1. Introduction
This paper is concerned with relations for the time-harmonic electromagnetic scattering of a multi-layered scatterer. An obstacle of this type is made up of a nested body consisting of a finite number of homogeneous layers. On the surfaces of the layers the dielectric transmission conditions are imposed. The scatterer’s core may be a perfect conductor, a dielectric, or has an impedance surface. In the time-harmonic scattering theory [
1] some theorems between the scattered fields of two problems which correspond to two distinct incident waves for the same scatterer appear. The incident waves can have a plane or line source. These properties are referred to as scattering theorems. In this work, we state and prove some scattering theorems and, in particular, a reciprocity, a general, a mixed reciprocity, and an optical theorem for a multi-layered scatterer which is excited by line source waves.
The reciprocity principle for line sources connects the total fields in two layers. The total field at 
 due to a source at 
 relates the total field at 
 due to a source at 
. A mixed reciprocity theorem associates the scattered field corresponding to a plane wave incident on a scatterer with the far-field pattern corresponding to a line source. The general theorem connects the far-field pattern generators (which are defined in [
2] Formula (15)) at 
 and 
. The optical theorem for a plane incident field with direction of propagation 
 states the total energy (the sum of the scattered and the absorption cross-section) that the obstacle removes for the incident field is proportional to the far-field pattern, which is valued in 
. In case of line source waves, the far-field pattern is replaced by a far-field pattern generator. This theorem is obtained from the general theorem for 
.
Using the reciprocity theorem, we can prove that the far-field operator corresponding to particular scattering problems is injective, with a dense range. These properties lead to a solution for the inverse scattering problems, Refs. [
3,
4,
5]. The general scattering theorem is utilized in order to found the eigenvalues and eigenfunctions of the far-field operator and then to solve the inverse scattering problem using the linear sampling method or the factorization method [
3,
4,
6]. The optical theorem can be used to evaluate the energy scattered by an obstacle, Refs. [
1,
7]. It is worth mentioning that these scattering relations can be mainly used in solving inverse scattering problems. In [
8] near-field inverse scattering problems for ellipsoid scatterers have been solved.
A multi-layered scatterer for acoustic waves in three dimensions is described in [
9]. Three-dimensional scattering relations have been proved for various kinds of scatterers. Indicatively, we refer to [
1] for electromagnetic and [
10] for elastic waves. Reciprocity, scattering, and optical theorems for acoustic and electromagnetic waves have been proven by Twersky in [
11,
12,
13]. In view of these results, and by applying the low-frequency theory, he calculated an approximation of the real part of the far-field pattern. Reciprocity theorems for acoustic and electromagnetic waves are contained in the books [
4,
14] by Colton and Kress. In the book [
1] written by Dassios and Kleinman, reciprocity, general, and optical theorems have also been recorded. In [
15], Angell et al. have proven a reciprocity relation for electromagnetic scattering corresponding to a scatterer with an impedance boundary. Recently, a general optical theorem for an arbitrary multipole excitation is proven in [
7] and scattering cross sections are evaluated in [
16,
17]. Corresponding theorems for elastic waves have been proven in [
10] by using spherical coordinates.
Furthermore, in the case of three-dimensional multi-layered scatterers, scattering relations have been proven for acoustic waves in [
9]. Scattering theorems for spherical waves have been proven in [
2] (acoustic and electromagnetic). Moreover, in [
2], a mixed reciprocity relation has been proven. In [
18] Potthast has studied an inverse acoustic scattering problem using a mixed reciprocity relation for a piecewise constant inhomogeneous medium.
Following the procedure which is described by Colton and Cakoni in [
3], the three dimensional electromagnetic scattering problem is reduced to a system of Helmholtz equations for a two-dimensional multi-layered scatterer. This kind of scatterer appears when a multi-layered infinitely long cylinder (all the layers have a common axis) is intersected vertically by a plane. Two-dimensional scattering theorems have been proven in [
3,
19]. In [
20], plane electromagnetic scattering is studied, whereas in [
21] line source elastic waves are studied. In [
22], a mixed reciprocity principle is proven in two dimensions.
In the majority of the papers, the source which generates the incident wave is located in the exterior of the scatterer. In cases where the source is located in the interior of the scatterer, there are some important applications. Specifically, the detection of buried objects in a layered background medium requires the solution of an inverse scattering problem in which the source is located in an interior layer. Moreover, for the study of seismic waves, it is necessary to put the sources inside the layered media. In [
23], inversion algorithms for determining the geometrical and physical characteristics of a spherical scatterer, which is excited by a line source, are developed. In [
6] an inverse scattering problem for an object which is buried in a layered medium is studied. There are also applications in medicine, Ref. [
24], such as implantation inside the human head for hyperthermia or biotelemetry purposes, Ref. [
25], results (scattering theorems) are used for a scatterer, which is either a layered ellipsoid or sphere with internal sources.
First, in 
Section 2, we formulate two-dimensional scattering problems for a multi-layered scatterer for three different imposed boundary conditions on its core. In 
Section 3, we state and prove a two-dimensional reciprocity principle in the case of line source waves. Defining the line source generator, a general scattering theorem in classical form is proven in 
Section 4. In both 
Section 3 and 
Section 4, a multi-layered scatterer is excited by two line source waves located at any two layers. In 
Section 5, a mixed theorem is proven when the scatterer is excited by a line source and a plane wave. In 
Section 6, we formulate an optical theorem for the corresponding scattering problems.
  2. Formulation
We consider electromagnetic scattering by a piecewise homogeneous obstacle 
 in 
 with a 
-boundary 
. The exterior 
 of the obstacle is an infinite homogeneous isotropic medium with electric permittivity 
, magnetic permeability 
, and vanishing conductivity. The interior of 
 is divided by means of closed and nonintersecting 
-surfaces 
, 
, 
 into layers 
, 
, with 
. The surface 
 surrounds 
 and there is one normal unit vector 
 at each point 
 of any surface 
 pointing into 
. The region 
, within which lies the origin, is the core of the obstacle, which may be a dielectric, a perfect conductor, or an imperfect conductor. The layer 
 is a homogeneous isotropic medium with electric permittivity 
, magnetic permeability 
, and vanishing conductivity. All the physical parameters 
 and 
 are positive constants. The geometry of the electromagnetic scattering problem in 
 is presented in 
Figure 1.
The total electric and magnetic fields 
 and 
 in 
 satisfy the time-harmonic Maxwell equations [
1],
      
