On isotropic cloaking and interior transmission eigenvalue problems
Abstract.
This paper is concerned with the invisibility cloaking in acoustic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. It is shown that an interior transmission eigenvalue problem arises in our study, which is the one considered theoretically in [13]. Based on such an observation, 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 if a certain nontransparency condition is satisfied, then 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 Herglotzapproximation of the associated interior transmission eigenfunctions. We provide both theoretical and numerical justifications.
Keywords. Acoustic wave scattering, invisibility cloaking, isotropic and regular, interior transmission eigenvalues
Mathematics Subject Classification (2010): Primary 35J57, 35P25, 35Q60, 78A46; Secondary 35R30, 65N25, 65N30
1. Introduction
1.1. Mathematical formulation and motivation
In this paper, we are concerned with the scattering of acoustic waves in the timeharmonic regime. Let , , be a bounded Lipschitz domain with a connected complement . Let , . It is assumed that , , are realvalued and measurable functions such that
(1.1) 
and
where is the Kronecker delta function, . We also let be a complexvalued function such that
(1.2) 
It is further assumed that for . denotes an acoustic medium in with its inhomogeneity supported in . signifies the density tensor of the acoustic medium, whereas and are, respectively, associated with the modulus and the loss of the acoustic material parameters. We remark that (1.1) and (1.2) are the physical conditions satisfied by a regular medium. In what follows, and are said to be uniformly elliptic with constants and , if they respectively, satisfy (1.1) and (1.2). The inhomogeneity of the acoustic medium is located in an isotropic and homogeneous background/matrix medium whose material parameters are normalised to be and , where denotes the identity matrix. is referred to as a scatterer in the following. The scatterer is said to be isotropic if , , for a scalar function , otherwise it is called anisotropic.
Next, we consider the timeharmonic acoustic wave propagation in the space . Let , be an entire solution to the following Helmholtz equation
(1.3) 
where denotes a wavenumber. The propagation of the acoustic wave will be interrupted due to the presence of the inhomogeneity , and this leads to the socalled scattering. We let denote the perturbed/scattered wave field and denote the total wave field. The total wave field is governed by the following Helmholtz system
(1.4) 
In (1.4), a function is said to satisfy the Sommerfeld radiation condition if the following limit
(1.5) 
holds uniformly for , . It is known that admits the following asymptotic expansion as (cf. [19, 43, 45]),
(1.6) 
which holds uniformly in . is known as the farfield pattern or the scattering amplitude.
An important inverse scattering problem arising from practical application is to recover by knowledge of the corresponding farfield pattern . is said to be invisible under the wave interrogation by if , which corresponds to the nonidentification in the inverse scattering problem mentioned above. 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
(1.7) 
In (1.7), denotes the target object being cloaked, and denotes the cloaking shell. For a practical cloaking device of the form (1.7), there are several crucial ingredients that one should incorporate into the design:
(i). 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.
(ii). 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.
(iii). For an ideal cloaking device, one can expect the following invisibility performance,
(1.8) 
However, in practice, especially in order to fulfil the requirements in items (i) and (ii) above, one can relax the ideal cloaking requirement (1.8) to be
(1.9) 
where is a set of incident waves consisting of entire solutions to the Helmholtz equation (1.3). 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 aim to explore as much as possible the three issues listed above for a practical cloaking scheme.
1.2. Existing developments and discussion
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. 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. Due to its practical importance, the invisibility cloaking has received great attentions in the literature in recent years, and several cloaking schemes have been proposed, including one based on transformation optics [30, 31, 48, 37] and another one based on plasmon resonances [1, 44].
