Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency

# Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency

Xiaoxu Xu Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China (xuxiaoxu14@mails.ucas.ac.cn)    Bo Zhang LSEC, NCMIS and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China (b.zhang@amt.ac.cn)    Haiwen Zhang NCMIS and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China (zhanghaiwen@amss.ac.cn)
###### Abstract

This paper is concerned with uniqueness results in inverse acoustic and electromagnetic scattering problems with phaseless total-field data at a fixed frequency. Motivated by our previous work (SIAM J. Appl. Math. 78 (2018), 1737-1753), where uniqueness results were proved for inverse acoustic scattering with phaseless far-field data generated by superpositions of two plane waves as the incident waves at a fixed frequency, in this paper, we use superpositions of two point sources as the incident fields at a fixed frequency and measure the modulus of the acoustic total-field (called phaseless acoustic near-field data) on two spheres enclosing the scatterers generated by such incident fields on the two spheres. Based on this idea, we prove that the impenetrable bounded obstacle or the index of refraction of an inhomogeneous medium can be uniquely determined from the phaseless acoustic near-field data at a fixed frequency. Moreover, the idea is also extended to the electromagnetic case, and it is proved that the impenetrable bounded obstacle or the index of refraction of an inhomogeneous medium can be uniquely determined by the phaseless electric near-field data at a fixed frequency, that is, the modulus of the tangential component with the orientations and , respectively, of the electric total-field measured on a sphere enclosing the scatters and generated by superpositions of two electric dipoles at a fixed frequency located on the measurement sphere and another bigger sphere with the polarization vectors and , respectively. As far as we know, this is the first uniqueness result for three-dimensional inverse electromagnetic scattering with phaseless near-field data.

Key words. Uniqueness, inverse acoustic scattering, inverse electromagnetic scattering, phaseless near-field, obstacle, inhomogeneous medium

AMS subject classifications. 78A46, 35P25

## 1 Introduction

Inverse scattering problems occur in many applications such as radar, remote sensing, geophysics, medical imaging and nondestructive testing. These problems aim at reconstructing the unknown scatterers from the measurement data of the scattered waves. In the past decades, inverse acoustic and electromagnetic scattering problems with phased data have been extensively studied mathematically and numerically. A comprehensive account of these studies can be found in the monographs [10, 14].

In many practical applications, it is much harder to obtain data with accurate phase information compared with just measuring the intensity (or the modulus) of the data, and thus it is often desirable to study inverse scattering with phaseless data (see, e.g., [10, Chapter 8] and the references quoted there). In fact, inverse scattering problems with phaseless data have also been widely studied numerically over the past decades (see, e.g. [2, 3, 10, 11, 12, 13, 18, 26, 32, 38, 37, 43, 44, 45] and the references quoted there).

Recently, uniqueness and stability results have also been established for inverse scattering with phaseless data (see, e.g. [1, 19, 20, 24, 25, 27, 29, 30, 31, 35, 39, 40, 41, 46]). For example, for point source incidence uniqueness results have been established in [24, 25] for inverse potential and acoustic medium scattering with the phaseless near-field data generated by point sources placed on a sphere enclosing the scatterer and measured in a small ball centered at each source position for an interval of frequencies, and in [30] for inverse acoustic medium scattering with the phaseless near-field data measured on an annulus surrounding the scatterer at fixed frequency.

The purpose of this paper is to propose a new approach to establish uniqueness results for inverse acoustic scattering problems with phaseless total-field data at a fixed frequency. Motivated by our previous work [39], where uniqueness results have been proved for inverse acoustic scattering with phaseless far-field data corresponding to superpositions of two plane waves as the incident fields at a fixed frequency, we consider to utilize the superposition of two point sources at a fixed frequency as the incident field. However, the idea of proofs used in [39] can not be applied directly to the inverse scattering problem with phaseless near-field data. This is due to the fact that our proofs in [39] are based essentially on the limit of the normalized eigenvalues of the far-field operators. To overcome this difficulty, we consider to use two spheres, which enclose the scatterers, as the locations of such incident fields and the measurement surfaces of the modulus of the acoustic total-field (the sum of the incident field and the scattered field). In fact, many phase retrieval algorithms have been developed for inverse scattering problems with phaseless near-field data measured on two surfaces to ensure the reliability of the near-field phase reconstruction algorithms (see, e.g. [17, 33, 34]). Based on this idea, we prove that the impenetrable bounded obstacle or the index of refraction of the inhomogeneous medium can be uniquely determined from the phaseless total-field data at a fixed frequency. Note that the superposition of two point sources was also used in [35] as the incident field to study uniqueness for phaseless inverse scattering problems. Some related uniqueness results can be found in [47, 48].

