| Popis: |
In recent years twisted bilayers of two-dimensional (2D) materials have become very popular in the field due to the possibility to totally change their electronic properties by simple rotation. At the same time, in the wide field of photonic crystals, this idea still remains almost untouched, and only some particular problems have been considered. One of the reasons is the computational difficulty of accurate consideration of moir\'e superlattices that appear due to the superimposition of misaligned lattices. Indeed, the unit cell of the complex lattice is typically much larger than the original crystals and requires many more computational resources for the computations. Here, we propose a moir\'e-adapted Fourier modal method (MA-FMM) in the form of scattering matrices for the description of twisted one-dimensional (1D) gratings' stacks. We demonstrate that MA-FMM allows us to consider sublattices in close vicinity to each other and account for their interaction via the near field. In the developed numerical scheme, we utilize the fact that each sublattice is only 1D periodic and therefore simpler than the resulting 2D superlattice, as well as the fact that even a small gap between the lattices filters out high Fourier harmonics due to their evanescent origin. Such approach accelerates the computations from 1 up to 3 or more orders of magnitude for typical structures depending on the number of harmonics. In turn, the high computational speed paves the way for rigorous study of almost any photonic crystals of the proposed geometry and demonstration of specific moir\'e-associated effects. |
