Exploring the limits of magnetic field focusing: Simple planar geometries

This work explores the possibility to arbitrarily shape in space low-frequency magnetic fields using a recently introduced synthesization technique (Choi et al., 2016). We investigate the ability to focus a magnetic field on a two-dimensional region using magnetic field sources distributed on a parallel plane. In agreement with the recent work, arbitrarily tight focusing is demonstrated possible. However, our results indicate that this comes at the cost of exponentially large power requirements and also leads to exponentially large fields in the region between the source and target planes. This imposes strict limitations on the application of the technique. In addition, we also demonstrate that arbitrary steering of the magnetic field focus within the target region is possible, without any additional cost in terms of power requirement. In exploring the potential for magnetic field synthesis, our findings highlight limits to be considered for practical applications, as well as promising capabilities not identified before.


Introduction
The low-frequency limit of electromagnetism -rigorously defined as the magnetoquasistatic limit [1] -has been recently attracting a growing interest, due to the wealth of related applications, such as magnetic induction tomography (MIT) [2], contactless power transfer [3,4] and underwater communication [5,6], which involve the use of low-frequency oscillating magnetic fields, typically from a few Hz to a few MHz. However, all these applications ideally require a source which generates a focused low-frequency magnetic field. In fact, in MIT a focused B-field is required to excite eddy currents in the wanted portion of the object, so to increase the in-plane spatial resolution and to facilitate three-dimensional image reconstruction. In contactless power transfer and underwater communication, focused B-fields are essential to retain a non-vanishing strength at a useful distance from the source. The unavailability of suitable sources of focused low-frequency magnetic fields has hampered the progress in the development of the above mentioned applications. Indeed, the standard oscillating magnetic dipole is unfitting for the task, as the B-field vanishes at a short range from the source. Furthermore, the extension of phased-array to the low-frequency limit is unpractical, as it would lead to devices with very large dimensions. Recently a promising approach to fill this gap was introduced by Choi et al. [7,8]. A planar array of adjacent coils -thus not relying on the principles underlying phased arrays -was considered. By appropriately tailoring the currents in the individual coils, it was shown that it is possible to synthesize a focused magnetic field.
This work aims to explore the limits of the technique for the synthesization of focused magnetic fields in two dimensions, for the case of a simple planar geometry of the magnetic sources. In particular, we investigate the number of sources required to obtain a wanted focusing, and the required power, which will turn out to be the most important limiting factor for the technique.
This paper is organized as follows. In Section 2 the problem and the notations are introduced. In Section 3 we present numerical results for magnetic field focusing and steering in one-and two-dimensions, discussing the power requirements in all considered cases, and the limitations imposed on practical applications. The behavior of the field in the direction perpendicular to the target plane is then studied, and the related additional constraints discussed. Conclusions are finally drawn.

The problem and notations
Our goal is to investigate to what extent it is possible to synthetize a focused low-frequency magnetic field using a set of current-carrying wires, with the current distribution across the wires controlling both the focus size and its position in space.
A target region is introduced, which can be one-dimensional (1D) or two-dimensional (2D). The aim is to focus the magnetic field at an arbitrary point = within this region.  Focusing the magnetic field within 1D and 2D target regions has different requirements for the magnetic field sources. Following Refs. [7,8], we consider planar arrangements, which include straight wires and square coils, as sketched in Fig. 1. For 1D target regions, we consider two possible geometries. In the first one, represented in Fig. 1(a), magnetic fields are generated by an array of infinitely long straight wires carrying electric currents, all parallel to the -axis and crossing the XZ plane at = − through equispaced points in the interval between = − and = . In the second one, represented in Fig. 1(b), a 1D array of square coils constitute the source of magnetic fields. The coils, with side length , lie on a plane parallel to the XY plane and intercepting the axis at = − . The coils are uniformly distanced one from the other, with their centers spaced of a distance . For 2D target regions, the considered magnetic field source, sketched in Fig. 1(c), consists of a 2D array of square coils. As for the 1D case, the coils have side length , lie on a plane parallel to the XY plane and intercept the axis at = − . The coils are uniformly distanced one from the other, with their centers spaced of a distance .
The extension of the target region is taken to be 2 for the 1D case and 4 2 for the 2D case. The dimension of the magnetic field ''spot size'' at the target region (line or plane) will be indicated by 2 for the 1D case and 4 for the 2D case.

