{'fraction_non_alphanumeric': 0.045237202325145325, 'fraction_numerical': 0.01701668854303394, 'mean_word_length': 4.231240804315841, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Identification of Electromagnetic Dipoles from Multi-frequency Sparse Electric Far Field Patterns

The inverse electromagnetic source scattering problem from multi-frequency sparse electric far field patterns is considered. The underlying source is a combination of electric dipoles and magnetic dipoles. We show that the locations and the polarization strengths of the dipoles can be uniquely determined by the multi-frequency electric far field patterns at sparse observation directions. The unique arguments rely on some geometrical discussions and ingenious integrals of the electric far field patterns with properly chosen functions. Motivated by the uniqueness proof, we introduce two indicator functions for locating the magnetic dipoles and the electric dipoles, respectively. Having located all the dipoles, the formulas for computing the corresponding polarization strengths are proposed. Finally, some numerical examples are presented to show the validity and robustness of the proposed algorithm.


Introduction
The inverse scattering of acoustic and electromagnetic waves plays an important role in many different areas such as radar, medical imaging, nondestructive testing and geophysical prospection. In this work, we focus on imaging the unknown objects from the measurements taken at sparse sensors. Such a problem arises from many applications where the measurements are difficult or even impossible to be taken all around the unknown objects. Although at a fixed sensor, we can vary the frequency to obtain more data, this is still a small set of data, which brings many difficulties to the problem.
The first work in this direction is given by Sylvester and Kelly in [13], where they showed a convex polygon containing the unknown source can be uniquely determined from the far field patterns taken at all frequencies, but only at a few observation directions. We refer the reader to [6] and [2] for a factorization method and a direct sampling method, respectively, using multi-frequency far field patterns taken at finitely many observation directions. The direct sampling method has also been applied for locating a perfect conductor with multi-frequency sparse backscattering far field data [3]. If the multi-frequency sparse scattered fields are measured, the corresponding uniqueness discussion and direct sampling method can be found in a recent work [9].
Note that with the multi-frequency sparse measurements, the uniqueness of an extended source is still an open problem. If it is known a priori that the object is a combination of point sources, based on some novel geometrical discussions, uniqueness for the locations and the scattering strengths can be established [7,8,9]. These uniqueness results also motivates novel direct sampling methods for locating the point sources and the formulas for computing the scattering strengths. A quality-Bayesian approach is proposed in [11,12] to reconstruct the locations and intensities of the unknown acoustic sources. We also refer to [5] for an MUSIC-type reconstruction method for small inhomogeneities by multi-frequency sparse far field patterns.
In this paper we consider the inverse source scattering problem for time harmonic electromagnetic waves using multi-frequency sparse electric far field patterns. For an extended source, it is shown in [10] that the smallest strip containing the source support with the observation direction as the normal can be uniquely determined by the multi-frequency far field pattern at a single observation direction. Similar to the acoustic case, a convex support of the extended source is then expected to be reconstructed with the data in finitely many observation directions. This paper focus on the case that the source to be determined is a combination of electric dipoles and magnetic dipoles. Different to the general source functions, the dipoles are characterized by their locations and polarization strengths. This enables us to study the uniqueness analyses and numerical algorithms with less data. This is also a nontrivial extension of the results [7,8] for inverse acoustic source scattering problems. Actually, in contrast to the acoustic case, the electric field generated by different types of dipoles are coupled together, which makes the problem more complicated. Inspired by [7], to distinguish the dipoles, we introduce two weighted integral using electric far field pattern at two opposite observation directions. Another difficulty is due to the fact that the electric far field pattern vanishes if the observation direction is parallel to the polarization direction of the dipole. Hence, more properly chosen directions are necessary to determine the dipoles when the polarization directions are unknown. Note that the electromagnetic fields generated by the electric dipoles and magnetic dipoles can be viewed as the fundamental solution of the Maxwell equations [4], thus the results obtained in this paper are expected to be helpful for understanding the inverse scattering problems with extended sources or scatterers.
The remaining part of this paper is organized as follows. In the next section, we fix the notations and introduce the scattering of the mixed electric dipoles and magnetic dipoles. Some uniqueness results are shown in section 3. Motivated by the uniqueness arguments, Section 4 is devoted to introduce two indicator functions for locating the two types of dipoles, respectively, and the numerical formulas for computing the corresponding polarization strengths. Numerical examples are presented in section 5 to verify the proposed numerical algorithm.

