Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking

# Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking

Jingzhi Li Faculty of Science, South University of Science and Technology of China, 518055, Shenzhen, P. R. China (lijz@sustc.edu.cn, lixf@sustc.edu.cn).    Xiaofei Li22footnotemark: 2    Hongyu Liu Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR (hongyu.liuip@gmail.com, jadelightking@qq.com).    Yuliang Wang33footnotemark: 3
###### Abstract

This paper is concerned with the invisibility cloaking in electromagnetic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. Our study is based on an interior transmission eigenvalue problem. We propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are both regular and isotropic. We establish that there exists an infinite set of incident waves such that the cloaking device is nearly-invisible under the corresponding wave interrogation. The set of waves is generated from the Maxwell-Herglotz approximation of the associated interior transmission eigenfunctions. We provide the mathematical design of the cloaking device and sharply quantify the cloaking performance.

Key words. electromagnetic scattering, invisibility cloaking, interior transmission eigenvalues

## 1 Introduction

Invisibility cloaking has received significant attentions in recent years in the scientific community due to its practical importance; see [1, 2, 3, 4, 5, 6, 13, 14, 15, 18, 19, 21] and the references therein for the relevant mathematical literature. The crucial idea is to coat a target object with a layer of artificially engineered material with desired properties so that the electromagnetic waves pass through the device without creating any shadow at the other end; namely, invisibility cloaking is achieved. Invisibility cloaking could find striking applications in many areas of science and technology such as radar and sonar, medical imaging, earthquake science and, energy science and engineering, to name just a few.

Generally speaking, a region of space is said to be cloaked if its contents, together with the cloak, are invisible to a particular class of wave measurements. In the literature, most of the existing works are concerned with the design of certain artificial mechanisms of controlling wave propagation so that invisibility is achieved independent of the source of the detecting waves; that is, for any generic wave fields that one uses to impinge on the cloaking device, there will be invisibility effect produced. In this paper, we shall develop a novel cloaking scheme where the invisibility is only achieved with respect to detecting waves from a particular set. In doing so, one can achieve the invisibility cloaking by completely regular and isotropic mediums. Next, we first present the mathematical setup and then discuss the main results of the current study.

Consider the time-harmonic electromagnetic (EM) wave scattering in a homogeneous space with the presence of an inhomogeneous scatterer. Let us first characterize the optical properties of an EM medium with the electric permittivity , magnetic permeability , and electric conductivity . We recall that is the space of real-valued symmetric matrices and that, for any Lipschitz domain , we say that is a tensor in satisfying the uniform ellipticity condition if and there exists such that

 c0|ξ|2≤γ(x)ξ⋅ξ≤c−10|ξ|2 for a.e. x∈Ω and every ξ∈R3.

shall be referred to as the ellipticity constant of the tensor . It is assumed that both and , , belong to , and are uniform elliptic with constant ; whereas it is also assumed that satisfies

 0≤σ(x)ξ⋅ξ≤λ0|ξ|2 for a.e. x∈R3 and every ξ∈R3,

where . Denote the medium associated with , and it is said to be regular if the material parameters fulfill the conditions described above. Moreover, , , is said to be isotropic if there exists such that , where signifies the identity matrix. Suppose is located in an isotropic and homogeneous background/matrix medium whose material parameters are given by

 ϵ(x)=I3×3, μ(x)=I3×3, σ(x)=0, for x∈R3∖¯Ω.

Let denote an EM wavenumber, corresponding to a certain EM spectrum. Consider the EM radiation in this frequency regime in the space

 (R3;ϵ,μ,σ)=(Ω;ϵ,μ,σ)∧(R3∖¯Ω;I3×3,I3×3,0).

Let be a pair of entire electric and magnetic fields, modeling the illumination source. They verify the time-harmonic Maxwell equations,

 curlEi−iωHi=0,  curlHi+iωEi=0  in R3. (1)

The presence of the inhomogeneous scatterer interrupts the propagation of the EM waves and , leading to the so-called wave scattering. We let and denote, respectively, the scattered electric and magnetic fields. Define

 E:=Ei+Es,  H:=Hi+Hs,

to be the total electric and magnetic fields, respectively. Then the EM scattering is governed by the following Maxwell system

