Approximate cloaking for the Helmholtz equation via transformation optics and consequences for perfect cloaking

TL;DR

This paper investigates approximate cloaking for the Helmholtz equation using transformation optics, analyzing its effectiveness, stability issues, and implications for achieving perfect invisibility in 2D and 3D spaces.

Contribution

It provides a detailed assessment of the cloaking scheme's invisibility degree, stability limitations, and conditions for near-perfect cloaking, extending understanding of transformation optics-based cloaking.

Findings

Quantifies the degree of invisibility achievable.

Identifies instability with respect to material parameters.

Derives properties necessary for perfect cloaking.

Abstract

In this paper, we study approximate cloaking of active devices for the Helmholtz equation in the whole space of dimension 2 or 3 using the scheme introduced by Kohn, Shen, Vogelius, and Weinstein in \cite{KohnShenVogeliusWeinstein}. More precisely, we assess the degree of invisibility, determine the limit of the field inside the cloaked and cloaking regions, and show that the scheme is unstable with respect to the material parameters. As a consequence, we obtain some feasible properties of "perfect" cloaking.