The plasmonic cloaking uses metamaterials with negative parameters, and can be used to cloak an active source. We refer to [2, 6, 32, 33, 38] and the references therein for the existing developments in this direction. The transformationoptics approach uses the transformation properties of optical parameters via a socalled pushforward to form the cloaking mediums. The transformationoptics mediums for an ideal cloak are nonnegative, but anisotropic and singular, possessing degeneracy and/or blowup singularities. In order to avoid the singular structures, various regularised cloaking schemes have been proposed and investigated in the literature, and instead of ideal cloaking, one considers approximate/nearinvisibility cloaking for the regularised constructions. We refer to[3, 4, 5, 8, 9, 34, 25, 39, 40, 41, 42] and the references therein for the existing developments in this direction. Though the regularised transformationoptics mediums are nonsingular, they retain the anisotropy, which still poses server difficulties to the practical fabrication. In [24, 27, 26], the authors propose to further approximate the nonsingular anisotropic cloaking mediums by isotropic ones using the theory of effective medium and inverse homogenisation. This is closely related to the issues of practical feasibility and relaxation/approximation discussed in Section 1.1. However, the isotropic cloaking medium obtained in [24, 27, 26] through the inverse homogenisation are still nearlysingular in the sense that the ellipticity constants of the cloaking material parameters are asymptotic depending on a regularisation parameter.
In this paper, we investigate the nonsingular and isotropic cloaking issue through a different perspective. We consider the cloaking device (1.7) in the ideal case directly for a given cloaking medium. This would lead to an interior transmission eigenvalue problem, which is the one considered in [13]. In [13], the authors consider an interior transmission eigenvalue problem for inhomogeneous media containing soundsoft obstacles. We connect the theoretical study in [13] with some important practical application on the invisibility cloaking, and propose a generalised interior transmission eigenvalue problem as well. The interior transmission eigenvalue problem arising from the study of inverse scattering theory [19, 20] has also received significant attentions in the literature in recent years [12]. Indeed, the invisibility cloaking problem is naturally connected to a certain interior transmission eigenvalue problem, as shall be discussed in Section 2. Such an observation was also made in a recent paper [11] where the authors show that a generic inhomogeneous scatterer with a rectangular support cannot be ideally invisible to every incident wave; that is, it scatters every interrogating wave field. This idea was further picked up in [7] where the authors numerically show that ideal invisibility can be achieved for certain wavenumbers with respect to a discrete and finite set of farfield measurement data. In [22, 23], from a different perspective, the authors show that if the support of a generic inhomogeneous scatterer possesses certain irregularities including a corner, an edge or a circular cone, then it scatters every interrogating wave field. The corner scattering problem has been quantified in [10] with stability estimates; see our remarks after Theorem 2.3 for more relevant discussions. Our current study shall indicate that an acoustic scatterer, satisfying a certain nontransparency condition, is nearlyinvisible with respect to certain incident wave fields. Those incident wave fields are generated from the Herglotzapproximation to certain interior transmission eigenfunctions. Furthermore, based on this study, we propose a novel cloaking scheme using nonsingular and isotropic cloaking mediums. The proposed cloaking device takes a threelayer structure with 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 nonsingular and isotropic. To our best knowledge, the results obtained in the current article are new to the literature. Finally, we refer to [17, 28, 29, 46, 50] for comprehensive surveys on the theoretical and experimental progress on invisibility cloaking in the literature; and we also refer to [11, 12, 14, 15, 16, 18, 19, 20, 21, 35, 47, 49] and the references therein for relevant theoretical and computational studies on the interior transmission eigenvalues.
The rest of the paper is organised as follows. In Section 2, we connect the interior transmission eigenvalue problem with the invisibility cloaking, along with some relevant discussions. In Section 3, we consider the isotropic invisibility cloaking, and establish the nearinvisibility results. Section 4 is devoted to numerical verification and demonstration.
2. Interior transmission eigenvalue problem and Herglotzapproximation
Let us consider the scattering problem associated with a cloaking device of the form (1.7). We first assume that the cloaked region is isolated from the outer space . That is, we consider an idealised case that the scattering problem is described by the following Helmholtz system for ,
(2.1) 
where or with
(2.2) 
and the exterior unit normal vector to . Similar as before, we let and signify the corresponding scattered wave field and the farfield pattern, respectively. If perfect invisibility is achieved for the scattering system (2.1), namely, , then by the Rellich theorem (cf. [19]), one has that
(2.3) 
Next, by using the standard transmission condition across for the solution to (2.1), one has
(2.4) 
where signify the limits taken from outside and inside of , respectively. Applying (2.3) to (2.4) and by setting and for , one can readily show that the following PDE system holds for ,
(2.5) 
If for a certain , there exists a nontrivial pair of solutions to (2.5), then is called an interior transmission eigenvalue associated with , and is called the corresponding pair of transmission eigenfunctions. It is pointed out that the interior transmission eigenvalue problem (2.5) with on was first proposed and investigated in [13] from a theoretical perspective. In this paper, we shall find some important application of (2.5) to invisibility cloaking.
