by SIAM. Unauthorized reproduction of this article is prohibited.
2009 Society for Industrial and Applied Mathematics
Vol. 51, No. 1, pp. 3–33
and Transformation Optics
We describe recent theoretical and experimental progress on making objects invisible to
detection by electromagnetic waves. Ideas for devices that would once have seemed fanciful
may now be at least approximately implemented physically using a new class of artiFcially
structured materials called
. Maxwell’s equations have transformation laws
that allow for the design of electromagnetic material parameters that steer light around a
hidden region, returning it to its original path on the far side. Not only would observers
be unaware of the contents of the hidden region, they would not even be aware that
something was being hidden. An object contained in the hidden region, which would have
no shadow, is said to be
. Proposals for, and even experimental implementations of,
such cloaking devices have received the most attention, but other designs having striking
e±ects on wave propagation are possible. All of these designs are initially based on the
transformation laws of the equations that govern wave propagation but, due to the singular
parameters that give rise to the desired e±ects, 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.
cloaking, transformation optics, electromagnetic wormholes, invisibility
AMS subject classiFcations.
78A40, 35P25, 35R30
Invisibility has been a subject of human fascination for millen-
nia, from the Greek legend of Perseus versus Medusa to the more recent
and the Harry Potter series. Over the years, there have been occasional sci-
entiFc prescriptions for invisibility in various settings, e.g., [46, 6]. However, since
Received by the editors ²ebruary 27, 2008; accepted for publication (in revised form) Septem-
ber 5, 2008; published electronically ²ebruary 5, 2009.
Department of Mathematics, University of Rochester, Rochester, NY 14627 (allan@math.
rochester.edu). The work of this author was partially supported by NS² grant DMS-0551894.
Department of Mathematics, University College London, Gower Street, London, WC1E 5BT,
The work of this author was partially supported by EPSRC grant
Helsinki University of Technology, Institute of Mathematics, P.O. Box 1100, ²IN-02015, ²inland
(Matti.Lassas@hut.F). The work of this author was partially supported by Academy of ²inland CoE