1. Introduction
In this work, the inverse electromagnetic scattering problem of determining the surface impedance and the shape of a buried partially coated scattering object in chiral media is studied. In order to do this, we need information of the value of the electric and magnetic fields on the surface of the earth.
A chiral material is one that displays optical activity such that, when the plane of vibration of a linearly polarized light passes through an opticaly active medium is rotated. Over the last few years, chiral materials have been studied more intensely and there are more studies on the subject, covering both their applications and their theoretical background. Furthermore, various papers have been written on direct and inverse electromagnetic scattering problems for chiral media. Indicatively, we refer to Reference [
1,
2,
3,
4]. These materials are characterized by a set of two constitutive equations, in which electric and magnetic fields are connected via a physical variable or constant, known as chirality.
In this work, Drude-Born-Fedorov constitutive equations are used, as they are symmetric under time reversality and duality transformation [
5]. In homogeneous and isotropic chiral media, the electric and magnetic fields are a combination of Left Circularly Polarized and Right Circularly Polarized components that have different phase speeds. So, in the applications, we can use the Bohren decomposition [
6] of electric and magnetic fields into suitable Left and Right Circularly Beltrami fields. Such fields have been employed in Reference [
2] for the definition of the chiral Herglotz wave functions and in Reference [
6] for formulation of the electric dipole, which both play an important role in the present work. In Reference [
7], the measure of chirality for a certain class of chiral scatterers has been calculated, while, in Reference [
8], properties of chiral metamaterials are described. In addition, we note that Ammari and N
d
lec in Reference [
1] have proved that the well-known Silver-M
ller radiation condition remains valid in chiral media. In Reference [
4,
9], the direct and the inverse electromagnetic scattering problems by a mixed impedance screen in a chiral environment are investigated, respectively. Beltrami fields have been used for the uniqueness and a variational method for the existence of the direct problem. In the inverse problem, a modified linear sampling method, originated from a factorization of the chiral far field operator, has been employed.
In this work, the inverse scattering problem of specifying the shape and the surface impedance of a buried coated scattering object in a chiral environment is studied. A qualitative method [
10], which is based on the chiral reciprocity gap operator, is used in Reference [
11]. In fact, this procedure is a modified type of the linear sampling method (LSM). The classical LSM, which was first established by Colton and Kirsch [
12], is simple, relatively quick and does not need any a priori information of the material parameters of the scattering object. However, in the electromagnetic imaging of a buried object via LSM, the computation of the Green’s function of the background material is necessary. Sometimes, this computation is practically impossible. The reciprocity gap functional method helps us to overcome this difficulty. The combination of the LSM and the reciprocity gap functional method was established by Colton and Haddar in Reference [
13] for acoustic waves and by Cakoni, Fares, and Haddar in Reference [
14] for electromagnetic waves. In Reference [
15], a reciprocity gap functional for elastic waves has been used for solving an inverse mixed impedance scattering problem. In Reference [
11], a reciprocity gap functional for chiral media has been defined, in order for an inverse scattering problem for a perfect conductor to be solved.
The present paper extends this method to study inverse scattering problems for buried partially coated objects in a chiral environment. In Reference [
16], the shape and the surface impedance of a buried coated scattering object have been determined. In Reference [
17], the same method has been applied to solve an electromagnetic inverse scattering problem for a partially coated anisotropic dielectric, which is in the inner of the earth. Using this method in Reference [
18], an inverse electromagnetic scattering problem for a perfectly conducting cavity, using measurements from the interior, has been solved. In Reference [
19], an interior inverse acoustic scattering problem for a cavity with an inhomogeneous medium inside has been studied. The same method has been employed in Reference [
20] in order for a sound field to be reconstructed in a spherical harmonic domain. In Reference [
21], the reciprocity gap functional method is applied to calculate the boundary and the permittivity of the scattering object in radar imaging. Recently, the reciprocity gap functional method has been employed in order to study an inverse scattering problem in electrical tomography [
22], in seismology [
23] and in source identification [
24]. For more details on the linear sampling and reciprocity gap functional method, we refer to Reference [
12], while, for general aspect in scattering theory, we refer to Reference [
25,
26].
In
Section 2 of this paper, the electromagnetic waves in chiral media are described and the chiral mixed impedance scattering problem is formulated. In
Section 3, the chiral reciprocity gap operator is defined, proved that it is injective and it has a dense range. In
Section 4, the main result of the paper is proved. In
Section 5, the surface impedance is determined. Finally, a conclusion is given in
Section 6.