Numerical simulations
Our numerical approach to determine the current distribution required for the desired focusing follows the line of Ref. [7]. Details of the procedure are in Appendices A and B. A discrete 1D or 2D grid (with = 1, 2, … , ) is superimposed on the target region as applicable, and the currents (with = 1, 2, … , ) in the magnetic field sources (straight wires or coils) are calculated by demanding that the three components of the synthesized field at the discretized points are all zero, except at = 0 . The solution for the currents can then be found by matrix inversion, provided that the number of magnetic field sources (straight wires or coils) is at least three times the number of grid points. We notice that we are considering the quasi-static low-frequency limit of electromagnetism. We thus perform all the calculations as in the static case (dc currents and dc magnetic fields). Results for the low-frequency magnetic fields of interest are then obtained by taking the currents found in the static limit as amplitudes of the ac currents required for the generation of the desired ac magnetic fields.

1D magnetic field focusing
We first consider the case of 1D magnetic field focusing produced by an array of equally spaced infinitely long wires, with the corresponding set-up shown in Fig. 1(a). The target region is the segment [− , ] along the -axis ( = = 0). The currents in the wires are calculated by matrix inversion. The field at = 0 is taken as ( 0 , 0, 0) = 0 = 0 0 ∕(2 ), the field created by a single source at a distance . For this configuration, the resolution is imposed by = 2 ∕( − 1). Fig. 2(a, b, c) illustrates 1D magnetic field focusing for the case of straight wires and a line of coils. Here = 3 to guarantee that the three components of the magnetic field take the prescribed values at the target points. By appropriately tailoring the current in the different wires (coils), it is possible to synthetize a field which has a maximum around 0 = 0. Furthermore, by increasing by about a factor 2 the number of sources, it is possible to decrease by the same factor the spot size of the magnetic field.
Magnetic field focusing for the case of a magnetic field source consisting of a line of square coils, as sketched in Fig. 1 Fig. 2(d, e, f). Also in this case, it is possible to reduce the magnetic field spot size at the target region by increasing the number of magnetic sources.
A central issue of the present discussion is the scaling of the current with the focus size. For a given desired spot size, we calculated by matrix inversion the required currents circulating in the different coils, and extracted the maximum value. Fig. 3 shows such a maximum value as a function of the inverse of the spot size, and evidences an exponentially increasing dependence. So, while focusing of the magnetic field down to a desired value is in principle possible, the current required makes focusing below a certain level impractical.

1D magnetic field steering
We consider here the possibility of steering the magnetic field focus within the target region for a fixed configuration of the magnetic field sources, i.e. for fixed number of the wires/coils and their spatial arrangement. This is very important for magnetic induction tomography, where it would allow to selectively address different areas of the object of interest, thus increasing the resolution.
Steering of the magnetic field focus can be achieved by varying the current distribution among the different wires depending on the D. Cubero et al.  desired position of the magnetic field focus. This is demonstrated in Fig. 4, where the current distribution among different wires/coils is determined by matrix inversion for different position of the magnetic field focus 0 within the target region. It can be seen that the focus position can be varied, while maintaining the focus size. We notice that there is a distortion whenever the desired focus position approaches the target region boundaries, see Fig. 4. In this case, while the field is forced to equal 0 at 0 by the matrix inversion procedure, it goes above the desired value (white region in the figure) between 0 and the nearest boundary, before decaying to a smaller value. We found that steering with a high degree of control can be achieved in about half of the target region. Obviously, whenever steering has to be achieve on a larger distance, a larger extension of the magnetic field source arrangement has to be considered. A remarkable and important finding is that steering the magnetic field focus away from the center of the target region does not require a significant increase of the maximum current across the wires/coils (see caption of Fig. 4), at variance with the requirements required for decreasing the focus waist, obtained at the cost of an exponential increase in current.