The idea is also applied to phaseless inverse electromagnetic scattering which is more complicated than the acoustic case. In this case, the electric total field is a complex vector-valued function, so we need to define the phaseless data used in this paper. In many applications (see, e.g. [4, 32, 36]), the phaseless near-field data are based on the measurement of the modulus of the tangential component of the electric total field on the measurement surface. Further, it has been elaborated in [16] that the measurement data are based on two tangential components of the electric field on the measurement sphere (see [16, p.100]). Therefore, the phaseless near-field data used is the modulus of the tangential component in the orientations and , respectively, of the electric total field measured on a sphere enclosing the scatters and generated by superpositions of two electric dipoles at a fixed frequency located on the measurement sphere and another bigger sphere with the polarizations and , respectively. Following a similar idea as in the acoustic case, we prove that the impenetrable bounded obstacle or the refractive index of the inhomogeneous medium (under the condition that the magnetic permeability is a positive constant) can be uniquely determined by the phaseless total-field data at a fixed frequency. To the best of our knowledge, this is the first uniqueness result for three-dimensional inverse electromagnetic scattering with phaseless near-field data. It should be mentioned that our uniqueness results in this paper are based on parts of the PhD thesis [42].

The outline of this paper is as follows. The acoustic and electromagnetic scattering models considered are given in Section LABEL:direct. Sections LABEL:acoustic and LABEL:em are devoted to the uniqueness results for phaseless inverse acoustic and electromagnetic scattering problems, respectively. Conclusions are given in Section LABEL:con.

## 2 The direct scattering problems

We will introduce the acoustic and electromagnetic scattering models considered in this paper. To this end, assume that is an open and bounded domain in with a boundary such that the exterior is connected. Assume further that , where is a ball centered at the origin with radius large enough.

### 2.1 The acoustic case

In this paper, we consider the problem of acoustic scattering by an impenetrable obstacle or an inhomogeneous medium in . We need the following fundamental solution to the three-dimensional Helmholtz equation in with :

For arbitrarily fixed consider the time-harmonic ( time dependence) point source

 wi:=wi(x,y)=Φk(x,y),x∈R3∖¯¯¯¯¯D,

which is incident on the obstacle from the unbounded part , where is the wave number, and are the wave frequency and speed in the homogeneous medium in the whole space. Then the problem of scattering of the point source by the impenetrable obstacle is formulated as the exterior boundary value problem:

 Δxws(x,y)+k2ws(x,y)=0, x∈R3∖¯¯¯¯¯D, \hb@xt@.01(2.1) Bw=0 on∂D, \hb@xt@.01(2.2) limr→∞r(∂ws∂r−ikws)=0, r=|x|, \hb@xt@.01(2.3)

where is the scattered field, is the total field, and (LABEL:rc) is the Sommerfeld radiation condition imposed on the scattered field . The boundary condition in (LABEL:bc) depends on the physical property of the obstacles :

 ⎧⎪⎨⎪⎩Bw:=won∂Dif D is a sound-soft obstacle,Bw:=∂w/∂ν+ηwon∂Dif D is an impedance obstacle,Bw:=wonΓD,Bw:=∂w/∂ν+ηwonΓIif D is a partially % coated obstacle,

where is the unit outward normal to the boundary and is the impedance function on satisfying that for all or . We assume that or , that is, is continuous on or . When , the impedance boundary condition becomes the Neumann boundary condition (a sound-hard obstacle). For a partially coated obstacle, we assume that the boundary has a Lipschitz dissection , where and are disjoint, relatively open subsets of and having as their common boundary in (see, e.g., [8]).

The problem of scattering of the point source by an inhomogeneous medium is modeled as follows:

 Δxws(x,y)+k2n(x)ws(x,y)=k2(1−n(x))wi(x,y), x∈R3, \hb@xt@.01(2.4) limr→∞r(∂ws∂r−ikws)=0, r=|x|, \hb@xt@.01(2.5)

where is the scattered field and in (LABEL:he-n) is the refractive index characterizing the inhomogeneous medium. We assume that has compact support and with for all .

The existence of a unique (variational) solution to the problems (LABEL:he)-(LABEL:rc) and (LABEL:he-n)-(LABEL:rc-n) has been proved in [14, 22, 7, 21]. In particular, the scattered-field has the asymptotic behavior:

 ws(x,y)=eik|x||x|{w∞(^x,y)+(1|x|)},|x|→∞

uniformly for all observation directions , where is the unit sphere in and is the far-field pattern of which is an analytic function of for each (see, e.g., [14, (2.13)]).

