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

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.


Introduction
The inverse source problem is concerned with the reconstruction of an unknown/inaccessible 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 [22,23] 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.

Problem formulation
Consider the following time-harmonic Maxwell system in R 3 , with the Silver-Müller radiation condition Throughout the rest of the paper, we use non-bold and bold fonts to signify scalar and vectorial quantities, respectively. In (2.1), E denotes the electric filed, H denotes the magnetic filed, J is an electric current density, ω denotes the frequency, ε 0 denotes the electric permittivity and µ 0 denotes the magnetic permeability of the isotropic homogeneous background medium. By eliminating H or E in (2.1), we obtain where k := ω √ µ 0 ε 0 . With the help of the vectorial Green function [19], the radiated field can be written as and respectively, where I is the 3 × 3 identity matrix and is the fundamental solution to the Helmholtz equation. The radiating fields E, H to the Maxwell system have the following asymptotic expansion [7] E and by using the integral representations (2.2) and (2.3), we have In what follows, we always assume that the electromagnetic source is a volume current that is supported in D. As mentioned earlier, there exists non-radiating sources that produce no radiating field outside D. 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 J only consists of radiating source, which is independent of the wavenumber k and of the form where D is a cube. Furthermore, the current density distribution J satisfies the transverse electric (TE) and transverse magnetic (TM) decomposition; that is, the source can be expressed in the form where f ∈ L 2 (D) and g ∈ H 1 (D). We also refer to [20] for more details on the TE/TM decomposition. Here, p is the polarization direction which is assumed to be known and yields the following admissible set From (2.4) and (2.5), it is clear that 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 S 2 + , say x 3 ≥ 0. 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 p ∈ R 3 and a finite number of wavenumbers {k}, we intend to recover the electromagnetic source J defined in (2.6) from the electric far-field data {E ∞ (x k ; k, p)} or the magnetic far-field data {H ∞ (x k ; k, p)}, wherex k depends on the wavenumber k andx k ∈ S 2 + .

Uniqueness
Prior to our discussion, we introduce some notations and relevant Sobolev spaces. Without loss of generality, we let Introduce the Fourier basis functions that are defined by By using the Fourier series expansion, the scalar functions f ∈ L 2 (D) and g ∈ H 1 (D) can be written as where the Fourier coefficients are given bŷ Therefore the Fourier expansion of the current density J is The proposed reconstruction scheme in the current article is based on determining the Fourier coefficientsf l andĝ l of the current density by using the corresponding electric or magnetic far-field data. For the subsequent use, we introduce the Sobolev spaces with In addition, the wavenumber cannot be zero in (2.4) and (2.5). Following [23], we introduce the following definition of wavenumbers.
Definition 3.1 (Admissible wavenumbers). Let λ be a sufficiently small positive constant and the admissible wavenumbers can be defined by Correspondingly, the observation direction is given bŷ By virtue of Definition 3.1, the Fourier basis functions defined in (3.1) could be written as Next we state the uniqueness result.
Proof. Let J be the electromagnetic source that produces the electric far-field data First, we consider the recovery of J by the magnetic far-field data. For every l ∈ Z 3 \{0}, using (2.5) and (3.4), we have From (2.7) and (3.6), we see that {x l , p ×x l ,x l × (p ×x l )} forms an orthogonal basis of R 3 . Multiplyingx l × p on the both sides of (3.7), and using the orthogonality, we obtainf Similarly, multiplyingx l × (p × l) on the both sides of (3.7), we havê For l = 0, we have Multiplyingx 0 × p on the both side of the last equation, and also using the orthogonal property, we obtain Thus, Next, we consider the recovery of J by the electric far-field data. For every l ∈ Z 3 \{0}, using (2.4) and (3.4), we have Through straightforward calculations, one can verify that Combining the last two equations, one can show that Multiplying p on the both sides of (3.12), and using the orthogonality, we obtain Similarly, multiplying p × l on the both sides of (3.12), we obtain (3.14) For l = 0, we have Multiplying p on the both sides of the last equation, and also using the orthogonality, we obtain .
The proof is complete.
In practical computations, we have to truncate the infinite series by a finite order N ∈ N to approximate J by wheref 0 could be represented by magnetic far-field 16) or electric far-field

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 H δ ∞ (x l ; k l ) such that where δ > 0. We first present two auxiliary results.
where constants C 1 , C 2 , C 3 , C 4 and C 5 depend on f, g, a and λ.
Lemma 4.1. [22] Let J be a vector function in (H σ p (D)) 3 and 0 ≤ µ ≤ σ, then the following estimate holds The stability result is contained in the following theorem. 3 and 0 ≤ µ ≤ σ, then the following estimate holds where C 6 , C 7 , C 8 depend only on f, g, a and λ.
Proof. It is readily seen that where C 6 = max {C 3 , C 4 , C 5 }. Hence, from (4.4) and Lemma 4.1 , we obtain which completes the proof.

