Finite-Difference Time-Domain Analysis of Microresonators and Metallic Subwavelength Aperture Structures

本篇論文採用有限差分時域法為數值模型模擬研究數種不同的元件，並以數學動機推導之完美匹配層作為吸收邊界的處理。首先分析波導耦合圓形微共振腔以及方形微共振腔。在分析微共振腔的同時我們亦針對格點配置的問題做了探討，期能使模型更接近真實情形。討論中發現，將格點中的電場安置在邊界上並將該格點的折射率取為邊界兩邊的平均值，在我們分析的例子中能得到最好的結果。其次，我們討論了表面電漿和金屬次波長孔隙的電磁波穿透問題，之後便著手分析一些光波段以及微波段的次波長孔隙的元件。其中有限差分數值分析使用了Drude色散模型來模擬金屬，在模擬中可以觀察到當表面電漿被激發時，波穿過孔隙穿透率的增強以及指向性。除了元件的研究，我們也提出了以非線性色散模型修改的完美匹配層。當模擬波在非線性色散介質中的行為時，此修改過的完美匹配層可以安置在計算空間的周圍，以有效地吸收往外擴散的波以模擬無限大的空間。In this research, the finite-difference time-domain (FDTD) method is employed to simulate several categories of devices with the appearance of the mathematic motivated perfectly matching layer (PML) around the computational domain. First, the micro-ring and square micro-ring resonators are analyzed. The issue of proper grid arrangement over the modelled structures for efficient numerical convergence is discussed. We discover that putting the electric field grid just at the boundary of dielectric interfaces and taking the index average of the grid provide the best results in the cases we concern. Two geometries of micro-ring resonator coupled by straight waveguides are simulated with the slab index of 3.2 and the cladding index of 1.0. Second, the surface plasmons (SPs) and the metallic subwavelength apertures are discussed. Several structures are analyzed by imposing the Drude model for material dispersion into the FDTD scheme to model the metal in the optical and microwave ranges. The enhancement of transmittance and the directional property are shown through the simulations. Beside simulation of various devices, we also implement a PML modified by nonlinear and dispersive model. The modified PML can be used to truncate the computational domain of modelling a nonlinear-dispersive medium to properly absorb the outgoing waves.1 Introduction 1 1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Mathematical Formulation 5 2.1 The Finite-Difference Time-Domain Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 The Yee Algorithm . . . . . . . . . . . . . . . . . . . . 6 2.1.3 Numerical Dispersion, Numerical Stability, and Other Characteristics . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Modelling of Frequency Dispersive Material . . . . . . . . . . 10 2.2.1 The Auxiliary Differential Equation Method . . . . . . 11 2.3 Absorbing Boundary Conditions . . . . . . . . . . . . . . . . . 13 2.3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . 13 2.3.2 Mathematically Motivated Perfectly Matched Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Mathematically Motivated PMLs for Nonlinear-Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.2 Nonlinear-Dispersive Formulations . . . . . . . . . . . 16 2.4.3 Nonlinear-Dispersive PML . . . . . . . . . . . . . . . . 18 2.4.4 Numerical Experiments . . . . . . . . . . . . . . . . . . 19 2.4.5 Modelling the Nonlinear-Dispersive Phenomenon . . . 20 2.5 Field Extension Techniques in FDTD simulations . . . . . . . 21 3 Modelling of Microresonators 32 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Grid Arrangement Issue . . . . . . . . . . . . . . . . . . . . . 33 3.2.1 Grid Arrangement in a Slab Waveguide . . . . . . . . . 34 3.2.2 Grid Arrangement in a 90-Degree Bend Waveguide . . 36 3.3 Micro-Ring Resonators . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Square Micro-Ring Resonators . . . . . . . . . . . . . . . . . . 38 4 Modelling of Subwavelength Aperture 57 4.1 Surface Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Subwavelength Apertures . . . . . . . . . . . . . . . . . . . . . 58 4.3 The Drude Model . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4 Modelling of Subwavelength Apertures in Microwave Range . . 61 4.4.1 The Reference Sample . . . . . . . . . . . . . . . . . . 61 4.4.2 The Sinusoidal Grating Sample . . . . . . . . . . . . . 62 4.4.3 The Symmetric Rectangular Grating Sample . . . . . . 64 4.4.4 The Asymmetric Rectangular Grating Sample . . . . . 64 4.4.5 The Directivity/Beaming Property . . . . . . . . . . . 65 4.5 Modelling of Subwavelength Apertures in Optical Frequency Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.6 Highly Directional Emission Properties . . . . . . . . . . . . . 67 5 Conclusion 8