2. Electromagnetic Waves in Chiral Media
We consider the scattering of a time-harmonic electromagnetic wave by an object embedded in a chiral medium. The Drude-Born-Fedorov constitutive relations [
6] are employed:
where
are the electric and magnetic fields,
the electric displacement,
the magnetic induction,
is the chirality measure,
the electric permittivity, and
the magnetic permeability. Then, applying the source-free Maxwell curl postulates:
where
is the angular frequency, we get the following relations:
where
and
. We point out that
k is not a wave number and does not have any particular physical significance. We assume that the physical parameters
are positive constants and
, (Reference [
6] (p. 87)). The fields
and
satisfy:
We eliminate the magnetic field
in Equations (
1) and (2) and obtain
In isotropic homogeneous chiral media, the electric and magnetic fields are composed of Left Circularly Polarized (LCP) and Right Circularly Polarized (RCP) waves with different phase speeds. So, for
and
, we make use of the Bohren decomposition into Beltrami fields
and
[
5], and we get
and hence
The Beltrami fields satisfy the differential equations:
which show that the homogeneous isotropic chiral media are circularly birefringent. The wave numbers
and
for the LCP and RCP Beltrami fields, respectively, are given by:
and satisfy:
For further information on the physical background for chiral media, we refer to Reference [
5,
6,
27].
We assume that a scatterer D with -boundary, is embedded in a piecewise isotropic homogeneous chiral material with ∖ to be connected. It is assumed that is divided into two open sets and , such that and . On (Dirichlet part), a perfectly conducting boundary condition is satisfied and (impedance part) is covered from a very thin dielectric layer. We consider to be a bounded domain, which contains , with - boundary . Let and be the chirality, the electric permittivity, and the magnetic permeability, respectively, that characterize the medium ∖, which will be referred to as the background medium. In addition, let , and be the corresponding parameters in the exterior ∖ of . We suppose that the physical parameters are positive constants. Finally, denotes the outward normal unit vector on the corresponding surface.
The incident field is a chiral electric dipole with polarization
located at
in a chiral environment. We assume that
lies on an auxiliary close surface
contained in
∖
. The electric incident field in a chiral medium is given by the formula [
5,
6]:
where
is the identity dyadic in
, and
are the wave numbers for the LCP and RCP Beltrami fields, respectively, in
∖
with
where
,
.
The incident on the scatterer
D electric wave
has the form:
where
is the scattered field due to the background material. In addition, the wave
in
∖
is given by:
where
is the dyadic Green’s function of the chiral background material. If
for
∖
,
for
∖
,
for
∖
and
for
∖
, then
satisfies the equation:
with respect to
x. Let
be the incident on
D electric field and
be the corresponding scattered field. Then, the total electric field
E is given by
and is the solution of the mixed impedance scattering problem:
where
is the unit sphere in
,
,
, with
and
.
The direct scattering problem can be studied as in Reference [
9]. The uniqueness of solution has been proved via the Beltrami fields, while, for the existence of solution, the variational method has been employed, using a Calderon type operator [
28] for chiral media. The corresponding inverse scattering problem is the determination of the unknown boundary of
D and the evaluation of surface impedance
from the information of the tangential components
and
on the boundary
for all points
. In chiral media, a Stratton-Chu type exterior integral representation for a radiating solution of Equation (
3) is the following:
We define the function spaces:
where
is a ball of radius
R containing
D, as well as
The space
is equipped with the norm
For the trace
of
, we have
and for
of
The trace space of
on
is defined by:
Finally, for the exterior domain ∖, we define the spaces ∖ and ∖ considering the domain ∖.
The exterior mixed impedance boundary value problem in chiral media is the following problem: Let
and
, find
such that:
If
and
, then the problem (
11)–(14) is the mixed impedance scattering problem (
6)–(9).
Let
and
. We consider the following chiral interior mixed impedance boundary value problem corresponding to (
6)–(9); given
and
, we find
such that:
The values of parameter
k for which the corresponding homogeneous interior mixed impedance scattering problem admits a nontrivial solution will be referred to as chiral Maxwell eigenvalues for
D. This problem in the achiral case has been solved in Reference [
29]. A similar scattering problem for a mixed impedance screen has been studied in Reference [
9]. In particular, a Calderon type operator for chiral media and a variational method have been employed to prove uniqueness and existence of solution. The present scattering problem is to find the shape of
D and the surface impedance
from the knowledge of electric and magnetic fields on
. In what follows, a brief description of the solvability of the interior mixed impedance problem (
15)–(17) is given.
