Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking
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 threelayer 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 nearlyinvisible under the corresponding wave interrogation. The set of waves is generated from the MaxwellHerglotz 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 timeharmonic 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 realvalued 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
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
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
Let denote an EM wavenumber, corresponding to a certain EM spectrum. Consider the EM radiation in this frequency regime in the space
Let be a pair of entire electric and magnetic fields, modeling the illumination source. They verify the timeharmonic Maxwell equations,
(1) 
The presence of the inhomogeneous scatterer interrupts the propagation of the EM waves and , leading to the socalled wave scattering. We let and denote, respectively, the scattered electric and magnetic fields. Define
to be the total electric and magnetic fields, respectively. Then the EM scattering is governed by the following Maxwell system
(2) 
The last limit in (2) is known as the SilverMller radiation condition. The Maxwell system (2) is wellposed 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:
(3) 
endowed with the scalar product
and the corresponding norms and . Moreover, define
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]:
where , . and are, respectively, referred to as the electric and magnetic farfield patterns, and they satisfy
The medium is said to be invisible under the electromagnetic wave interrogation by if
(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
(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 objectdependent. 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
and , where is a set of incident waves consisting of entire solutions to the Maxwell equation (1). That is, nearinvisibility 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 socalled interior transmission eigenvalue problem associated with the Maxwell system as follows,
(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 MaxwellHerglotz functions (cf. (44)), there exists such that
(7) 
Consider the scattering problem (2) by taking the incident electric and magnetic wave fields
then there hold
(8) 
and
(9) 
where is a positive constant independent of and .
Proposition 1.1 is kind of a folktelling result in the literature on the study of interior transmission eigenvalues, and it shall be needed in our cloaking study.
Motivated by the twolayer cloaking device we then consider a threelayer cloaking device. Let be bounded Lipschitz domain such that is connected and consider an EM medium configuration as follows (see Figure 2),
(10) 
where is isotropic and the lossy layer is chosen to be
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.
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 MaxwellHerglotz approximation. In section 5, we consider the isotropic invisibility cloaking, and establish the nearinvisibility 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
(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,
(14) 
where . Let and signify the corresponding scattered wave field and the farfield pattern, respectively. If perfect invisibility is achieved for the scattering system (14), namely, then one has that
(15) 
By using the standard transmission conditions across for the solution and , one has
(16) 
where
The second condition in (16) can also be written as
We assume that the the magnetic permeability in . The above condition becomes
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
(17) 
has nontrivial solutions, where . Note that the interior transmission eigenvalue problem (17) is nonself 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
(18) 
We also get the boundary conditions
and
Then the interior transmission problem (17) can be reformulated in terms of and as follows:
With continuity of the data and across , we have from (18) that
(19) 
and
(20) 
Multiplying both sides of (18) by and applying operator on both sides, we get a fourth order equation for in :
Note that is only defined in . We define
Then , where is defined by
which is equipped with the norm
(21) 
Now we are able to formulate a fourth order system of as follows,
(22) 
With the aid of the vector identity
(25) 
by Green’s second vector theorem, together with the boundary conditions in (22), equation (23) becomes
(26) 
By (24), the fourth and fifth transmission conditions in (22) can be written in terms of
Use the above boundary conditions, (26) becomes
By Green’s first vector theorem, and with the aid of the vector identity (25), we obtain
Therefore, the variational formulation of the interior transmission problem (22) becomes: find such that
(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
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 :
and
where is defined by (21). Then the variational form (27) of the interior transmission is equivalent to
By the Riesz representation theorem there exist two bounded linear operators and such that
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
(28) 
The Poincaré inequality gives us that: