Multilevel Fast Multipole Algorithm for Electromagnetic Scattering by Large Metasurfaces Using Static Mode Representation

Metasurfaces, consisting of large arrays of interacting sub-wavelength scatterers, pose significant challenges for general-purpose computational methods due to their large electrical size and multiscale nature. In this paper, leveraging the Poggio-Miller-Chang-Harrington-Wu-Tsai formulation, we combine the Multilevel Fast Multipole Algorithm (MLFMA) with a Static Mode Representation (SMR) of the unknown equivalent surface current density. These static modes are entire-domain basis functions that depend solely on the object’s shape and are independent of material properties and frequency (as introduced in IEEE Trans. Antennas Propag. 71 (8), 6779–6793, 2023). By compressing the number of unknowns using SMR and exploiting the efficient <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i>(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> log<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>) scaling of the MLFMA matrix-vector products, our MLFMA-SMR method substantially reduces CPU time and memory requirements compared to traditional MLFMA implementations using Rao-Wilton-Glisson basis functions. We assess the method’s accuracy and computational cost (time and memory) through several test cases, including the full-wave simulation of a 100λ × 100λ canonical metalens. Overall, the MLFMA-SMR method offers substantial benefits for the analysis and optimization of large-scale metasurfaces and metalenses.