inverse obstacle scattering for Maxwell’s equations in an unbounded structure

# inverse obstacle scattering for Maxwell’s equations in an unbounded structure

Peijun Li Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA (lipeijun@math.purdue.edu).    Jue Wang School of Science, Harbin Engineering University, Harbin, 150001, China (wangjue3721@163.com). The research was supported in part by a National Natural Science Foundation of China (No. 11801116) and Fundamental Research Funds for the Central Universities (No. GK2110260213).    Lei Zhang School of Mathematical Sciences, Heilongjiang University, Harbin, 150080, China (zl19802003@163.com). The work was supported partially by a National Natural Science Foundation of China (No. 11871198), the Special Funds of Science and Technology Innovation Talents of Harbin (No. 2017RAQXJ099) and the Fundamental Research Funds for the Universities of Heilongjiang Province - Heilongjiang University Special Fund Project (No. RCYJTD201804).
###### Abstract

This paper is concerned with analysis of electromagnetic wave scattering by an obstacle which is embedded in a two-layered lossy medium separated by an unbounded rough surface. Given a dipole point source, the direct problem is to determine the electromagnetic wave field for the given obstacle and unbounded rough surface; the inverse problem is to reconstruct simultaneously the obstacle and unbounded rough surface from the electromagnetic field measured on a plane surface above the obstacle. For the direct problem, a new boundary integral equation is proposed and its well-posedness is established. The analysis is based on the exponential decay of the dyadic Green function for Maxwell’s equations in a lossy medium. For the inverse problem, the global uniqueness is proved and a local stability is discussed. A crucial step in the proof of the stability is to obtain the existence and characterization of the domain derivative of the electric field with respect to the shape of the obstacle and unbounded rough surface.

Key words. Maxwell’s equations, inverse scattering problem, unbounded rough surface, domain derivative, uniqueness, local stability

AMS subject classifications. 78A46, 78M30

## 1 Introduction

Consider the electromagnetic scattering of a dipole point source illumination by an obstacle which is embedded in a two-layered medium separated by an unbounded rough surface in three dimensions. An obstacle is referred to as an impenetrable medium which has a bounded closed surface; an unbounded rough surface stands for a nonlocal perturbation of an infinite plane surface such that the perturbed surface lies within a finite distance of the original plane. Given the dipole point source, the direct problem is to determine the electromagnetic wave field for the known obstacle and unbounded rough surface; the inverse problem is to reconstruct both of the obstacle and the unbounded rough surface, from the measured wave field. The scattering problems arise from diverse scientific areas such as radar and sonar, geophysical exploration, nondestructive testing, and medical imaging. In particular, the obstacle scattering in unbounded structures has significant applications in radar based object recognition above the sea surface and detection of underwater or underground mines.

As a fundamental problem in scattering theory, the obstacle scattering problem, where the obstacle is embedded in a homogeneous medium, has been examined extensively by numerous researchers. The details can be found in the monographs [6, 27] and [5, 7, 16] on the mathematical and numerical studies of the direct and inverse problems, respectively. The unbounded rough surface scattering problems have also been widely examined in both of the mathematical and engineering communities. We refer to [8, 12, 15, 25, 28, 29, 30, 31, 33] for various solution methods including mathematical, computational, approximate, asymptotic, and statistical methods. The scattering problems in unbounded structures are quite challenging due to two major issues: the usual Silver–Müller radiation condition is no longer valid; the Fredholm alternative argument does not apply due to the lack of compactness result. The mathematical analysis can be found in [10, 11, 18, 22, 32] and [13, 20, 23] on the well-posedness of the two-dimensional Helmholtz equation and the three-dimensional Maxwell equations, respectively. The inverse problems have also been considered mathematically and computationally for unbounded rough surfaces in [1, 2, 3, 24].

In this paper, we study the electromagnetic obstacle scattering for the three-dimensional Maxwell equations in an unbounded structure. Specifically, we consider the illumination of a time-harmonic electromagnetic wave, generated from a dipole point source, onto a perfectly electrically conducting obstacle which is embedded in a two-layered medium separated by an unbounded rough surface. The obstacle is located either above or below the surface and may have multiple disjoint components. For simplicity of presentation, we assume that the obstacle has only one component and is located above the surface. The free spaces are assumed to be filled with some homogeneous and lossy materials accounting for the energy absorption. The problem has received much attention and many computational work have been done in the engineering community [14, 17, 19]. However, the rigorous analysis is very rare, especially for the three-dimensional Maxwell equations.

