Two dimensional invisibility cloaking via transformation optics

TL;DR

This paper explores two-dimensional invisibility cloaking using transformation optics, addressing the mathematical challenges of singular parameters and establishing the properties of finite energy solutions in weighted Sobolev spaces.

Contribution

It introduces novel characterizations of finite energy solutions for singular cloaking PDEs and reveals hidden boundary conditions at the cloaking interface.

Findings

Singular parameters in 2D cloaking are more complex than in 3D.

Finite energy solutions are characterized in weighted Sobolev spaces.

Hidden boundary conditions are identified at the cloaking interface.

Abstract

We investigate two-dimensional invisibility cloaking via transformation optics approach. The cloaking media possess much more singular parameters than those having been considered for three-dimensional cloaking in literature. Finite energy solutions for these cloaking devices are studied in appropriate weighted Sobolev spaces. We derive some crucial properties of the singularly weighted Sobolev spaces. The invisibility cloaking is then justified by decoupling the underlying singular PDEs into one problem in the cloaked region and the other one in the cloaking layer. We derive some completely novel characterizations of the finite energy solutions corresponding to the singular cloaking problems. Particularly, some `hidden' boundary conditions on the cloaking interface are shown for the first time. We present our study for a very general system of PDEs, where the Helmholtz equation underlying acoustic cloaking is included as a special case.