Full-wave invisibility of active devices at all frequencies

There has recently been considerable interest in the possibility, both theoretical and practical, of invisibility (or “cloaking”) from observation by electromagnetic (EM) waves. Here, we prove invisibility with respect to solutions of the Helmholtz and Maxwell’s equations, for several constructions of cloaking devices. The basic idea, as in the papers [GLU2, GLU3, Le, PSS1], is to use a singular transformation that pushes isotropic electromagnetic parameters forward into singular, anisotropic ones. We define the notion of finite energy solutions of the Helmholtz and Maxwell’s equations for such singular electromagnetic parameters, and study the behavior of the solutions on the entire domain, including the cloaked region and its boundary. We show that, neglecting dispersion, the construction of [GLU3, PSS1] Department of Mathematics, University of Rochester, Rochester, NY 14627, USA. Partially supported by NSF grant DMS-0551894. Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK Helsinki University of Technology, Institute of Mathematics, P.O.Box 1100, FIN02015, Finland. Partially supported by Academy of Finland CoE Project 213476. Department of Mathematics, University of Washington, Seattle, WA 98195, USA. Partially supported by the NSF and a Walker Family Endowed Professorship.