By adopting the scaling functions of the B-spline wavelet on the interval (BSWI) as interpolation functions, a wavelet-based boundary element method (BEM) model is constructed to compute the band structures of two-dimensional photonic crystals (2D PCs), which are composed of square or triangular lattices with arbitrarily shaped inclusions. The boundary integral equations of both the matrix and inclusion are developed in a unit cell because of the structural periodicity. In order to make the curve boundary be compatible well, geometric boundaries are interpolated by employing second order BSWI scaling functions while arbitrary order scaling functions are used to approximate boundary variables. For any given angular frequency, an effective technique is provided to generate matrix values related to the boundary shape. Moreover, singular integral problem involved in the presented wavelet-based BEM is considered. Then, combining the Bloch theorem and the interface conditions, a linear eigenvalue equation related to the Bloch wave vector is obtained. Numerical examples are given to verify the performance of the wavelet-based BEM developed herein compared with the conventional BEM.

通过采用B的缩放函数以区间上的样条小波 (BSWI) 作为插值函数，构建了基于小波的边界元法 (BEM) 模型来计算由方形或三角形晶格组成的二维光子晶体 (2D PC) 的能带结构带有任意形状的夹杂物。由于结构周期性，矩阵和夹杂物的边界积分方程都在一个晶胞中展开。为了使曲线边界具有良好的兼容性，几何边界采用二阶BSWI标度函数进行插值，同时使用任意阶标度函数来逼近边界变量。对于任何给定的角频率，提供了一种有效的技术来生成与边界形状相关的矩阵值。而且，考虑了所提出的基于小波的边界元法中涉及的奇异积分问题。然后，结合布洛赫定理和界面条件，得到与布洛赫波矢量相关的线性特征值方程。给出了数值例子来验证本文开发的基于小波的边界元与传统边界元方法相比的性能。