Showing papers by "Yaroslav Kurylev published in 2007"

Full-Wave Invisibility of Active Devices at All Frequencies

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.

Electromagnetic wormholes and virtual magnetic monopoles from metamaterials.

Abstract: We describe new configurations of electromagnetic (EM) material parameters, the electric permittivity $ϵ$ and magnetic permeability $\ensuremath{\mu}$, which allow one to construct devices that function as invisible tunnels These allow EM wave propagation between the regions at the two ends of a tunnel, but the tunnels themselves and the regions they enclose are not detectable to lateral EM observations Such devices act 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, invisible cables, and scopes for MRI-assisted surgery

Improvement of cylindrical cloaking with the SHS lining

Abstract: 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 epsilon and mu. 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.

Effectiveness and improvement of cylindrical cloaking with the SHS lining

Abstract: We analyze, both analytically and numerically, 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 tensors tend towards the singular ideal cloaking fields, so that the anisotropy ratio tends to infinity, the $D$ and $B$ fields blow up near the cloaking surface. 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.

Rigidity of broken geodesic flow and inverse problems

Abstract: Consider a broken geodesics $\alpha([0,l])$ on a compact Riemannian manifold $(M,g)$ with boundary of dimension $n\geq 3$. The broken geodesics are unions of two geodesics with the property that they have a common end point. Assume that for every broken geodesic $\alpha([0,l])$ starting at and ending to the boundary $\partial M$ we know the starting point and direction $(\alpha(0),\alpha'(0))$, the end point and direction $(\alpha(l),\alpha'(l))$, and the length $l$. We show that this data determines uniquely, up to an isometry, the manifold $(M,g)$. This result has applications in inverse problems on very heterogeneous media for situations where there are many scattering points in the medium, and arises in several applications including geophysics and medical imaging. As an example we consider the inverse problem for the radiative transfer equation (or the linear transport equation) with a non-constant wave speed. Assuming that the scattering kernel is everywhere positive, we show that the boundary measurements determine the wave speed inside the domain up to an isometry.

Stability of boundary distance representation and reconstruction of Riemannian manifolds

Abstract: A boundary distance representation of a Riemannian manifold with boundary (M, g, partial derivative M) is the set of functions {r(x) is an element of C(partial derivative M) : x is an element of M}, where r(x) are the distance functions to the boundary, r(x)(z) = d(x, z), z is an element of partial derivative M. In this paper we study the question whether this representation determines the Riemannian manifold in a stable way if this manifold satisfies some a priori geometric bounds. The answer is affermative, moreover, given a discrete set of approximate boundary distance functions, we construct a finite metric space that approximates the manifold ( M, g) in the Gromov-Hausdorff topology. In applications, the boundary distance representation appears in many inverse problems, where measurements are made on the boundary of the object under investigation. As an example, for the heat equation with an unknown heat conductivity the boundary measurements determine the boundary distance representation of the Riemannian metric which corresponds to this conductivity.

Inverse spectral problems on a closed manifold

Abstract: In this paper we consider two inverse problems on a closed connected Riemannian manifold $(M,g)$. The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that $M$ is divided by a hypersurface $\Sigma$ into two components and we know the eigenvalues $\lambda_j$ of the Laplace operator on $(M,g)$ and also the Cauchy data, on $\Sigma$, of the corresponding eigenfunctions $\phi_j$, i.e. $\phi_j|_{\Sigma},\partial_ u\phi_j|_{\Sigma}$, where $ u$ is the normal to $\Sigma$. We prove that these data determine $(M,g)$ uniquely, i.e. up to an isometry. In the second problem we are given much less data, namely, $\lambda_j$ and $\phi_j|_{\Sigma}$ only. However, if $\Sigma$ consists of at least two components, $\Sigma_1, \Sigma_2$, we are still able to determine $(M,g)$ assuming some conditions on $M$ and $\Sigma$. These conditions are formulated in terms of the spectra of the manifolds with boundary obtained by cutting $M$ along $\Sigma_i$, $i=1,2$, and are of a generic nature. We consider also some other inverse problems on $M$ related to the above with data which is easier to obtain from measurements than the spectral data described.

Inverse boundary spectral problem for Riemannian polyhedra

Abstract: We consider an admissible Riemannian polyhedron with piece-wise smooth boundary. The associated Laplace defines the boundary spectral data as the set of eigenvalues and restrictions to the boundary of the corresponding eigenfunctions. In this paper we prove that the boundary spectral data prescribed on an open subset of the polyhedron boundary determine the admissible Riemannian polyhedron uniquely.

Mathematics of Invisibility

Abstract: We will describe recent some of the recent theoretical progress on making objects invisible to electromagnetic waves based on singular transformations.

Electromagnetic wormholes via handlebody constructions

Abstract: Cloaking devices are prescriptions of electrostatic, optical or electromagnetic parameter fields (conductivity $\sigma(x)$, index of refraction $n(x)$, or electric permittivity $\epsilon(x)$ and magnetic permeability $\mu(x)$) which are piecewise smooth on $\mathbb R^3$ and singular on a hypersurface $\Sigma$, and such that objects in the region enclosed by $\Sigma$ are not detectable to external observation by waves. Here, we give related constructions of invisible tunnels, which allow electromagnetic waves to pass between possibly distant points, but with only the ends of the tunnels visible to electromagnetic imaging. Effectively, these change the topology of space with respect to solutions of Maxwell's equations, corresponding to attaching a handlebody to $\mathbb R^3$. The resulting devices thus function as electromagnetic wormholes.

Time reversal methods in unknown medium and inverse problems

Abstract: A novel method to solve inverse problems for the wave equation is introduced The method is a combination of the boundary control method and an iterative time reversal scheme, leading to adaptive imaging of coefficient functions of the wave equation using focusing waves in unknown medium The approach is computationally effective since the iteration lets the medium do most of the processing of the data The iterative time reversal scheme also gives an algorithm for constructing boundary controls for which the corresponding final values are as close as possible to the final values of a given wave in a part of the domain, and as close as possible to zero elsewhere The algorithm does not assume that the coefficients of the wave equation are known