以全波向量邊界元素法 (Full-wave vector boundary element method, VBEM)為基礎，本論文提出一個嚴謹的功率耦合模型，此模型適用於三核心光波導結構。除了可以模擬各種極化狀態的輸入光對三核心光波導之耦合行為的影響，結合米勒矩陣法 (Mueller matrix method) 此模型亦可以用以研究三核心光波導之極化相關損失 (Polarization dependent loss, PDL)。在本論文中，此功率耦合模型被應用於兩種功率分歧器的研究。第一種功率分歧器是由步階折射係數之單模光纖所建構而成，稱為燒熔式3 3光纖耦合器。對於燒熔式3 3光纖耦合器，本論文之研究內容包含了輸入光之極化狀態對燒熔式3 3光纖耦合器的耦合行為之影響與製造參數，例如:燒熔程度與加熱長度對耦合器之極化相關損失的影響。第二種功率分歧器是由光子晶體光纖 (Photonic crystal fibers, PCFs) 所建構而成。在本論文中，我們研究了三角型結構之三核心光子晶體光纖的重要基本特性，例如:耦合長度、頻寬與極化相關損失。我們發現三核心光子晶體光纖可以被應用於實現超小型功率分歧器。此外，選擇某一個適當的耦合點來製造三核心光子晶體光纖將可以提高功率分歧器的良率與效能穩定度。除了對功率分歧器的耦合行為之研究以外，在本論文中我們同時也以時域有限差分法 (finite-difference time-domain method, FDTD) 對二維光子晶體 (Photonic crystals, PCs) 進行研究與分析。對於以多重平面波作為FDTD的初始條件來求解二維光子晶體之光子能帶結構所產生的相位干涉現象將在本論文中被探討與分析。分析的結果顯示當相位干涉接近於互相抵消且發生於特徵頻率點時，原本應該存在的模態將會被抵銷掉。為了克服這個問題，我們提出了一個新的FDTD求解程序，此一求解程序可以避免掉模態被抵銷的現象並且在被考慮的頻帶上得到完整的模態。

A rigorous power coupling model for three-core optical waveguides is proposed based on a full-wave vector boundary element method (VBEM). In addition to the influence of the state of the polarization (SOP) of the input light on the coupling behavior of the three-core optical waveguides can be simulated, the polarization dependent loss (PDL) of the three-core optical waveguides can also be investigated by combining the Mueller matrix method into the power coupling model. In this dissertation, the power coupling model is applied to investigate two kinds of power splitters. The first power splitters are constructed of step-index single mode fibers called triangular 3 3 fused tapered couplers. The influence of the SOP of the input light on the coupling behavior of the triangular 3 3 fused tapered couplers and the effect of fabricating parameters of the coupler, fusion degree, and heated length on the PDL of the coupler are investigated in this dissertation. The second kind of power splitters are constructed of photonic crystal fibers (PCFs). And, several fundamental coupling properties of three-core photonic crystal fibers (PCFs) with equilateral triangular cores are investigated numerically included coupling length, bandwidth, and polarization dependent loss (PDL). It is found the three-core PCFs are good candidate to be realized as an ultra-compact power splitter. And, for three-core PCFs that chose a proper coupling point can raise the yield and performance stability of the power splitter. In addition to the coupling behavior of the power splitters, two-dimensional photonic crystals (PCs) are also studied in this dissertation based on finite-difference time-domain (FDTD) method. The phase interference phenomenon due to the multiple plane-wave signals as initial conditions of the FDTD method for computing band structure of two-dimensional PCs is studied in this dissertation. It is found some normal modes supposed to exist could be lost if the phase interference is nearly out of phase at eigenfrequency. To overcome this problem, we proposed a new solving procedure based on FDTD algorithm which can avoid mode loss phenomenon and obtain complete normal modes over interested frequency range.