In this work, we introduce an energy decaying condition to replace the Silver–Müller radiation condition in order to ensure the uniqueness of the solution. The asymptotic behaviour of dyadic Green’s function is analyzed and plays an important role in the analysis for the well-posedness of the direct problem. A new boundary integral equation is proposed for the associated boundary value problem. Based on some energy estimates, the uniqueness of the solution for the scattering problem is established. For the inverse problem, we intend to answer the following question: what information can we extract about the obstacle and the unbounded rough surface from the tangential trace of the electric field measured on the plane surface above the obstacle? The first result is a global uniqueness theorem. We show that any two obstacles and unbounded rough surfaces are identical if they generate the same data. The proof is based on a combination of the Holmgren uniqueness, unique continuation, and a construction of singular perturbation. The second result is concerned with a local stability: if two obstacles are “close” and two unbounded rough surfaces are also “close”, then for any , the measurements of the two tangential trace of the electric fields being -close implies that both of the two obstacles and the two unbounded rough surfaces are -close. A crucial step in the stability proof is to obtain the existence and characterization of the domain derivative of the electric field with respect to the shape of the obstacle and unbounded rough surface.

The paper is organized as follows. In Section 2, we introduce the model problem and present some asymptotic analysis for dyadic Green’s function of the Maxwell equations. Section 3 is devoted to the well-posedness of the direct scattering problem. An equivalent integral representation is proposed for the boundary value problem. A new boundary integral equation is developed and its well-posedness is established. In Sections 4 and 5, we discuss the global uniqueness and local stability of the inverse problem, respectively. The domain derivative is studied. The paper is concluded with some general remarks in Section 6.

## 2 Problem formulation

Let us first specify the problem geometry which is shown in Figure LABEL:pg. Let be an unbounded rough surface given by

 S={x=(x1,x2,x3)∈R3:x3=f(x1,x2)},

where . The surface divides into and , where

 Ω+1={x∈R3:x3>f(x1,x2)},Ω2={x∈R3:x3

Let be a bounded obstacle with boundary . The obstacle is assumed to be a perfect electrical conductor which is located either in or in . For instance, we may assume that . Define . The domain is assumed to be filled with some homogeneous, isotropic, and absorbing medium which may be characterized by the dielectric permittivity , the magnetic permeability , and the electric conductivity , .

In , the electromagnetic waves satisfy the time-harmonic Maxwell equations (time dependence ):

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩∇×Ej=iωμjHj,∇×Hj=−iωεjEj+Jj,∇⋅(εjEj)=ρj,∇⋅(μjHj)=0,

where is the angular frequency, , , denote the electric field, the magnetic field, the electric current density, respectively, and is the electric charge density. The external current source is assumed to be located in . The relation between the electric current density and the electric field is given by

 {J1=σ1E1+Jcsin Ω1,J2=σ2E2in Ω2,

where stands for the current source.

Using the above constitutive relation, we obtain coupled systems

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∇×E1=iωμ1H1,∇×H1=−iω(ε1+iσ1ω)E1+Jcs,(ε1+iσ1ω)∇⋅E1=1iω∇⋅Jcs,∇⋅(μ1H1)=0,in Ω1, \hb@xt@.01(2.1)

and

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∇×E2=iωμ2H2,∇×H2=−iω(ε2+iσ2ω)E2,(ε2+iσ2ω)∇⋅E2=0,∇⋅(μ2H2)=0,in Ω2. \hb@xt@.01(2.2)

Eliminating the magnetic field in (LABEL:EHeq1), we obtain a decoupled equation for the electric field :

 ∇×(∇×E1(x))−κ21E1(x)=iωμ1Jcs(x),x∈ Ω1. \hb@xt@.01(2.3)

Similarly, it follows from (LABEL:EHeq2) that we may deduce a decoupled Maxwell system for the electric field :

 ∇×(∇×E2(x))−κ22E2(x)=0,x∈ Ω2. \hb@xt@.01(2.4)

Here is the wave number in . Since are positive constants, is a complex constant with which accounts for the energy absorption.

By the perfect conductor assumption for the obstacle, it holds that

 νΓ×E1=0on Γ, \hb@xt@.01(2.5)

where denotes the unit normal vector on the boundary directed into the exterior of . The usual continuity conditions need to be imposed, i.e., the tangential traces of the electric and magnetic fields are continuous across :

 νS×E1=νS×E2,νS×H1=νS×H2on S, \hb@xt@.01(2.6)

where denotes the unit normal vector on pointing from to .

