# Beyond local effective material properties for metamaterials

## TL;DR

This paper investigates nonlocal effective material properties of metamaterials, emphasizing the importance of accounting for strong spatial dispersion to accurately describe their physical behavior and improve modeling accuracy.

## Contribution

It introduces a general formulation for nonlocal constitutive relations in metamaterials, extending beyond local responses to better match actual dispersion properties.

## Key findings

- Isofrequency surfaces match actual metamaterials better with nonlocal models
- Strong spatial dispersion significantly affects metamaterial behavior
- Nonlocal laws are necessary for accurate effective descriptions

## Abstract

To discuss the properties of metamaterials on physical grounds and to consider them in applications, effective material parameters are usually introduced and assigned to a given metamaterial. In most cases, only weak spatial dispersion is considered. It allows to assign local material properties, i.e. a permittivity and a permeability. However, this turned out to be insufficient. To solve this problem, we study here the effective properties of metamaterials with constitutive relations beyond a local response and take strong spatial dispersion into account. The isofrequency surfaces of the dispersion relation are investigated and compared to those of an actual metamaterial. The significant improvement provides evidence for the necessity to use nonlocal material laws in the effective description of metamaterials. The general formulation we choose here renders our approach applicable to a wide class of metamaterials.