Fast Large-Scale Electromagnetic Simulation of Doubly Periodic Structures in Layered Media

Abstract

This work focuses on the electromagnetic simulation of doubly periodic structures embedded in layered media, which can be commonly found in extreme ultraviolet (EUV) lithography, metasurfaces, and frequency selective surfaces. Such problems can be solved by rigorous numerical methods like finite-difference time-domain (FDTD) method and finite element method (FEM). However, FDTD and FEM are universal methods far from achieving the best efficiency for the target problem. To exploit the problem property and facilitate the problem solving of large size in low complexity, two approaches are proposed. The first approach, Calder\'{o}n preconditioned spectral-element spectral-integral (CP-SESI) method, is an improvement over the existing finite-element boundary-integral method. By introducing the Calder\'{o}n preconditioner, domain decomposition and the fast Fourier transform technique, the time and memory complexity of CP-SESI is reduced to O(N\textsuperscript{1.30}) and O(N\textsuperscript{1.07}), respectively. The second approach, based on modified U-Net, introduces two stages of problem solving: the training stage and the inference stage. In the training stage, accurate data generated by the rigorous CP-SESI solver is fed to the U-Net. In the inference stage, the U-Net can be applied to solve unseen problems in real time. Particularly, the EUV problem with mask size of 4 um by 4 um can be solved on a personal desktop within 5 min on CPU or 30 s on GPU. Besides, two types of equivalent boundary conditions to replace thin structures are developed and incorporated into the framework of CP-SESI. The first one, surface current boundary condition, has better accuracy for resistive materials. The second one, impedance transition boundary condition, is more accurate for conductive materials. The accuracy comparison between the above two boundary conditions are compared.