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

Abstract

We deal with the scattering of an acoustic medium modeled by an index of refraction n varying in a bounded region Ω of double-struckR3 and equal to unity outside Ω. This region is perforated with an extremely large number of small holes Dm'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(aβ − 2), the minimum distance between the holes is d ∼ at, and the surface impedance functions of the form λm∼λm,0a−β with β > 0 and λm,0 being constants and eventually complex numbers. Under some natural conditions on the parameters β,t, and λm,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→0. As applications of these results, we discuss the following findings: 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 \upOmega $ñ(x)=−n(x),x∈Ω and $\tilde {n}(x)=1,\; x \in \mathbb {R}^{3}\setminus {\overline {\upOmega }}$ñ(x)=1,x∈R3∖Ω¯. This means that with this process, we can switch the sign of the index of the refraction from positive to negative values. We can choose the surface impedance functions attached to each surface of the holes so that the equivalent index of refraction $\tilde {n}$ñ is $\tilde {n}(x)=1,\; x \in \mathbb {R}^{3}$ñ(x)=1,x∈R3. This means that the region Ω modeled by the original index of refraction n is approximately cloaked. Copyright © 2015 John Wiley & Sons, Ltd.

Multiscale analysis of the acoustic scattering by many scatterers of impedance type

“…With this freedom of taking the surface impedance, we can generate a large class of indices of refraction with possible applications to the acoustic metamaterials. Details on this issue are reported in [1] and [2]. Let us emphasize that we provide those results with explicit error estimates.…”