Improved Finite-Difference Frequency-Domain Method for Modal Analysis of Optical Waveguides and Photonic Crystal Devices

本論文提出改善之有限差分頻域法來處理光波導及光子晶體元件之模態分析。藉由內差及外差來近似介質接面兩側的電磁場，及滿足介面連續的條件來處理波導結構中介質不連續之問題，並以完全匹配層作為計算視窗之邊界條件來分析光波導的損耗特性。 由一維平板波導及二維光纖波導的計算分析結果，顯示了我們所提出之改善有限差分頻域法可以有效地增加數值計算的準確性並且可以降低因為波導結構中斜線介面或弧線介面所產生的數值誤差。即使波導結構中有介質尖角存在，如通道波導和脊狀波導，我們所提出的方法還是可以提供足夠準確的數值分析結果。同時，利用這個有限差分頻域模型，我們成它a得到一維反共振反射型波導的損耗特性。而二維波導的計算結果，更顯示了這個模型具有降低計算視窗尺寸的能力。我們並且成它a應用此一方法於計算分析平板光子晶體波導、光子晶體光纖及光子晶體共振腔的特性，並且與其他數值方法得到相同的結果。 為了分析二維光子晶體的能帶結構，我們另提出了一個具有快速收斂性的精簡有限差分頻域演算法來計算二維光子晶體之能帶圖、光能隙及手指圖，數值計算結果相當準確，且較其他數值方法為佳。An improved finite-difference frequency-domain (FDFD) method is proposed for the modal analysis of optical waveguides and photonic crystal (PC) devices. For dealing with the dielectric discontinuity in the waveguide structures, the electromagnetic fields at both sides of the dielectric interface are approximated by interpolation and extrapolation and the continuity boundary conditions (BCs) are fulfilled. To obtain the lossy properties of optical waveguides, the perfectly matched layers (PMLs) are employed as the BCs of the computing window in this FDFD model. The calculated results of 1-D slab waveguides and 2-D optical fibers demonstrate that this improved FDFD formula can efficiently increase the numerical accuracy and reduce the numerical errors caused by the oblique and curved dielectric interfaces. Utilizing this FDFD model, the leaky properties of 1-D ARROWs structures can be successfully obtained. For channel and rib waveguides with dielectric corners in the structures, it can also provide sufficiently accurate numerical results. This FDFD model is also demonstrated possessing the ability in reducing the demand of the size of the computing window. Besides, the FDFD model is successfully applied to analyze the characteristics of PC related devices, including planar PC waveguides, photonic crystal fibers, and PC resonant cavities. The properties of the guided and leaky modes of the PC devices are obtained and a good agreement with other numerical simulations is achieved. As for the analysis of band structures of 2-D PCs, a compact FDFD eigenvalue algorithm with rapid numerical convergence is proposed. The band diagrams, photonic band gaps (PBGs), and the finger plots of 2-D PCs are obtained. The proposed method can perform better than other popularly used numerical methods.1 Introduction 1 1.1 Photonic Crystal Devices . . . . . . . . . . . . . . . . . . . . . 1 1.2 Numerical Schemes for Waveguide Analysis . . . . . . . . . . . 5 1.3 Numerical Schemes for the Analysis of Photonic Crystal Devices 7 1.4 Yee-mesh-based Mode Solvers . . . . . . . . . . . . .. . . . . . . 8 1.5 Overview and Organization of the Dissertation . . . . . . . . . . . 9 1.6 Contributions of the Present Work . . . . . . . . . . . . . . . . . 12 2 The Conventional Finite Difference Method 19 2.1 Overview . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 19 2.2 Central Difference Scheme . . . . . . . . . . . . . . . . . . . . . . 20 2.3 The E-Formulation FDM . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 Derivation for Two-Dimensional Problems . . . . . . . 21 2.3.2 Approximation at Dielectric Interfaces . . . . . . . . . 26 2.4 The H-Formulation FDM . . . . . . . . . . . . . . . . . . . . . 27 2.4.1 Derivation for Two-Dimensional Problems . . . . . . . 27 2.5 Numerical Errors . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5.1 Truncation Errors . . . . . . . . . . . . . . . . . . . . . 31 2.5.2 Modelling Errors and Round-o® Errors . . . . . . . . . 32 2.6 Boundary Conditions of Computing Domains . . . . . . . . . 32 2.6.1 Physically Lossy Medium . . . . . . . . . . . . . . . . . 33 2.6.2 Perfectly Matched Layers . . . . . . . . . . . . . . . . 34 2.6.3 Transparent Boundary Condition . . . . . . . . . . . . 35 3 Finite-Difference Frequency-Domain Method 41 3.1 Mode Solver for 1-D Problems . . . . . . . . . . . . . . . . . . 41 3.1.1 The TE Polarized Wave . . . . . . . . . . . . . . . . . 41 3.1.2 The TM Polarized Wave . . . . . . . . . . . . . . . . . 43 3.2 Formulae for Two-Dimensional Problems . . . . . . . . . . . . 44 3.3 FDFD Method with PMLs . . . . . . . . . . . . . . . . . . . . 48 3.4 Improvements for the Dielectric Interface . . . . . . . . . . . . 51 3.4.1 Index Average scheme . . . . . . . . . . . . . . . . . . 51 3.4.2 Proper Boundary Condition Matching . . . . . . . . . 52 3.5 Formulae for Band Structures of 2-D Photonic Crystals . . . . 54 3.5.1 The FDFD Method . . . . . . . . . . . . . . . . . . . . 55 3.5.2 Periodic Boundary Conditions for 2-D Photonic Crystals 58 3.6 Compact Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Numerical Results for Optical Waveguide Analysis 73 4.1 One-Dimensional Waveguides . . . . . . . . . . . . . . . . . . 74 4.1.1 Slab Waveguides . . . . . . . . . . . . . . . . . . . . . 74 4.1.2 Antiresonant Reflecting Optical Waveguides . . . . . . 75 4.2 Two-Dimensional Waveguides . . . . . . . . . . . . . . . . . . 78 4.2.1 Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . 78 4.2.2 Channel Waveguides . . . . . . . . . . . . . . . . . . . 81 4.2.3 Rib Waveguides . . . . . . . . . . . . . . . . . . . . . . 82 5 Numerical Results for Band Structures of Two-Dimensional Photonic Crystals 112 5.1 Other Algorithms for 2-D Photonic Crystals . . . . . . . . . . 112 5.2 Photonic Crystals with Square Lattices . . . . . . . . . . . . . 115 5.3 Photonic Crystals with Triangular Lattices . . . . . . . . . . . 118 5.4 Finger Plots of Two-Dimensional Photonic Crystals . . . . . . 123 6 Numerical Results for Photonic Crystal Devices 152 6.1 Planar Photonic Crystal Waveguides . . . . . . . . . . . . . . 152 6.2 Photonic Crystal Fibers . . . . . . . . . . . . . . . . . . . . . 157 6.2.1 Holey Fibers . . . . . . . . . . . . . . . . . . . . . . . . 158 6.2.2 Two-core Photonic Crystal Fibers . . . . . . . . . . . . 162 6.2.3 Honeycomb Photonic Crystal Fibers . . . . . . . . . . 165 6.2.4 Hollow Photonic Crystal Fibers . . . . . . . . . . . . . 167 6.3 Photonic Crystal Cavities . . . . . . . . . . . . . . . . . . . . 170 7 Summary and Conclusion 21