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.