# Numerical modeling of time-modulated metasurfaces with a Discontinuous Galerkin method

M. Elsawy¹, R. Gelly¹, S. Lanteri¹,

¹Atlantis project-team, Université Côte d'Azur, Inria, CNRS, LJAD, Sophia Antipolis, France

The overarching goal of this work is to propose and develop a novel numerical methodology for the design of time-modulated metasurfaces. Here, we present a first step in this direction with the introduction of a Discontinuous Galerkin (DG) method for the solution of time-domain Maxwell's equations for time-varying materials.

## Introduction

*Metasurfaces* [1] are thin artificial materials composed of an array of light-scattering nanometric elements, called *nano-resonators*. They have been proven to be a flexible medium to manipulate the light properties for a given optical function and have been applied to the fields of imaging, polarimetry and holography. Metasurfaces have generally a fixed optical function. However in the recent years, the concept of *time-modulated* metasurfaces [2] emerged, which relies on external stimulation to leverage a time-varying optical response thus allowing novel physical effects and applications.

## Methodology

To numerically study time-modulated metasurfaces, we model their interaction with electromagnetic waves by solving Maxwell's equations in heterogeneous and time-varying media. Most of the works dedicated to the subject are based on the Finite-Difference Time-Domain (FDTD) method which is simple to implement, although it can lose accuracy when dealing with complex geometries. In our case, we use the Discontinuous Galerkin Time-Domain (DGTD) method [3] which is similar to the Finite Element (FE) method and allows to account for the discontinuities in the electromagnetic fields.

## Preliminary results

We first validated our time-domain solver using a manufactured solution (Fig. 1a). We are currently considering the extension of the proposed DGTD method to deal with time-modulated diffraction gratings (Fig. 1b) proposed by [4].

<figure><img src="image_1.png" /><figcaption>Fig. 1.: (a): Manufactured solution: time-modulated resonant cavity. (b): Time-modulated diffraction grating.</figcaption></figure>

## References

[1] N. Yu et al., Science, vol. 334, no. 6054, pp. 333–337, 2011.

[2] E. Mikheeva et al., ACS photonics, vol. 9, no. 5, pp. 1458–1482, 2022.

[3] J. Viquerat, PhD Thesis, 2015.

[4] S. Inampudi et al., Opt. Mater. Express, vol. 9, no. 1, p. 162, 2019.