N ov 2 00 7 A Rigorous Analysis of High Order Electromagnetic

There is currently a great deal of interest in the invisibility cloaks recently proposed by Pendry et al. that are based in the transformation approach. They obtained their results using first order transformations. In a recent paper 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 can not 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 can not leave the concealed regions and viceversa, the electromagnetic waves outside the cloaked objects can not go inside the concealed regions. PACS classification scheme 2006: 41.20.Jb, 02.30.Tb,02.30.Zz, 02.60.Lj. AMS 2000 classification 35L45, 35L50, 35L80, 35P25, 35Q60, 78A25, 78A45. Research partially supported by CONACYT under Project P42553F. .On leave of absence from Departamento de Métodos Matemáticos y Numéricos. Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas. Universidad Nacional Autónoma de México. Apartado Postal 20-726, México DF 01000. Fellow Sistema Nacional de Investigadores. 1 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.