The incident electromagnetic fields satisfy Maxwell’s equations

 {∇×(∇×Ei(x))−κ21Ei(x)=iωμ1Jcs(x),∇×(∇×Hi(x))−κ21Hi(x)=∇×Jcs(x),x∈Ω1. \hb@xt@.01(2.7)

In , the total electromagnetic fields consist of the incident fields and the scattered fields . In , the electromagnetic fields are called the transmitted fields.

In addition, we propose an energy decaying condition

 limr→+∞∫∂B+r|Es|2ds=0,limr→+∞∫∂B+r|Hs|2ds=0 \hb@xt@.01(2.8)

and

 limr→+∞∫∂B−r|E2|2ds=0,limr→+∞∫∂B−r|H2|2ds=0, \hb@xt@.01(2.9)

where denotes the hemisphere of radius above or below .

The dyadic Green function is defined by the solution of the following equation

 ∇x×(∇x×Gj(x−y))−κ2jGj(x−y)=δ(x−y)Iin Ωj, \hb@xt@.01(2.10)

where is the unitary dyadic and is the Dirac delta function. It is known that the dyadic Green function is given by

 Gj(x−y)=[I+∇y∇yκ2j]exp(iκj|x−y|)4π|x−y|. \hb@xt@.01(2.11)

We assume that the dipole point source is located at and has a polarization , . Induced by this dipole point source, the incident electromagnetic fields are

 Ei(x)=G1(x−xs)q,Hi(x)=1iωμ1(∇×Ei(x)),x∈Ω1. \hb@xt@.01(2.12)

Hence the current source satisfies

 iωμ1Jcs(x)=qδ(x−xs),x∈Ω1.

Denote by the set of functions . The direct scattering problem can be stated as follows.

###### Problem 2.1

Given the incident field in (LABEL:PSEi), the direct problem is to determine and such that

1. The electric fields and satisfy (LABEL:Eeq1) and (LABEL:Eeq2), respectively;

2. The electric field satisfies the boundary condition (LABEL:EBCS1);

3. The electromagnetic fields satisfy (LABEL:EBCS);

4. The scattered fields and the transmitted fields satisfy the radiation conditions (LABEL:EsIRC) and (LABEL:E2IRC), respectively.

It requires to study the dyadic Green function in order to find the integral representation of the solution for the scattering problem. The details may be found in [4] on the general properties of the dyadic Green function.

###### Lemma 2.2

For each fixed , the dyadic Green function given in (LABEL:G) admits the asymptotic behaviour

 Gj(x−y) =O(exp(−I(κj)|x|)|x|)^Ias |x−y|→∞, ∇x×Gj(x−y) =O(exp(−I(κj)|x|)|x|)^Ias |x−y|→∞,

where and .

Proof. Following

 |x−y|=√|x|2−2x⋅y+|y|2=|x|−^x⋅y+O(1|x|)as |x|→∞,

where , we have

 exp(iκj|x−y|)|x−y|= exp(iκj|x|)|x| ×{exp(−iκj^x⋅y)+O(1|x|)}as |x|→∞ \hb@xt@.01(2.13)

uniformly for all satisfying . By (LABEL:EL02-2), for , we obtain for that

 Gj(x−y) =[I+∇y∇yκ2j]exp(iκj|x|)4π|x|{exp(−iκj^x⋅y)+O(1|x|)} =exp(iκj|x|)4π|x|{[I−^x^x]exp(−iκj^x⋅y)+O(1|x|)^I} =O(exp(−I(κj)|x|)|x|)^I

and

 ∇x×Gj(x−y) =−∇y×Gj(x−y) =exp(iκj|x|)4π|x|{−∇y×[(I−^x^x)exp(−iκj^x⋅y)]+O(1|x|)^I} =iκjexp(iκj|x|)4π|x|{^x×[(I−^x^x)exp(−iκj^x⋅y)]+O(1|x|)^I} =O(exp(−I(κj)|x|)|x|)^I,

which completes the proof.

We introduce some Banach spaces. For , denote by the set of bounded and continuous functions on , which is a Banach space under the norm

 ∥ϕ∥∞=supx∈V|ϕ(x)|.