 ⎧⎪⎨⎪⎩curlE(x)−iωμ(x)H(x)=0,x∈R3,curlH(x)+iωϵ(x)E(x)=σ(x)E(x),x∈R3,lim|x|→+∞(μ1/2(x)Hs(x)×x−|x|ϵ1/2(x)Es(x))=0. (2)

The last limit in (2) is known as the Silver-Mller radiation condition. The Maxwell system (2) is well-posed and there exists a unique pair of solutions . Here and also in what follows, for any open set , we make use of the following Sobolev spaces:

 H(curl,Ω):={u∈L2(Ω)3|curlu∈L2(Ω)3},H0(curl,Ω):={u∈H(% curl,Ω):ν×u=0, ν×curlu=0 on ∂Ω},H2(curl,Ω):={u∈H(% curl,Ω):curlu∈H(curl,Ω)},H20(curl,Ω):={u∈H0(curl,Ω):curlu∈H(curl,Ω)}, (3)

endowed with the scalar product

 (u,v)H(curl,Ω) =(u,v)L2(Ω)+(curlu,curlv)L2(Ω), (u,v)H2(curl,Ω) =(u,v)H(curl,Ω)+(curlu,curlv)H(curl,Ω),

and the corresponding norms and . Moreover, define

 TH−1/2Div(∂Ω):={U∈TH−1/2(∂Ω):DivU∈H−1/2(∂Ω)},

where Div is the surface divergence operator on , is the subspace of all those which are orthogonal to , and is the usual -based Sobolev space of order .

For the solutions to (2), we have that as [11, 23]:

 Es(x)=eiω|x||x|E∞(^x)+O(1|x|2),
 Hs(x)=eiω|x||x|H∞(^x)+O(1|x|2),

where , . and are, respectively, referred to as the electric and magnetic far-field patterns, and they satisfy

 H∞(^x)=^x×E∞(^x)  and  ^x⋅E∞(^x)=^x⋅H∞(^x)=0,  ∀^x∈S2.

The medium is said to be invisible under the electromagnetic wave interrogation by if

 E∞(^x,(Ei,Hi),(Ω;ϵ,μ,σ))≡0 and H∞(^x,(Ei,Hi),(Ω;ϵ,μ,σ))≡0,  ∀^x∈S2. (4)

In the current work, we shall consider the cloaking technique in achieving the invisibility. Let be a bounded Lipschitz domain. Consider a cloaking device of the following form

 (R3;ϵ,μ,σ)=(D;ϵc,μc,σc)∧(Ω∖¯D;ϵm,μm,σm)∧(R3∖¯Ω;I3×3,I3×3,0), (5)

where denotes the target object being cloaked, and denotes the cloaking shell medium (see Figure 1).

For a practical cloaking device of the form (5), there are several crucial ingredients that one should incorporate into the design:

• The target object can be allowed to be arbitrary (but regular). That is, the cloaking device should not be object-dependent. In what follows, this issue shall be referred to as the target independence for a cloaking device.

• The cloaking medium should be feasible for construction and fabrication. Indeed, it would be the most practically feasible if is uniformly elliptic with fixed constants and isotropic as well. In what follows, this issue shall be referred to as the practical feasibility for a cloaking device.

• For an ideal cloaking device, one can expect the invisibility performance (4). However, in practice, especially in order to fulfill the above two requirements, one can relax the ideal cloaking requirement (4) to be

 |E∞(^x,(Ei,Hi),(Ω;ϵ,μ,σ))|≪1  and  |H∞(^x,(Ei,Hi),(Ω;ϵ,μ,σ))|≪1,

and , where is a set of incident waves consisting of entire solutions to the Maxwell equation (1). That is, near-invisibility can be achieved for scattering measurements made with interrogating waves from the set . In what follows, this issue shall be referred to as the relaxation and approximation for a cloaking device.

