Invisibility and Inverse Problems

We describe recent theoretical and experimental progress on making objects invisible. Ideas for devices that would have once seemed fanciful may now be at least approximately realized physically, using a new class of artificially structured materials, metamaterials. The equations that govern a variety of wave phenomena, including electrostatics, electromagnetism, acoustics and quantum mechanics, have transformation laws under changes of variables which allow one to design material parameters that steer waves around a hidden region, returning them to their original path on the far side. Not only are observers unaware of the contents of the hidden region, they are not even aware that something is being hidden; the object, which casts no shadow, is said to be cloaked. Proposals for, and even experimental implementations of, such cloaking devices have received the most attention, but other devices having striking effects on wave propagation, unseen in nature, are also possible. These designs are initially based ∗Department of Mathematics, University of Rochester, Rochester, NY 14627, USA. Partially supported by NSF grant DMS-0551894. †Department of Mathematics, University College London, Gower Street, London, WC1E 5BT, 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. 1 ar X iv :0 81 0. 02 63 v1 [ m at hph ] 1 O ct 2 00 8 on the transformation laws of the relevant PDEs, but due to the singular transformations needed for the desired effects, care needs to be taken in formulating and analyzing physically meaningful solutions. We recount the recent history of the subject and discuss some of the mathematical and physical issues involved.