Fourier method for identifying electromagnetic sources with multi-frequency far-field data

# Fourier method for identifying electromagnetic sources with multi-frequency far-field data

Xianchao Wang , Department of Mathematics, Harbin Institute of Technology, Harbin, China. Email: xcwang90@gmail.com    Minghui Song, Department of Mathematics, Harbin Institute of Technology, Harbin, China. Email: songmh@hit.edu.cn    Yukun Guo , Department of Mathematics, Harbin Institute of Technology, Harbin, China. Email: ykguo@hit.edu.cn    Hongjie Li , Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, China. Email:hongjieli@yeah.net    Hongyu Liu Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, China. Email: hongyuliu@hkbu.edu.hk
###### Abstract

We consider the inverse problem of determining an unknown vectorial source current distribution associated with the homogeneous Maxwell system. We propose a novel non-iterative reconstruction method for solving the aforementioned inverse problem from far-field measurements. The method is based on recovering the Fourier coefficients of the unknown source. A key ingredient of the method is to establish the relationship between the Fourier coefficients and the multi-frequency far-field data. Uniqueness and stability results are established for the proposed reconstruction method. Numerical experiments are presented to illustrate the effectiveness and efficiency of the method.

Keywords:  inverse source problem, Maxwell’s system, Fourier expansion, multi-frequency, far-field

2010 Mathematics Subject Classification:  35R30, 35P25, 78A46

## 1 Introduction

The inverse source problem is concerned with the reconstruction of an unknown/inaccess-ible active source from the measurement of the radiating field induced by the source. The inverse source problem arises in many important applications including acoustic tomography [3, 6, 15, 16], medical imaging[2, 4, 12] and detection of pollution for the environment[10]. In this paper, we are mainly concerned with the inverse source problem for wave propagation in the time-harmonic regime. In the last decades, many theoretical and numerical studies have been done in dealing with the inverse source problem for wave scattering. The uniqueness and stability results can be found in [5, 14]. Several numerical reconstruction methods have also been proposed and developed in the literature. For a fixed frequency, we refer the reader to [2, 9, 13]. However, with only one single frequency, the inverse source problem lacks of stability and it leads to severe ill-posedness. In order to improve the resolution, multi-frequency measurements should be employed in the reconstruction [5, 11, 21].

The goal of this paper is to develop a novel numerical scheme for reconstructing an electric current source associated with the time-harmonic Maxwell system. Due to the existence of non-radiating sources [8, 17], the vectorial current sources cannot be uniquely determined from surface measurements. Albanese and Monk [1] showed that surface currents and dipole sources have a unique solution, but it is not valid for volume currents. Valdivia[21] showed that the volume currents could be uniquely identified if the current density is divergence free. Following the spirit of our earlier work [23, 22] by three of the authors of using Fourier method for inverse acoustic source problem, we develop a Fourier method for the reconstruction of a volume current associated with the time-harmonic Maxwell system. The extension from the scalar Hemholtz equation to the vectorial Maxwell system involves much subtle and technical analysis. First, we establish the one-to-one correspondence between the Fourier coefficients and the far-field data, so that the Fourier coefficients can be directly calculated. Second, the proposed method is stable and robust to measurement noise. This is rigorously verified by establishing the corresponding stability estimates. Finally, compared to near-field Fourier method, our method is easy to implement with cheaper computational costs.

The rest of the paper is organized as follows. Section 2 describes the mathematical setup of the inverse source problem of our study. The theoretical uniqueness and stability results of proposed Fourier method are given in Section 3 and Section 4, respectively. Section 5 presents several numerical examples to illustrate the effectiveness and efficiency of the proposed method.

## 2 Problem formulation

Consider the following time-harmonic Maxwell system in ,

 {∇×E−iωμ0H=0,∇×H+iωε0E=J, (2.1)

 lim|x|→+∞|x|(√μ0H×^x−√ε0E)=0,

where and . Throughout the rest of the paper, we use non-bold and bold fonts to signify scalar and vectorial quantities, respectively. In (2.1), denotes the electric filed, denotes the magnetic filed, is an electric current density, denotes the frequency, denotes the electric permittivity and denotes the magnetic permeability of the isotropic homogeneous background medium. By eliminating or in (2.1), we obtain

 ∇×∇×E−k2E=iωμ0J,

and

 ∇×∇×H−k2H=∇×J,

where . With the help of the vectorial Green function [19], the radiated field can be written as

 E(x)=iωμ0(I+1k2∇∇⋅)∫R3Φ(x,y)J(y)dy, (2.2)

and

 H(x)=∇×∫R3Φ(x,y)J(y)dy, (2.3)

respectively, where is the identity matrix and

 Φ(x,y)=eik|x−y|4π|x−y|,x≠y,

is the fundamental solution to the Helmholtz equation. The radiating fields to the Maxwell system have the following asymptotic expansion [7]

