利用代理模型尋找最大能隙之光子晶體結構

Abstract

光子晶體的能帶結構，可用馬克思威方程式描述，是大個大型矩陣的特徵值問題。計算光子晶體的能帶結構相當耗時，找尋不同幾何形狀中，有最佳能隙的結構更是一件困難的工作。 在這篇文章中我們探討簡單立方結構，對固定的球、柱半徑，我們需要改變不同的入射光路徑找尋第五及第六特徵值間隙，再改變不同的球、柱半徑，找尋最大的特徵值差。這是個二層的最佳化問題，我們把能隙看成一個未知的函數，欲找尋其最大值，而找尋最大特徵值差又是個最佳化問題。傳統的方法如基因演算法，並不適合，此函數無法承受成千上萬次的取值。 我們利用Kriging 方法，利用少許取到的樣本模擬未知函數的行為，再使用期望進步法(expected improvement) 選取最可能發生最大值的位置，可在幾十次函數取值之下，得到不錯的結果。

Finding the best configuration of photonic crystals, which has maximum bandgapmidgap ratio is a time consuming process. The bandgap structures can be described as Maxwell’s equations, involving solving eigenvalue problems of large matrices. In this article, we focus on maximizing the bandgap structure of simple cubic photonic crystals. The problem is a two stage optimization problem. For one fixed sphere radius and cylindar radius configuration, we need to find 5th and the 6th eigenvalues along one specific path and the bandgap between them. Next, vary the radii, and find the overall maximum bandgap-midgap ratio. Traditional methods such as genetic algorithm does not work properly due to massive function evaluations. We use surrogate modelling method, in particular, the Kriging method to deal with this problem. We use just a few samples to build a model to mimic the behavior of the unknown bandgap function, and then using expected improvement method to choose points may have true maximum. In our experiments, we get impressive results within fifty radii selecting.