Direct Method for Reconstructing the Radiating Part of a Planar Source from its Far-Fields

A planar current is generally divided into a radiating part that generates propagation ﬁelds and a non-radiating part that generates evanescent ﬁelds. This paper proposes a direct method to reconstruct the radiating part of a planar source from its far ﬁelds based on their exact relationship. A standard reconstruction process is provided in which the far-ﬁelds are sampled at the peaks of each propagation mode. Analysis shows that the achievable reconstruction resolution of the source distribution is about half a wavelength. The paper also demonstrates that it is possible to reconstruct the source by sampling the far-ﬁelds on a plane or along a linear path. The performance of the reconstruction algorithm is illustrated with numerical examples.


I. INTRODUCTION
N many practical applications, such as in antenna synthesis and diagnosis [1]- [5], imaging of defects [6], radar imaging [7], it is required to reconstruct the source from its fields.This kind of problems are referred to as inverse electromagnetic source problems.Unlike in the inverse electromagnetic scattering problems [8] [9], it is not necessary to recover the property of the media but only the source distributions.Inverse source problems are usually linear problems.However, they are generally ill-posed and have to be solved with some kind of optimization algorithm or regularization techniques, or with the help of priori information [10]- [18].The introduction of the concept of the number of degrees of freedom (NDF) of the fields helps to overcome the ill-posedness of the problem [19]- [22].Although the NBF of a practical inverse source problem can be effectively obtained by performing singular value decomposition (SVD) to the corresponding operator, the calculation is time consuming.The point spread function (PSF) has been applied successfully in solving inverse source problems [23]- [25].However, it is very complicated to find the exact evaluation of the PSF.For most geometries, PSF can only be performed numerically, or with approximate analytical assessment.
It is well known that a source can be divided into a radiating part and a non-radiating part [26]- [28].The radiating source generates propagation fields and contributes to the far-fields of the source, while the non-radiating part generates evanescent fields and does not contribute to the far-fields.Therefore, from the far-fields of a source, we can only reconstruct its radiating Gaobiao Xiao is with the Key Laboratory of Ministry of Education of Design and Electromagnetic Compatibility of High-Speed Electronic Systems, the part with achievable reconstruction spatial resolution subject to the Rayleigh diffraction limit, typically half wavelength [33]- [35].If additional information, like the near fields of the source, is available, the diffraction limit may be broken and super resolution can be obtained using proper algorithm [29]- [32].
We have recently proposed a method for synthesis of largescale antenna arrays [36].By expanding the sources with Fourier series, a linear source or a planar source on a rectangular sheet can be expressed with superposition of its harmonic components.These components have further been divided into two groups based on the property of the fields generated by them.The propagation group consists of all harmonic components that generate propagation fields, and the evanescent group includes all other components that generate evanescent fields.The first group is apparently the radiating part of the source, while the latter one is the non-radiating part.The exact relationship between the far-fields and the radiating part of the current is derived in [36] and is found to be very efficient for synthesis of large-scale antenna arrays.This paper shows that the relationship can also be applied for efficiently and accurately reconstructing the radiating part of the source from its far-fields.The achievable spatial resolution is about half wavelength, the same as the Rayleigh diffraction limit.The source pictures can also be reconstructed with moderate quality by sampling on a plane or along a straight line, which demonstrates the potential application of the method in practical applications like remote sensing.
This paper is organized as follows.In Section II, the standard reconstruction process is described, and numerical examples are provided to reveal the reconstruction performance of the algorithm.The non-standard reconstruction method by sampling on a plane or along a linear path is discussed in Section III, and a brief conclusion is given in Section IV.