In this paper, we also consider the superposition of two point sources

 wi=wi(x;y1,y2)=wi(x,y1)+wi(x,y2)=Φk(x,y1)+Φk(x,y2) \hb@xt@.01(2.6)

as the incident field, where are the locations of the two point sources. It then follows by the linear superposition principle that the corresponding scattered field

 ws(x;y1,y2)=ws(x,y1)+ws(x,y2) \hb@xt@.01(2.7)

and the corresponding total field

 w(x;y1,y2)=w(x,y1)+w(x,y2), \hb@xt@.01(2.8)

where and are the scattered field and the total field corresponding to the incident point source , respectively, .

The inverse acoustic obstacle (or medium) scattering problem we consider in this paper is to reconstruct the obstacle and its physical property (or the index of refraction of the inhomogeneous medium) from the phaseless total field for on some spheres enclosing and the inhomogeneous medium.

### 2.2 The electromagnetic case

In this paper, we consider two electromagnetic scattering models, that is, scattering by an impenetrable obstacle and scattering by an inhomogeneous medium. We will consider the time-harmonic ( time dependence) incident electric dipole located at and described by the matrices and defined by

 Ei(x,y)p:=ikcurlxcurlx[pΦk(x,y)],Hi(x,y)p:=curlx[pΦk(x,y)],x≠y,

for , where is the polarization vector, is the wave number, and are the wave frequency and speed in the homogeneous medium in , respectively, and and are the electric permittivity and the magnetic permeability of the homogeneous medium, respectively. A direct calculation shows that for ,

 Ei(x,y) = ikΦk(x,y)I+ik∇x∇xΦk(x,y) = ik{[k2+(ik−1|x−y|)1|x−y|]I+ˆx−y⋅ˆx−y⊤f(|x−y|)}Φk(x,y),

where is a identity matrix, , and . Then the problem of scattering of the electric dipole and by the impenetrable obstacle can be modeled as the exterior boundary value problem:

 curlxEs−ikHs=0 inR3∖¯¯¯¯¯D, \hb@xt@.01(2.10) curlxHs+ikEs=0 inR3∖¯¯¯¯¯D, \hb@xt@.01(2.11) BE=0 on∂D, \hb@xt@.01(2.12) limr→∞(Hs×x−rEs)=0, r=|x|, \hb@xt@.01(2.13)

where is the scattered field, and are the electric total field and the magnetic total field, respectively, and (LABEL:ele_rc) is the Silver–Müller radiation condition which holds uniformly for all and ensures the uniqueness of the scattered field. The boundary condition in (LABEL:ele_bc) depends on the physical property of the obstacle , that is, on (called as the PEC condition) if is a perfect conductor, where is the unit outward normal to the boundary , on if is an impedance obstacle, where is the impedance function on , and

 BE:=ν×EonΓD,BE:=ν×curlE−iλ(ν×E)×νonΓI

if is a partially coated obstacle, where has a Lipschitz dissection with and being disjoint and relatively open subsets of and having as their common boundary in and is the impedance function on . We assume throughout this paper that with for all or with for all .

The problem of scattering of an electric dipole by an inhomogeneous medium is modeled as the medium scattering problem:

 curlxEs−ikHs=0 inR3 \hb@xt@.01(2.14) curlxHs+ikn(x)Es=ik(1−n(x))Ei inR3 \hb@xt@.01(2.15) limr→∞(Hs×x−rEs)=0, r=|x|, \hb@xt@.01(2.16)

where is the scattered field and is the total field. The refractive index in (LABEL:ele_em2) is given by

 n(x):=1ε0(ε(x)+iσ(x)ω).

In this paper, we assume the magnetic permeability to be a positive constant in the whole space. We assume further that has a compact support and for with and for all .

The existence of a unique (variational) solution to the problems (LABEL:ele_e1)–(LABEL:ele_rc) and (LABEL:ele_em1)–(LABEL:ele_rcm) has been established in [8, 9, 14]. In particular, it is well known that the electromagnetic scattered field has the asymptotic behavior:

 Es(x,y)p=eik|x||x|{E∞(^x,y)p+(1|x|)},|x|→∞

uniformly for all observation directions , where is the electric far-field pattern of which is an analytic function of for each (see, e.g., [14, (6.23)]). Because of the linearity of the direct scattering problem with respect to the incident field, we can express the scattered waves by matrices and , the total waves by matrices and , and the far-field patterns by and , respectively.