In this paper, we shall develop a cloaking scheme that addresses all of the issues discussed above. Our study connects to a so-called interior transmission eigenvalue problem associated with the Maxwell system as follows,

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩curlEm(x)−iωμmHm(x)=0 in Ω∖¯D,curlHm(x)+iωϵmEm(x)=0 % in Ω∖¯D,curlE0(x)−iωH0(x)=0 in Ω,curlH0(x)+iωE0(x)=0 in Ω,ν×Em=0 on ∂D,ν×Em=ν×E0,  ν×Hm=ν×H0 on ∂Ω. (6)

Concerning the problem (6), some remarks are in order. The interior transmission eigenvalue problem is associated with an isotropic EM medium that contains a PEC (perfectly electric conducting) obstacle . A similar interior transmission eigenvalue problem arising from the acoustic scattering governed by the Helmholtz system associated with an isotropic acoustic medium containing an impenetrable obstacle was considered in [7]. The corresponding result was applied to the invisibility cloaking study for acoustic waves in [19]. In the current article, we shall extend those studies to the much more technical and complicated Maxwell system governing the electromagnetic scattering. We shall first prove the discreteness and existence of the interior transmission eigenvalues of the system (6). To our best knowledge, those results are new to literature on the study of interior transmission eigenvalue problems. Then we shall apply the obtained results to the invisibility cloaking study. The first result we can show concerning the invisibility cloaking is as follows.

###### Proposition 1.1.

Consider the EM configuration in (5) with and replaced to be a PEC obstacle. Let be an interior transmission eigenvalue associated with , and , be a corresponding pairs of eigenfunctions of (6). For any sufficiently small , by the denseness property of Maxwell-Herglotz functions (cf. (44)), there exists such that

 ∥Egω−E0∥H(curl,Ω)<ε,  ∥Hgω−H0∥H(curl,Ω)<ε. (7)

Consider the scattering problem (2) by taking the incident electric and magnetic wave fields

 Ei=Egω,  Hi=Hgω,

then there hold

 |E∞(^x,(Egω,Hgω),(Ω∖¯D;ϵm,μm,0))|≤Cε,  ∀^x∈S2, (8)

and

 |H∞(^x,(Egω,Hgω),(Ω∖¯D;ϵm,μm,0))|≤Cε,  ∀^x∈S2, (9)

where is a positive constant independent of and .

Proposition 1.1 is kind of a folk-telling result in the literature on the study of interior transmission eigenvalues, and it shall be needed in our cloaking study.

Motivated by the two-layer cloaking device we then consider a three-layer cloaking device. Let be bounded Lipschitz domain such that is connected and consider an EM medium configuration as follows (see Figure 2),

 (R3;ϵ,μ,σ)=(Σ;ϵa,μa,σa)∧(D∖¯Σ;ϵl,μl,σl)∧(Ω∖¯D;ϵm,μm,0)∧(R3∖¯Ω;I3×3,I3×3,0), (10)

where is isotropic and the lossy layer is chosen to be

 ϵl=α1τ−1⋅I3×3,  μl=α2τ⋅I3×3,  σl=α3τ−1⋅I3×3,

where is an asymptotically small parameter, and are constants in . The target medium in the cloaked region can be arbitrary but regular.

For such a cloaking construction, our main theorem is as follows:

###### Theorem 1.2.

Let be described in (10). Let and be the same as those in Proposition 1.1. Consider the scattering system (2) corresponding to with . Then we have

 |E∞(^x,(Egω,Hgω),(D∖¯Σ;ϵl,μl,σl)∧(Ω∖¯D;ϵm,μm,0))|≤C(ε+τ1/2∥Egω∥H(curl,Ω)+τ1/2∥Hgω∥H(curl,Ω)), (11)
 |H∞(^x,(Egω,Hgω),(D∖¯Σ;ϵl,μl,σl)∧(Ω∖¯D;ϵm,μm,0))|≤C(ε+τ1/2∥Egω∥H(curl,Ω)+τ1/2∥Hgω∥H(curl,Ω)), (12)

where is positive constant independent of and .

By Theorem 1.2, the cloaking layer makes an arbitrary object located in the cloaked region nearly invisible to the wave interrogation by .

The rest of the paper is organized as follows. In section 2, we introduce the interior transmission eigenvalue problem for inhomogeneous media containing obstacles, and derive its variational form. In section 3, we investigate the spectral property of the interior transmission eigenvalue problem. We prove the discreteness and the existence of the interior transmission eigenvalues. In section 4, we introduce the Maxwell-Herglotz approximation. In section 5, we consider the isotropic invisibility cloaking, and establish the near-invisibility results. This paper is ended with a short discussion.

