近年來以波導核心高折射率材料及覆蓋層之較低折射率材料相互交錯並週期性排列而組成的次波長光柵波導(Subwavelength Grating, SWG)被提出並被廣泛地研究，我們不但可以藉由改變次波長光柵波導的幾何結構與組成材料來調變其光學特性，且因其擁有delocalized mode，而能有效地降低傳統絕緣層上覆矽(Silicon on Insulator, SOI)波導結構因側壁粗糙所面臨的散射損耗問題。
本論文研究使用二維有限元素法分析長直與彎曲次波長光柵波導之光學特性，證實delocalized mode能在長直次波長光柵波導近乎無損耗傳遞。而為了減少彎曲次波長光柵波導所面臨的高彎曲損耗問題，我們也提出改良型彎曲次波長光柵波導結構以降低彎曲區域所產生的彎曲損耗。我們進一步將次波長光柵波導應用於馬赫-桑德耳干涉器中，利用次波長光柵波導與均勻波導模態等效折射率不同之特性產生干涉現象，且能以不縮減波導寬度之高製程容忍度的方式提高其感測特性。我們發現在固定次波長光柵波導週期下，馬赫-桑德耳干涉器對待測折射率變化之敏感度會隨著參數a0的增加而提高，其最高感測能力為1850nm/RIU。此外，我們也將次波長光柵波導應用於跑道式微環形共振器之耦合區域中，利用次波長光柵波導之delocalized mode特性，來提高長直波導與環形共振器之間的耦合效率，並找出參數a0與d分別為0.15μm以及0.4μm時擁有最好之共振特性。我們也計算出此共振器結構對待測折射率變化之感測能力為62nm/RIU，且發現加入次波長光柵波導之共振器結構擁有對溫度變化較不敏感之特性，其敏感度為75pm/oC。

Recently, subwavelength grating (SWG) waveguides have been proposed to guide light in silicon on insulator (SOI) structures with more extended field in the cladding regions. SWG waveguides are formed by periodically interlacing silicon segments at the subwavelength scale buried in a material with smaller refractive index. By varying the size of silicon segments or the index of the surrounding material, we can easily modulate the optical characteristics of the SWG waveguides.
In this thesis, we numerically investigate the optical characteristics of straight and bent SWG waveguides by utilizing a 2-D finite element method (FEM). In order to suppress the high bending losses resulted from the delocalization of mode field in SWG waveguides, we propose a improved bent SWG waveguide to reduce the bending loss. We have also employed SWG waveguides into SOI-based Mach-Zehnder interferometers. Due to the effective modal index difference between the SWG waveguide and the conventional SOI waveguide, we can successfully obtain the interference spectrum from the Mach-Zehnder interferometer. The simulation results show that the refractive index sensitivity of the proposed Mach-Zehnder interferometer is raised by the delocalized mode of the SWG waveguide and increased with a0 for a fixed Λ. The highest sensitivity is 1850nm/RIU as a0=0.2μm. Besides, we also apply SWG waveguides into the SOI-based micro-racetrack resonators. The coupling efficiency between the bus and the ring resonator can be improved by the delocalized mode. The results show that the refractive index sensitivity is 62nm/RIU, and the SWG micro-racetrack resonators possess a smaller temperature sensitivity of 75pm/oC due to the composite structure of the SWG waveguide.

誌謝............................................................................................................i
中文摘要.....................................................................................................iii
Abstract......................................................................................................iv
目錄............................................................................................................v
表目錄.........................................................................................................vii
圖目錄.........................................................................................................viii
第一章 緒論..............................................................................................1
1.1 Silicon on Insulator光波導......................................................................1
1.2 次波長光柵波導介紹及應用....................................................................3
1.2.1 方向耦合器...........................................................................................4
1.2.2 模態轉換器...........................................................................................7
1.2.3 交錯波導...............................................................................................8
1.3 研究動機...............................................................................................10
第二章 有限元素法.....................................................................................12
2.1 與邊界值相等的變分問題........................................................................12
2.2 區域剖分和插值函數...............................................................................14
2.3 元素分析...............................................................................................18
2.4 總體合成................................................................................................21
2.5 引入邊界條件.........................................................................................24
第三章 次波長光柵波導數值模擬分析...........................................................24
3.1 次波長光柵波導之光學特性......................................................................24
3.2 彎曲次波長光柵波導................................................................................32
第四章 次波長光柵波導應用於馬赫-桑德耳干涉器 ............................................. 37
4.1 馬赫-桑德耳干涉器結構與原理 ..................................................... 37
4.2 次波長光柵波導應用於馬赫-桑德耳干涉器之光學特性 ............. 39
4.3 次波長光柵波導應用於馬赫-桑德耳干涉器之感測特性 ............. 43
第五章 次波長光柵波導應用於跑道式微環形共振器 .......................................... 50
5.1 微環形共振器結構與原理 .............................................................. 50
5.2 次波長光柵波導應用於跑道式微環形共振器之光學特性 .......... 53
5.3 次波長光柵波導應用於跑道式微環形共振器之感測特性 .......... 58
5.4 全次波長光柵跑道式微環形共振器與傳統跑道式微環形共振器
之比較 .............................................................................................. 62
第六章 結論與未來展望 .......................................................................................... 65
參考文獻 ...................................................................................................................... 67