According to our discussion above, if perfect invisibility is obtained for the scattering system (2.1), then one has the eigenfunctions for the interior transmission eigenvalue problem (2.5). However, the existence of interior transmission eigenfunctions for (2.5) does not necessarily imply perfect invisibility for the scattering system (2.1). Nevertheless, we shall show that nearinvisibility can still be achieved under certain cirumstances. To that end, we first need to extend the interior transmission eigenfunction to the whole space by the socalled Herglotzapproximation to form an incident wave field for (2.1). Starting from now and throughout the rest of the paper, we assume that is of class . Define
(2.6) 
is called a Herglotz wave function. We have
Theorem 2.1 (Theorem 2 in [51]).
Let denote the space of all Herglotz wave functions of the form (2.6). Define, respectively,
and
Then is dense in with respect to the topology induced by the norm.
We shall also need to introduce the following nontransparency condition for . Consider the following PDE,
(2.7) 
It is assumed that is not an eigenvalue to (2.7) in the sense that if , then there exists only a trivial solution to (2.7). Hence one has a welldefined DtN map as follows,
(2.8) 
where is the unique solution to (2.7). Moreover, we assume that the PDE system (2.7), with the Dirichlet boundary condition replaced by a Neumann boundary condition , is also wellposed. Hence, the NtD map , is also welldefined. Next, we consider the following exterior scattering problem
(2.9) 
Define the exterior DtN map by
(2.10) 
where is the unique solution to (2.9). It is known that and its inverse are both welldefined (cf. [19, 45]). Then is said to satisfy the nontransparency condition associated with if there holds
(2.11) 
Remark 2.1.
It can be shown that one must have
(2.12) 
for any unless . Indeed, for , we let be the solution to (2.7), and be the solution to (2.9). Set . If , then one readily verifies that is the solution to the following system
(2.13) 
Hence, one must have that in (see Section 8, [19]), which readily implies . Hence, it is unobjectionable to claim that the nontransparency condition (2.11) is a generic condition, when is an interior transmission eigenvalue for . Nevertheless, we would like to remark that it seems, at least according to our numerical observations, that the nontransparency condition is not satisfied for an inhomogeneous scatterer with an irregular/nonsmooth support, say a corner in its support; see our remarks after Theorem 2.3 for more relevant discussions.
Now, we are in a position to present one of the main theorems of this paper which connects the interior transmission eigenvalue problem (2.5) to the invisibility cloaking.
Theorem 2.2.
Let be an interior transmission eigenvalue associated with , and be a corresponding pair of eigenfunctions. For any sufficiently small , by Theorem 2.1, we let be such that
(2.14) 
Consider the scattering problem (2.1) by taking the incident wave field
(2.15) 
If satisfies the nontransparency condition (2.11) with respect to , then there holds
(2.16) 
where is a positive constant depending only on , and .
Proof.
Since is an interior transmission eigenvalue associated with and are the corresponding eigenfunctions, we see from (2.5) that
(2.17) 
By (2.1) and setting , we clearly have
(2.18) 
and moreover
(2.19) 
By subtracting (2.17) from (2.18), and using the transmission conditions on in (2.17), we then obtain
(2.20) 
Next, by noting the nontransparency condition (2.11), we first treat the case by assuming that
(2.21) 
By (2.20), we have
(2.22) 
and hence
(2.23) 
which then yields
(2.24) 
Combining (2.14), (2.21) and (2.24), together with straightforward calculations, one readily has
(2.25) 
where is a positive constant depending only on , and .
