The Boundary Conditions for Point Transformed Electromagnetic Invisibility Cloaks ∗ †

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 point transformed electromagnetic 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 cloaking boundary conditions that the solutions of Maxwell equations have to satisfy at the outside, ∂K+, and at the inside, ∂K−, of the boundary of the cloaked object K in the case where the permittivity and the permeability are bounded below and above in K. Namely, that the tangential components of the electric and the magnetic fields have to vanish at ∂K+ what is always trueand that the normal components of the curl of the electric and the magnetic fields have to vanish at ∂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 ∂K we verify by an explicit calculation that our 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 passive and active devices contained in general anisotropic media, in particular to objects with passive and active devices contained inside general crystals. 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. Fellow Sistema Nacional de Investigadores.