Contents

Abstract i
Contents iv
List of Figure vi
List of Tables xi

1 Introduction 1
1.1 Motivation and Literature Survey 1
1.2 Chapter Outline 3

2 The Vector Power Coupling Model for Three-Core Optical Waveguides 5
2.1 The Vector Boundary Element Method 6
2.1.1 Theory 6
2.1.2 Numerical Procedure 10
2.2 Power Coupling Formulation 13
2.3 Polarization Dependent Loss 20
2.4 Validity Check 21
2.5 Summary 22

3 Analysis of Three-core Fused Fiber Power Splitter 24
3.1 Propagation Constants and Field Patterns of the Guiding Modes 24
3.2 Power Coupled Model for Scalar Input Lights 29
3.3 Power Coupling Analysis for Polarized Input Lights 33
3.4 Polarization Dependent Loss of Three-core Fused Fiber Power Splitter 36
3.5 Summary 38

4 Analysis of Three-core Photonic Crystal Fiber Power Splitter 40
4.1 Single-core Photonic Crystal Fibers 41
4.2 Power Coupling Analysis of Three-core Photonic Crystal Fibers 45
4.3 Applications of Three-core Photonic Crystal Fibers 54
4.4 Summary 56

5 Conclusion 58

Appendix:
Two-Dimensional Photonic Crystal Analysis through Finite-Difference Time-Domain Method 60
A.1 Two-Dimensional Photonic Crystals (PCs) 64
A.2 The Finite-Difference Time-Domain (FDTD) Algorithm 71
A.3 The FDTD Method for Computing Band Structure of Two-Dimensional Photonic Crystals 81
A.3.1 The Conventional FDTD Method for Computing Band Structure of 2-D Photonic Crystals 82
A.3.2 Initial Condition Analysis of the FDTD Method for Computing Band Structure of Photonic Crystals 85
A.3.3 The Modified FDTD Method for Computing Band Structure of 2-D Photonic Crystals 98
A.3.4 The FDTD Method for Computing the Out-of-Plane Band Structure of 2-D Photonic Crystals 101
A.4 Summary 107

