Electromagnetic Metamaterials and their
Approximations:
Practical and Theoretical Aspects
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Existence and
Nonexistence Results for Invisibility Cloaking
Professor
Matti Lassas
Helsinki University of Technology

Abstract:
There has been considerable interest in the
possibility, both theoretical and practical, of
invisibility of objects to different types of waves.
We consider several examples of invisibility
cloaking by coating the enclosures with anisotropic
materials.
In particular we consider:
(A) Examples for the conductivity equation [1,2].
These examples are important in electrical impedance
tomography and are based on singular transformations
that push isotropic electromagnetic parameters
forward into singular, anisotropic ones.
The original motivation was to provide
counterexamples for Calderon's inverse problem.
Calderon's problem is the question whether the
boundary measurements map determines uniquely the
conductivity inside a body.
These counterexamples give theoretical instructions
how to cover an object so that it appears
``invisible' in zero frequency measurements.
(B) Cloaking constructions for Helmholtz and
Maxwell's equations, where the material parameters
are the same as in the examples in (A), see [3].
Physical implementations of such an invisibility
cloak using metamaterials was suggested in 2006 by
Pendry, Schurig, and Smith.
At the same time, other type of invisibility cloaks
based on metamaterials were suggested by Leonhardt.
As the electromagnetic parameters in examples (A)
and (B) are singular, we consider how the solutions
could be defined rigorously. In particular, we will
consider in detail the existence of the finite
energy solutions when cloaking a ball or an infinite
cylinder.
Interestingly, the mathematical existence and
nonexistence results suggest the use of linings at
the surface at which the material parameters become
singular, and analysis of approximate cloaking
constructions shows that such linings improve
greatly the behaviour of approximative invisibility
cloaks [4].
The results have been done in collaboration with
A. Greenleaf, Y. Kurylev and G. Uhlmann.
References:
[1] A. Greenleaf, M. Lassas, G. Uhlmann: On
nonuniqueness for Calderon's inverse problem,
Mathematical Research Letters 10 (2003), 685693.
[2] A. Greenleaf, M. Lassas, G. Uhlmann: Anisotropic
conductivities that cannot detected in Electrical
Impedance Tomography. Physiological Measurement, 24
(2003), 413420.
[3] A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann:
Fullwave invisibility of active devices at all
frequencies, Communications in Mathematical Physics
275 (2007), 749789.
[4] A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann:
Improvement of cylindrical cloaking with the SHS
lining. Optics Express 15 (2007), 1271712734. 