      where 
 is the angular frequency.
The corresponding scattered electromagnetic field 
, 
 satisfies the Silver–Müller radiation condition
      
      uniformly in all directions 
. The total electromagnetic field 
, 
 in 
 satisfies
      
      where 
, 
 is the incident wave field. All the fields 
, 
, 
, 
, 
, 
 are divergence-free.
The transmission conditions on the surface 
 for 
 are
      
On the surface 
 of the core, we impose either the perfect conductor boundary condition
      
      or the dielectric transmission conditions
      
      or the impedance boundary condition
      
      where 
 is the surface impedance.
In the present work, we are interested in proving scattering relations that both play an important role in studying inverse scattering problems, see Ref. [
3]. Specifically, the general scattering theorem is applied to the computation of the far-field pattern in the low-frequency theory (see [
1]). The study of these relations will be done in two dimensions for the line source waves of the following scattering problems. The first problem, which corresponds to the perfect conductor core, is defined by Equations (
1)–(
6). The second one, which corresponds to the dielectric transmission condition of the core, is defined by Equations (
1)–(
5), (
7) and (
8). The third one, which corresponds to the impedance boundary condition of the core, is defined by Equations (
1)–(
5) and (
9).
We suppose that 
 is an infinitely long cylinder which is oriented parallel to the 
-axis. The cross-section 
D of 
 in the 
-plane will be referred to as a two-dimensional scatterer. We also suppose that 
, 
, are infinitely long cylindrical surfaces. The 
-plane intersects the surfaces 
 at the 
-curves 
, see 
Figure 2.
Following the process specified by Cakoni and Colton in Ref. [
3] (pp. 47 and 83), for the electric fields, we make the additional assumption that 
, 
 where 
 is the total field in 
. The incident field can be either a plane wave
      
      where 
 is the wave number in 
 and 
 is the incident direction with 
, [
3], or a line source wave, [
26],
      
      where 
 is the zero order Hankel function of the first kind. Using these assumptions and applying the Maxwell Equation (
1), we obtain 
, 
, 
 and the following equations in 
Equations (
12) and (
13) imply that the 
-component 
 of 
 satisfies the Helmholtz equation
      
      for 
 for the perfect conductor or the impedance boundary condition on the core core and 
 for the dielectric core. The symbol 
 stands for the two-dimensional Laplace operator. The 
-component 
 of the total electric field 
 in 
 is given by
      
      where 
 is the 
-component of the scattered field 
. Specifically, 
, 
 satisfies the Sommerfeld radiation condition
      
The transmission condition Equations (
4) and (
5) are written in terms of the 
-component 
 of the total electric field 
, 
, as
      
The boundary condition Equation (
6) for the perfect conductor core is reformulated as
      
The transmission condition Equations (
7) and (
8) for the dielectric conductor are rewritten as
      
      and for the impedance boundary condition Equation (
9), we have
      
Summarizing the above analysis, we formulate the following three two-dimensional scattering problems. The first problem is defined by Equations (
14)–(
19) and is denoted by 
, the second one is defined by Equations (
14)–(
18), (
20) and (
21) and is denoted by 
, and the third one is defined by Equations (
14)–(
18) and (
22) and is denoted by 
. The well-posedness of these problems can be obtained by extending the techniques of [
27].
In all the above problems, we assume that the multi-layered scatterer is excited by two line source waves located at any two layers. In case of the mixed reciprocity, the scatterer is excited by a line source and a plane wave.
We will denote by 
, 
 and 
 the dependence of the scattered field, the far-field pattern, and the total field in 
 on the incident direction 
. For incident line source wave at 
, we utilize the fundamental solution
      
      where 
 is the zero-order Hankel function of the first kind. We will denote by 
, 
, and 
 to represent the dependence of the total field in 
, 
, the scattered field, and the far-field pattern on the position of the source 
.
The scattered line source field has the following asymptotic behaviour in two dimensions
      
      uniformly in all directions 
 [
3]. The far-field pattern 
 is given by [
3],
      
In the rest of the paper, we will use Twersky’s notation [
11],
      
As we can see, the far-field pattern is expressed through the Twersky’s notation as
      