2D magnetic field focusing and steering
Magnetic field focusing and steering can be directly generalized to a 2D target region by using the 2D magnetic field source configuration of Fig. 1(c) consisting of an array of square coils of length . Fig. 5 shows numerical results for focusing and steering for the 2D configuration.
The reported results validate for the 2D case what already observed for 1D: by varying the number of coils and target fixed points , here taken to be = 2 2 , it is possible to decrease the spot size, although this is at the cost of an exponential increase in the current required, as shown in Fig. 6 -like in the 1D case, the field at the focusing point 0 is required to be that of a single, symmetrically placed coil, here (see Appendix A). For a fixed number of coils, it is possible to steer the focus within the target region without a significant increase in the current required.
As an explicit example, we consider a 2D set-up with = 10 cm, and aim to focus the magnetic field to a spot size of = ∕2 = 5 cm with peak amplitude 0 = 0.5 × 10 −6 T. From the expression for the field a single coil, Eq. (1), which we take as a reference, we derive that the requirement of a field peak amplitude of 0 = 0.5 × 10 −6 T implies a reference current of 0 = 0.38 A. Our results of Fig. 6 show that for ∕ = 2 we have max ∕ 0 = 30, thus the maximum current required in a single coil is max = 11.4 A. On the other hand, if we require a 2-fold reduction in the spot size for the same distance and field peak amplitude 0 , the current required becomes extremely large, and equal to = 31025 A.

Behavior outside the target plane
The discussion so far dealt uniquely with the control of the magnetic field within the target plane. This was according to the fact that in this work we do not address the issue of 3D field shaping. Nevertheless, it is interesting to study the field behavior also in the region between the  source plane and the target plane, as this may introduce limitations in practical uses of 2D field shaping in its present conception. To this purpose, consider the configuration presented in the panels (b) and (c) of Fig. 5 as case study. The corresponding behavior of the field in the transverse direction, i.e. along with 0 = 0 = 0, is represented in Fig. 7. The field turns out to be extremely large in proximity of the sources, and decays rapidly to reach the wanted value at the target plane. For the weaker focusing case, the variation in field amplitude spans five orders of magnitude, while for the more focused configuration a larger variation spanning ten orders of magnitude is observed.
The large values required for the field amplitude between source and target plane should be taken into account for applications which require 2D shaping, as such large fields may not be compatible with the materials between the two planes. Specifically, large fields are compatible with remote power transfer where no conductive material is present in the gap. Similarly, magnetic induction tomography of the surface of nanostructures would not find large fields in the gap region a limiting factor. On the contrary, for biomedical [9,10] and security [11,12] applications of electromagnetic induction imaging, the impact of such large fields would have to be taken into account in designing the imaging systems. Excessively large fields may cause damage to the tissues in between the source and the target tissue, certainly an important limiting factor for biomedical imaging. For through-barrier imaging of relevance to security applications, large fields would lead to large unwanted eddy currents in the material shielding the target object, potentially compromising the imaging approach.

Conclusions
Following a recently introduced synthesis technique [7,8], we have investigated the possibility of focusing and steering the magnetic field in 1D and 2D regions by using planar arrays of wires or coils. In agreement with previous work, we found that focusing to any desired level on a target plane is in principle possible by appropriately tailoring the magnetic field source. However, our results also identified important constraints to be taken into account for practical implementations. Firstly, an increase of the focusing level, i.e. a reduction of the magnetic field waist, requires an exponential increase in the current amplitude, which rapidly lead to impracticable power requirements. Secondly, a tight focusing on the target plane produces very large fields in the region between source and target regions -a factor whose impact has to be carefully considered in the application at hand. In addition, we also demonstrated that steering the magnetic focus within the target plane is possible by changing the distribution of currents in the coils, importantly at no additional cost in terms of power requirements.
Despite the limitations highlighted in our work, the synthesis technique considered here is promising for magnetic field beam shaping so to overcome the severe limitations imposed by the standard dipole pattern of a single coil. This paves the way for numerous applications, from efficient transmission of power to high resolution probing and imaging.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
where ( ) ( , , ) is the field created by the coil , divided by its dimensionless current parameter .
The -component ( = , , or ) of the generated field ( ) is required to satisfy specific values 0 ( ) at the nodes of a lattice that maps the entire target region. Thus, from (B.1), the currents are determined by the linear problem where are real constants. In view of practical implementations, we are interested in the minimum current solution, that is, the solution that minimizes the quantity ∑ =1 2 = ⋅ , which can be found by demanding to be perpendicular to , i.e. The solution 0 can be found, and (B.6) solved, using standard linear solving algorithms [15].