楊照彥臺灣大學：應用力學研究所許惠貞Hsu, Chei-ChenChei-ChenHsu2007-11-292018-06-292007-11-292018-06-292005http://ntur.lib.ntu.edu.tw//handle/246246/62459本研究旨在利用時域頻譜配點補償法求解無限域中含複雜形狀散射體受電磁波入射所引致的散射現象，並且利用時域頻譜配點方法在設計與模擬繞射光學元件上。此法可用於分析材料含有不連續折射率分佈的電磁場問題，利用區域分割的概念，將整個計算域分割為數個子區域，尤其在不連續折射率處。每個子區域的場值先以Legendre多項式為基函數展開，在不同區域的邊界條件上利用特徵值傳輸。其中，補償項可以控制整個計算系統的穩定性與精確性。 於數值算例中，首先我們比較了幾個已有解析解的散射問題，例如：圓柱型的完美導體、圓柱型的介電材質、多層的圓柱型完美導體與介電材質，透過場等效原理所獲得的結果與解析解比對其準確性非常高。另外，我們也應用在較複雜的幾何形狀的散射體，也都得到良好的模擬結果。此外，我們利用時域頻譜配點方法模擬與設計繞射光學波導元件及繞射光學鏡片得到繞射光學元件的光學特性來驗證本方法的正確性與實用性。The purpose of this study is to solve scattering problems where there is a complicated shape scatter interacting with incident electromagnetic wave in an infinite spatial domain. The method can be applied to analyze electromagnetic problems with material containing discontinuous refractive indices. Employing the domain decomposition, one can divide the whole computational domain into several sub-domains, particularly at the interfaces of discontinuous refractive indices. The electromagnetic fields in each sub-domain can be expanded in terms of Legendre polynomial as basis functions and at the boundary interface between different regions, the characteristics are constructed and employed to establish the penalty terms which are closely related to the stability and flexibility of the method. Application to scattering by a perfectly conducting and electric circular cylinder has been carried out and compared with solution from field equivalent principle. Extensions to more complicated geometries such as perfectly electric square cylinder conductor and conducting with dielectric layer where no analytical solution is available are also simulated. Applications of the pseudospectral penalty method to more practical optoelectronic devices such as diffractive optic elements including diffractive optical waveguide and diffractive optical lens are computed. All the present results are in good agreement with available data in the literatures. With all the results obtained, it can be concluded that the present multi-domain pseudospectral penalty method in time domain provides an alternative simulation method to the finite difference time domain method for the electromagnetic wave propagation problems. Although, the present work is restricted to two-dimensional Maxwell equations, the extension to three-dimensional problems is of no conceptual difficulty and will be the subject of future work.Contents 1 Introduction 14 1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2 Literature Review and Motivations . . . . . . . . . . . . . . . . . . . . 15 1.3 Content of The Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Electromagnetics Preliminaries 19 2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.1 Differential Form of Maxwell’s Equations . . . . . . . . . . . . . 19 2.1.2 Constitutive Relations for Isotropic Medium . . . . . . . . . . . 20 2.2 Time-Harmonic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Electromagnetic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Magnetic Vector Potential ~A . . . . . . . . . . . . . . . . . . . 23 2.4.2 Electric Potential ~F . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Near to Far Field Transformation . . . . . . . . . . . . . . . . . . . . . 26 2.6 Radar Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Pseudospectral Penalty Methods for Time Domain Maxwell’s Equations 29 3.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Pseudospectral Approximation Methods . . . . . . . . . . . . . . . . . 31 3.2.1 Legendre Pseudospectral Method . . . . . . . . . . . . . . . . . 31 3.2.2 Chebyshev Pseudospectral Method . . . . . . . . . . . . . . . . 33 3.3 Pseudospectral Penalty Method for Model PDE . . . . . . . . . . . . . 34 3.4 Pseudospectral Penalty Method and Two-Dimensional Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4.1 Characteristic Variables and Well-posed Boundary Operator of 2-D Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . 36 3.4.2 Characteristic Representation of Physical Boundary Conditions 39 3.4.3 The Maxwell’s Equations with Pseudospectral Penalty Method 41 3.5 Absorbing Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . 43 3.6 Total Field and Scattered Field . . . . . . . . . . . . . . . . . . . . . . 44 3.7 Nearfield and Farfield Calculations . . . . . . . . . . . . . . . . . . . . 45 4 Two Dimensional Electromagnetic Wave Scattering Problems 49 4.1 Incident Plane Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.1 Normal Incident Plane Wave : TEy Polarization . . . . . . . . . 49 4.1.2 Normal Incident Plane Wave : TMy Polarization . . . . . . . . 50 4.2 Scattering by Perfect Electric Conducting Medium . . . . . . . . . . . 50 4.2.1 Scattering by Circular Cylinder : TEy Polarization . . . . . . . 50 4.2.2 Scattering by Circular Cylinder : TMy Polarization . . . . . . . 53 4.2.3 Scattering by Square Cylinder : TEy Polarization . . . . . . . . 56 4.2.4 Scattering by Square Cylinder : TMy Polarization . . . . . . . . 57 4.3 Scattering by Dielectric Medium . . . . . . . . . . . . . . . . . . . . . 57 4.3.1 Scattering by Circular Cylinder : TEy Polarization . . . . . . . 57 4.3.2 Scattering by Circular Cylinder : TMy Polarization . . . . . . . 60 4.4 Scattering by Conducting and Dielectric Layered Objects . . . . . . . 64 4.4.1 Square Conductor with Layered Dielectric : TEy Polarization . 64 4.4.2 Square Conductor with Layered Dielectric : TMy Polarization . 65 5 Diffractive Optical Elements 121 5.1 Diffractive Optical Waveguide . . . . . . . . . . . . . . . . . . . . . . . 122 5.2 Diffractive Optical Lens . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6 Conclusion 152 A Two Dimensional Full Space Green’s function 154 B Near to Far Field Field Transformation 1573551858 bytesapplication/pdfen-US電磁散射問題繞射光學元件Electromagnetic Scattering ProblemsDiffractive Optical Elementsthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/62459/1/ntu-94-D87543002-1.pdf