II. THE STANDARD RECONSTRUCTION ALGORITHM
, in free space.The electric field at the position r in far region can be generally expressed as The primes for the source coordinates in the integrands are omitted for the sake of simplicity.In spherical coordinate system, sin cos  We deliberately put the minus sign to the right hand side of (3).
Similarly, the y-component of the current can be equated to its y-polarized far-field alone.Therefore, it is possible for us to reconstruct the two components of the current separately.
At first, we consider the problem of reconstructing the xcomponent of the source current from the x-polarization field.Discarding the factor relating to the distance r , the x-polarized far-field of the current sheet can be expressed as, where the 2-D mode function As explained in [36], each mode function describes a beam in the space with its peak at the direction of   The wavevector of the   , m n mode at its peak direction In order to analyze the property of these modes in a simple way, we choose , where  is the wavelength, x N and y N are two integers.It has been shown in [36] that only those modes belong to the propagation set  can contribute to the far-field, where  is defined as 1 , A typical constellation of the propagation group in k-space is shown in Fig. 2, where each circle denotes the wave vector  given by (12).Note that a qualified far-field must be zero at the direction of 0 xmn   .We may simply exclude this mode from the propagation group to avoid possible abnormal data at this direction.
In a practical reconstruction task, it is neither necessary nor easy to know the exact sizes of the source area.We can estimate the source area at first, and then choose a proper reconstruction area to cover it.The dimensions of the reconstruction area must be selected as It can be seen from ( 13) that the spatial frequency of the highest radiating component is x x y y N N k     .Therefore, the spatial resolution in both direction is 2  , which agrees with the Rayleigh diffraction limit.In other word, we cannot expect to get higher spatial resolution than 2


with the proposed reconstruction algorithm.
To demonstrate the effectiveness of the method, we are now to reconstruct the pattern of a 3 3  digit array formed by 364 Hertzian dipoles, as shown in Fig. 3 The numerical experiments are carried out in an ideal situation that all far fields are accurately calculated at the required sampling directions.No noises are included.The reconstructed radiating part of the x-component of the source currents can be calculated with (13).The results are plotted in Fig. 4. The 9 digits can be recognized in all these four cases.In Case-1 and Case-2, almost all dipoles can be distinguished as the smallest spacing between them is larger than the achievable spatial resolution ( 2 ).However, in Case-3 and Case-4, the dipoles cannot be distinguished anymore because their small spacings obviously exceed the range of achievable spatial resolution.From (10) we observe that larger reconstruction area generates larger number of propagation modes, and requires more sampling data of the far-fields.As a result, more information can be recovered.However, it is not necessary to use a reconstruction area much larger than the estimated source area.As shown in Fig. 4(c), (d), there is almost no useful information in the blue area outside the source region.
To exploit the effect of the reconstruction area, we choose 20 In many situations, there may have no priori information that we can use to choose the direction and the center of the reconstruction area.However, their choices have inevitable impact on the reconstruction results.We will use Case-3 to reveal this effect.First, we rotate the source area with an angle of 45 and 80 in the xoy plane, respectively.The polarization of all dipoles is rotated in the same way.Assume that we still reconstruct only the x-component with the reconstruction area of 40 Generally, it requires to reconstruct both the x-component and the y-component of the planar current in order to recover the total radiating part of the source.We still use the Case-3 as the source to recover, but the polarization of the middle raw of digits, i.e., "456", is changed from x-polarization to ypolarization.The reconstructed x-component source picture and y-component source picture are shown in Fig. 7.Each reconstruction process can accurately reconstruct the corresponding component of the source current.However, we have to combine them together to get a complete source picture.

