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There is currently a great deal of interest in the invisibility cloaksrecently proposed by Pendry et al. that are based in the transformationapproach. They obtained their results using first order transformations. Inrecent papers Hendi et al. and Cai et al. considered invisibility cloaks withhigh order transformations. In this paper we study high order electromagneticinvisibility cloaks in transformation media obtained by high ordertransformations from general anisotropic media. We consider the case wherethere is a finite number of spherical cloaks located in different points inspace. We prove that for any incident plane wave, at any frequency, thescattered wave is identically zero. We also consider the scattering of finiteenergy 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 theincoming one. This proves that the invisibility cloaks can not be detected inany scattering experiment with electromagnetic waves in high ordertransformation media, and in particular in the first order transformation mediaof Pendry et al. We also prove that the high order invisibility cloaks, as wellas the first order ones, cloak passive and active devices. The cloaked objectscompletely decouple from the exterior. Actually, the cloaking outside isindependent of what is inside the cloaked objects. The electromagnetic wavesinside the cloaked objects can not leave the concealed regions and viceversa,the electromagnetic waves outside the cloaked objects can not go inside theconcealed regions. As we prove our results for media that are obtained bytransformation from general anisotropic materials, we prove that it is possibleto cloak objects inside general crystals.

A rigorous analysis of high order electromagnetic invisibility cloaks