# Analytical and Numerical Analysis of Linear and Nonlinear Properties of an rf-SQUID Based Metasurface

## TL;DR

This paper develops a comprehensive model for rf-SQUID based metasurfaces interacting with electromagnetic waves, deriving analytical solutions in the linear regime and numerically exploring nonlinear effects like bistability.

## Contribution

It introduces a coupled differential equation model for rf-SQUID metasurfaces, providing analytical solutions and numerical methods to study nonlinear electromagnetic interactions.

## Key findings

- Analytical expressions for reflection, transmission, and absorption in the linear regime.
- Numerical demonstration of bistability in nonlinear response.
- Convergence and error analysis of the numerical scheme.

## Abstract

We derive a model to describe the interaction of an rf-SQUID (radio frequency superconducting quantum interference device) based metasurface with free space electromagnetic waves. The electromagnetic fields are described on the base of Maxwell's equations. For the rf-SQUID metasurface we rely on an equivalent circuit model. After a detailed derivation, we show that the problem that is described by a system of coupled differential equations is wellposed and, therefore, has a unique solution. In the small amplitude limit, we provide analytical expressions for reflection, transmission, and absorption depending on the frequency. To investigate the nonlinear regime, we numerically solve the system of coupled differential equations using a finite element scheme with transparent boundary conditions and the Crank-Nicolson method. We also provide a rigorous error analysis that shows convergence of the scheme at the expected rates. The simulation results for the adiabatic increase of either the field's amplitude or its frequency show that the metasurface's response in the nonlinear interaction regime exhibits bistable behavior both in transmission and reflection.