We will also consider the following superposition of two electric dipoles as the incident field:

 Ei = Ei(x,y1,p1,τ1,y2,p2,τ2):=τ1Ei(x,y1)p1+τ2Ei(x,y2)p2 = ikcurlxcurlx[τ1p1Φk(x,y1)+τ2p2Φk(x,y2)], Hi = Hi(x,y1,p1,τ1,y2,p2,τ2):=τ1Hi(x,y1)p1+τ2Hi(x,y2)p2 = curlx[τ1p1Φk(x,y1)+τ2p2Φk(x,y2)],

where , , , , and . By the linear superposition principle, the electric total field and scattered field corresponding to the superposition of two electric dipoles as the incident field satisfy

 Es(x,y1,p1,τ1,y2,p2,τ2):=τ1Es(x,y1)p1+τ2Es(x,y2)p2

and

 E(x,y1,p1,τ1,y2,p2,τ2):=τ1E(x,y1)p1+τ2E(x,y2)p2, \hb@xt@.01(2.17)

where and are the electric scattered field and the electric total field corresponding to the incident field , respectively, .

Following [16, 32, 36], we measure the modulus of the tangential component of the electric total field on a sphere centered at the origin with radius . To represent the tangential components, we introduce the following spherical coordinate

 ⎧⎨⎩x1=rsinθcosϕ,x2=rsinθsinϕ,x3=rcosθ,

with and . For any , the spherical coordinate gives an one-to-one correspondence between and . Here, and denote the north and south poles of , respectively. If we define

 eϕ(x):=(−sinϕ,cosϕ,0),eθ(x):=(cosθcosϕ,cosθsinϕ,−sinθ),

then and are two orthonormal tangential vectors of at . Now, we can represent our phaseless measurement data by

 |em(x)⋅E(x,y1,en(y1),τ1,y2,el(y2),τ2)|

with , , , and .

The inverse electromagnetic obstacle or medium scattering problem we consider in this paper is to reconstruct the obstacle and its physical property or the index of refraction of the inhomogeneous medium from the modulus of the tangential component of the electric total field, , for all in some spheres enclosing or the inhomogeneous medium, and . The purpose of this paper is to prove uniqueness results for the above inverse acoustic and electromagnetic scattering problems.

## 3 Inverse acoustic scattering with phaseless total-field data

This section is devoted to the uniqueness results for inverse acoustic scattering with phaseless total-field data at a fixed frequency measured on two spheres enclosing the scatterers (see Figures LABEL:nearo and LABEL:nearm).

Denote by and the scattered field and the total field, respectively, associated with the impenetrable obstacle (or the refractive index ) and corresponding to the incident field , . Let denote the ball centered at the origin with radius with denoting the boundary of . By appropriately choosing , it can be ensured that is not a Dirichlet eigenvalue of in . Here, is called a Dirichlet eigenvalue of in a bounded domain if the the interior Dirichlet boundary value problem

 {Δu+k2u=0inV,u=0on∂V

has a nontrivial solution . The above assumption on and can be easily satisfied since the Dirichlet eigenvalues of in a bounded domain are discrete and satisfy the strong monotonicity property [28, Theorem 4.7] (see also the arguments in the proof of [14, Theorem 5.2]). Let denote the unbounded component of the complement of . Then we have the following global uniqueness results for the phaseless inverse scattering problems.

###### Theorem 3.1

Let be two bounded domains and let be large enough so that . Assume that is not a Dirichlet eigenvalue of in .

(a) Assume that and are two impenetrable obstacles with boundary conditions and , respectively. If the corresponding total fields satisfy

 |w1(x,y)|=|w2(x,y)|,∀(x,y)∈(∂BR1×∂BR1)∪(∂BR2×({y0}∪∂BR2)), x≠y \hb@xt@.01(3.1)

and

 |w1(x;y,y0)|=|w2(x;y,y0)|,∀(x,y)∈(∂BR1×∂BR1)∪(∂BR2×∂BR2), x≠y,y0 \hb@xt@.01(3.2)

for an arbitrarily fixed , then and .

(b) Assume that are the indices of refraction of two inhomogeneous media with supported in . If the corresponding total fields satisfy (LABEL:eq1) and (LABEL:eq2), then .

To prove Theorem LABEL:tt, we need the following lemmas on the property of the total field.

