在本論文中，我們首先使用二維有限時域差分(2D-FDTD)法結合等效折射率法模擬SOI微環形共振器之穿透頻譜做數值擬合的分析，得到微環形共振器之能量耦合係數與光學損耗，我們以此數值分析方法取代，當耦合間隙寬度變窄時出現多模的複雜模態計算。
我們發現當耦合間隙寬度變窄時輻射損耗急遽上升，由模擬結果我們可以得到TE、TM極化下不同半徑之本質彎曲損耗，TE極化下半徑為1.5、2.5、5μm本質彎曲損耗分別為20dB/cm、3dB/cm、1dB/cm。TM極化下半徑為1.5、2.5、5μm本質彎曲損耗分別為2573dB/cm、64dB/cm、0.8dB/cm。接著，在相同耦合間隙下之不同耦合器的SOI微環形共振器中，彎曲波導耦合至微環形共振器之能量耦合係數較大，其中TM模態光學損耗較小。由以上數值分析的光學參數結果可提供於高密度整合積體光路中，設計微環形共振器的耦合區域與彎曲半徑所決定的光學損耗。這些結果將有助於提升在未來製作光學被動元件之品質。

In this thesis, we have analyzed the coupling gap dependent micro-ring loss in a single ring all-pass filter configuration using the two dimension (2D) finite difference time domain (FDTD) and EIM (effective index method). We utilized a new analysis scheme by calculating the transmission signal as a function of input wavelength and fitting the transmission spectrum with a phenomenological ring loss parameter. This novel scheme circumvents the complex waveguide mode analysis process, when the coupling gap is narrow and the all-pass coupling region becomes multi-mode.
We first find that the radiation loss increases rapidly with decreasing coupling gap width. Our results show that the intrinsic bending losses of silicon micro-rings (on oxide) with the radius of 1.5 μm, 2.5μm, and 5μm are about 20 dB/cm, 3 dB/cm, 1 dB/cm for TE polarization modes, respectively. For TM modes, the intrinsic bending losses with the radius of 1.5μm, 2.5 μm, and 5μm are about 2573 dB/cm, 64 dB/cm, and 0.8 dB/cm, respectively. Next, we find that power coupling coefficients of the single ring all-pass filter configuration using the ring to ring couplers are much higher than the bus to ring couplers. The radiation losses of the ring to ring couplers for TM modes are improved significantly in all coupling gap widths. In a high-density integrated optics circuit, specially designed ring coupling region device structure is needed to address this serious optical loss issue.

第一章 緒論.............. 1
1.1簡介................ 1
1.2研究動機....... 2
1.3論文概述...... 5
第二章 數值模擬方法..... 6
2.1有限時域差分法... 6
2.2等效折射率法...... 10
第三章 環形共振器基本理論與模態耦合理論.......12
3.1微環形共振器與耦合器.............12
3.1.1品質因子....... 12
3.1.2微環形共振器之特性...........14
3.1.3自由光譜範圍..........15
3.1.4 光學損耗與品質因子........17
3.1.5 彎曲波導.........18
3.2 環形共振器波導間耦合.......22
3.2.1 時域模態耦合理論...............23
3.2.2利用時域模態耦合理論推導數值擬合公式.....26
3.3 耦合區域......27
第四章 數值模擬結果與分析........ 30
4.1 長直波導耦合至環形共振器.... 30
4.1.1相同半徑下不同耦合間隙之模擬結果分析....30
4.1.2元件包覆層為空氣在相同半徑下不同耦合間隙之 模擬結果分析..............55
4.2 彎曲波導耦合至環形共振器........ 58
4.2.1相同半徑下不同耦合間隙之模擬結果分析...59
4.3總結........65
第五章 結論........71
參考文獻.......72

[1] 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.
[2] M. A. Popovic, “Theory and design of high-index-contrast microphotonic circuits,” PhD thesis, MIT, 2008.
[3] B. Liu, A. Shakouri, and J. E. Bowers “Wide Tunable Double Ring Resonator Coupled Lasers,” IEEE Photon. Technol. Lett., vol. 14, pp. 600-602, 2002.
[4] T. Barwicz, M. A. Popovic, P. Rakich, M. Watts, H. Haus, E. Ippen, and H. Smith, "Microring-resonator based add-drop filters in SiN: fabrication and analysis," Opt. Express., vol. 12, pp. 1437-1442, 2004.
[5] Y. Vlasov, W. M. J. Green,and F. Xia, "High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks," Nature Phot., vol. 2, pp. 242-246, 2008.
[6] Q. Xu, B. Schmidt, J. Shakya, and M. Lipson, “Cascaded silicon micro-ring modulators for WDM optical interconnection,” Opt. Express., vol. 14, pp. 9431-9435, 2006.
[7] F. Xia, L. Sekaric, and Y. A. Vlasov, "Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators," Opt. Express., vol. 14, pp. 3872-3886, 2006.
[8] RSoft Design Group, “BeamPROP 8.2 User Guide,” Ossining, NY, 2010.
[9] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. AP-14, pp. 302-307, 1966.
[10] J.P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves”, Journal of Computational Physics, vol. 114, pp. 185-200, 1994
[11] RSoft Design Group, “FullWAVE 6.2 User Guide,” Ossining, NY, 2010.
[12] K. Okamoto, “Fundamentals of Optical Waveguides,” 2nd ed. Academic Press, NY, 2006.
[13] E. A. J. Marcatili and S. E. Miller, “Improved relationships describing directional control in electromagnetic wave guidance,” Bell Syst. Tech.J., vol. 48, pp. 2161-2188, 1969.
[14] R. G. Hunsperger (1991) Integrated Optics: Theory and Technology, 3rd edn, Springer-Verlag, Berlin.
[15] User manual Phoenix Software, Phoenix BV.
[16] S. Xiao, M. H. Khan, H. Shen, and M. Qi, “Modeling and measurement of losses in silicon-on-insulator resonators and bends,” Opt. Express, vol. 15, pp. 10553-10561, 2007.
[17] A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. vol.36, pp. 321-322, 2000.
[18] C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filtering,” IEEE J. Lightwave Technol., vol.35, pp. 1322-1331, 1999.
[19] B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” IEEE J. Lightwave Technol., vol. 15, pp. 998-1005, 1997.
[20] A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett., vol. 14, pp. 483-485, 2002.