Scattering due to electromagnetic dipoles
We consider the scattering of the electromagnetic point sources in a homogeneous isotropic medium in R 3 . The point sources are a combination of magnetic dipoles and electric dipoles.
Let k > 0 be the wave number. For a magnetic dipole located at z ∈ R 3 , the corresponding electromagnetic field is given by [4] E mag (x, z, q, k) := curl x qΦ k (x, z), H mag (x, z, q, k) : is the fundamental solution to the Helmholtz equation ∆u + k 2 u = 0, q := τ p is the polarization strength given by product of the strength τ ∈ C and the polarization direction p ∈ S 2 := {x ∈ represent the electromagnetic field generated by an electric dipole. As mentioned in page 230 of [4], the fields (2.1) and (2.2) can be viewed as the fundamental solution of the Maxwell equations By straightforward calculations, it can be seen that and Here, E ∞ mag and E ∞ elec defined on the unit sphere S 2 are known as the electric far field patterns for the magnetic dipole and the electric dipole, respectively.
We consider an array of M dipoles at locations z 1 , z 2 , · · · , z M ∈ R 3 in the homogeneous isotropic space R 3 . Denote by q m := τ m p m ∈ C 3 the polarization strength vector of the m-th dipole. Without loss of generality we assume that the first M 1 dipoles are magnetic dipoles and the other M 2 := M − M 1 dipoles are electric dipoles. Then, the electric field due to M dipoles is given by Correspondingly, the electric far field pattern is given by Denote by the set of L properly chosen observation directions. The inverse scattering problem is to identify the dipoles from the multi-frequency electric far field patterns E ∞ (x, k) at sparse observation directionsx ∈ Θ L . More specifically, we want to determine the locations z 1 , z 2 , · · · , z M , to reconstruct the polarization strength vectors q 1 , q 2 , · · · , q M and to clarify the dipole type of each dipole.

Uniqueness of inverse electromagnetic point sources problem
In this section, we investigate under what conditions the dipoles can be identified by the multifrequency sparse electric far field patterns.
(1) With the help of the vector identity a × (b × c) = (a · c)b − (a · b)c, noting the fact thatx 1 ∈ S 2 , we have Taking the vector product of (3.7) withx 2 yields that Noting thatx 1 andx 2 are non-collinear, we havex 1 ×x 2 = 0. Therefore, With the help of (2.3), we havê Inserting (3.8) and (3.9) into (3.7) we derive that which is actually the formula for computing q 1 from the electric far field patterns E ∞ (x j , k), j = 1, 2.
(2) With the help of the representation (2.3), we have This implies thatx · z 1 is uniquely determined by E ∞ (x, k) when |kx · z 1 | < π. The uniqueness of z 1 then follows by noting that the three observation directionsx j , j = 1, 2, 3 are linearly independent.
Note that, in the above local uniqueness results, we have only used the electric far field pattern for a fixed wave number k. The following theorem shows that both the location z 1 and polarization strength q 1 can be uniquely determined when multiple frequencies are used.