 E(x)=eik|x||x|{E∞(^x)+O(1|x|)},|x|→+∞, H(x)=eik|x||x|{H∞(^x)+O(1|x|)},|x|→+∞,

and by using the integral representations (2.2) and (2.3), we have

 E∞(^x)=iωμ04π(I−^x^x⊤)∫R3e−ik^x⋅yJ(y)dy, (2.4) H∞(^x)=ik4π^x×∫R3e−ik^x⋅yJ(y)dy. (2.5)

In what follows, we always assume that the electromagnetic source is a volume current that is supported in . As mentioned earlier, there exists non-radiating sources that produce no radiating field outside . Hence, without any a prior knowledge, one can only recover the radiating part of the current density distribution. In order to formulate the uniqueness result, we assume that the current density distribution only consists of radiating source, which is independent of the wavenumber and of the form

 J∈(L2(R3))3,supp J⊂D,

where is a cube. Furthermore, the current density distribution satisfies the transverse electric (TE) and transverse magnetic (TM) decomposition; that is, the source can be expressed in the form

 J=pf+p×∇g, (2.6)

where and . We also refer to [20] for more details on the TE/TM decomposition. Here, is the polarization direction which is assumed to be known and yields the following admissible set

 P:={p∈S2∣p×l≠0,∀ l∈Z3∖{0}}. (2.7)

From (2.4) and (2.5), it is clear that

 E∞(−^x)=−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯E∞(^x), H∞(−^x)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯H∞(^x),

where and also in what follows, the overbar stands for the complex conjugate in this paper. Therefore, for our inverse problem, the measurements of the far-field data could be from an upper hemisphere , say . Figure 1 provides a schematic illustration of the geometric setting of the measurements. With the above discussion, the inverse source problem of the current study can be stated as follows,

###### Inverse Problem.

Given a fixed polarization direction and a finite number of wavenumbers , we intend to recover the electromagnetic source defined in (2.6) from the electric far-field data or the magnetic far-field data , where depends on the wavenumber and .

## 3 Uniqueness

Prior to our discussion, we introduce some notations and relevant Sobolev spaces. Without loss of generality, we let

 D=(−a2, a2)3,a∈R+.

Introduce the Fourier basis functions that are defined by

 ϕl(x)=exp(i2πal⋅x),l∈Z3, x∈R3. (3.1)

By using the Fourier series expansion, the scalar functions and can be written as

 f=∑l∈Z3^flϕl,g=∑l∈Z3∖{0}^glϕl,

where the Fourier coefficients are given by

 ^fl=1a3∫Df(x)¯¯¯¯¯¯¯¯¯¯¯¯ϕl(x)dx, (3.2) ^gl=1a3∫Dg(x)¯¯¯¯¯¯¯¯¯¯¯¯ϕl(x)dx. (3.3)

Therefore the Fourier expansion of the current density is

 J=pf+p×∇g=p∑l∈Z3^flϕl+2πia∑l∈Z3∖{0}(p×l) ^glϕl. (3.4)

The proposed reconstruction scheme in the current article is based on determining the Fourier coefficients and of the current density by using the corresponding electric or magnetic far-field data. For the subsequent use, we introduce the Sobolev spaces with

 (Hσp(D))3:={pf+p×∇g∣f∈Hσ(D),g∈Hσ+1(D),p∈S2},

equipped with the norm

 ∥G∥p,σ=⎛⎜⎝∑l∈Z3(1+|l|2)σ|^fl|2+4π2a2∑l∈Z3∖{0}(1+|l|2)σ|p×l|2|^gl|2⎞⎟⎠1/2.

In addition, the wavenumber cannot be zero in (2.4) and (2.5). Following [23], we introduce the following definition of wavenumbers.

Let be a sufficiently small positive constant and the admissible wavenumbers can be defined by

 kl:=⎧⎪ ⎪⎨⎪ ⎪⎩2πa|l|,l∈Z3∖{0},2πaλ,l=0. (3.5)

Correspondingly, the observation direction is given by

 ^xl:={^l,l∈Z3∖{0},(1,0,0),l=0. (3.6)

By virtue of Definition 3.1, the Fourier basis functions defined in (3.1) could be written as

 ϕl(x)=exp(ikl^l⋅x),l∈Z3, x∈R3.

Next we state the uniqueness result.

###### Theorem 3.1.

Let and be defined in (3.5) and (3.6), then the Fourier coefficients and in (3.2) and (3.3) could be uniquely determined by or , where .

###### Proof..

Let be the electromagnetic source that produces the electric far-field data and the magnetic far-field data on .