## 2 Interior transmission eigenvalue problem for inhomogeneous media containing obstacles

### 2.1 Physical background of the interior transmission eigenvalue problem

Let us consider the EM configuration (5). We assume that and first consider the case that is replaced by a PEC obstacle. Then the scattering problem is described by the following Maxwell system for

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩curlE(x)−iωμmH(x)=0 % in Ω∖¯D,curlH(x)+iωϵmE(x)=0 in Ω∖¯D,curlE(x)−iωH(x)=0 in R3∖¯Ω,curlH(x)+iωE(x)=0 in R3∖¯Ω,ν×E=0 on ∂D,E−Ei,H−Hi satisfy the Silver-M% ¨uller radiation condition, (13)

where is the unit outward normal vector to boundary . By straightforward manipulations, we can also write the Maxwell system (13) in terms of as follows,

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩curlcurlE(x)−ω2n(x)E(x)=0 in Ω∖¯D,curlcurlE(x)−ω2E(x)=0 in R3∖¯Ω,ν×E(x)=0 on ∂D,E−Ei satisfies the Silver-M¨uller % radiation condition, (14)

where . Let and signify the corresponding scattered wave field and the far-field pattern, respectively. If perfect invisibility is achieved for the scattering system (14), namely, then one has that

 Es(x)=0  for x∈R3∖¯Ω. (15)

By using the standard transmission conditions across for the solution and , one has

 ν×E|−=ν×E|+  and  ν×H|−=ν×H|+  on ∂Ω, (16)

where

 ν×E|±(x):=limh→+0ν×E(x±hν(x)),  x∈∂Ω,
 ν×H|±(x):=limh→+0ν×H(x±hν(x)),  x∈∂Ω.

The second condition in (16) can also be written as

 ν×μ−1mcurlE|−=ν×curlE|+  on ∂Ω.

We assume that the the magnetic permeability in . The above condition becomes

 ν×curlE|−=ν×curlE|+  on ∂Ω.

Applying (15) to (16) and by setting for and for , one can readily show that if perfect invisibility is achieved, then the following interior transmission eigenvalue problem

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩curlcurlEm−ω2n(x)Em=0 in Ω∖¯D,curlcurlE0−ω2E0=0 in Ω,ν×Em=0 on ∂D,ν×Em=ν×E0,  ν×curlEm=ν×curlE0 on ∂Ω (17)

has nontrivial solutions, where . Note that the interior transmission eigenvalue problem (17) is non-self adjoint. The equivalent interior transmission eigenvalue system of (17) for the pair is (6).

### 2.2 Variational formulation of the interior transmission eigenvalue problem

In this subsection we are going to derive the variational form of the interior transmission problem (17).

Let in . By the first and second identities in (17) we have that satisfies

 curlcurl~E−ω2n~E=ω2(n−1)E0 in Ω∖¯D. (18)

We also get the boundary conditions

 ν×~E=0,  ν×curl~E=0 on ∂Ω,

and

 ν×~E=−ν×E0 on ∂D.

Then the interior transmission problem (17) can be reformulated in terms of and as follows:

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩curlcurl~E−ω2n~E=ω2(n−1)E0 in Ω∖¯D,curlcurlE0−ω2E0=0 in Ω,ν×~E=−ν×E0 on ∂D,ν×~E=0, ν×curl~E=0 on ∂Ω,

With continuity of the data and across , we have from (18) that

 ν×(1ω2(n−1)−1(curl curl−ω2n)~E)∣∣+=ν×E0|−  % on ∂D, (19)

and

 ν×curl(1ω2(n−1)−1(curl curl−ω2n)~E)∣∣+=ν×curlE0|−  on ∂D. (20)

Multiplying both sides of (18) by and applying operator on both sides, we get a fourth order equation for in :

 (curl curl−ω2)(n−1)−1(curl curl−ω2n)~E=0 in Ω∖¯D.

Note that is only defined in . We define

 E={~Ein Ω∖¯D,−E0in D.