Proof. Under the assumption that
Using the far field representation (2.3), we have where · denotes the complex conjugate. Integrating the above equation with respect to k over (k − , k + ), in terms of (3.11), we obtain Note also that (3.11). Thereforê , j = 1, 2, 3 by the fact thatx 1 ,x 2 andx 3 are three linearly independent directions. Finally, the uniqueness of q 1 is obtained by the first result in Theorem 3.1.
After slight modifications of the proof, the analogous uniqueness results of Theorem 3.1 and Theorem 3.2 can be formulated for a single electric dipole. An interesting question is how to distinguish the dipole types from the electric far field patterns. We will give an answer in the next subsection for more complex multiple mixed dipoles.

Uniqueness for multiple mixed dipoles
In this subsection we study the much more complicated case with multiple mixed dipoles. As shown in (2.3)-(2.5), an obvious difficulty is how to decouple the electric dipoles and the magnetic dipoles. Besides, for multiple dipoles, the proper choice of the observation directions plays an important role in locating the dipoles and clarifying the dipole type.
Recall the observation direction set Θ L given in (2.6). Define by the plane passing through z m with normalx l ∈ Θ L . For any point z ∈ R 3 , denote by Assume that any three directions in Θ L are not coplanar, then Then, by the pigeonhole principle, there exists a point z m such that at least three planes Π l 1 ,m , Π l 2 ,m , Π l 3 ,m pass through z * , i.e., However,x l j , j = 1, 2, 3 are not coplanar, which implies z * − z m = 0. This is a contradiction to z * ∈ R 3 \ {z 1 , z 2 , · · · , z M }. Lemma 3.3 shows that dipoles can be uniquely determined when L ≥ 2M + 1 if f (z) can be calculated for all z ∈ R 3 from the electric far field patterns. By (2.5), for dipole with polarization direction p m , the electric far field at the observation direction ±p m is zero, which means they contain no information about the dipole. Therefore, given dipole (z m , τ m , p m ) and observationx l , it might be impossible to calculatex l · z m and to obtain Π l,m from the electric far field pattern. Fortunately, the number of observation directions that don't work is finite, which indicates uniqueness is possible when the number of observation directions L is large enough.
Proof. Givenx ∈ Θ L and z ∈ R 3 , define We have For any K > 0, in view of (2.5) we have Letting K → ∞, we see that Obviously, fx ,mag (z) ≥fx ,mag (z). By Lemma 3.3, we have On the other hand, for fixed z m , 1 ≤ m ≤ M 1 , there are at most 2( In terms of (3.19) and (3.20), we deduce that {z m |1 ≤ m ≤ M 1 } = {z ∈ R 3 | x∈Θ Lfx ,mag (z) > L/2} is uniquely determined. For each magnetic dipole (z m , q m ), 1 ≤ m ≤ M 1 , we can always find at least two linearly independent observation directionsx 1 andx 2 such that F mag (z m ,x j ) =x j × q m , j = 1, 2.
Following the arguments in the proof of Theorem 3.1 (1) we have For electric dipoles, we define

Inserting (2.5) into (3.23) and letting K → ∞ we have
The uniqueness of the locations {z M 1 +1 , z M 1 +2 , · · · , z M } for the electric dipoles follows from similar arguments for the magnetic dipoles. For each z m ∈ {z M 1 +1 , z M 1 +2 , · · · , z M }, similar to the case of magnetic dipoles, we have at least two linearly independent observation directionsx 1 andx 2 such that (3.25) In view of (3.7) we have From this and (3.25) we obtain that Therefore,

Inserting this into (3.26), we derive that
which is a formula for computing q m from F elec (z m ,x 1 ) and F elec (z m ,x 2 ). This completes the proof of the theorem.
Finally, we want to remark that less data is needed provided a priori information on the physically property of the dipoles. For example, assume that all the point sources (z m , q m ) are magnetic dipoles with q m ∈ R 3 , 1 ≤ m ≤ M . Then Therefore, the magnetic dipoles are uniquely determined from only half of the data used in Theorem 3.4, i.e, the multi-frequency electric far field patterns E ∞ (x, k),x ∈ Θ L , k ∈ (0, ∞).

