Realizability of metamaterials with prescribed electric permittivity and magnetic permeability tensors

TL;DR

This paper demonstrates that any pair of real symmetric tensors for electric permittivity and magnetic permeability can be realized as the effective properties of a carefully constructed metamaterial at a fixed frequency.

Contribution

It provides a method to construct metamaterials with prescribed effective tensors using hierarchical laminates of low-loss materials with specific eigenvalue properties.

Findings

Any symmetric tensor pair can be realized as effective properties.

Hierarchical laminate structures enable precise control of effective tensors.

Construction works at fixed frequency with sub-wavelength microstructures.

Abstract

We show that any pair of real symmetric tensors $\BGve$ and $\BGm$ can be realized as the effective electric permittivity and effective magnetic permeability of a metamaterial at a given fixed frequency. The construction starts with two extremely low loss metamaterials, with arbitrarily small microstructure, whose existence is ensured by the work of Bouchitt{\'e} and Bourel and Bouchitt\'e and Schweizer, one having at the given frequency a permittivity tensor with exactly one negative eigenvalue, and a positive permeability tensor, and the other having a positive permittivity tensor, and a permeability tensor having exactly one negative eigenvalue. To achieve the desired effective properties these materials are laminated together in a hierarchical multiple rank laminate structure, with widely separated length scales, and varying directions of lamination, but with the largest length scale still much shorter than the wavelengths and attenuation lengths in the macroscopic effective medium.