The Fourier modal method for gratings with bi-anisotropic materials

We report an advanced formulation of the Fourier modal method developed for two-dimensionally periodic multilayered structures containing materials with non-zero macroscopic magneto-electric coefficients (also known as coefficients of chirality and bi-anisotropy) represented as arbitrary 3 by 3 tensors. We consider two numerical schemes for this formulation: with and without Lifeng Li factorization rules. For both schemes, we provide explicit expressions for the Fourier tensors of macroscopic material parameters and demonstrate that, in the absence of magneto-electric coupling, they reduce to conventional Li operators. We show that the scheme employing factorization rules facilitates improved convergence, even when the macroscopic chirality coefficient is large. The described formulation represents the fast and rigorous technique for theoretical studies of periodic structures with chiral, bi-anisotropic, or non-reciprocal materials in the widely used framework of the Fourier modal method.