On optimal cloaking-by-mapping transformations

TL;DR

This paper investigates the design of cloaking transformations, focusing on minimizing anisotropy in meta-materials, and provides evidence that non-radial transformations do not reduce anisotropy for circular cloaks but may be beneficial for non-circular ones.

Contribution

It formulates the anisotropy minimization as an energy problem and explores the potential advantages of non-radial transformations for complex boundary shapes.

Findings

Radial transformations do not reduce anisotropy for circular cloaks.

Non-radial cloaks may reduce anisotropy for non-circular boundaries.

Energy minimization approach supports conjectures on optimal cloaking transformations.

Abstract

A central ingredient of cloaking-by-mapping is the diffeomorphisn which transforms an annulus with a small hole into an annulus with a finite size hole, while being the identity on the outer boundary of the annulus. The resulting meta-material is anisotropic, which makes it difficult to manufacture. The problem of minimizing anisotropy among radial transformations has been studied in [4]. In this work, as in [4], we formulate the problem of minimizing anisotropy as an energy minimization problem. Our main goal is to provide strong evidence for the conjecture that for cloaks with circular boundaries, non-radial transformations do not lead to lower degree of anisotropy. In the final section, we consider cloaks with non-circular boundaries and show that in this case, non-radial cloaks may be advantageous, when it comes to minimizing anisotropy.