FENG LI-XIN AND LI YUAN

(School of Mathematical Sciences,Heilongjiang University,Harbin,150080)

Electromagnetic Scattering in a Two-layered Medium?

FENG LI-XIN AND LI YUAN

(School of Mathematical Sciences,Heilongjiang University,Harbin,150080)

The object of this paper is to investigate the three-dimensional electromagnetic scattering problems in a two-layered background medium.These problems have an important application in today’s technology,such as to detect objects that are buried in soil.Here,we model both the exterior impedance problem and the inhomogeneous medium problem inR3.We establish uniqueness and existence for the solution of the two scattering problems,respectively.

electromagnetic,scattering problem,layered medium,integral equation,uniqueness and existence,Green function

1 Introduction

Recently,considerable attention has been devoted to the analysis for the electromagnetic scattering problems in a layered medium(see[1]–[10]).These problems have an important application in today’s engineering and physics,such as to detect objects that are buried in soil.Here,we are interested in the three-dimensional electromagnetic scattering problems in a two-layered background medium.Among the extensive literatures we refer to the publications[1],[3]and[7].The paper[3]is concerned with the solution of Maxwell equations in the modeling of the scattering of a time-harmonic electromagnetic wave by a perfect conductor (obstacle)located in a two-layered medium.The use of the Silver-Mller radiation condition in each layer was shown to provide a well-posed scattering problem.The analysis was based on the study of the Green tensor.The analyticity properties of the scattering problem with respect to the frequency were also investigated.In the paper[7],the author studied also the electromagnetic scattering problems in a two-layered medium from a perfectly conductingobstacle.In contrast with the paper[3],in one layer the Silver-Mu¨ller radiation condition was used,and in the other layer an exponential decay condition was used.The scattered field was modeled via a boundary layer approach and for its kernel the Green’s matrix for the two-layered medium was constructed.The author established uniqueness and existence for the solution of the corresponding boundary integral equation.In this paper,we are concerned with both the exterior impedance problem and the inhomogeneous medium problem in a two-layered background medium.We establish uniqueness and existence for the solution of the two scattering problems,respectively.The idea of our proof is inspired by the paper [7].

We model the obstacle(or inhomogeneous part)by some domain D in the lower halfspace.For simplicity,we assume that D has a sufficiently smooth boundary,i.e.,we assume a C2boundary.However,in principle,D can be a domain with corners and edges.We denote byD1=D1={x=(x1,x2,x3)∈R3:x3＞0}the upper half-space and byD2= {x=(x1,x2,x3)∈R3:x3＜0}the lower half-space.S={x=(x1,x2,x3)∈R3:x3=0} denotes the interface betweenD1andD2.We assume D?D2to be a bounded domain with connected complement and de fi ne D2=D2.

We consider the propagation of electromagnetic waves with frequency ω in the twolayered medium consisting of the isotropic half spacesDjwith electric permittivity ?j, magnetic permeabilityμjand electric conductivity σjfor j=1,2.First,we describe the exterior impedance problem.The electromagnetic wave is described by the electric fields Ejand the magnetic fields Hjin Djthat satisfy the Maxwell equations

We consider the harmonic time-dependence in the form

Then the space-dependent parts Ej,Hj∈C1(Dj)∩C(ˉDj),j=1,2,satisfy the timeharmonic Maxwell equations in the symmetric form

with wave numbers

The square roots for the wave numbers are chosen so that Rekj≥0 and Imkj≥0,j=1,2. For some applications,such as in state of the air-soil,forD1,we have air which is nonconducting,i.e.,σ1=0,and consequently,Imk1=0.However,inD2we have soil with a conductivity σ2＞0,and therefore Imk2＞0.According to the continuity of the tangential components of the electric field Ejand the magnetic field Hjcross the interface,we can show that at the interface betweenD1andD2,the fields satisfy the transmission conditions

with the constants aEand aHgiven by

where ν is the upward unit normal to S.

We describe the obstacle D which is not perfectly conducting but which does not allow the electric and magnetic field to penetrate deeply into the body D by the impedance boundary condition

where ν is the outward unit normal to D,zris the relative impedance of the boundary. For physical reasons,this impedance must have a non-positive real part,i.e.,Rezr≤0.In particular,zr=?iδ corresponds to a coated scatterer by a thin dielectric layer for some δ＞0,which depends on both the coating thickness and the dielectric relative permittivity (see[8]).

