A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index

TL;DR

This paper analyzes how a large number of small holes with negative imaginary surface impedances can create an acoustic metamaterial with negative refraction index or achieve cloaking, by modeling and approximating the scattered waves.

Contribution

It characterizes the effective medium generated by many small impedance holes and shows how to switch the refraction index sign or achieve cloaking through specific impedance choices.

Findings

Negative imaginary surface impedances can produce negative refraction index.

The effective medium can be designed to be cloaking or have negative refraction.

Explicit error estimates are provided for the approximation.

Abstract

We deal with the scattering of an acoustic medium modeled by an index of refraction $n$ varying in a bounded region $\Omega$ of $\mathbb{R}^3$ and equal to unity outside $\Omega$. This region is perforated with an extremely large number of small holes $D_m$'s of maximum radius $a$, $a<<1$, modeled by surface impedance functions. Precisely, we are in the regime described by the number of holes of the order $M:=O(a^{\beta-2})$, the minimum distance between the holes is $d\sim a^t$ and the surface impedance functions of the form $\lambda_m \sim \lambda_{m,0} a^{-\beta}$ with $\beta >0$ and $\lambda_{m,0}$ being constants and eventually complex numbers. Under some natural conditions on the parameters $\beta, t$ and $\lambda_{m,0}$, we characterize the equivalent medium generating, approximately, the same scattered waves as the original perforated acoustic medium. We give an explicit error estimate between the scattered waves generated by the perforated medium and the equivalent one respectively, as $a \rightarrow 0$. As applications of these results, we discuss the following findings: 1. If we choose negative valued imaginary surface impedance functions, attached to each surface of the holes, then the equivalent medium behaves as a passive acoustic medium only if it is an acoustic metamaterial with index of refraction $\tilde{n}(x)=-n(x),\; x \in \Omega$ and $\tilde{n}(x)=1,\; x \in \mathbb{R}^3\setminus{\overline{\Omega}}$. This means that, with this process, we can switch the sign of the index of the refraction from positive to negative values. 2. We can choose the surface impedance functions attached to each surface of the holes so that the equivalent index of refraction $\tilde{n}$ is $\tilde{n}(x)=1,\; x \in \mathbb{R}^3$. This means that the region $\Omega$ modeled by the original index of refraction $n$ is approximately cloaked.