First, we consider the recovery of by the magnetic far-field data. For every , using (2.5) and (3.4), we have

 H∞(^xl;kl) (3.7) =ikl4π^xl×∫D⎛⎜⎝p^f0e−ikl^xl⋅y+∑~l∈Z3∖{0}(p^f~l+2πia(p×~l)^g~l)ei(k~l^~l−kl^xl)⋅y⎞⎟⎠dy =ikla34π(^xl×p^fl+2πia^xl×(p×l)^gl).

From (2.7) and (3.6), we see that forms an orthogonal basis of . Multiplying on the both sides of (3.7), and using the orthogonality, we obtain

 ^fl=4π^xl×p⋅H∞(^xl;kl)ikla3|^xl×p|2. (3.8)

Similarly, multiplying on the both sides of (3.7), we have

 ^gl=−2^xl×(p×l)⋅H∞(^xl;kl)kla2|^xl×(p×l)|2. (3.9)

For , we have

 H∞(^x0;k0) =ik04π^x0×∫D⎛⎜⎝p^f0e−ik0^x0⋅y+∑l∈Z3∖{0}(p^fl+2πia(p×l)^gl)ei(kl^l−k0^x0)⋅y⎞⎟⎠dy.

Multiplying on the both side of the last equation, and also using the orthogonal property, we obtain

 ^x0×p⋅H∞(^x0;k0) =ik04π|^x0×p|2⎛⎜⎝a3^f0sinλπλπ+∑l∈Z3∖{0}^fl∫Dei(kl^l−k0^x0)⋅ydy⎞⎟⎠.

Thus,

 ^f0=λπa3sinλπ⎛⎜⎝4π^x0×p⋅H∞(^x0;k0)ik0|^x0×p|2−∑l∈Z3∖{0}^fl∫Dei(kl^l−k0^x0)⋅ydy⎞⎟⎠. (3.10)

Next, we consider the recovery of by the electric far-field data. For every , using (2.4) and (3.4), we have

 E∞(^xl;kl)=iωμ0a34π(I−^xl^x⊤l)(p^fl+2πia(p×l)^gl). (3.11)

Through straightforward calculations, one can verify that

 ^xl×(^xl×A)=−(I−^xl^x⊤l)A,∀A∈R3.

Combining the last two equations, one can show that

 E∞(^xl;kl)=iωμ0a34π(−^xl×(^xl×p)^fl+2πia(−^xl×(^xl×(p×l))^gl). (3.12)

Multiplying on the both sides of (3.12), and using the orthogonality, we obtain

 p⋅E∞(^xl;kl) =iωμ0a34π(−p⋅^xl×(^xl×p)^fl+2πia(−p⋅^xl×(^xl×(p×l))^gl) =iωμ0a34π((^xl×p)⋅(^xl×p)^fl+2πia(^xl×p)⋅(^xl×(p×l))^gl) =iωμ0a34π|^xl×p|2^fl.

Thus,

 ^fl=4πp⋅E∞(^xl;kl)iωμ0a3|^xl×p|2. (3.13)

Similarly, multiplying on the both sides of (3.12), we obtain

 ^gl=−2(p×l)⋅E∞(^xl;kl)ωμ0a2|^xl×(p×l)|2. (3.14)

For , we have

 E∞(^x0;k0) =iωμ04π(I−^x0^x⊤0)∫D⎛⎜⎝p^f0e−ik0^x0⋅y+∑l∈Z3∖{0}(p^fl+2πia(p×l)^gl)ei(kl^l−k0^x0)⋅y⎞⎟⎠dy.

Multiplying on the both sides of the last equation, and also using the orthogonality, we obtain

Thus,

 ^f0=λπa3sinλπ⎛⎜⎝4πp⋅E∞(^x0;k0)iωμ0|^x0×p|2−∑l∈Z3∖{0}^fl∫Dei(kl^l−k0^x0)⋅ydy⎞⎟⎠.

The proof is complete. ∎

In practical computations, we have to truncate the infinite series by a finite order to approximate by

 JN=p^f0+∑1≤|l|∞≤N(p^fl+2πia(p×l)^gl)ϕl, (3.15)

where could be represented by magnetic far-field

 ^f0≈λπa3sinλπ⎛⎝4π^x0×p⋅H∞(^x0;k0)ik0|^x0×p|2−∑1≤|l|∞≤N^fl∫Dei(kl^l−k0^x0)⋅ydy⎞⎠, (3.16)

or electric far-field

 ^f0≈λπa3sinλπ⎛⎝4πp⋅E∞(^x0;k0)iωμ0|^x0×p|2−∑1≤|l|∞≤N^fl∫Dei(kl^l−k0^x0)⋅ydy⎞⎠. (3.17)

## 4 Stability

In this section, we derive the stability estimates of recovering the Fourier coefficients of the electric current source by using the far-field data. We only consider the stability of using the magnetic far-field data, and the case with the electric far-field data can be treated in a similar manner. In what follows, we introduce such that

 |Hδ∞(^xl;kl)−H∞(^xl;kl)|≤δ|H∞(^xl;kl)|,

where . We first present two auxiliary results.

