Anisotropic conductivities that cannot be detected by EIT

Greenleaf, Allan; Lassas, Matti; Uhlmann, Günther

doi:10.1088/0967-3334/24/2/353

Cited by 347 publications

(386 citation statements)

References 9 publications

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“…Even though this map does not determine the coefficients (see, e.g. [10]), one may define cloaking as the fact that certain perturbations of the coefficients do not lead to relevant changes of the Dirichlet-to-Neumann map [9].…”

Section: Introductionmentioning

confidence: 99%

Cloaking of Small Objects by Anomalous Localized Resonance

Bouchitté

Schweizer

2010

The Quarterly Journal of Mechanics and Applied Mathematics

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We investigate operators L η u = ∇ · (a η ∇u) and solutions u η of L η u η = 0 to various boundary conditions. The coefficients a η are assumed to have a real part with changing sign and a small, non-negative imaginary part. We investigate a ring geometry with radii 1 and R in two space dimensions and use Fourier expansions in polar coordinates to analyze the qualitative behavior of solutions when boundary conditions on a small inclusion B ε (x 0 ) are imposed. Our result is that u η depends qualitatively on the position of the inclusion. If |x 0 | is larger than the cloaking radius R * := R 3/2 , then u η behaves as if no ring were present. If, instead, |x 0 | is smaller than R * , then the small inclusion is invisible in the limit η → 0.

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Section: Introductionmentioning

confidence: 99%

Cloaking of Small Objects by Anomalous Localized Resonance

Bouchitté

Schweizer

2010

The Quarterly Journal of Mechanics and Applied Mathematics

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“…This means that all boundary measurements for the homogeneous conductivity γ = 1 and the degenerated conductivity σ are the same. The result above was proven in [64,65] for the case of dimension n ≥ 3. The same basic construction works in the two dimensional case [112].…”

Section: Invisibility For Electrostaticsmentioning

confidence: 73%

“…Theoretical models for this have been found by applying diffeomorphisms having singularities. These were first introduced in the framework of electrostatics, yielding counterexamples to the anisotropic Calderón problem in the form of singular, anisotropic conductivities in R n , n ≥ 3, indistinguishable from a constant isotropic conductivity in that they have the same Dirichlet-to-Neumann map [64,65]. The same construction was rediscovered for electromagnetism in [153], with the intention of actually building such a device with appropriately designed metamaterials; a modified version of this was then experimentally demonstrated in [171].…”

Section: Invisibility For Electrostaticsmentioning

confidence: 99%

“…Roughly speaking, mapping the manifold M smoothly to the set M\B M (x 0 , ρ), where B M (x 0 , ρ) is a metric ball of M, and by putting an object in the obtained hole B M (x 0 , ρ), one could hide it from detection at the boundary. This observation was used in [64,65], where "undetectability" results were introduced in three dimensions, using degenerations of Riemannian metrics, whose singular limits can be considered as coming directly from singular changes of variables. The degeneration of the metric (see Fig.…”

Section: Invisibility For Electrostaticsmentioning

confidence: 99%

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Inverse problems: seeing the unseen

2014

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This survey article deals mainly with two inverse problems and the relation between them. The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography and also Calderón's problem since the famous analyst proposed it in the mathematical literature (Calderón in On an inverse boundary value problem. Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980), Soc Brasil Mat. Rio de Janeiro, pp. [65][66][67][68][69][70][71][72][73] 1980). The second is on travel time tomography. The question is whether one can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometry problem, the boundary rigidity problem. Can we determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points? These two inverse problems concern visibility, that is whether we can determine the internal properties of a medium by making measurements at the boundary. The last topic of this paper considers the opposite issue: invisibility: Can one make objects invisible to different types of waves, including light?Communicated by Ari Laptev.

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“…There exist various cloaking mechanisms, among which is the transformation optics approach [1][2][3][4][5][6][7][8][9][10][11][12][13]. The fundamental idea is the invariance of Maxwell's equations under a space-deforming transformation if the material properties are altered accordingly.…”

Section: Introductionmentioning

confidence: 99%

Polarization-Invariant Directional Cloaking by Transformation Optics

Agarwal¹,

Chen²,

Li³

et al. 2011

PIER

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Abstract-We propose a three-dimensional directional cloak for arbitrarily polarized incoming electromagnetic waves, motivated by the fact that there will be negligible scattering when the direction of impinging wave coincides with the axial direction of a very thin and elongated perfectly electric conducting (PEC) scatterer. The performance of the cloak under different polarizations of incoming waves are numerically investigated. The case in which the direction of incoming wave is perturbed off the ideal direction is also quantitatively studied. Numerical simulations show that the directional cloaking device is able to tolerate a large range of tilted angles of incoming waves.

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Anisotropic conductivities that cannot be detected by EIT

Abstract: We construct anisotropic conductivities in dimension 3 that give rise to the same voltage and current measurements at the boundary of a body as a homogeneous isotropic conductivity. These conductivities are non-zero, but degenerate close to a surface inside the body.

Cited by 347 publications

References 9 publications

Cloaking of Small Objects by Anomalous Localized Resonance

Cloaking of Small Objects by Anomalous Localized Resonance

Inverse problems: seeing the unseen

Polarization-Invariant Directional Cloaking by Transformation Optics

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