An efficient numerical method for simulating two-dimensional non-periodic metasurfaces

TL;DR

The paper presents an NtD-based full-wave method for large 2D non-periodic metasurfaces that contain many unit cells but only a few distinct elements. By enclosing the structure with PMLs, decomposing it into unit cells, and using Neumann-to-Dirichlet operators together with local boundary expansions, it reduces the number of unknowns and yields a block-tridiagonal system for the interface normal derivative $\partial_x u$. NtD matrices are computed via FEM and small boundary expansions, then assembled into a compact linear system whose solution reconstructs the field in each unit cell and the far-field via a plane-wave expansion. Numerical results demonstrate second-order convergence and substantial gains in memory and compute time compared to full FEM, enabling simulations of $10^5$ subwavelength elements on a personal computer. The approach is well-suited for rapid metasurface design and optimization where large-scale full-wave simulations were previously prohibitive.

Abstract

Metasurfaces are extremely useful for controlling and manipulating electromagnetic waves. Full-wave numerical simulation is highly desired for their design and optimization, but it is notoriously difficult, even for two-dimensional metasurfaces, when they comprise a huge number of subwavelength elements. This paper focuses on two-dimensional non-periodic metasurfaces that contain only a relatively small number of distinct subwavelength elements. We develop an efficient numerical method based on Neumann-to-Dirichlet operators, the finite element method and local function expansions. Our method drastically reduces the total number of unknowns and is capable of simulating two-dimensional metasurfaces with $10^{5}$ subwavelength elements on a personal computer. Numerical examples demonstrate that the method maintains high accuracy while offering significant advantages in both computational time and memory usage compared to the classical full-domain finite element method, making it particularly suited for the analysis of large metasurfaces.

Figures (5)

• Figure 1: Sketch of a non-periodic 2D metasurface, which can be divided into $p-1$ unit cells.
• Figure 2: The sketches of the $j$-th cell and the left and right PML.
• Figure 3: A sketch of 2D phase gradient metasurface.
• Figure 4: Relative errors of the numerical solutions obtained by our method and FEM.
• Figure 5: Simulations for the phase gradient metasurface solved by our method (Left) and FEM (Right).