Finite Element Analysis of Dielectric and Plasmonic Periodic Structures

藉由在有限元素法中加入週期性結構邊界條件，成功的建立了帶有一個與兩個方向的週期結構之模型。利用嶄新的邊界條件演算法，可以快速與準確的分析由頻率相關與不相關材料組成的兩個方向的週期結構之色散曲線圖與模態特性。當表面電漿在磁場偏極化下被激發出時會出現有趣且帶有實用利益的現象。本色散關係的呈現始於一維的極化與金屬光子晶體，從無能量損耗與有能量損耗的光子晶體發現進而檢驗明顯的差異。與解析解的比較下可以驗證計算分析的準確性與收斂性。然後，我們分析由圓柱與橢圓柱構成之奈米電漿子波導，此結構可利用表面電漿共振，在低於繞射極限的情況下傳遞電磁波。從實數頻率作為輸入參數的有限元素法可得到此奈米結構的高解析度色散曲線圖，亦在此系統中發現複數布拉赫波向量的複數模態並廣泛的討論。後，使用新開發之兩個週期性的邊界條件演算法去分析二維介質與金屬光子晶體，可得到關於傳導、複數與消逝模態的完整資訊。在介質光子經典的分析展現出卓越的高準度，與Dirichlet-to-Neumann map和mutiple multipole method的計算結果亦呈現很好的吻合。並且，我們詳細的展示了與表面電漿子相關的模態特性。Periodic structures are successfully modeled by the implementations of periodic boundary conditions (PBCs) in the finite element method (FEM) for single and double periodicity. With a novel algorithm of PBCs, fast and precise calculations can be executed to investigate dispersion diagrams and modal characteristics ofoubly periodic structures composed of either frequency-dependent and frequency-independent material dielectric constants. Interesting and advantageous phenomena are discovered for the H-polarization situations for which surface plasmons are meantime excited. Starting from one-dimensional polaritonic and metallic photonic crystals, essential characters of dispersion relations are presented. Apparent dissimilarities between lossless and lossy photonic crystals are revealed and examined. In comparisonith analytical solutions, the correctness and behavior of numerical convergence can be accurately verified. Afterward we analyze the nano-plasmonic waveguides in the forms of circular and elliptical cylinders for guiding electromagnetic waves with plasmon resonances below the diffraction limit. Outstanding high-resolution dispersioniagrams of such subwavelength structures are performed by the real-ω FEM. The complex modes possessing complex Bloch-wave vectors are as well discovered in these systems and have been extensively discussed. The developed algorithm of PBCs for doubly periodic systems is then employedo analyze two-dimensional dielectric and metallic photonic crystals. Complete information about the propagating, complex, and evanescent modes are disclosed. Eminently high precision is shown in the calculations of dielectric photonic crystals. And excellent agreement between the Dirichlet-to-Neumann map and the multiple multipole method are shown for metallic photonic crystals. Furthermore, weemonstrate modal characteristics correlated with surface plasmons in detail.1 Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Finite ElementMethod . . . . . . . . . . . . . . . . . . . . . . . 2.3 Plasmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Brillouin Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Contributions of the PresentWork . . . . . . . . . . . . . . . . . . . 6.6 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Mathematical Formulation and Related Techniques 11.1 Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The PerfectlyMatched Layers . . . . . . . . . . . . . . . . . . . . . . 12.3 Generalized Scalar FEM Based Algorithm with the In-Plane Waveropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 The Governing Equation . . . . . . . . . . . . . . . . . . . . . 14.3.2 Finite Element Discretization . . . . . . . . . . . . . . . . . . 16.4 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 18.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.2 Periodic Boundary Conditions for 2-D Structures with Singleeriodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.3 Periodical Boundary Conditions for 2-D Structures with Doubleeriodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.5 Finite Element Method BasedMatrix Eigenvalue Equation . . . . . . 28.5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.5.2 The Real-k Method Based on Finite-Element Formulation . . 29.5.3 The Real-ω Method Based on Finite-Element Formulation . . 30.6 Comparison and Further Discussion of PBCs . . . . . . . . . . . . . . 36 Analyzing Characteristics of Singly Periodic Systems 45.1 Introduction to ComplexModes . . . . . . . . . . . . . . . . . . . . . 45.2 Analysis of One-Dimensional Photonic Crystals . . . . . . . . . . . . 48.2.1 Polaritonic Photonic Crystal . . . . . . . . . . . . . . . . . . . 50.2.2 Metallic Photonic Crystal . . . . . . . . . . . . . . . . . . . . 52.3 Nano-PlasmonicWaveguides . . . . . . . . . . . . . . . . . . . . . . . 53.3.1 Nano-plasmonicWaveguides: Circular Cylinders . . . . . . . . 55.3.2 Nano-plasmonic Waveguides: Elliptical Cylinders . . . . . . . 58.4 Further Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Analyzing Characteristics of Doubly Periodic Systems 100.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100.2 Band Structure Analysis for Isotropic Dielectric Photonic Crystals . . 102.3 Band Structure Analysis for Isotropic Metallic Photonic Crystals . . . 105.3.1 Lossless Photonic Crystals . . . . . . . . . . . . . . . . . . . . 105.3.2 Lossy Photonic Crystals . . . . . . . . . . . . . . . . . . . . . 110.4 Further Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Conclusion 141 List of Acronyms 14