For , denote by the Banach space of functions which are uniformly Hölder continuous with exponent . The norm is defined by

 ∥ϕ∥C0,α(V)=∥ϕ∥∞+supx,y∈Vx≠y|ϕ(x)−ϕ(y)||x−y|α.

Let which is a Banach space under the norm

 ∥ϕ∥C1,α(V)=∥ϕ∥∞+∥∇ϕ∥C0,α(V).

## 3 Well-posedness of the direct problem

In this section, we show the existence and uniqueness of the solution to Problem LABEL:SP by using the boundary integral equation method. First we derive an integral representation for the solution of Problem LABEL:SP using dyadic Green’s theorem combined with the radiation conditions (LABEL:EsIRC) and (LABEL:E2IRC).

###### Theorem 3.1

Let the fields be the solution of Problem LABEL:SP, then have the integral representations

 E1(x)=Ei(x)+ ∫S{[iωμ1G1(x−y)]⋅[νS(y)×H1(y)]+ [∇x×G1(x−y)]⋅[νS(y)×E1(y)]}dsy + ∫Γ{[iωμ1G1(x−y)]⋅[νΓ(y)×H1(y)]}dsy,x∈Ω1, \hb@xt@.01(3.1)

and

 E2(x)= −∫S{[iωμ2G2(x−y)]⋅[νS(y)×H2(y)] +[∇x×G2(x−y)]⋅[νΓ(y)×E2(y)]}dsy,x∈Ω2. \hb@xt@.01(3.2)

Proof. Let . Denote with the boundary , where and . For each fixed , applying the vector dyadic Green second theorem to and in the region , we obtain

 ∫Ωr{E1(y)⋅[∇y×∇y×G1(y−x)]−[∇y×∇y×E1(y)]⋅G1(y−x)}dy =−∫∂Ωr{[ν(y)×(∇y×E1(y))]⋅G1(y−x) +[ν(y)×E1(y)]⋅[∇y×G1(y−x)]}dsy, \hb@xt@.01(3.3)

where stands for the unit normal vector at pointing out of .

It follows from (LABEL:Eeq1) and (LABEL:G1eq) that

 ∫Ωr{E1(y)⋅[∇y×∇y×G1(y−x)]−[∇y×∇y×E1(y)]⋅G1(y−x)}dy =∫Ωr[E1(y)]⋅[∇y×∇y×G1(y−x)−κ21G1(y−x)]dy −∫Ωr[∇y×∇y×E1(y)−κ21E1(y)]⋅[G1(y−x)]dy =∫Ωr[E1(y)⋅(δ(y−x)I)]dy−∫Ωr[iωμ1Jcs(y)⋅G1(y−x)]dy =E1(x)−∫Ωr[iωμ1Jcs(y)⋅G1(y−x)]dy,

where

 limr→+∞∫Ωr[iωμ1Jcs(y)⋅G1(y−x)]dy =∫Ω1[qδ(y−xs)⋅G1(y−x)]dy =G1(x−xs)q=Ei(x). \hb@xt@.01(3.4)

Hence, letting , with the aid of (LABEL:ET1-1)–(LABEL:ET1-3), we have

 E1(x)−Ei(x) =−∫∂Ω1{[ν(y)×(∇y×E1(y))]⋅G1(y−x) +[ν(y)×E1(y)]⋅[∇y×G1(y−x)]}dsy =−(∫S+∫Γ+limr→+∞∫∂B+r){[ν(y)×(∇y×E1(y))]⋅G1(y−x) +[ν(y)×E1(y)]⋅[∇y×G1(y−x)]}dsy. \hb@xt@.01(3.5)

Following Lemma LABEL:EL02 and (LABEL:EsIRC)–(LABEL:E2IRC), we obtain for that

 ∣∣∣∫∂B+r{[ν(y)×(∇y×Es(y))]⋅G1(y−x) +[ν(y)×Es(y)]⋅[∇y×G1(y−x)]}dsy∣∣∣ ≤[ω2μ21∫∂B+r|Hs(y)|2dsy]12⋅[∫∂B+r|G1(y−x)|2dsy]12 +[∫∂B+r|Es(y)|2dsy]12⋅[∫∂B+r|∇y×G1(y−x)|2dsy]12→0. \hb@xt@.01(3.6)

