A Novel Structure-Preserving Algorithm with Green’s Function Approach for Computing Band Structures of Photonic Quasicrystals

Abstract

A novel bi-infinite approach to compute the band structures of 2D photonic superlattices with 1D quasicrystal sequences is devised. Leveraging strategically the bi-infinite characteristic, the approach first transforms the infinite-dimensional eigenvalue problem into a finite-dimensional nonlinear eigenvalue problem (NEVP) on a single cell for efficient numerical solution. Challengingly, the NEVP is built upon the solutions to two systems of cyclic nonlinear matrix equations (NMEs) that have to be solved repeatedly during iteratively solving the NEVP. The solutions are efficiently calculated by a newly developed highly efficient coalescing technique followed by a structure-preserving doubling algorithm. It is showed that the cost of coalescing is proportional to the logarithm of $N,$ the length of the truncated quasicrystal sequence, which is significant as the cost of coalescing becomes more noticeable as $N$ gets bigger for highly accurate simulations. Finally, through mathematical analysis, inclusion intervals for eigenvalue of interest are estimated so as to significantly narrow down the scope of search, and that significantly contributes to the overall efficiency of the approach, as the NEVP is nonlinear in nature and has to be solved iteratively.