從蕈狀結構反射相位圖取得表面波帶隙之近似方法

Abstract

本論文主要是在蕈狀結構(mushroom structures)的分析上，提供一個近似方法，可從反射相位圖(reflection phase plot)來取得表面波帶隙(surface-wave band gap)的資訊。由研究結果可知，表面波帶隙的上界可從掠射角入射(grazing incidence)的TE平面波反射相位圖中的第一個等相位頻率(in-phase frequency)得出，而表面波帶隙的下界可利用本論文所提供的近似公式並代入TM反射相位圖中三個特定頻率來得出近似值，三個頻率分別是(1) TM平面波垂直入射相位圖中的第一個等相位頻率、(2) TM平面波掠射角入射相位圖中的第一個等相位頻率以及(3) TM平面波垂直入射相位圖中的第一個+90˚相位差所對應的頻率。 我們利用商用電磁模擬軟體CST，比較從色散圖(dispersion plot)以及用近似方法從反射相位圖得出的表面波帶隙之差異。從模擬結果可得知，在蕈狀結構中的金屬片寬與週期比在0.5到0.95之間，誤差能小於10%。另外，近似方法的概念也可以引入蕈狀結構的等效電路模型中，從中可推導出不需模擬與數值分析的近似公式解來求得表面波帶隙。藉由使用電路模型，在模型對結構參數限制之下，誤差可低於12%。 若這近似方法在設計上可以被採用的話，我們可以從模擬或模型計算出的反射相位圖當中，同時得到等相位頻帶(in-phase band)與表面波帶隙，如此可以加快計算兩頻帶的速度。除此之外，藉由電路模型所推導出的公式解可以運用到蕈狀結構的初始設計上，並提供一個可能的線索來確立蕈狀結構上等相位頻帶與表面波帶隙的關係。

In this thesis, we introduce an approximate method to find the surface-wave band gap from reflection phase plots for mushroom structures. The results show that the upper bound of the surface-wave band gap can be directly obtained by looking for the first in-phase frequency on the reflection phase plot of TE plane waves under grazing incidence, and the lower bound of the surface-wave band gap can be estimated by using the approximate formula introduced in this thesis with three specific frequencies on the reflection phase plots of TM plane waves. These three frequencies are (1) the first in-phase frequency under normal incidence, (2) the first in-phase frequency under grazing incidence, and (3) the frequency where the first +90˚ reflection phase occurs under normal incidence. This study compares the surface-wave band gaps from the dispersion plot and from the reflection phase plots through the approximate method in the commercial electromagnetic software CST. According to numerical results, if the ratio of the patch width and the period for mushroom structures is between 0.5 and 0.95, the errors can be smaller than 10%. Additionally, we also add the concepts of this approximate method to an analytical circuit model of mushroom structures and derive the approximate explicit formulas for finding surface-wave band gap without simulation and numerical analysis. By using the analytical circuit model, within the limitation of parameters of mushroom structures for the circuit model, the errors can be smaller than 12%. If this approximate method can be adopted for designs, we can find the in-phase band and the surface-wave band gap simultaneously on the reflection phase plots either in simulation or in analytical models, so we can find the information of the two bands faster. Furthermore, the formulas derived from the analytical circuit model could be used for initial designs of mushroom structures, and may give a possible key to solve the problem of the relationship between the in-phase band and the surface-wave band gap for mushroom structures.