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 \cite{glu1, glu2, le},\cite{pss1} is to use anisotropic {\it transformation media} whose permittivity and permeability $\var^{λν}, μ^{λν}$, are obtained from the ones, $\var_0^{λν}, μ^{λν}_0$, 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. This formalism 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 {\it transformation media} on the boundary of the cloaked objects. 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. Moreover, the unique self-adjoint extension is unitarily equivalent to the electromagnetic propagator in the medium $\var_0^{λν}, μ^{λν}_0$. Using this fact, and since the coordinate transformation is the identity outside of a ball, we prove that the scattering operator is the identity. Our results give a rigorous proof that the construction of \cite{glu1, glu2, le}, \cite{pss1} perfectly cloaks passive and active devices from observation by electromagnetic waves. Furthermore, we prove cloaking for general anisotropic materials. In particular, our results prove that it is possible to cloak objects inside general crystals.