F eb 2 00 8 A Rigorous Time - Domain Analysis of Full – Wave Electromagnetic Cloaking ( Invisibility ) ∗ †

There is currently a great deal of interest in the theoretical and practical possibility of cloaking objects from the observation by electromagnetic waves. The basic idea of these invisibility devices [8, 9, 13], [18] is to use anisotropic transformation media whose permittivity and permeability ε , μ, are obtained from the ones, ε 0 , μ λν 0 , of isotropic media, by singular transformations of coordinates. In this paper we study electromagnetic cloaking in the time-domain using the formalism of time-dependent scattering theory [23]. This formalism provides us with a rigorous method to analyze the propagation of electromagnetic wave packets with finite energy in transformation media. In particular, it allows us to settle in an unambiguous way the mathematical problems posed by the singularities of the inverse of the permittivity and the permeability of the transformation media on the boundary of the cloaked objects. Von Neumann’s theory of self-adjoint extensions of symmetric operators plays an important role on this issue. We write Maxwell’s equations in Schrödinger form with the electromagnetic propagator playing the role of the Hamiltonian. We prove that the electromagnetic propagator outside of the cloaked objects is essentially self-adjoint. This means that it has only one self-adjoint extension, AΩ, and that this self-adjoint extension generates the only possible unitary time evolution, with constant energy, for finite energy electromagnetic waves, propagating outside of the cloaked objects. PACS classification scheme 2006: 41.20.Jb, 02.30.Tb,02.30.Zz, 02.60.Lj. Research partially supported by CONACYT under Project P42553F. Fellow Sistema Nacional de Investigadores. 1 Moreover, AΩ is unitarily equivalent to the electromagnetic propagator in the medium ε 0 , μ λν 0 . Using this fact, and since the coordinate transformation is the identity outside of a ball, we prove that the scattering operator is the identity. This implies that for any incoming finite-energy electromagnetic wave packet the outgoing wave packet is precisely the same. In other words, it is not possible to detect the cloaked objects in any scattering experiment where a finite-energy wave packet is sent towards the cloaked objects, since the outgoing wave packet that is measured after interaction is the same as the incoming one. Our results give a rigorous proof that the construction of [8, 9, 13], [18] cloaks passive and active devices from observation by electromagnetic waves. Actually, the cloaking outside is independent of what is inside the cloaked objects. As is well known, self-adjoint extensions can be understood in terms of boundary conditions. Actually, for the electromagnetic fields in the domain of AΩ the component tangential to the exterior of the boundary of the cloaked objects of both, the electric and the magnetic field have to be zero. This boundary condition is self-adjoint in our case because the permittivity and the permeability are degenerate on the boundary of the cloaked objects. Furthermore, we prove cloaking for general anisotropic materials. In particular, our results prove that it is possible to cloak objects inside general crystals.