Abstract. There is currently a great deal of interest in the invisibility cloaks recently proposed by Pendry et al that are based on the transformation approach. They obtained their results using first-order transformations. In recent papers, Hendi et al and Cai et al considered invisibility cloaks with high-order transformations. In this paper, we study high-order electromagnetic invisibility cloaks in transformation media obtained by high-order transformations from general anisotropic media. We consider the case where there is a finite number of spherical cloaks located in different points in space. We prove that for any incident plane wave, at any frequency, the scattered wave is identically zero. We also consider the scattering of finite-energy wave packets. We prove that the scattering matrix is the identity, i.e., that for any incoming wave packet the outgoing wave packet is the same as the incoming one. This proves that the invisibility cloaks cannot be detected in any scattering experiment with electromagnetic waves in high-order transformation media, and in particular in the first-order transformation media of Pendry et al. We also prove that the high-order invisibility cloaks, as well as the first-order ones, cloak passive and active devices. The cloaked objects completely decouple from the exterior. Actually, the cloaking outside is independent of what is inside the cloaked objects. The electromagnetic waves inside the cloaked objects cannot leave the concealed regions and vice versa, the electromagnetic waves outside the cloaked objects cannot go inside the concealed regions. As we prove our results for media that are obtained by transformation from general anisotropic materials, we prove that it is possible to cloak objects inside general crystals.

PACS numbers: 41.20.Jb, 02.30.Tb, 02.30.Zz, 02.60.Lj

Mathematics Subject Classification: 35L45, 35L50, 35L80, 35P25, 35Q60, 78A25, 78A45

^* Research partially supported by CONACYT under Project P42553-F.

Print publication: Issue 6 (15 February 2008)
Received 4 November 2007, in final form 20 December 2007
Published 29 January 2008