Numerical examples
In this section, we carry out a series of numerical experiments to illustrate that the proposed Fourier reconstruction method is effective and efficient. First, we briefly describe some parameters setting of our numerical experiments. Let D = [−0.5, 0.5] 3 , namely, a = 1. Assume that the wave propagates in the vacuum space, where µ 0 = 4π × 10 −7 and ε 0 = 8.8541 × 10 −12 . Synthetic electromagnetic farfield data are generated by solving the direct problem of (2.1) by using the quadratic finite elements on a truncated spherical domain enclosed by a PML layer. The mesh of the forward solver is successively refined till the relative error of the successive measured electromagnetic wave data is below 0.1%. To show the stability of our proposed method, we also add some random noise to the synthetic far-field data by considering where r 1 and r 2 are two uniform random numbers, both ranging from −1 to 1, and δ > 0 represents the noise level. From Remark 4.1, the truncation N is given by where [X] denotes the largest integer that is smaller than X + 1. Next, we specify details of obtaining the artificial multi-frequency electromagnetic far-field data. Let then the wavenumber set is given by and the observation directions are given by Thus, every wavenumber and observation direction can be denoted by k j ∈ K N and x j ∈ X N , respectively, where j = 1, 2, · · · , (2N + 1) 3 . Correspondingly, the frequency ω j is chosen as ω j = k j / √ µ 0 ε 0 . With the admissible wavenumbers defined earlier, the artificial electromagnetic far-field data with noise can be written as Finally, we specify details of the numerical inversion via the Fourier method. We reconstruct the electric current source J (x), x ∈ D by the truncated Fourier expansion Given the noisy far-field data defined above, if we use the electric far-field data {E δ ∞ (x j ; k j )}, then the Fourier coefficientsf l ,ĝ l , 1 ≤ |l| ∞ ≤ N andf 0 are computed by (3.13), (3.14) and (3.17), respectively. If we use the magnetic far-field data {H δ ∞ (x j ; k j )}, then the Fourier coefficientsf l ,ĝ l , 1 ≤ |l| ∞ ≤ N andf 0 are computed by (3.8), (3.9) and (3.16), respectively. Divide the domain D into a mesh with a uniform grid of size 50×50×50. The approximated Fourier series J δ N (z) are computed at the mesh nodes z j , j = 1, 2, · · · , 50 3 by (3.15). The relative error is defined as Unless specified otherwise, we use the magnetic far-field data to reconstruct the electric current source. Based on the above discussion, we formulate the reconstruction scheme by the Fourier method in Algorithm S as follows.
Algorithm S: Fourier method for reconstructing the electromagnetic source Step 1 Choose the parameters λ, N , the wavenumber set K N and observation direction set X N .
Step 2 Collect the measured electric far-field data E δ ∞ (x j ; k j ) or the magnetic far-field data H δ ∞ (x j ; k j ) forx j ∈ X N and k j ∈ K N .
Step 4 Select a sampling mesh T h in a region D. For each sampling point z j ∈ T h , calculate the imaging functional J N defined in (3.15), then J N is the reconstruction of J .
Example 1. In this example, we numerically estimate the stability of the proposed method. We consider the following smooth source function   Figure 2 shows the comparison between the exact and the reconstructed source function at the plane x 3 = 0 with the additional noise δ = 2%. We observe that the reconstructions are very close to the exact one. To exhibit the accuracy quantitatively, we list the relative errors in L 2 in table 1. Meanwhile, table 1 illustrates that the stability and CPU time increase as the truncation order N (δ) increases.
Example 2. In this example, we use the electric far-field data to recover the source. We aim to recover a smooth source as follows J = pf + p × ∇g; . Figure 3 presents the iso-surface plots of the exact source and the reconstruction with noise 2% , which demonstrate clearly that our proposed method performance nicely.
Example 3. In this example, we consider a discontinuous source function. For simplicity, the source function is given by    Figure 5 shows the Gibbs phenomenon of the reconstructions over the line x 2 = x 3 = 0 with the truncation order N = 5, 15, 25, respectively.