For the other case with
(2.26) 
that is
(2.27) 
by a completely similar argument, one can show that
(2.28) 
where is a positive constant depending only on , and .
That is, one either has (2.25) or (2.28) for the exterior scattering system (2.19). Finally, by the wellposedness of the scattering problem from a soundhard or soundsoft obstacle (see [19, 43]), one readily has (2.16).
The proof is complete.
∎
Now, we let denote the set of all the interior transmission eigenvalues associated with , and set
(2.29) 
Clearly, is a subspace of , and by Theorem 2.1, for a sufficiently small , we let be an net of in in the sense that for any , there exists a such that
(2.30) 
By Theorem 2.2, one clearly has that
Theorem 2.3.
For any associated with an interior transmission eigenvalue , if is nontransparent with respect to , then there holds
(2.31) 
where is a positive constant depending only on , and .
Therefore, by Theorem 2.3, as long as the nontransparency condition holds, the cloaked region with the coating is nearly invisible to the wave interrogation for any incident field from , under the assumption that the cloaked region is isolated from the outer space. Some remarks and discussions are in order.

Clearly, the existence and distribution of interior transmission eigenvalues and eigenfunctions for (2.5) shall be of crucial importance. The existence, discreteness and infiniteness of the interior transmission eigenvalues for (2.5) under general assumptions on and for the case with was established in [13]. We believe that the case with on can be treated similarly. Inversely, for certain specific wavenumbers and entire wave fields, the design of such that those wavenumbers and wave fields are the respective interior transmission eigenvalues and eigenfunctions shall also be of great interest for customised invisibility cloaking constructions. We leave the theoretical investigation on these issues for our future study. In Section 4, we present extensive numerical experiments to illustrate the discreteness and infiniteness of , and the validity of Theorems 2.2 and 2.3.

The nontransparency condition (2.11) is critical for the nearinvisibility result in Theorem 2.3. Our numerical examples in Section 4 indicate that if is smooth/regular, then nearinvisibility can be achieved at almost all the computed interior transmission eigenvalues; whereas if is irregular, say possessing a corner, then nearinvisibility generically cannot be achieved. The numerical observation is consistent with the theoretical studies in [10, 11, 22, 23]. Indeed, as discussed in Section 1.2, it is shown in [11, 22, 23] that if the support of the scatterer possesses certain irregularities, then it scatters every interrogating wave field. In [10], it is further quantified that the scattered wave field from a corner possesses a positive lower bound. Hence, it might be justifiable to conclude that the nontransparency condition (2.11) holds true for generic acoustic mediums with smooth/regular supports. More quantitatively, according to (2.24),
(2.32) should be a regular number when and, both and are regular/smooth; whereas (2.32) either blows up or becomes very large when and, or are irregular. Providing more justifications for the above conclusion is fraught with difficulties, we shall also leave it for further investigation.

Theorem 2.3 is based on the idealised assumption that the cloaked region is completely isolated from the outer space. In Section 3, we shall present a finite realisation of on by using properly designed isotropic mediums with loss. The loss mediums are also regular and isotropic, and moreover for the targetindependence consideration of the cloaking device, it should enable the object being cloaked to be arbitrary.
3. Isotropic invisibility cloaking
Let be the one considered in Theorem 2.3. It is assumed that is of class and, that is real and . Let be a domain of Lipschitz class. Let be an asymptotically small parameter. Set
(3.1) 
if in (2.5); and
(3.2) 
if in (2.5), where and are positive constants. Consider an acoustic medium configuration as follows,
(3.3) 
where is a regular acoustic medium. Then, we have
Theorem 3.1.
Proof.
Let us consider the scattering system (1.4) corresponding to described in (3.3). By multiplying both sides of the equation in (1.4) by the complex conjugate of , namely , and integrating over , together with the use of integration by parts, we have
(3.5) 
By taking the imaginary parts of both sides of (3.5), one can easily verify that
(3.6) 
Hence, if is given in (3.1), one has
(3.7) 
whereas if is given in (3.2), then one has
(3.8) 
Next, we first consider the case with