Reference 108
Publication List 115

[1] D. O. Culverhouse, T. A. Birks, S. G. Farwell, and P. St. J. Russel,“3 3 all-fiber routing switch,” IEEE Photon.Tech.Lett., vol. 9, pp. 333-335,1997.
[2] Y. H. Ja, “Analysis of optical fiber ring resonator using a collinear 3 3 fiber coupler” J. Quantum Electron., vol. 30, pp. 2639-2644, 1994.
[3] Y. H. Ja, “Analysis of four-port optical fiber ring and loop resonators using a 3 3 fiber coupler and degenerate two-wave mixing,” IEEE J. Quantum Electron., vol. 28, pp. 2749-2757, 1992.
[4] M. Eisenmann, and E. Weidel, “Single-mode fused biconical couplers for wavelength division multiplexing with channel spacing between 100 and 300 nm,” J. Lightwave Technol., vol. 6, pp. 113-119, 1988.
[5] D. B. Mortimore, “Theory and fabrication of 4 4 single-mode fused optical fiber couplers,” Appl. Opt., vol 29, pp. 371-374, 1990.
[6] S. W. Yang, T. L. Wu, C.W. Wu, and H. C. Chang, “Numerical modeling of weakly fused fiber-optic polarization beamsplitters—part II: the three-dimensional electromagnetic model,” J. Lightwave Technol., vol. 16, pp. 691-696, 1998.
[7] Y. H. Ja, “Avernier fiber double-ring resonator with a 3 3 fiber coupler and degenerate two-wave mixing,” IEEE Photon.Tech.Lett., vol. 4, pp. 743-745, 1992.
[8] Y. H. Ja, “Single-mode optical-fiber double-ring resonator with a planar 3 3 fiber coupler,” Opt. Lett., vol. 18, pp. 1502-1504, 1993.
[9] Y. H. Ja, “A single-mode optical fiber ring resomator using planar 3 3 fiber coupler and a sagnac loop,” J. Lightwave Technol., vol. 12, pp. 1348-1354, 1994.
[10] T. L. Wu, and H. C. Chang, “Rigorous analysis of form birefringence of fused fibre couplers,” Electron. Lett., vol. 30, pp. 998-999, 1994.
[11] K. Morishita, and K. Takashina, “Polarization properties of fused fiber couplers and polarizing beamsplitters,” J. Lightwave Technol., vol. 9, pp. 1503-1507, 1991.
[12] K. A. Murphy, M. F. Gunther, and A. M. Vengsarkar, “Variable-ratio polarization-insensitive, 3 3 fused biconical tapered couplers,” Electron. Lett., vol. 27, pp. 1336-1337, 1991.
[13] A. Niu, C. M. Fitzgerald, T. A. Birks, and C. D. Hussey, “1 3 linear array singlemode fibre couplers,” Electron. Lett., vol. 28, pp. 2330-2332, 1992.
[14] K. Okamoto, “Theoretical investigation of light coupling phenomena in wavelength-flattened couplers,” J. Lightwave Technol. Vol. 8, pp. 678-683, 1990.
[15] C. W. Wu, T. L. Wu, H. C. Chang, “A novel fabrication method for all-fiber, weakly fused, polarization beamsplitters,” IEEE Photon.Tech.Lett., vol. 7, pp. 786-788, 1995.
[16] S. W. Yang, and H. C. Chang, “Numerical modeling of weakly fused fiber-optic polarization beamsplitters—part I: accurate calculation of coupling coefficients and form birefringence,” J. Lightwave Technol., vol. 16, pp. 685-690, 1998.
[17] F. P, Payne, C. D. Hussey, and M. S. Yataki, “Polarisation analysis of strongly fused and weakly fused tapered couplers,” Electron. Lett., vol. 21, pp. 561-563, 1985.
[18] A. W. Snyder, and X. H. Zheng, “Fused couplers of arbitrary cross-section,” Electron. Lett., vol. 21, pp. 1079-1080, 1985.
[19] M. N. McLandrich, R. J. Orazi, and H. R. Marlin “Polarization independent narrow channel wavelength division multiplexing fiber couplers for 1.55 m,” J. Lightwave Technol., vol. 9, pp. 442-447, 1991.
[20] Y. H. Ja, “Performance parameters of a wavelength-division demultiplexer made with a single 3 3 coupler optical fiber ring or loop resonator,” J. Lightwave Technol., vol. 11, pp. 1337-1343, 1993.
[21] T. A. Birks, “Effect of twist in 3 3 fused tapered couplers,” Appl. Opt., vol. 31, pp. 3004-3014, 1992.
[22] Y. H. Chew, T. T. Tjhung, and F. V. C. Mendis, “Performance of single- and double-ring resonators using 3 3 optical-fiber coupler,” J. Lightwave Technol., vol. 11, pp. 1998-2008, 1993.
[23] P. A. Davies, and G. Abd-el-Hamid, “Four-port fiber-optic ring resonator,” Electron. Lett., vol. 24, pp. 662-663, 1988.
[24] A. J. Stevenson, and J. D. Love, “Vector modes of six-port couplers,” Electron. Lett., vol. 23, pp. 1011-1013, 1987.
