Analysis of Frequency Selective Surfaces by Spectral Galerkin Method

「頻率選擇表面」（frequency selective surfaces）是由一群在空間中呈週期性排列的金屬所組成的二維陣列。由於其濾波的特性，多年來，吸引了大量研究工作的投入。研究中發現，頻率選擇平面在電磁波的許多頻段中，具有廣泛的應用。例如在微波中可用於雷達或天線系統，在紅外光的頻段中則可用於極化器或提升分子雷射的效率。能夠影響頻率選擇表面之頻率響應的因素有許多，諸如金屬排列的幾何形狀、週期長度及其間隙大小；而作為其基底之介電材料的電磁特性及厚度也是需要考量的因素。 當入射波通過頻率選擇表面時會在金屬的表面誘發出表面電流，進而由這些表面電流激發出散射場。在本文中，我們採用頻率域的Galerkin方法來分析有關頻率選擇表面散射場的問題。在頻率域裡，我們引進Floquet理論，使得誘發表面電流能以傅立葉級數的形式來表示，而計算域也從無限延伸的陣列減少至單一週期。在建立散射場和誘發表面電流的關係式後，帶入金屬表面的邊界條件，就可針對誘發表面電流求解。對於含有介電材料或多層結構的頻率選擇表面，則必須先採用Itoh所提出的頻域阻抗法（spectral domain immitance approach），以求得用來聯繫表面電流和散射場的Green函數，然後再進行求解。 為了方便分析幾何排列較為複雜的結構，我們採用小區域（subdomain）的基底函數將作為未知數的誘發表面電流展開。依循Galerkin法的步驟便可求得電流的分佈情形。當未知數的數目很大時，可仰賴適當的疊代法（本文採用共軛梯度法conjugate gradient method），並配合快速傅立葉的運算法來提高求解的效率。根據誘發表面電流的分佈情形，位於整個頻率選擇表面結構之上表面及下表面的散射場即可以頻域的形式求得。最後，在不同Floquet模態下的反射和穿透係數，皆可由這些位於上、下表面的散射場來表示，對於頻率選擇表面之頻率響應的分析就完成了。 本文針對不同結構的頻率選擇表面，分析其於不同極化方向的入射波在不同角度入射下所對應的頻率響應，並與已知的結果作比對，來驗證理論和計算過程的正確性。之後再改變用來描述頻率選擇表面結構的參數，如介電值的厚度、介電常數或金屬排列的方式，以觀察其對於頻率響應的影響。Because of the filtering property suggested, two-dimensional periodic screens which were named the frequency selective surfaces (FSS) have attracted a great deal of attention for many years and have been found various applications, such as band-pass radomes, reflectors of antenna system, polarizers and so on. The frequency response of FSS highly depends on the configurations and spacing of the elements as well as on the thickness and permittivity of dielectric layers that may be part of the screens. When an incident field propagates through FSS, surface currents will be induced on the conducting screens and then, in turn, radiate a scattered field. In this thesis, we employ the spectral Galerkin method to analyze the scattering phenomena of the FSS. In the spectral domain, Floquet’s theorem allows the induced surface currents to be expressed in terms of a Fourier series and reduces the computation domain from an infinite array into a single cell. For the FSS with multilayered structures, we also employ the spectral immitance approach to derive the spectral dyadic Green’s functions which relate the induced surface currents to the scattered field. Moreover, to be more feasible for analyzing FSS with complex configurations, the subdomain basis functions are adopted to expand the induced currents. Although that will increase the number of unknowns, the computation speed can be improved by using a fast Fourier transform based iterative approach (the conjugate gradient method, FFTCG). After the distribution of the induced surface currents is determined, the spectral scattered fields can be found. Finally, we can express the reflection and transmission coefficients at different Floquet modes in terms of the spectral scattered fields at the top and bottom surfaces of the FSS. Results for the free-standing and the single-layered-dielectric FSS with various geometries are presented, and are compared with existing results to check the correctness of our programming. In addition, some parameters, such as the configurations of the conducting screens, the thickness and the permittivity of the dielectric layers, which describe the structure of the FSS are varied to investigate the resultant effects on the frequency response.Acknowledgements i Abstract in Chinese ii Abstract iii 1 Introduction 1 2 Formulations 6 2.1 Free-standing FSS 6 2.1.1 With Infinite Conductivity 7 2.1.2 With Finite Conductivity 13 2.2 FSS with Dielectrics 14 2.2.1 Determination of Incident Fields for Multilayered FSS 15 2.2.2 Spectral Green’s Functions for Multilayered FSS 19 2.2.2.1 Spectral Domain Immitance Approach 19 2.2.2.2 Transmission Line Theory 21 2.2.3 Brief Summary 24 2.3 A Simple Example – FSS Printed on a Single Layer of Dielectric 24 3 Solutions to Operator Equations – Galerkin Method 30 3.1 Method of Moments 30 3.2 Roof-top Basis Functions 33 3.3 Accelerating the Computations 41 3.3.1 Fast Fourier Transform 41 3.3.2 Conjugate Gradient Algorithm 42 4 Reflection and Transmission Coefficients 45 4.1 Reflection Coefficients 46 4.2 Transmission Coefficients 50 4.3 Brief Summary 53 5 Numerical Results 55 5.1 Square Plates 55 5.2 Thin Conducting Patches 58 5.2.1 Normal Incidence 58 5.2.2 Varying Angles of Incidence 61 5.3 Jerusalem Crosses 61 5.4 Slot Apertures 70 5.4.1 Effects of Dielectric Constant 70 5.4.2 Effects of Thickness of Dielectric Layer 70 5.4.3 Fanned-Out Slot Apertures 71 6 Summary and Future Works 77 A Floquet’s Theorem 79 B Derivations of Characteristic impedances 82 C Fast Fourier Transform 85 References 8