Logarithmically complex rigorous Fourier space solution to the 1D grating diffraction problem

TL;DR

This paper introduces a tensor train decomposition-based method that significantly reduces computational complexity in solving large-scale 1D grating diffraction problems, enabling efficient and rigorous electromagnetic analysis.

Contribution

It adapts tensor train algorithms to achieve logarithmic complexity growth, making large multiscale diffraction problems computationally feasible.

Findings

Achieves logarithmic complexity in Fourier space solutions

Enables rigorous analysis of large multiscale gratings

Reduces computational resources for electromagnetic simulations

Abstract

The rigorous solution to the grating diffraction problem is a cornerstone step in many scientific fields and industrial applications ranging from the study of the fundamental properties of metasurfaces to the simulation of photolithography masks. Fourier space methods, such as the Fourier Modal Method, are established tools for the analysis of the electromagnetic properties of periodic structures, but are too computationally demanding to be directly applied to large and multiscale optical structures. This work focuses on pushing the limits of rigorous computations of periodic electromagnetic structures by adapting a powerful tensor compression technique called the Tensor Train decomposition. We have found that the millions and billions of numbers produced by standard discretization schemes are inherently excessive for storing the information about diffraction problems required for computations with a given accuracy, and we show how to adapt the TT algorithms to have a logarithmically growing amount of information to be sufficient for reliable rigorous solution of the Maxwell's equations on an example of large period multiscale 1D grating structures.