Fast Calculations of Bands of Vector Electromagnetic Waves in 3D Periodic Structures using Broadband Green’s Function-Multiple Scattering Theory

We have extended the Fast method of BBGF_MST to the vector 3D case of 3D periodic structures for band diagrams and band field calculations. In the BBGF-MST approach, the broadband transformation to vector spherical waves for 3D is derived using the Broadband Green's function, and the band eigenvalue problem is expressed in terms of the single scatterer T matrix. For the first five bands, we use six vector spherical waves to calculate the KKR matrix equation. The BBGF-MST method can calculate the band eigenvalues in vector 3D problems with only 0.309 seconds for the first 5 bands on a standard laptop. The method outperforms commercial software COMSOL which requires 34 secs that was run on a latest desktop/server for 110 times. Taking account that the desktop server is 10 times faster than the standard laptop, this means that the Fast MST- BBGF method is several hundred times faster than COMSOL. In band field calculation, we use an extended coefficients approach to use the low order eigenvector of 6 to extend to 30 coefficients without the need of recalculating the eigen solution. The extended coefficients approach gives accurate band field results.

我们将BBGF_MST快速方法推广至适用于能带图与能带场计算的三维周期结构的三维矢量情形。在BBGF-MST方法框架下，借助宽带格林函数（Broadband Green's function）推导得到三维矢量球面波的宽带变换，并以单散射体T矩阵（single scatterer T matrix）表述能带本征值问题。针对前五个能带，我们采用六个矢量球面波求解KKR矩阵方程（KKR matrix equation）。在标准笔记本电脑上，BBGF-MST方法仅需0.309秒即可完成前五个能带的三维矢量问题能带本征值计算。该方法的性能优于商用软件COMSOL：在最新台式机/服务器上运行COMSOL时，单次计算耗时34秒，本方法的计算速度是其110倍。考虑到台式机/服务器的运算速度是标准笔记本电脑的10倍，由此可知BBGF-MST快速方法的计算速度比COMSOL快数百倍。在能带场计算环节，我们采用拓展系数法：将6阶低阶本征向量拓展为30阶系数，无需重新求解本征解。该拓展系数法可得到精确的能带场计算结果。