###### Theorem 4.1.

For , we have

 |^fδl−^fl|≤C1δ,1≤|l|∞≤N, (4.1) |^gδl−^gl|≤C2δ,1≤|l|∞≤N, (4.2) |^fδ0−^f0|≤C3δ+C4λNδ+C5λ√N, (4.3)

where constants and depend on and .

###### Proof..

For , from Schwarz inequality and (3.8), we have

 |^fδl−^fl| =∣∣ ∣∣4π^xl×pikla3|^xl×p|2⋅(Hδ∞(^xl;kl)−H∞(^xl;kl))∣∣ ∣∣ ≤4πikla3|^xl×p|δ|H∞(^xl;kl)| ≤δa3|^xl×p|∣∣∣^x×∫De−ik^xl⋅yJ(y)dy∣∣∣ ≤δa3|^xl×p||^x×p|∣∣∣∫De−ik^xl⋅y (f(y)+|∇g(y)|)dy∣∣∣ ≤δa3(∫D∣∣e−ik^xl⋅y∣∣2dy)1/2(∥f∥L2(D)+∥∇g∥L2(D)) ≤C1δ

where and it leads to estimate (4.1).

Correspondingly, from (3.9), we have

 |^gδl−^gl| ≤2kla2|^xl×(p×l)|δ|H∞(^xl;kl)| ≤δ2π|l|a2|^xl×p|∣∣∣^x×∫De−ik^xl⋅yJ(y)dy∣∣∣ ≤∥f∥L2(D)+∥g∥H1(D)2π|l|a1/2 δ ≤C2δ,

where and it verifies (4.2).

For , from Schwarz inequality and (3.16), we have

 |^fδ0−^f0|≤ λπa3sinλπ∣∣ ∣∣4π^x0×pik0|^x0×p|2⋅(Hδ∞(^x0;k0)−H∞(^x0;k0))∣∣ ∣∣ +λπa3sinλπ∑1≤|l|∞≤N∣∣∣(^fδl−^fl)∫Dei(kl^l−k0^x0)⋅ydy∣∣∣I1 +λπa3sinλπ∑|l|∞≥N∣∣∣^fl∫Dei(kl^l−k0^x0)⋅ydy∣∣∣I2 ≜ C3δ+I1+I2.

where .

Define , from (3.5) and (3.6), we find that

 ∫Dei(kl^l−k0^x0)⋅ydy=⎧⎪⎨⎪⎩a3sin(l1−λ)π(l1−λ)π,|l|=|l1|,0,|l|≠|l1|,

which together with (4.1) gives

 I1≤ λπa3sinλπ∑1≤|l|∞≤N∣∣∣^fδl−^fl∣∣∣∣∣∣a3sin(l1−λ)π(l1−λ)π∣∣∣ ≤ λπsinλπ2N∑j=1(C1δsinλπ(j−λ)π) ≤ C4λNδ,

where . On the other hand, one can deduce that

 I2≤ λπa3sinλπ∑|l|∞>N|^fl|∣∣∣a3sin(l1−λ)π(l1−λ)π∣∣∣ ≤ λπsinλπ⎛⎝∑|l|∞>N|^fl|2⎞⎠1/2⎛⎝∑|l|∞>N∣∣∣sin(l1−λ)π(l1−λ)π∣∣∣2⎞⎠1/2 ≤ λπsinλπ1a3∥f∥L2(D)(2∞∑j=N+1∣∣∣sinλπ(j−λ)π∣∣∣2)1/2 ≤ 2λa3√N∥f∥L2(D) = C5λ√N,

where . Finally, we obtain

 |^fδ0−^f0|≤C3δ+C4λNδ+C5λ√N.

The proof is complete. ∎

###### Lemma 4.1.

[22] Let be a vector function in and , then the following estimate holds

 ∥JN−J∥p,μ≤Nμ−σ∥J∥p,σ,0≤μ≤σ.

The stability result is contained in the following theorem.

###### Theorem 4.2.

Let and , then the following estimate holds

 ∥JδN−J∥p,μ≤C6δ+C6λNδ+C6λ√N+C7Nμ+3/2δ+C8Nμ+5/2δ+Nμ−σ∥J∥p,σ,

where depend only on and .