  3. Reciprocity for Line Source Waves
For incident plane waves, similar scattering relations have already been proven in [
9] for three dimensions and in [
19] for two dimensions. Now, we will emphasize proving the corresponding relations for line source waves in two dimensions. For this purpose, we consider two line source waves at 
 and 
, with 
. Then, we prove the following lemma.
Lemma 1. Let  be an incident line source wave at . Let  be an incident line source wave at  with corresponding scattered field  and total field  in . Then, it holdswhere  is a disc centered at  with radius ,  is a disc centered at  with radius , and  is a disc centered at the origin with radius R surrounding the points  and .  Proof.  Since 
 and 
 are solutions of the Helmholtz equation for 
 and satisfy the Sommerfeld radiation condition, we obtain relation Equation (
28).
Using the asymptotic forms, Ref. [
3],
        
        as 
, applying the mean value theorem and letting 
 we obtain relation Equation (
29).
For relation Equation (
30), we replace 
 and we take
        
The first integral of the right-hand side of Equation (
33) is equal to 
, due to relation Equation (
29). For the last integral, taking into account the interior elliptic regularity, see Refs. [
5,
22,
28], we have
        
        and by using the Cauchy–Schwarz inequality we conclude that
        
□
 Remark 1. Relation Equation (29) is also valid if  is replaced by . In general,  can be replaced by bounded fields.  Next, we prove the following reciprocity theorem for line source waves and multi-layered scatterers in two dimensions.
Theorem 1. Let  and  be two incident line source waves at  and , respectively. Then the corresponding total fields  and  for the scattering problem , , satisfy the reciprocity relation  Proof.  Due to the bilinearity of Equation (
26), and taking into account tion 
, we have
        
We initially consider the case 
, i.e., the two sources are outside of the scatterer. For the first integral of the right-hand side of relation Equation (
37), since 
 and 
 are regular solutions of the Helmholtz equation in 
, an application of the scalar Green’s second theorem gives
        
For the computation of the next integral, we consider two small disjoint discs  and  centered at  and  with radius  and , respectively. We also consider a large disc  centered at the origin with radius R surrounding the points ,  and the scatterer.
Since 
 and 
 are solutions of the Helmholtz equation for 
, we apply the scalar Green’s second theorem and obtain
        
Letting 
, 
, using Lemma 1, and taking into account that the last integral of Equation (
39) is zero, due to the application of the scalar Green’s second theorem in 
 and since 
 and 
 are regular solutions of the Helmholtz equations for 
, we obtain
        
Since 
, 
 are radiating solutions of the Helmholtz equation in 
, we have
        
The total fields 
 and 
, 
, are regular solutions of the Helmholtz equation in 
. Thus, the successive application of the scalar Green’s second theorem and the use of the transmission conditions on 
 imply that
        
By using the imposed boundary conditions on 
 for 
 and 
, we directly obtain that the integral is equal to zero. For 
, we apply the transmission conditions, we use the scalar Green’s second theorem, and we obtain
        
Substituting relations Equations (
38), (
40)–(
42) and (
44) in Equation (
37), we derive
        
From definition Equation (
23) of the line source, we see that 
, and due to relation Equation (
45), we have
        
Next, we consider the case , i.e., one source is inside and the other one is outside the scatterer.
Taking into account that 
, 
 are solutions of the Helmholtz equation in 
D for 
 and by using Lemma 1, we obtain
        
For the next integral of Equation (
37), working as in the previous case, we obtain
        
Moreover, it holds
        
        since the involved functions are radiating solutions of the Helmholtz equation in 
. For the total fields we have
        
The first integral of the right-hand side is equal to zero due to the imposed boundary and transmission conditions on the surface 
 of the core. Applying Lemma 1, we obtain
        
Substituting relation Equations (
47)–(
49) and (
51) in Equation (
37), we obtain
        
We finally consider the case , i.e., both of the sources are inside the scatterer.
It holds
        
        since 
 and 
 are radiating solutions of the Helmholtz equation. Moreover, 
 and 
 are radiating solutions of the Helmholtz equation, and thus we obtain
        
Concerning the total fields, by successively applying the scalar Green’s second theorem, we have
        
Due to the boundary and the transmission conditions on the core and via Lemma 1, we obtain
        
The substitution of relation Equations (
53), (
54) and (
56) in Equation (
37) implies the generation of relation Equation (
36).    □
 Remark 2. If the positions of the line source waves are located at the same layer or both of them are outside the scatterer, i.e., , then the reciprocity relation Equation (36) is reformulated as