[25] H. C. Huang, and H. C. Chang, “Analysis of equilateral three-core fibers by circular harmonics expansion method,” J. Lightwave Technol., vol. 8, pp 945-952, 1990.
[26] T. L. Wu, and H. C. Chang, “Rigorous analysis of form birefringence of weakly fused fiber-optic couplers,” J. Lightwave Technol., vol. 13, pp. 687-691, 1995.
[27] C. C. Su, “A surface integral equations method for homogeneous optical fibers and coupled image lines of arbitrary cross sections,” IEEE Trans. Microwave Theory Tech., vol. MTT-33, pp. 1114-1120, 1985.
[28] T. L. Wu and C. H. Chao, “Photonic Crystal Fiber Analysis through the Vector Boundary-Element Method: Effect of Elliptical Air Hole,” IEEE Photonics Technology Letters, Vol. 16, No. 1, pp. 126 - 128, Jan. 2004
[29] N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol., vol. 21, pp. 1787-1792, 2003.
[30] Y. Zhu, E. Simova, P. Berini, and C. P. Grover, "A comparison of wavelength dependent polarization dependent loss measurements in fiber gratings," IEEE Trans. Instrumentation and Measurement, vol. 49, pp.1231-1239, 2000.
[31] S. Schmidt, and C. hentschel, "PDL measurements using the HP8169A polarization controller," Hewlwtt-Packard publication no. 5964-9937E, 1995.
[32] J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-sillica single-mode optical fiber with photonic crystal cladding,” Opt. Lett., vol. 21, pp. 1547-1549. 1996.
[33] T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett., vol. 22, pp. 961-963, 1997.
[34] J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J.-P. de Sandro, “Large mode area photonic crystal fiber,” Electron. lett., vol. 34, pp.1347-1348, 1998.
[35] M. J. Gander, R. McBride, J. D. C. Jones, D. Mogilevtsev, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Experimental measurement of group velocity in photonic crystal fibers,” Electron. Lett., vol. 35, pp. 63-64, 1999.
[36] J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. S. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett., vol. 12, pp. 807-809, 2000.
[37] A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett., vol. 25, no. 18, pp. 1325-1327, 2000.
[38] T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knuders, A. Bjarklev, J. R. Jensen, and H. Simonsen, “ Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett., vol. 13, pp. 588-590, 2001.
[39] M. J. Stell and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. lightwave technol., vol. 19, pp.495-503, 2001.
[40] K. Saitoh, and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” J. lightwave technol., vol. 38, pp.927-933, 2002.
[41] M. Koshiba, and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett., vol. 13, pp. 1313-1315, 2001.
[42] M. J. Steel, T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett., vol. 26, pp. 488-490, 2001.
[43] A. peyrilloux, T. Chartier, A. Hideur, L. Berthrlot, G. Melin, S. Lempereur, D. Pagnoux, and P.Roy, “Theoretical and experimental study of the birefringence of a photonic crystal fiber,” J. lightwave technol., vol. 21, pp.536-539, 2003.
[44] M. J. Steel, and R. M. Osgood, Jr. “Elliptical-hole photonic crystal fibers,” Opt. Lett., vol. 26, pp. 229-231, 2001.
[45] A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Perturbation analysis of disperdion properities in photonic crystal fibers through the finite element method,” J. lightwave technol., vol. 20, pp.1433-1442, 2002.
[46] A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Vector description of higher-order modes in photonic crystal fibers,” J.Opt. Soc. Am. A., vol. 17, pp.1333-1340, 2000.
[47] R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science, vol. 285, pp. 1537-1539, 1999.
[48] J. C. Knight, J. Broeng, T. A. Birks, P. St. Russell, "Photonic band gap guidemce in optical fibers," Science, vol. 282, pp. 1476-1478, 1998.
[49] P. Russell, "Photonic crystal fibers," Science, vol. 299, pp. 358-362, 2003.
