CiteULike: norris's Lassas

Analysis of the PML equations in general convex geometry

Matti Lassas — 2008-03-04T16:25:34-00:00

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 131, No. 05. (2007), pp. 1183-1207.

In this work, we study a mesh termination scheme in acoustic scattering, known as the perfectly matched layer (PML) method. The main result of the paper is the following. Assume that the scatterer is contained in a bounded and strictly convex artificial domain. We surround this domain by a PML of constant thickness. On the peripheral boundary of this layer, a homogenous Dirichlet condition is imposed. We show in this paper that the resulting boundary-value problem for the scattered field is uniquely solvable for all wavenumbers and the solution within the artificial domain converges exponentially fast toward the full-space scattering solution when the layer thickness is increased. The proof is based on the idea of interpreting the PML medium as a complex stretching of the coordinates in R<sup><em>n</em></sup> and on the use of complexified layer potential techniques.

Electromagnetic wormholes and virtual magnetic monopoles

Allan Greenleaf — 2007-05-08T00:28:54-00:00

Comm. Math. Phys. (20 Mar 2008)

We describe new configurations of electromagnetic (EM) material parameters, the electric permittivity $ε$ and magnetic permeability $μ$, that allow one to construct from metamaterials objects that function as invisible tunnels. These allow EM wave propagation between two points, but the tunnels and the regions they enclose are not detectable to EM observations. Such devices function as wormholes with respect to Maxwell's equations and effectively change the topology of space vis-a-vis EM wave propagation. We suggest several applications, including devices behaving as virtual magnetic monopoles.

On nonuniqueness for Calderon's inverse problem

Allan Greenleaf — 2008-03-04T16:01:37-00:00

Math. Res. Lett., Vol. 10 (1 Jul 2003), pp. 685-693.

We construct anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity. These conductivities are singular close to a surface inside the body.

Anisotropic conductivities that cannot be detected by EIT.

A Greenleaf — 2008-03-04T15:59:22-00:00

Physiol Meas, Vol. 24, No. 2. (May 2003), pp. 413-419.

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.

Improvement of cylindrical cloaking with the SHS lining

A Greenleaf — 2008-02-11T17:27:28-00:00

Opt. Express, Vol. 15, No. 20. (1 October 2007), pp. 12717-12734.

We analyze the effectiveness of cloaking an infinite cylinder from observations by electromagnetic waves in three dimensions. We show that, as truncated approximations of the ideal permittivity and permeability material parameters tend towards the singular ideal cloaking values, the D and B fields blow up near the cloaking surface. Since the metamaterials used to implement cloaking are based on effective medium theory, the resulting large variation in D and B poses a challenge to the suitability of the field-averaged characterization of ε and μ. We also consider cloaking with and without the SHS (soft-and-hard surface) lining. We demonstrate numerically that cloaking is significantly improved by the SHS lining, with both the far field of the scattered wave significantly reduced and the blow up of D and B prevented.

Full-Wave Invisibility of Active Devices at All Frequencies

Allan Greenleaf — 2008-02-11T16:09:58-00:00

Communications in Mathematical Physics, Vol. 275, No. 3. (4 November 2007), pp. 749-789.

Abstract There has recently been considerable interest in the possibility, both theoretical and practical, of invisibility (or “cloaking”) from observation by electromagnetic (EM) waves. Here, we prove invisibility with respect to solutions of the Helmholtz and Maxwell’s equations, for several constructions of cloaking devices. The basic idea, as in the papers [GLU2, GLU3, Le, PSS1], is to use a singular transformation that pushes isotropic electromagnetic parameters forward into singular, anisotropic ones. We define the notion of finite energy solutions of the Helmholtz and Maxwell’s equations for such singular electromagnetic parameters, and study the behavior of the solutions on the entire domain, including the cloaked region and its boundary. We show that, neglecting dispersion, the construction of [GLU3, PSS1] cloaks passive objects, i.e., those without internal currents, at all frequencies k. Due to the singularity of the metric, one needs to work with weak solutions. Analyzing the behavior of such solutions inside the cloaked region, we show that, depending on the chosen construction, there appear new “hidden” boundary conditions at the surface separating the cloaked and uncloaked regions. We also consider the effect on invisibility of active devices inside the cloaked region, interpreted as collections of sources and sinks or internal currents. When these conditions are overdetermined, as happens for Maxwell’s equations, generic internal currents prevent the existence of finite energy solutions and invisibility is compromised. We give two basic constructions for cloaking a region D contained in a domain $$Ω⊂\mathbb R^n, n≥ 3$$ , from detection by measurements made at $$∂Ω$$ of Cauchy data of waves on Ω. These constructions, the single and double coatings, correspond to surrounding either just the outer boundary $$∂ D^+$$ of the cloaked region, or both $$∂ D^+$$ and $$∂ D^-$$ , with metamaterials whose EM material parameters (index of refraction or electric permittivity and magnetic permeability) are conformal to a singular Riemannian metric on Ω. For the single coating construction, invisibility holds for the Helmholtz equation, but fails for Maxwell’s equations with generic internal currents. However, invisibility can be restored by modifying the single coating construction, by either inserting a physical surface at $$∂ D^-$$ or using the double coating. When cloaking an infinite cylinder, invisibility results for Maxwell’s equations are valid if the coating material is lined on $$∂ D^-$$ with a surface satisfying the soft and hard surface (SHS) boundary condition, but in general not without such a lining, even for passive objects.

Comment on "Scattering Theory Derivation of a 3D Acoustic Cloaking Shell"

Allan Greenleaf — 2008-02-11T16:04:03-00:00

(21 Jan 2008)

In a recent Letter, Cummer et al. give a description of material parameters for acoustic wave propagation giving rise to a 3D spherical cloak, and verify the cloaking phenomenon on the level of scattering coefficients. A similar configuration has been given by Chen and Chan. In this Comment, we show that these theoretical constructions follow directly from our earlier work <a href="http://arxiv.org/abs/math/0611185">this http URL</a> on full wave analysis of cloaking for the Helmholtz equation with respect to Riemannian metrics. Furthermore, the analysis there covers the case of acoustically radiating objects being enclosed in the cloaked region.

Anisotropic conductivities that cannot be detected by EIT

A Greenleaf — 2008-01-25T02:27:39-00:00

Physiological Measurement, Vol. 24, No. 2. (2003), pp. 413-419.

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.