Uniqueness when dipoles are in a plane
In this subsection, we consider an interesting case that all of the dipoles are located in a known plane. Less data is then needed to identify the dipoles. Without lose of generality, suppose that the known plane is It's interesting that the observation directions set Θ L can be a subset of Π, while we require any three directions are not coplanar in subsection 3.2.
Using Lemma 3.5 and following the proof of Theorem 3.4, we immediately have the following unique result with less data.

Numerical algorithms
Following the idea in the proof of uniqueness results in the previous section, we introduce some numerical algorithms for locating the dipoles and reconstructing the corresponding polarization strengths.

Magnetic dipoles
As the discussion in the uniqueness analyses, we begin with identifying the magnetic dipoles. and

Indicator for locating the magnetic dipoles
By the analysis in the proof of Theorem 3.4, I mag (z, K, ) goes to zero outside the planes Π l,M , 1 ≤ l ≤ L, 1 ≤ m ≤ M 1 when K tends to infinity. Furthermore, with the increase of the number L of the properly chosen observation directions, I mag (z, K, ) tends to one for z ∈ {z 1 , z 2 , · · · , z M 1 }. Therefore, the indicator (4.28) can be used to locate the magnetic dipoles with sufficient many properly chosen observation directions and sufficiently large K. Finally, we want to give a remark on the auxiliary function (4.29). From the point of view of numerical computations, the strength τ m plays the role of weight for the m−th dipole, therefore the dipoles with quite small strengths are difficult to be located. The introduction of the auxiliary function (4.29) is used to balance these weights and to avoid missing the dipoles with small strengths. Practically, we may choose the cut-off value such that |q m | > for all m = 1, 2, · · · , M 1 .

Identifying the polarization strengths of magnetic dipoles
Having located all the magnetic dipoles, we now consider the corresponding polarization strengths. For a magnetic dipole (z m * , q m * ), we take two linearly independent directionŝ x m * ,ŷ m * ∈ Θ L such that (4.31) In terms of (3.21), we have (4.32) Numerically, we replace F mag (z m * ,x), F mag (z m * ,ŷ) by F mag (z m * ,x, K), F mag (z m * ,ŷ, K) and set (4.33) Proof. With the help of (3.16) we have From this, comparing (4.32) and (4.33) we have This completes the proof.
Letting K → ∞ we see that Therefore, q m * ,K can be an approximation to q m * for large K.

Electric dipoles
Having identified the magnetic dipoles, we look for the electric dipoles. We omit the analysis since it's similar to the case of magnetic dipoles.

Indicator for the locations of electric dipoles
We define as the indicator function for locating the electric dipoles, where T defined in (4.29) and Moreover, we define

Identifying the polarization strengths of electric dipoles
Given all the location of electric dipoles {z M 1 +1 , z M 1 +2 , · · · , z M }, for the m * th electric dipole, we take two linearly independent directionsx m * ,ŷ m * such that In view of (3.27), we have (4.38) Replacing F elec (z m * ,x m * ), F elec (z m * ,ŷ m * ) by F elec (z m * ,x m * , K), F elec (z m * ,ŷ m * , K), respectively, we get the approximation of q m * by

Algorithm for all dipoles
We combine all the indicators and formulas introduced in the previous subsections to form the following imaging algorithm. Imaging Algorithm for multiple mixed type dipoles.
• Select a sampling region in R 3 with a fine mesh containing all the dipoles.
• Locate all the magnetic dipoles by plotting the indicator I mag (z, K, ) given in (4.28) with properly chosen > 0. Reconstruct the corresponding polarization strengths by the formula (4.33).
• Locate all the electric dipoles by plotting the indicator I elec (z, K, ) given in (4.34) with properly chosen > 0. Reconstruct the corresponding polarization strengths by the formula (4.39).
If not otherwise stated, the observation directionsx l = (x 1 l , x 2 l , x 3 l ) in Θ L are selected by Fibonacci lattices: is the golden ratio to ensure that the lattices are evenly spaced [1]. Note that we consider the Fibonacci lattices just because we guess any three Fibonacci lattices are not coplanar. However, to our best knowledge, this is still not be proved.