The total electromagnetic field(Ej,Hj)is such that

uniformly for all direction in the upper half-spaceD1.For the wave number k2with Imk2＞0,we assume an exponential decay as follows:

for x∈D2and some constant M＞0.

Next,we describe the inhomogeneous medium problem.We assume also that the electromagnetic field is time-harmonic,and then the space-dependent parts Ej,Hjsatisfy the time-harmonic Maxwell equations

where n1(x)=1 inD1.We assume that n2(x)∈C1,α(D2)for some 0＜α＜1,Ren2≥0, Imn2＞0 and m:=1?n2(x)has compact support D?D2,i.e.,

The total electromagnetic fields(Ej,Hj)are such that

This paper is organized as follows.In Section 2 we establish the uniqueness of solutions to the above two scattering problems.Section 3 is devoted to prove the existence of the solutions via the integral equation approach.We provide an explicit form of the fundamental solution in a two-layered background medium in Section 4.

2 Uniqueness

The main task of this section is to establish uniqueness for the solutions to the exterior impedance problem and the inhomogeneous medium problem in a two-layered background medium.

Theorem 2.1The exterior impedance problem(1.2)–(1.9)in a two-layered background medium has at most one solution.

We introduce the ball BR={x∈R3:|x|＜R}and the disk SR={x∈S:|x|＜R}with radius R＞0 centered at the origin.In addition,for j=1,2,we denote by BR,jthe half ball BR,j={x∈Dj:|x|＜R}and by?R,jthe half sphere?R,j={x∈Dj:|x|=R}. Furthermore,by ν we denote the outward unit normal to?BR,jand the upward unit normal to SR(see Fig.2.1).

Fig.2.1

Theorem 2.2The inhomogeneous medium problem in a two-layered background medium has at most one solution.

Similarly to the proof of Theorem 2.1,using the Gauss divergence theorem in the domain BR,2,with the help of the vector identity

In view of this condition,adding the equation(2.9)to(2.10)we get

By our assumptions on kjand nj,j=1,2,simple computation yields

i.e.,the constant in front of the volume integral in(2.11)is positive.Therefore,we can conclude that E2=0 in D2and by the first of the Maxwell equations(1.2)we also have H2=0 in D2.Then,by the transmission conditions(1.4),Holmgren’s theorem implies E1=H1=0 inD1.This completes our proof.

3 Existence

The main task of this section is to establish existence for the solutions to the exterior impedance problem and the inhomogeneous medium problem in a two-layered background medium by using integral equation approach.LetΨj(x,y),j=1,2 be a fundamental solution to the Helmholtz equation for the layered medium inR3,which satis fies

We now study the existence of the exterior impedance problem.We denote by C0,α(?D) the Hlder continuous function space de fi ned on?D and by T0,α(?D)the Hlder continuous tangential fields de fi ned on?D(see[11]). We de fi ne the single-layer operator S:C0,α(?D)→C0,α(?D)by

Theorem 3.1The exterior impedance problem(1.2)–(1.9)has a solution of the form

whereS0is the single-layer operator in the potential theoretic limit casek2=0and the densitiesb∈T0,α(?D)andφ∈C0,α(?D).

Proof.The proof is completely analogous to that of the section 9.5 in[11],but for the convenience of the reader,we brie fl y sketch the proof below.By the de fi nition of the fundamental solutionΨj(x,y),the vector fields Ejs,Hjsgiven in(3.1)clearly satisfy the Maxwell equation(1.2),the radiation condition and the transmission conditions(1.4).Next, we must show that there exist densities b∈T0,α(?D)and φ∈C0,α(?D)such that the vector fields

satisfy the impedance boundary condition(1.6).To this end,inserting(3.1)into(1.6)and using the jump,we conclude that b and φ must satisfy the integral equations

where P stands for the orthogonal projection of a vector field de fi ne on?D onto the tangent plane,that is,Pa:=(ν×a)×ν.We can prove that the operators M11,M12and M22are all compact.Writing the system(3.2)in the form

we now see that the first of the two matrix operators has a bounded inverse because of its triangular form,and the second is compact.Hence,applying the Riesz-Fredholm theory, with the help of Theorem 2.1,yields the existence of the densities b and φ.This completes our proof.

Under the assumptions given in the previous section for the refractive index nj,j=1,2, next we turn to the inhomogeneous medium problem.