Then , where is defined by

 W:={E∈H0(curl,Ω):curlcurlE|Ω∖¯D∈L2(Ω∖¯D)},

which is equipped with the norm

 ∥E∥2W=∥E∥2H(curl,Ω)+∥curlcurlE∥2L2(Ω∖¯D). (21)

Now we are able to formulate a fourth order system of as follows,

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(curl curl−ω2)(n−1)−1(curl curl−ω2n)E=0 in Ω∖¯D,curlcurlE−ω2E=0 in D,ν×E|+=ν×E|− on ∂D,ν×(1ω2(n−1)−1(curl curl−ω2n)E)∣∣+=−ν×E|− on ∂D,ν×curl(1ω2(n−1)−1(curl curl−ω2n)E)∣∣+=−ν×curlE|− on ∂D,ν×E=0, ν×curlE=0 on ∂Ω. (22)

Take a test vector function . Multiplying the first equation in (22) by we have

 0=∫Ω∖¯D(curl curl−ω2)(n−1)−1(% curl curl−ω2n)E⋅¯Φdx. (23)

Denote

 Ψ=(n−1)−1(curl curl−ω2n)E  in%  Ω∖¯D. (24)

With the aid of the vector identity

 curlcurlE=−ΔE+∇divE, (25)

by Green’s second vector theorem, together with the boundary conditions in (22), equation (23) becomes

 0=∫Ω∖¯DΨ⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(curl curl−ω2)Φdx−∫∂D(ν×Ψ⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯curlΦ−curlΨ⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯ν×Φ)ds. (26)

By (24), the fourth and fifth transmission conditions in (22) can be written in terms of

 ν×Ψ|+=−ω2ν×E|−,  ν×curlΨ|+=−ω2ν×curlE|−  on ∂D.

Use the above boundary conditions, (26) becomes

 0=∫Ω∖¯D(n−1)−1(curlcurlE−ω2E)⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(curlcurlΦ−ω2Φ)dx−∫Ω∖¯Dω2E⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯curlcurlΦdx+∫Ω∖¯Dω4E⋅¯Φdx+ω2∫∂D(ν×E|−⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯curlΦ+ν×curlE|−⋅¯Φ)ds.

By Green’s first vector theorem, and with the aid of the vector identity (25), we obtain

 0=∫Ω∖¯D(n−1)−1(curlcurlE−ω2E)⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(curlcurlΦ−ω2Φ)dx−ω2∫ΩcurlE⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯curlΦdx+ω4∫ΩE⋅¯Φdx.

Therefore, the variational formulation of the interior transmission problem (22) becomes: find such that

 ∫Ω∖¯D(n−1)−1(curlcurlE−ω2E)⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(curlcurlΦ−ω2Φ)dx−ω2∫ΩcurlE⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯curlΦdx+ω4∫ΩE⋅¯Φdx=0, (27)

for all . By taking appropriate test function it is easy to see that a solution of the variational problem (27) defines a week solution to (22) and therefore to the interior transmission problem (17).

## 3 Spectral property of the interior transmission eigenvalue problem

In this section, we investigate the spectral property of the interior transmission eigenvalue problem (17). We prove the discreteness and the existence of the interior transmission eigenvalues by considering in and in , respectively.

### 3.1 Discreteness of the spectrum when n−1<0

###### Theorem 3.1.

Assume that in . And moreover, satisfies . Then the set of transmission eigenvalues is discrete.

###### Proof.

Suppose in . We define two sesquilinear forms on :

 Aω(E,Φ)=−∫Ω∖¯D(n−1)−1(curlcurlE−ω2E)⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(curlcurlΦ−ω2Φ)dx+ω2∫ΩcurlE⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯curlΦdx+ω4∫ΩE⋅¯Φdx

and

 B(E,Φ)=2∫ΩE⋅¯Φdx,

where is defined by (21). Then the variational form (27) of the interior transmission is equivalent to

 Find  E∈W such that
 Aω(E,Φ)−ω4B(E,Φ)=0  for all Φ∈W.

By the Riesz representation theorem there exist two bounded linear operators and such that

 (AωE,Φ)W:=Aω(E,Φ) and (BE,Φ)W:=B(E,Φ).

By Lemma 3.2 and 3.3 in the sequel we have is coercive and is compact. Hence the operator is Fredholm with index zero. The transmission eigenvalues are the values of for which has a nontrivial kernel. To apply the analytic Fredholm theorem, it remains to show that or is injective for at least one .

For all we have that

 Aω(E,E)−ω4B(E,E)=∫Ω∖¯D(1−n)−1|curlcurlE−ω2E|2dx+ω2∥curlE∥2L2(Ω)−ω4∥E∥2L2(Ω). (28)

The Poincaré inequality gives us that:

 ∥E∥2L2(Ω)≤C(∥curlE∥2L2(Ω)+∥divE∥2L2(Ω)),