Identifying the locations and the polarization strengths of the dipoles
In the first example, we consider the mixed type point sources with 3 magnetic dipoles and 3 electric dipoles. The locations and polarization strengths are shown in the second and third columns of Table 1 Table 1, in the hyperplane z 1 = 0, there are two magnetic dipoles located at (0, −1, 0) and (0, 0, −1), respectively. Obviously, Figure 1(a) shows that these two magnetic dipoles are well captured by the indicator I 4 mag . Meanwhile, as shown in Figure 1(b), the two electric dipoles located in the hyperplane z 1 = 0 are clearly reconstructed by the indicator I 4 elec . Considering 10% relative noise in the measurement data, the location reconstructions are quite stable. This can also be seen in Figure 1(c-f) for the other reconstructions in the coordinate planes.
Having located all the dipoles, we compute the corresponding polarization strengths by the formulas (4.33) and (4.39). For each strength reconstruction, we have used the electric far field patterns with K = 200 at two selected observation directions satisfying (4.31) or (4.37). The fourth column of Table 1 shows the reconstructed polarization strengths. Considering 10% relative noise in the measurements again, the relative errors for the strengths shown in the fifth column of Table 1 are acceptable.

Locating multiple magnetic dipoles using different observation directions
In the second example, we consider locating M 1 = 19 magnetic dipoles. The positions and polarization strengths vector are presented in Table 2. All the dipoles are located in the hyperplane Π := {(z 1 , z 2 , z 3 ) ∈ R 3 |z 3 = 0}. We consider the following three observation direction sets:  • A single pair of directions ±x.
• L directions in the unit sphere S 2 chosen by the Fibonacci lattices (5.41); • L equally distributed directions in the hyper-plane Π, i.e., Due to the a priori information that all the dipoles are located in Π, we consider the research domain [−2, 2] × [−2, 2] × 0 with 41 × 41 equally spaced sampling points. Figure 2 shows the reconstructions with a single pair of directions ±x. The highlighted lines show that there must be at least one dipole located in the lines. Conversely, the line passing through the dipoles may be missed. As shown in the Figure 2(a), we have observe only five lines z 1 = −0.8, −0.2, 0, 0.8 and 1. The lines passing through the dipoles with polarization strengths paralleling tox = (1, 0, 0) are missed. This is obvious by noting the representation (2.5) of the electric far field pattern. Figure 3 shows the results with more observation directions in S 2 . Obviously, with the increase of the number of the observation directions, the unknown dipoles are clearly located and the false positions are removed. Note that L = 40 < 4M 1 = 76, i.e., we have used less data than those needed in the uniqueness Theorem 3.4.
Once we know that all the dipoles are in the plane Π, by Theorem 3.6, the locations of magnetic dipoles can be determined from electric far field pattern in the directions in Π. Figure  4 shows the reconstruction using the third direction set. Obviously, with the same direction number L, the reconstructions in Figure 4 are better than those in Figure 3.

Conclusion and remark
We have studied the uniqueness and numerical algorithm for identifying the mixed type dipoles. The novel geometrical arguments and ingenious integrals of the multi-frequency sparse electric far field patterns with properly chosen functions are key to the analyses. The numerical examples further verify our theoretical analyses and numerical algorithm. Precisely, the dipoles are well located and distinguished by the proposed indicator functions. Furthermore, the corresponding polarization strengths can also be reconstructed by the proposed formulas. Similar ideas can also be applied to the multi-frequency electric fields taken at sparse sensors. However, more complex geometrical discussion will be involved. As mentioned in the introduction part, the dipoles can be viewed as the fundamental solution to the Maxwell equations. Thus we expect that our indicator functions are also applicable for determining the extended sources.