Lemma 3.1Let(Ej,Hj)∈C(Dj),j=1,2be a solution to the inhomogeneous medium problem.ThenEj,j=1,2satisfy the integral equation

Conversely,letEj∈C(Dj),j=1,2be a solution of the equation(3.3).ThenEjandHj:=curlEj/(ikj),j=1,2are a solution of the inhomogeneous medium problem.

Proof.The proof is completely similar to that of Theorems 9.1 and 9.2 in[11]withΦreplaced byΨj.

We are now in a position to show that there exists a unique solution to the inhomogeneous medium problem.

Theorem 3.2The electromagnetic scattering problem for the inhomogeneous medium in a two-layered background medium has a unique solution.

Proof.By Lemma 3.1,it suffices to prove the existence of a solution Ej∈C(Dj)to(3.3). We de fi ne an electromagnetic operator Te:

Since Tehas a weakly singular kernel it is a compact operator.Hence,we can apply the Riesz-Fredholm theory and must show that the homogenous equation corresponding to(3.3) has only the trivial solution.This can be done,by using the unique Theorem 2.2.Therefore, equation(3.3)can be solved and the inverse operator(I?Te)?1is bounded.The theorem is now proved.

Remark 3.1The regularity of the solution follows from the regularity results for surface vector potentials analogous to the theorems in[11].

4 The Fundamental Solution in Two-layered Medium

The aim of this section is to construct the fundamental solution to the three dimensional Helmholtz equation in two-layeredbackground medium,i.e.,to look for the functionΨj(x,y), j=1,2 such that

5 Conclusions

Using construction method we have proved the existence of solutions to electromagnetic scattering problems in the layered medium.The construction process gives a method for solving the problems.For the exterior impedance problem,we can obtain φ and b by solving a system of equations(3.2),and get electric field and magnetic field by(3.1).For the inhomogeneous medium problem,we can obtain the electric field and magnetic field by solving an integral equation(3.3).In this paper,we have studied the existence of solutions by using integral equation approach in which the kernel is a Green’s function in the layered medium.Alternative useful tool for solving electromagnetic problems is Green matrix.To determine the existence of solutions by using Green matrix in the layered medium and draw corresponding conclusions will be the topic of a subsequent paper.

[1]Angell,T.S.and Kirsch,A.,The conductive boundary condition for Maxwell’s equations,SIAM J.Appl.Math.,52(1992),1597–1610.

[2]Ben Hassen,M.F.,Erhard,K.and Potthast,R.,The point-source method for 3D reconstructions for the Helmholtz and Maxwell equations,Inverse Problems,22(2006),331–353.

[3]Cutzach,P.M.and Hazard,C.,Existence,uniqueness and analyticity properties for electromagnetic scattering in a two-layered medium,Math.Methods Appl.Sci.,21(1998),433–461.

[4]Chen,J.,Haber,E.and Oldenburg,D.,Three-dimensional numerical modeling and inversion of magnetometric resistivity data,Geophys.J.Int.,149(2002),679–697.

[5]Collins,L.,Gao,P.,Scho field,D.and Moulton,J.,A statistical approach to landmine detection using broadband elctromagnetic data,IEEE Trans.Geosci.Remote Sens.,40(2002),950–962.

[6]Coyle,J.,Locating the support of objects contaied in a two-layered background medium in two dimensionals,Inverse Problems,16(2000),275–292.

[7]Delbary,F.,Erhard,K.,Kress,R.,Potthast,R.and Schulz,J.,Inverse electromagnetic scattering in a two-layered medium with an application to mine detaction,Inverse Problems, 24(2008),1–18.

[8]Kirsch,A.and Monk,P.,A fi nte element method for the approximating electromagnetic scattering from a conducting object,Numer.Math.,92(2002),501–534.

[9]Liu,E.,Geophysical application of a new surface integral equation methods for EM modeling,Geophys.,63(1998),411–423.

[10]Piperno,S.,Remaki,M.and Fezou,L.,A non-di ff usive finite volume scheme for the threedimensional Maxwell’s equations on unstructured meshes,SIAM J.Numer.Anal.,39(2002), 2089–2108.

[11]Colton,D.and Kress,R.,Inverse Acoustic and Electromagnetic Scattering Theory,2nd ed., Springer,Berlin,1998.

Communicated by Ma Fu-ming

65N21,78A46

A

1674-5647(2011)04-0349-11

date:July 2,2009.

The NSF(10801046)of China,the Heilongjiang Education Committee Grant(11551362, 11551364)and the Heilongjiang University Grant(Hdtd2010-14).