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Electromagnetic Metamaterials and their Approximations:
Practical and Theoretical Aspects

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Existence and Non-existence 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 non-existence 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.

[1] A. Greenleaf, M. Lassas, G. Uhlmann: On nonuniqueness for Calderon's inverse problem, Mathematical Research Letters 10 (2003), 685-693.
[2] A. Greenleaf, M. Lassas, G. Uhlmann: Anisotropic conductivities that cannot detected in Electrical Impedance Tomography. Physiological Measurement, 24 (2003), 413-420.
[3] A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann: Full-wave invisibility of active devices at all frequencies, Communications in Mathematical Physics 275 (2007), 749-789.
[4] A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann: Improvement of cylindrical cloaking with the SHS lining. Optics Express 15 (2007), 12717-12734.

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