The Fourier modal method for gratings with bi-anisotropic materials

TL;DR

This work extends the Fourier modal method (FMM) to 2D periodic multilayer structures containing magneto-electric bi-anisotropic media described by 3×3 tensors ε, μ, χ, and ξ. It introduces two numerical schemes—one with naive Fourier representations and another leveraging generalized Li factorization rules—to improve convergence, deriving explicit Fourier representations for the material tensors and validating that the schemes converge to the conventional Li operators when cross-coupling vanishes. A convergence study demonstrates that the Li-based Scheme 2 markedly enhances accuracy, especially for large chirality, with convergence exponents up to 3 and quadratic scaling of compute time with harmonic number N_g. A numerical example shows chirality-induced shifts of guided resonances in a chiral metasurface, illustrating the practical impact for design of bi-anisotropic photonic devices. Overall, the paper provides a fast, rigorous framework for analyzing chiral, non-reciprocal, and bi-anisotropic periodic structures within the FMM, enabling robust exploration of advanced metamaterials and chiral polaritonics.

Abstract

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.

Figures (5)

• Figure 1: Scheme of a two-dimensional photonic crystal slab. The colors indicate different materials.
• Figure 2: One-dimensional grating for the demonstration of convergence. The colors indicate different materials.
• Figure 3: Convergence of the two numerical schemes to the analytical solution in a double logarithmic scale for different values of the refractive indices and chirality coefficients. The green and blue curves represent Scheme 1 (without factorization rules) and Scheme 2 (with factorization rules), respectively. $|\Delta k_z|$ denotes the difference between the $k_z$ value calculated by numerical Schemes 1 and 2 and the exact value, obtained analytically. The computation time for the main matrix $\mathbb{M}$ is shown by the blue and green dots for each number of harmonics. $\hbar\omega = 1320$ meV, $k_1 = k_2 = 0$. For calculations, we used a 14-core 12th Gen Intel(R) Core(TM) i7-12700H processor, with 16 GB of RAM.
• Figure 4: (a) Schematic illustration of the metasurface, comprising a periodic array of chiral elements on a substrate. The blue and grey colors represent materials with distinct dielectric permittivities $\varepsilon$ and chirality parameters $\chi$ (b) Absorption spectra calculated in $p$-polarization at $k_1=0.1$$\mu$m$^{-1}$. (c) and (d) left panels --- Photon energy and in-plane wavevector dependencies of the absorption coefficient calculated for the non-chiral case with $\chi=\xi^*=0$. (c) and (d) right panels --- The difference of the absorption spectra calculated for different signs of chiral coefficients. In (c) and (d) the positive and negative values of $k$ correspond to the $\Gamma-X$ and $\Gamma-M$ directions in reciprocal space. The calculation was performed by selecting 11 harmonics by $x$ direction and 11 harmonics by $y$ direction.
• Figure 5: Finding an exact value of $k_3$ in a slab with one-dimensional periodicity. Dashed lines denote boundaries of the considered unit cell.