[50] N. Florous, K. Saitoh, and M. Koshiba, “A novel approach for designing photonic crystal fiber splitters with polarization-independent propagation characteristics,” Opt. Express, vol. 13, pp. 7365-7373, 2005.
[51] K. Saitoh, Y. Sato, and M. Koshiba, “Polarization splitter in three-core photonic crystal fibers,” Opt. Express, vol. 12, pp. 3940-3946, 2004.
[52] L. Zhang, and C. Yang, “Polarization-dependent coupling in twin-core photonic crystal fibers,” J. lightwave technol., vol. 22, pp.1367-1373, 2004.
[53] L. Zhang, and C. Yang, “Anovel polarization splitter based on the photonic crystal fiber with nonidentical dual cores,” IEEE Photon. Technol. Lett., vol. 16, pp.1670-1672, 2004.
[54] J. Lǽgsgaard, O. Bang, and A. Bjarklev, “Photonic crystal fiber design for broadband directional coupling,” Opt. Lett., vol. 29, pp. 2473-2475, 2004.
[55] T. L. Wu, and H. J. Ou, “A vector power coupling model for analyzing polarization-dependent loss of equilater triangular 3 3 weakly fused fiber couplers,” Opt. Commun., vol. 224, pp. 81-88, 2003.
[56] E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. Vol. 58, pp. 2059-2062, 1987.
[57] S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. Vol. 58, pp. 2486-2489, 1987.
[58] K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a Photonic Gap in Periodic Dielectric Structures,” Phys. Rev. Lett. Vol. 65, pp. 3152-3155, 1990.
[59] E. Yablonovitch, T. J. Cmitter, and K. M. Leung, “Photonic Band Structure: The Face-Centered-Cubic Case Employing Nonspherical Atoms,” Phys. Rev. Lett. Vol. 67, pp. 2295-2298, 1991.
[60] R. Abhari, and G.V. Eleftheriades, “Metallo-dielectric electromagnetic bandgap structures for suppression and isolation of the parallel-plate noise in high-speed circuits,” IEEE Trans. Microwave Theory Tech.,Vol. 51, pp. 1629 – 1639, 2003.
[61] E. Yablonovitch, “Photonic Crystals: Semiconductors of Light,” Scientific American (International Edition), vol. 285, no. 6, pp. 47-55, December 2001.
[62] O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science, vol. 284, pp. 1819-1821, 1999.
[63] C. Y. Liu, and L. W. Chen, "Tunable photonic crystal waveguide coupler with nematic liquid crystals," IEEE Photon. Technol. Lett., vol. 16, pp. 1849-1851, 2004.
[64] P. R. Villeneuve and M. Piché, "Photonic band gap in two-dimensional square and hexagonal lattices," Phys. Rev. B, vol. 46, pp. 4969-4972, 1992.
[65] J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett., vol. 69, pp. 2772-2775, 1992.
[66] H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Tran. Microw. Theory Tech., vol. 44, pp. 2688-2695, 1996.
[67] K. Saitoh and M. Koshiba, "Confinement losses in air-guiding photonic bandgap fibers," IEEE Photon. Technol. Lett., vol. 15, pp. 236-238, 2003.
[68] C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B, vol. 51, pp. 16635-16642, 1995.
[69] M. Qiu, and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," J. Appl. Phys., vol. 87, pp. 8268-8275, 2000.
[70] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. Princeton, NJ: Princeton Univ. Press, 1995.
[71] E. Yablonovitch, "Photonic band-gap structures," J. Opt. Soc. Amer. B, vol. 10, no. 2, pp. 283-295, 1993.
[72] A. Taflove, Computational Electrodynamics: The Finite Difference Time Domain Method. Boston, MA: Artech House, 1995.
[73] 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, May 1966.
[74] G. Mur, “Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic Field Equations,” IEEE Trans. Electromagnetic Compatibility, Vol. 23, pp. 377-382, 1981.
[75] J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computational Physics, Vol. 114, pp. 185-200, 1994.
[76] R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Existence of a photonic band gap in two dimensions," Appl. Phys. Lett., vol. 61, pp.495-497, 1992.
[77] A. Asi, and L. Shafai, “Dispersion analysis of anisotropic inhomogeneous waveguides using compact 2D-FDTD,” Electron Lett., Vol. 28, pp. 1451-1452, 1992.
[78] K. Saitoh, and M. Koshiba, “Confinement lossed in air-guiding photonic bandgap fibers,” IEEE Photon. Technol. Lett., Vol. 15, pp.236-238, 2003.