By Lemma LABEL:EL02 and the definition of incident field , we have for that

 ∣∣∣∫∂B+r{[ν(y)×(∇y×Ei(y))]⋅G1(y−x) +[ν(y)×Ei(y)]⋅[∇y×G1(y−x)]}dsy∣∣∣ ≤[ω2μ21∫∂B+r|Hi(y)|2dsy]12⋅[∫∂B+r|G1(y−x)|2dsy]12 +[∫∂B+r|Ei(y)|2dsy]12⋅[∫∂B+r|∇y×G1(y−x)|2dsy]12→0. \hb@xt@.01(3.7)

Using (LABEL:ET1-4)–(LABEL:ET1-6) and conditions (ii), (iv) in Problem LABEL:SP, and letting , we have for each fixed that

 E1(x)−Ei(x)= −∫S{[ν(y)×(∇y×E1(y))]⋅G1(y−x) +[ν(y)×E1(y)]⋅[∇y×G1(y−x)]}dsy −∫Γ{[ν(y)×(∇y×E1(y))]⋅G1(y−x)}dsy = ∫S{[iωμ1G1(x−y)]⋅[νS(y)×H1(y)] +[∇x×G1(x−y)]⋅[νS(y)×E1(y)]}dsy +∫Γ{[iωμ1G1(x−y)]⋅[νΓ(y)×H1(y)]}dsy.

Similarly, for each fixed , we have

 E2(x)= −∫S{[G2(x−y)]⋅[νS(y)×(∇y×E2(y))] +[∇x×G2(x−y)]⋅[νS(y)×E2(y)]}dsy = −∫S{[iωμ2G2(x−y)]⋅[νS(y)×H2(y)] +[∇x×G2(x−y)]⋅[νS(y)×E2(y)]}dsy,

where

 νS(y)×Ej(y) =limh→+0νS(y)×Ej(y+(−1)jhνS(y)), νS(y)×[∇y×Ej(y)] =limh→+0νS(y)×[∇y×Ej(y+(−1)jhνS(y))]

are to be understood in the sense of uniform convergence on , and .

Finally, from the jump relations and (LABEL:EBCS1), we note that the integral representations (LABEL:E1)–(LABEL:E2) lead to the boundary integral equations:

 12νS(x)×E1(x)= νS(x)×Ei(x)+∫S{[iωμ1νS(x)×G1(x−y)]⋅[νS(y)×H1(y)] +[νS(x)×(∇x×G1(x−y))]⋅[νS(y)×E1(y)]}dsy +∫Γ{[iωμ1νS(x)×G1(x−y)]⋅[νΓ(y)×H1(y)]}dsy,x∈S, \hb@xt@.01(3.8)
 0= νΓ(x)×Ei(x)+∫S{[iωμ1νΓ(x)×G1(x−y)]⋅[νS(y)×H1(y)] +[νΓ(x)×(∇x×G1(x−y))]⋅[νS(y)×E1(y)]}dsy +∫Γ{[iωμ1νΓ(x)×G1(x−y)]⋅[νΓ(y)×H1(y)]}dsy,x∈Γ, \hb@xt@.01(3.9)

and

 12νS(x)×E2(x)= −∫S{[iωμ2νS(x)×G2(x−y)]⋅[νS(y)×H2(y)] +[νS(x)×(∇x×G2(x−y))]⋅[νS(y)×E2(y)]}dsy,x∈S. \hb@xt@.01(3.10)

Hence, the electric fields satisfy the boundary integral equations (LABEL:E1S)–(LABEL:E2S) and the continuity conditions

 νS×E1=νS×E2,νS×H1=νS×H2on S, \hb@xt@.01(3.11)

which completes the proof.

To show the well-posedness of the boundary integral equations (LABEL:E1S)–(LABEL:E2S), we introduce the normed subspace of continuous tangential fields

 T(S):={ψ∈C(S):νS⋅ψ=0},

and the normed space of uniformly Hölder continuous tangential fields

 T0,α(S):={ψ∈T(S)| ψ∈C0,α(S)}.

We consider the integral operator defined by

 (TΨ)(x) =∫S[iωμ1νS(x)×G1(x−y)]⋅[Ψ(y)]dsy =∫R2[iωμ1(νS(x)×G1(x−y))⋅Ψ(y)|y3=f(y1,y2)](1+f2y1+f2y2)1/2dy1dy2, \hb@xt@.01(3.12)

and the integral operator defined by

 (KΦ)(x) =∫S[νS(x)×(∇x×G1(x−y))]⋅[Φ(y)]dsy =∫R2[iωμ1(νS(x