Full and Partial Cloaking in Electromagnetic Scattering

Deng, Youjun; Liu, Hongyu; Uhlmann, Günther

doi:10.1007/s00205-016-1035-6

Cited by 35 publications

(21 citation statements)

References 37 publications

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“…We will construct isotropic matrices of the form (28) γ n (x) = γ(x, n|x|)I, x ∈ B 3 that H-converge to γ * , where, γ(x, y) is periodic in y with period 1. Recall that γ * can be computed from γ(x, y) using (10). From γ n , it is straightforward to construct isotropic electric permittivities and magnetic permeabilities for the approximate cloak.…”

Section: Construction Of the Approximate Cloakmentioning

confidence: 99%

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Approximate isotropic cloak for the Maxwell equations

Ghosh

Tarikere

2018

Journal of Mathematical Physics

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We construct a regular isotropic approximate cloak for the Maxwell system of equations. The method of transformation optics has enabled the design of electromagnetic parameters that cloak a region from external observation. However, these constructions are singular and anisotropic, making practical implementation difficult. Thus, regular approximations to these cloaks have been constructed that cloak a given region to any desired degree of accuracy. In this paper, we show how to construct isotropic approximations to these regularized cloaks using homogenization techniques, so that one obtains cloaking of arbitrary accuracy with regular and isotropic parameters.Though the regularized approximate cloak is nonsingular, its material parameters are still anisotropic, which poses another difficulty in practical implementation. Therefore, it would be useful if, as a second step these regularized approximate cloaks were further approximated by regular isotropic cloaks. It is a well-known phenomenon in homogenization theory [1,45,7] that homogenization of isotropic material parameters can lead, in the small scale limit, to anisotropic parameters.This approach, called isotropic transformation optics was introduced and performed in the case of electrostatic, acoustic and quantum cloaking by Greenleaf, Kurylev, Lassas and Uhlmann in [17] and [19]. It was shown in [19] that the for a fixed Dirichlet boundary value, the Neumann data for the approximate cloaking construction converge strongly. Moreover, Faraco, Kurylev and Ruiz [11] showed that the same construction in fact gives us strong convergence of the Dirichlet-to-Neumann maps in the operator norm. More recently, it has been implemented in a quasilinear model [12].In this paper, we will construct an approximate isotropic cloak for Maxwell's equations using similar homogenization techniques. We begin by summarizing the main results of the existence-uniqueness and cloaking theory for the Maxwell equations in Section 2. In Section 3, we present a brief overview of homogenization and present a proof of the homogenization result for the Maxwell equations, under the assumption that both ε(x) and µ(x) have positive imaginary parts. Section 4 will be devoted to an explicit construction of an approximate isotropic cloak using inverse homogenization techniques as in [17]. Since the magnetic permeability of the anisotropic approximate cloak does not have an imaginary part, we need to go through a two step process. In the first step, we alter permittivities and permeabilities by a small positive parameter δ so that the assumptions of Section 3 are satisfied, and then construct isotropic parameters ε n δ and µ n that homogenize to the altered paremetrs. In the second step, we let δ → 0. Finally in Section 5, we will prove our Main Theorem: that as we first let n → ∞ and then δ → 0, the electromagnetic boundary measurements corresponding to our approximate cloaking construction converge strongly to the boundary measurements in empty space. The order in which these limits are taken can...

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Section: Construction Of the Approximate Cloakmentioning

confidence: 99%

“…Also see [37] for a broad survey of approximate cloaking. In [10], the authors generalize this approximate cloaking construction to more general generating sets, such as curves and planar surfaces.…”

Section: Introductionmentioning

confidence: 99%

Approximate isotropic cloak for the Maxwell equations

Ghosh

Tarikere

2018

Journal of Mathematical Physics

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“…This counterexample was obtained using a blow-up map. Analogous non-uniqueness results have been studied in the invisibility cloaking, where an arbitrary object is hidden from measurements by coating it with a material that corresponds to a degenerate Riemannian metric [GLU03,GKLU07,GKLU09a,GKLU09b,DLU17].…”

Section: Introductionmentioning

confidence: 92%

The Poisson embedding approach to the Calderón problem

2019

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We introduce a new approach to the anisotropic Calderón problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calderón type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result of [LU02] solving the Calderón problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations. fn (J(x))} linearly independent. Write V = (u 1 f 1 , . . . , u 2 fn ). By the formula (8) we have V = U • J in B. Now U is bijective in some neighborhood Ω ⊂ M 2 of J(x), and since J is continuous there is a neighborhood W of x with J(W ) ⊂ Ω. Thus we haveSince the harmonic functions u j f k , j = 1, 2, are smooth, the smoothness of J near x follows.

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“…On the other hand, there are many researchers have considered another methods, called the small-inclusion-blowup construction, which regularized the singular medium which was induced by the one-point-blowup construction for the perfect cloaking. The small-inclusion-blowup method was studied by many researchers: [5,26] for the conductivity model, [1,4,11,25,28,29,30,31] for the acoustic model, [6,7,8,12] for the Maxwell model and [23] for the Lamé model. In [24], the authors also pointed out that the truncation of singularity construction and the small-inclusion construction are equivalent.…”

Section: Introductionmentioning

confidence: 99%

Nearly cloaking for the elasticity system with residual stress

Lin

2017

Asymptotic Analysis

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The nearly cloaking via transformation optics approach for the isotropic elastic wave fields is considered. This work extends the study of the nearly cloaking scheme to the elasticity system with residual stress in R N for N = 2, 3, which is an anisotropic elasticity system. It is worth to mention that there are no minor symmetric properties for the elastic tensor with residual stress and this system is invariant under a coordinate transformation. Therefore, we think the elasticity system residual stress model is more natural than the isotropic elasticity system in designing the elastic cloaking medium in the physical sense. In addition, the main difficulty of treating this problem lies in the fact that there are no layer potential theory for the residual stress system. Instead, we will derive suitable elliptic estimates for the elasticity system residual stress by comparing with the Lamé system to achieve desired results.

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Full and Partial Cloaking in Electromagnetic Scattering

Cited by 35 publications

References 37 publications

Approximate isotropic cloak for the Maxwell equations

Approximate isotropic cloak for the Maxwell equations

The Poisson embedding approach to the Calderón problem

Nearly cloaking for the elasticity system with residual stress

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