###### Lemma 3.2

Let and let be a bounded domain such that . Suppose is the total field of the obstacle scattering problem (LABEL:he)-(LABEL:rc) or the medium scattering problem (LABEL:he-n)-(LABEL:rc-n) associated with the point source . Then, for any fixed we have

 w(x,y0)≢0,x∈∂BR1,x≠y0, \hb@xt@.01(3.3) w(x,y0)≢0,x∈∂BR2, \hb@xt@.01(3.4) w(x,y)≢0,(x,y)∈∂BR2×∂BR2,x≠y. \hb@xt@.01(3.5)

Proof. Since is singular at or , we know that (LABEL:01) and (LABEL:03) are true.

We now prove (LABEL:02). Assume to the contrary that for , that is, for . Then, by the uniqueness of the exterior Dirichlet problem it follows that for all . Since the scattered field is analytic for and is analytic for , we have for all . This is a contradiction since has a singularity at and is analytic when is in a neighbourhood of . Thus, (LABEL:02) is true.

###### Lemma 3.3

Under the assumption of Lemma LABEL:l1, we have the following results.

There exist two open sets such that and for all .

There exist two open sets such that and for all , where .

Proof. We only prove (ii). The proof of (i) is similar.

By (LABEL:02) we know that for there exists such that . Since is continuous for with , there exists a neighbourhood of such that for all . Further, since is analytic with respect to and , respectively, when , then it follows from (LABEL:03) that there exist two points and such that with . Finally, again by the continuity of for with , there exists a neighbourhood of and a neighbourhood of such that and for all . Thus, for all . This completes the proof.

Proof of Theorem LABEL:tt. From (LABEL:lsp-t) it is easy to see that (LABEL:eq2) is equivalent to the equation

 |w1(x,y)+w1(x,y0)|=|w2(x,y)+w2(x,y0)|

for all with . This, together with (LABEL:eq1), implies that

 Re{w1(x,y)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯w1(x,y0)}=Re{w2(x,y)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯w2(x,y0)} \hb@xt@.01(3.6)

for all with . Define , . Then it follows from (LABEL:eq1) that , for all with , so we can write

 wj(x,y)=r(x,y)eiϑj(x,y),∀x,y∈∂BR1∪∂BR2,x≠y,j=1,2

with real-valued functions , .

Case 1. (LABEL:real) holds with .

Since are analytic functions of and , respectively, and has a singularity at , then, by Lemma LABEL:l2 we can choose two open sets small enough so that , for all , and are analytic with respect to and , respectively.

Now, by (LABEL:real) we have

 cos[ϑ1(x,y)−ϑ1(x,y0)]=cos[ϑ2(x,y)−ϑ2(x,y0)] \hb@xt@.01(3.7)

for all . Since are real-valued analytic functions of and , respectively, we have either

 ϑ1(x,y)−ϑ1(x,y0)=ϑ2(x,y)−ϑ2(x,y0)+2pπ,∀(x,y)∈U1×U2 \hb@xt@.01(3.8)

or

 ϑ1(x,y)−ϑ1(x,y0)=−[ϑ2(x,y)−ϑ2(x,y0)]+2pπ,∀(x,y)∈U1×U2, \hb@xt@.01(3.9)

where .

For the case when (LABEL:cos1) holds, we have

 ϑ1(x,y)−ϑ2(x,y)=ϑ1(x,y0)−ϑ2(x,y0)+2pπ,∀(x,y)∈U1×U2.

This implies that depends only on . Then it follows that

 w1(x,y)=r(x,y)eiϑ1(x,y)=r(x,y)eiα(x)+iϑ2(x,y)=eiα(x)w2(x,y)

for all and . By the analyticity of in with , we get

 w1(x,y)=eiα(x)w2(x,y),∀x∈U1,y∈∂BR1,x≠y. \hb@xt@.01(3.10)

Changing the variables and in (LABEL:*) gives

 w1(y,x)=eiα(y)w2(y,x),∀x∈∂BR1,y∈U1,x≠y. \hb@xt@.01(3.11)

Use (LABEL:*), (LABEL:eq3) and the reciprocity relation that for all (see [14, Theorem 3.17]) to give

 eiα(x)w2(x,y)=eiα(y)w2(x,y),∀x,y∈U1withx≠y. \hb@xt@.01(3.12)

Since has a singularity at , and by (LABEL:**) and the analyticity of with respect to and , respectively, with , it follows that for all with . This means that for all , where is a real constant. Substituting this formula into (LABEL:*) gives that for all with . Again, by the analyticity of with respect to with we have

 w1(x,y)=eiαw