure,” Phys. Rev. Lett. 65, 3152 (1990).
[10] M. Plihal, and A. A. Maradudin, “Photonic band structure of two-dimensional random dielectric systems,” Phys. Rev. B 44, 8565 (1991).
[11] E. Yablonovitch, and T. J. Gmitter, “Photonic band structure: The face-centered-cubic case,” Phys. Rev. Lett. 63, 1950 (1989).
[12] M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-dimensional photonic band structures,” Opt. Comm. 80, 199 (1991).
[13] W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023 (1992).
[14] C. Jin, S. Han, X. Meng, B. Cheng, and D. Zhang, “Demultiplexer using directly resonant tunneling between point defects and waveguides in a photonic crystal,” J. Appl. Phys. 91, 4771 (2002).
[15] X. F. Yu, and S. H. Fan, ”Bends and splitters for self-collimated beams in photonic crystal,” Appl. Phys. Lett. 83, 3251-3253 (2003).
[16] C. C. Chen, H. D. Chien, and P. G. Luan, “photonic crystal beam splitters,” Appl. Opt. 43, 6187 (2004).
[17] H. T. Chien, C. C. Chen, and P. G. Luan, “Photonic crystal beam splitters,” Opt. Comm. 259, 873 (2006).
[18] P. G. Luan, and K. D. Chang, “Periodic dielectric waveguide beam splitter based on co-directional coupling,” Opt. Express 15, 4536 (2007).
[19] T. Liu, A. R. Zakharian, M. Fallahi, and M. Mansuripur, “Multimode interference-based photonic crystal waveguide power splitter,” J. Lightwave Technol. 22, 2842 (2004).
[20] F. Monifi, M. Djavid, A. Ghaffari, and M. S. Abrishamian, “Photonic crystal bends and power splitters based on ring resonators,” Opt. Comm. 281, 5929 (2008).
[21] J. Zhou, Q. Chang, D. Mu, J. Yang, W. Han, and L. Wang, “Theoretical investigation of waveguide power splitters with parallel output ports in two dimensional square-lattice photonic crystals,” J. Opt. Soc. Am. B 26, 2469 (2009).
[22] A. Sharkawy, S. Shi, D. Prather, and R. Soref, “Electro-optical switching using coupled photonic crystal waveguides,” Opt. Express 10, 1048 (2002).
[23] B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring Resonator Channel Dropping Filters,” J. Lightwave Technol. 15, 998 (1997).
[24] Z. Qiang, W. Zhou, and R. A. Soref, “Optical add-drop filters based on photonic crystal ring resonators,” Opt. Express 15, 1823 (2007).
[25] M. Djavid, A. Ghaffari, F. Monifi, and M. S. Abrishamian, “T-shaped channel-drop filters using photonic crystal ring resonators,” Phys. E 40, 3151 (2008).
[26] F. Monifi, A. Ghaffari, M. Djavid, and M. S. Abrishamian, “Three output port channel-drop filter based on photonic crystal,” Appl. Opt. 48, 804 (2009).
[27] F. Monifi, M. Djavid, A. Ghaffari, and M. S. Abrishamian, “Design of efficient photonic crystal bend and power splitter using super defects,” J. Opt. Soc. Am. B 25, 1805 (2008).
[28] Z. Qiang, W. Zhou, and R. A. Soref, “Optical add-drop filters based on photonic crystal ring resonators,” Opt. Express 15, 1823 (2007).
[29] P. Andalib, and N. Granpayeh, “All-optical ultracompact photonic crystal and gate based on nonlinear ring resonators,” J. Opt. Soc. Am. B 26, 10 (2009).
[30] T. Sondergaard, and K. H. Dridi, “Energy flow in photonic crystal waveguides,” Phys. Rev. B. 61, 15688 (2000).
[31] S. Fan, S. G. Johnson, and J. D. Joannopoulos, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162 (2001).
[32] A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787 (1996).
[33] F. Monifi, M. Djavid, A. Ghaffari, and M. S. Abrishamian, “Design of efficient photonic crystal bend and power splitter using super defects,” J. Opt. Soc. Am. B 25, 1805 (2008).
[34] Z. Hu, and Y. Y. Lu, “Improved bends for two-dimensional photonic crystal waveguides,” Opt. Comm. 284, 2812 (2011).
[35] A. Chutinan, and S. Noda, “Highly confined waveguides and waveguide bends in three-dimensional photonic crystal, ” Appl. Phys. Lett. 75, 3739 (1999).
[36] S. Boscolo, and M. Midrio, “Y junctions in photonic crystal channel waveguides: high transmission and impedance matching,” Opt. Lett. 27, 1001 (2002).
[37] S. Y. Lin, E. Chow, J. Bur, S. G. Johnson, and J. D. Joannopoulous, “Low-loss, wide-angle Y spliiter at ~1.6-μm wavelengths built with a two-dimensional photonic crystal,” Opt. Lett. 27, 1400 (2002).
[38] A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 80, 1698 (2002).
[39] P. G. Luan, and K. D. Chang, “Transmission characteristics of finite periodic dielectric waveguides,” Opt. Express 14, 3263 (2006)
[40] P. G. Luan, and Z. Ye, “Two dimensional photonic crystal,” E-print, http://140.115.40.128/publication/P03.pdf.
[41] M. Plihal, and A. A. Maradudin, “Photonic band structure of two-dimensional system: The triangular lattice,” Phys. Rev. B 44, 8565 (1991).
[42] K. S. YEE, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE T. Antenn. Propag. 14, 302 (1966).
[43] A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech, London (1995).
[44] M. Qiu, and S. He, “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phys. Rev. B 61, 12871 (2000).
[45] P. G. Luan, and Z. Ye, “Acoustic wave propagation in a one-dimensional layered system,” Phys. Rev. E 63, 066611 (2001).
[46] B. C. Gupta, C. H. Kuo, and Z. Ye, “Propagation inhibition and localization of electromagnetic waves in two- dimensional random dielectric systems,” Phys. Rev. E 69, 066615 (2004).
[47] P. R. Villeneuve, F. Shanhui, and J. D. Joannopoulos, “Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837 (1996).
[48] M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express, 12, 1551 (2004).
[49] A. Yariv, X. Yong, K. L. Reginald, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711 (1999).
[50] P. Sanchis, J. Garcia, A. Martinez, F. Cuesta, A. Griol, and J. Marti, “Analysis of adiabatic coupling between photonic crystal single-line-defect and coupled-resonator optical waveguides,” Opt. Lett. 28, 1903 (2003).
[51] P. Sanchis, J. Marti, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Experimental results on adiabatic coupling into SOI photonic crystal coupled-cavity waveguides,” IEEE Photonics Tech. L. 17, 1199 (2005).
[52] J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 114, 185 (1994).
[53] K. S. YEE, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE T. Antenn. Propag. 14, 302 (1966).