The Boundary Conditions for Point Transformed Electromagnetic Invisibility Cloaks

TL;DR

This paper establishes the boundary conditions necessary for electromagnetic invisibility cloaks created by transformation media, ensuring energy conservation and effective cloaking of active and passive objects within anisotropic materials.

Contribution

It identifies specific boundary conditions for Maxwell's equations at cloak boundaries, applicable to various orders of cloaks and anisotropic media, advancing the theoretical understanding of electromagnetic invisibility.

Findings

Boundary conditions require vanishing tangential electric and magnetic fields at outer boundary.

Normal components of the curl of electric and magnetic fields must vanish inside the cloaked region.

Cloaking remains effective even with active devices at the boundary of the cloaked object.

Abstract

In this paper we study point transformed electromagnetic invisibility cloaks in transformation media that are obtained by transformation from general anisotropic media. We assume that there are several cloaks located in different points in space. Our results apply in particular to the first order invisibility cloaks introduced by Pendry et al. and to the high order invisibility cloaks introduced by Hendi et al. and by Cai et al.. We identify the appropriate {\it cloaking boundary conditions} that the solutions of Maxwell equations have to satisfy at the outside, $\partial K_+$, and at the inside, $\partial K_-$, of the boundary of the cloaked object $K$. Namely, that the tangential components of the electric and the magnetic fields have to vanish at $\partial K_+$ -what is always true- and that the normal components of the curl of the electric and the magnetic fields have to vanish at $\partial K_-$. These results are proven requiring that energy be conserved. In the case of one spherical cloak with a spherically stratified $K$ and a radial current at $\partial K$ we verify by an explicit calculation that our {\it cloaking boundary conditions} are satisfied and that cloaking of active devices holds even if the current is at the boundary of the cloaked object. As we prove our results for media that are obtained by transformation from general anisotropic media, our results apply to the cloaking of objects with active and passive devices contained in general anisotropic media, in particular to objects with active and passive devices contained inside general crystals.