III. SAMPLING ON A PLANE
In the standard reconstruction process discussed in Section II, the spatial samplings are performed at the peak directions of all propagation modes.We will show in this section that it is possible to reconstruct the source picture by sampling at other directions with the same number of sampling data.As the sampling points are not at the peaks of the mode functions, the coefficients of the current cannot be determined directly with (12).Instead, they have to be calculated by solving a matrix equation.For the sake of convenience, we reorder the coefficients of the current to form a single column vector I and number them from 1 to Npro , where Npro is the number of the total radiating harmonic component of the current, and is also the number of the total propagation modes.The sampled far-fields are arranged as a column vector far F in the same order with I .Then the current vector can be obtained by solving the following matrix equation, The sampling grid is not necessary to be arranged within a square-shaped area.In practical applications, like remote sensing, we may sample the far-field using aircraft along a linear path sam L , as shown in Fig. 10.In this situation, the information in the transversal direction will be lost, and only the source distribution in the direction of the sampling path can be reconstructed.We again take the source Case-3 as example, and all dipoles are x-polarized.The far-fields are sampled right above the source area, along a linear path in the x-direction and the y-direction, respectively.The reconstructed results are shown in Fig. 11

IV. CONCLUSIONS
Although it is well known that an electromagnetic source can be separated into a radiating part and a non-radiating part, there is no explicit expressions for them.Numerical methods for obtaining the radiating source are usually not quite convenient to implement.A very simple way to determine the radiating source of a linear source or a planar source on a rectangular sheet is proposed in [36], where a harmonic component of the source is considered to be radiating if it generates a propagating field in the space; otherwise, the component is non-radiating.The far-fields are then exactly, not approximately, expressed by the radiating part of the source.With the help of the explicit expression, the radiating part of the source can be reconstructed accurately from its far-fields.In the standard reconstruction process, the radiating components of the source are directly obtained from the sampled far-fields, which is much more efficient than any other formulations that require to perform optimization or solve equation.The proposed non-standard reconstruction process requires to solve a matrix equation.However, the computational cost is relatively low as the coefficient matrix is basically reversible and easy to fill.


is the angle between the position vector r and the x- axis, and y  is that with the y-axis.ˆx θ and ˆy θ are respectively the corresponding unit vectors, as shown in Fig. 1.The factor sin x  and sin y  come from the x-polarized infinitesimal dipole and the y-polarized infinitesimal dipole composing the current sheets.The x-component of the current can be separated to be related to the x-polarized far field alone,

Fig. 1
Fig.1 Current sheet and the unit vectors in the coordinate system.
mode.As all propagation modes are mirror symmetrical with respect to the current sheet, only the upper half of the constellation is illustrated in Fig.2.

Fig. 2 x
Fig.2 Constellation of the propagation group in the k-space.The far-field of the current sheet is completely determined by the associated propagation modes,     sin , , y x . All dipoles have unit amplitude and are located on a uniform grid with spacing dip d in the xoy plane.Four cases of source distributions are examined.They all have the same patterns as shown in Fig.3, but the spacings and the source areas are changed, as listed in below: Case-1: dip d   , resulting a source area of 21 39

Fig. 3
Fig.3 Source picture to reconstruct.364 Hertzian dipoles locate on the xoy plane, with each dot representing a dipole.
which are in proportion to their source area.The number of the corresponding propagation modes are then reduced to 1257 and 317, respectively.The reconstructed results are shown in Fig.5.It can be seen that the definition of the reconstructed source pictures is almost the same as that in Fig.4(c), (d).The reconstruction area mainly affects the range of the reconstructed picture, but has little effect on the spatial resolution.We may use smaller reconstruction area to alleviate the requirement for large amount of sampling data.(a) (b) Fig.5 The results with smaller reconstruction areas.(a) Case-3.(b) Case-4.

Fig. 6
Fig.6 The reconstructed currents when the source area rotates with angle of (a) 45 , (b) 80 , and shifts by 20 (c) in x-direction, (d) in y-direction.

Fig. 7
The reconstructed pictures of a source with two polarizations.(a) xcomponent, (b) y-component.
(a), (b).Actually, they are 1-D pictures.They are uniform in the transversal direction, while in the direction of the sampling path, the reconstructed source distribution at a certain point includes the contributions from all the sources on the transversal line.

Fig. 10 Fig. 11
Fig.10 The reconstructed results sampling on a linear grid right above the source.(a) in x-direction, (b) in y-direction.