Finite-Difference Time-Domain Modeling of Photonic Crystals and Dispersive Phenomena

本研究以無損及非色散有限差分時域法研究幾種二維光子晶體結構，使用的波源以全場散射場法來產生平面波或高斯波束，邊界條件則為同向性完美匹配層，其中垂直於平面的場需分裂為二個場來達到阻抗匹配。我們分析了L型光子晶體波導，T型分波光子晶體波導，十字交錯型光子晶體波導。我們也模擬了傳統型介質波導來相互比較其間的差異性及優缺點。這些結構均是以橫向磁場模擬為主。另外我們也模擬了微共振腔型濾波器，它分為二種基本型，分別為環型及盤型。其中我們又討論了以兩個相鄰很近的環型波導所組成的濾波器，其共振模態就類似一較粗的環型波導。同樣的以光子晶體為基礎的濾波器，包括單極型以及雙極簡併型共振腔。研究的第二部份以平面波展開法為主，探討頻帶結構及光子晶體波導的缺陷模態。在不同極化波的考量下，我們可利用此方法來建構不同光子晶體排列所對應的頻帶結構圖。它可方便我們日後查詢在不同的條件下適合的光子頻隙，或是光子頻隙最佳化的考量依據。我們也用此法來計算光子晶體共振腔的模態。最後，我們討論負折射率物質所形成的方型稜鏡，三角型稜鏡及凹透鏡其中多次折射的總合現象。探討其結構然後與正折射率物質來比較。很有趣的發現在負折射率物質相位速度與群速有相反的方向，而在正折射率物質，則兩個方向為相同。In this research, the lossless and nondispersive finite-difference time-domain (FDTD) method with staircase approximation and the split-field perfectly matched layer boundaries is employed to model several types of photonic devices in two dimensions. L-bends, T-branches, and waveguide crossings made in both conventional slab waveguides and photonic crystals for transverse magnetic modes are analyzed. Microcavity resonator based filters including micro-rings and micro-disks are also analyzed, as well as channel dropping filters in the forms of photonic crystal monopole cavity and doubly degenerate hexapole cavities. In the second part of this research, the plane wave expansion method is employed to model the photonic crystal band structures or defect modes in photonic crystal defect waveguides. We also build up the band gap maps in rectangular lattice for TE and TM modes. The resonant modes of a rectangular photonic crystal cavity are also investigated by this approach. Finally, we discuss the negative refractive index materials in different photonic components like square and triangular prisms. We compare them with the normal ones. Interestingly we find that the phase velocity is opposite to the group velocity and the refraction direction is different from those in the normal index materials.List of Figures iv CHAPTER 1 1 Introduction 1 1.1 Motivation 1 1.2 General Reviews of Photonic Crystals Waveguides and Couplers 3 1.3 Thesis Outline 5 CHAPTER 2 14 Computational Methods of Photonic Crystal 14 2.1 Plane Wave Expansion Method 14 2.1.1 Wave Equation and Eigenvalue Problems 14 2.2 Finite-Difference Time-Domain Method 17 2.2.1 3-D Maxwell’s Equations 17 2.2.2 Finite Differences and Notations for Discretization 19 2.2.3 Yee’s SCHEME 19 2.3 Maxwell’s Equations and Yee’s Scheme in 2D TEz and TMz Modes 21 2.3.1 The TMz and TEz Modes 21 2.3.2 TMz and TEz Modes in The FDTD Method 22 2.4 Numerical Phase Velocity and Phase Error Analysis 23 2.4.1 Numerical Dispersion Relation 23 2.4.2 Phase Error Analysis 24 2.5 Numerical Stability 26 2.6 Perfectly Matched Layers 28 2.6.1 Mathematically Motivated Perfectly Matched Layers for The TEz Field 28 2.7 The Total-Field/Scattered-Field Technique 37 2.7.1 Ideas and One-Dimensional Formulation 37 2.7.2 Two-Dimemsional Formulation of The TF/SF Technique 42 CHAPTER 3 54 2-D TMz Mode Integrated Optic Waveguide Simulation 54 3.1 Some Remarks Before Simulation 54 3.2 Simple 90-Degree-Bend Waveguide 55 3.3 90-Degree-Bend Waveguide Modified by A Square Resonator Indide The Corner 56 3.4 90-Degree-Bend Waveguide Modified by A Square Resonator With A 45-Degree-Cut at Its Outer Corner 57 3.5 90-Degree-Bend Waveguide Modified by A Quarter-Disk Resonator Cavity 58 3.6 Simple T-Junction 58 3.7 T-Junction Modified by A Resonator 59 3.8 Waveguide Crossings 60 3.9 Mircroring Cavity Channel Drop Filters 60 CHAPTER 4 83 Simulation Results of Photonic Crystal Band Structures 83 4.1 Atlas of Band Gaps for Two-Dimensional Photonic Crystals 83 4.1.1 Band Gap Maps Versus Radius of The Columns 84 4.2 Rectangular Photonic Crystal Cavity 85 4.3 The Shifted-Inverse Power Method (SIPM) 85 CHAPTER 5 106 Photonic Crystal Bends and Junctions of Various Forms 106 5.1 Sharp Bends In Photonic Crystal Form 106 5.2 T-Branches In Photonic CrystalS 110 5.3 Cross WaveguideS In Photonic Crystals 112 5.4 Channel Drop FilterS 115 5.5 Antenna 118 5.6 Coupled-Cavity Waveguide 120 5.7 Waveguide Couplers 122 CHAPTER 6 157 Negative Index Metamaterials 157 6.1 Electromagnetic Response of The Metamaterials 157 6.2 Negative Refraction 160 6.3 2D-FDTD Simulator for TM Case 163 6.4 Focusing In Negative Refractive Index 167 6.5 Negative Space 169 CHAPTER 7 182 Conclusion 182 References 18