[1] S. E. Miller, “Integrated optics: an introduction.,” Bell Syst. Tech. J., vol. 48, pp. 2059-2068, 1969.
[2] Y. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express, vol. 12, pp. 1622-1631, 2004.
[3] W. Bogaerts, R. Baets, P. Dumon, and V. Wiaux, “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” Journal of Lightwave Technology, vol. 23, pp. 401-412, 2005.
[4] F. Grillot, L. Vivien, S. Laval, D. Pascal, and E. Cassan, “Size influence on the propagation loss induced by sidewall roughness in ultrasmall SOI waveguides,” IEEE Photon. Technol. Lett., vol. 16, pp. 1661-1663, 2004.
[5] M. C. Lee, W. C. Chiu, T. M. Yang, and C. H. Chen, “Low-loss silicon wire waveguides with 3-D tapered couplers fabricated by self-profile transformation,” Appl. Phys. Lett., vol. 91, pp. 338-339, 2007.
[6] R. Pafchek, R. Tummidi, J. Li, M. A. Webster, E. Chen, and T. L. Koch, “Low-loss silicon-on-insulator shallow-ridge TE and TM waveguides formed using thermal oxidation,” Appl. Opt., vol. 17, pp. 958-963, 2009.
[7] P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, A. Delâge, S. Janz, G. C. Aers, D. X. Xu, A. Densmore, and T. J. Hall, “Subwavelength grating periodic structures in silicon-on-insulator: a new type of microphotonic waveguide” Opt. Express, vol. 18, pp. 20251-20262, 2010.
[8] Y. Quan, P. Han, Q. Ran, F. Zeng, L. Gao, and C. Zhao, “A photonic wire-based directional coupler based on SOI,” Optics Communications, vol. 281, pp. 3105-3110, 2008.
[9] A. J. Oliveira, M. S. Costa, C. E. Rubio-Mercedes, and V. F. Rodriguez-Esquerre, “Numerical analysis of periodic segmented waveguides directional couplers,” in Latin America Optics and Photonics Conference, paper LM2A.22, 2012.
[10] P. D. Trinh, S. Yagnanarayanan, and B. Jalali, “Integrated optical directional couplers in silicon-on-insulator,” Electron. Lett., vol. 31, pp. 2097-2098, 1995.
[11] P. Cheben, D-X. Xu, S. Janz, and A. Densmore, “Subwavelength waveguide grating for mode conversion and light coupling in integrated optics,” Opt. Express, vol. 14, pp. 4695-4702, 2006.
[12] S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett., vol. 23, pp. 1855-1857, 1998.
[13] P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, A. Delage, D.-X. Xu, S. Janz, A. Densmore, and T. J. Hall, “Subwavelength grating crossings for silicon wire waveguides,” Opt. Express, vol. 18, pp. 16146-16155, 2010.
[14] B. Guha, A. Gondarenko, and M. Lipson, “Minimizing temperature sensitivity of silicon Mach-Zehnder interferometers,” Opt. Express, vol. 18, pp. 1879-1887, 2010.
[15] L. Meng, D. Zhao, Z. Ruan, Q. Li, Y. Yang, and M. Qiu, “Optimized grating as an ultra-narrow band absorber or plasmonic sensor,” Opt. Lett., vol. 39, pp. 1137-1140, 2014.
[16] J. Niehusmann, A. Vörckel, and P. H. Bolivar, “Ultrahigh-quality-factor silicon-on-insulator microring resonator,” Opt. Lett., vol. 29, pp. 2861-2863, 2004.
[17] P. Monk, Finite Element Methods for Maxwell’s Equations, The Clarendon Press Oxford University Press, New York, 2003.
[18] R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, pp. 237–339, 2002.
[19] S. C. Brenner and L. R. Scott., The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 2nd edition, 2002.
[20] M. Koshiba, H. Saitoh, M. Eguchi, and K. Hirayama, “Simple scalar finite element approach to optical waveguides,” IEE PROC. J., vol. 139, pp. 166-171, 1992.
[21] J. M. Jin, The Finite Element Method in Electromagnetics, New York: Wiley, 1993.
[22] O. C. Zienkiewicz, The Finite Element Method, 3rd ed. McGraw-Hill, New York, 1973.
[23] M. Koshiba, Optical Waveguide Theory by the Finite Element Method, KTK Scientific Publishers and Kluwer Academic Publishers, Dordrecht, Holland, 1992.
[24] B. Szabo and I. Babuska, Finite Element Analysis, J. Wiley and Sons, New York, 1991.
[25] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. Wiley, New York, 2007.
[26] D. Dai and S. He, “A silicon-on-insulator photonic wire based evanescent field sensor,” IEEE. Photon. Technol. Lett., vol. 18, pp. 2520-2522, 2006.
[27] Y. Vlasov, W. M. J. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nature, vol. 2, pp. 242-246, 2008.
[28] S. Bidnyk, D. Feng, A. Balakrishnan, M. Gao, H. Liang, W. Qian, C.-C. Kung, J. Fong, J. Yin, and M. Asghari, “Silicon-on-insulator based planarcircuit for passive optical network application,” IEEE Photon. Technol. Lett., vol. 18, pp. 2392-2394, 2006.
[29] T. Claes, J. G. Molera, K. De Vos, E. Schacht, R. Baets, and P. Bienstman, “Label-free biosensing with a slot-waveguide-based ring resonator in silicon on insulator,” IEEE. Photon. Technol. Lett., vol. 1, pp. 197-204, 2009.
[30] R. Adler, L. J. Chu, and R. M. Fano, Electromagnetic Energy Transmission and Radiation, Wiley and Sons, New York, 1960.
[31] S. M. Nawrocka, T. Liu, X. Wang, and R. R. Panepucci, “Tunable silicon microring resonator with wide free spectral range,” Appl. Phys. Lett., vol. 89, 071110, 2006.