For the uniqueness of (
15)–(17), we consider the corresponding homogeneous problem
, and we multiply (
15) with
(complex conjugate of
E) and integrate over
D. Taking into account the boundary conditions we get
where
is the tangential component of
E. From (
18), taking the imaginary part and using the unique continuation principle as in Reference [
12,
29], we conclude that
in
D. For the existence, we consider the variational formulation for the problem (
15)–(17). For all test functions
with
we have
We look for solution
E of the form
, where
with
, which there exists from the definition of
. Substituting in (
19), we take:
where
In (
21),
denotes the
scalar product and
the
product. Equation (
20) has been studied in Reference [
28,
29] for the achiral case. With a similar process for the chiral case, the following theorem is proved.
Theorem 1. If then the chiral interior partially coated problem (15)–(17) has a unique solution. 3. The Chiral Reciprocity Gap Operator
The reciprocity gap operator for electromagnetic scattering in chiral media has been defined in Reference [
11], in order to study an inverse scattering problem for a perfectly conducting obstacle.
Let
be the solution of the scattering problem (
6)–(9). The chiral reciprocity gap functional is defined by
where
and the integrals are interpreted in the sense of the duality between
,
. In particular, if
, then the chiral reciprocity gap functional can be seen as an integral operator
, given by:
The reciprocity gap functional method is based on the solvability of an integral equation for
, which contains an appropriate family of solutions in
. Usually, we use a set of either single layer potentials or Hergotz wave functions. Here, for the determination of the boundary of
D, chiral Herglotz wave functions will be employed, because these functions satisfy density properties which will be used later. In Reference [
2], the electric
and magnetic
chiral Herglotz wave functions have been defined and are given by
where
are the LCP and the RCP Beltrami Herglotz fields, with kernels
and
, respectively, and
,
. In particular, for the kernels, we have
,
, where
In addition, we define the following space:
with the inner product:
where
,
,
, are the Beltrami fields of
b and
h, respectively, and
[
2]. Let
be the electric dipole with polarization
located at
z in a chiral medium. We study the solvability of the integral equation:
with respect to
g in
.
We will prove that the operator R, under appropriate conditions, is injective and has dense range.
Lemma 1. If is not empty then the operator , defined by (23) is injective. Proof. We assume that
. Then,
for all
and
. On (
22), we apply the second vector Green’s theorem for the first integral, Gauss’ theorem for the second integral for
E,
W in
∖
, which are both solutions of (
15), we use the boundary conditions on
to take
Let
be the unique solution of the boundary value problem:
Substituting
and
, from (29) and (30) into (
27), we take
The total electric field
E is given by:
Hence, using (
32) and the boundary conditions (7) and (8), we get
and taking into account that the fields
and
are both radiating solutions of (
28), we have:
From the Stratton-Chu type formula (
10) for chiral media, we take
for arbitrary polarization
p, and therefore
for
. Then, by the uniqueness of the electromagnetic scattering in a chiral environment for a perfect conductor [
1,
3], we conclude that
outside the surface
. Applying unique continuation, we have
in the domain between the boundary
and the surface
. Therefore,
and using the uniqueness of the interior partially coated chiral electromagnetic problem for
W, implying
. □
Lemma 2. If is not empty then the operator defined by (23) has dense range. Proof. Let
, such that
for all
. We will prove that
. In view of the bilinearity of functional
and the definition of operator
R, we get
where
. If we define
then, from (
22) and the assumption for
q, we have that
Using Green’s and Gauss’ theorems for
in
∖
as in Lemma 1 and taking into account the boundary conditions on
, we conclude that
for all
. The density of the chiral Herglotz wave functions has been used in order to prove that the set
is dense in
. This follows from the fact that
contains the chiral Herglotz wave functions, given by (
24) and (25) (see [
2,
29]), which satisfy the Equation (
3) and
,
. In addition, we have taken into account that the interior mixed impedance boundary value problem (
15)–(17) is well-posed. Therefore,
and
on
. Hence,
has zero Cauchy data on
and therefore
in the domain between
and
. Finally, taking into account the jump relations [
3] of
across
, we arrive at
on
. Therefore,
for all
, hence
. □