Spectral approach to the inverse problem for the field of arbitrary changing electric dipole

Abstract

The inverse problem for electromagnetic field produced by arbitrary altered charge distribution in dipole approximation is solved. The charge distribution is represented by its dipole moment. It is assumed that the spectral properties of magnetic field of the dipole are known. The position of the dipole and its Fourier components are considered as the unknown quantities. It is assumed that relative increments of amplitude and phase of magnetic field in the vicinity of the observation point are known. The derived results can be used for study of phenomena concerned with occurrence and variation of localized electric charge distribution, when the position and the dynamics of a localized source of electromagnetic field are to be defined.

Keywords:

• inverse problem
• dipole
• charge
• electromagnetic field
• spectrum

AMS Subject Classifications:

• 86A22
• 42-02
• 45Q05

Introduction

The early studies of the inverse problem in electrodynamics have been devoted to the inverse scattering problem for electromagnetic waves. In succeeding years, the focus was turned to the theory of inverse boundary values problem, see for examples.[1–3] A large variety of papers are devoted to inverse problem for the field of static magnetic or static electric dipole in connection with medical imaging. For example, the current sources in the brain produce external magnetic fields and scalp surface potentials that can be measured using magnetoencephalography and electroencephalography (see e.g. [4, 5]). The solution in a particular case of a static dipole field was obtained in [6].

Special field of research is constituted by the inverse problems for electromagnetic field of a point-like source like a moving charge or varying dipole. There was a number of papers which can be considered as devoted to the inverse problem of electromagnetic radiation of a charge moving in bending magnets and undulators. For example, some ways of solution of this problem was suggested in [7]. Solution of the inverse problem of radiation of a moving point-like charge in a far-field approximation was presented in [8]. A general solution of the inverse problem for the Liénard-Wiechert potentials of an arbitrary moving charged particle was given in [9].

Another elementary and widespread source of electromagnetic field is a variable electric or magnetic dipole. The electromagnetic field of an arbitrary distribution of charge at the distance much greater than the characteristic dimension of the area where the charge is distributed can be described in a certain approximation as a field of a point-like dipole. There are many phenomena in which electromagnetic field of localized distribution of charge is known or can be measured, and it is necessary to restore the properties of the source. For example, investigations of electric charge distribution in the thundery clouds, or emission of electromagnetic waves at the boundary of continental platforms which can be used for earthquakes forecasting (see e.g. [10, 11]), or investigation of development of cracks in crystals [12–14] (see also recent discussion on identification of the position and shape of cracks detected in elastic solid samples [15]).

In this paper, we assume that the source of the field is localized in an area which is much smaller than the distance between the source and the observer and can be replaced by an arbitrary varying dipole. This does not mean that we consider only the so-called far zone, because the far zone is defined as area at distances much greater than the wave length. But, a non-relativistic charged particle (which is the case of this paper) generates electromagnetic field with characteristic wave length much greater than the area in which the particle moves. Hence, we consider here both, the near field zone within the distance comparable or less than the wave length and the far field zone.

In our previous paper [16], we have solved the inversion problem for the field of an arbitrary changing electric dipole. It was assumed that at some point of the space, the electric and magnetic fields of a dipole are known as the functions of time. As a result of the solution, formulae were derived for the field source position and for the dynamics of the dipole. Such approach is not very convenient for practical applications, because the measurements of fast changing electromagnetic fields as the functions of time face some difficulties. Usually, it is much easier to measure the amplitude of electric or magnetic field at some frequency of spectrum. From this point of view, it can be of interest such definition of the inverse problem when the given quantities are the spectral characteristics of the field, and the desired solution is the position and spectral properties of the field source.

Moreover, many measurements made up to now in the field of practical solution of the inverse problems have been made in some definite range of frequencies. See, for example, the study of mechanoelectrical transformations in concretes [17] or measurements of electromagnetic precursors, generated before the occurrence of an earthquake.[11]

As we shall see further, the vector of magnetic field of the dipole is orthogonal to the line connecting the dipole with an observer. Hence, if we define the harmonic component of magnetic field vector, we can define the line between the source and the observer. In this paper, we present a solution of the inverse problem for the case when the spectral components of magnetic field at some point are given or we know the gradient of the spectral components at the point of observation. It is shown that the information only on the spectral properties of the magnetic field is not sufficient for solution of the inverse problem. But if the gradient of the field at the observation point is known, the full solution of the problem is possible.

Inverse problem for spectral components of the magnetic field

Suppose we know the magnetic field which is produced by some charge distribution at fixed frequency $\mathit{\omega }$ at some point of space. Suppose the distance between this point and some point inside the area where the charge is distributed is much greater than the dimensions of the area. In this case, we can replace the charge distribution by its electric dipole. Then, the problem is to find the position of the dipole and the Fourier amplitude of the dipole.

Fourier transform of the magnetic field is given by [18, Section 72] (Gaussian system of units is used):(1) $$\begin{array}{c}\hfill {\mathit{H}}_{\mathit{\omega }}(\mathit{r})=\frac{\mathrm{i}k}{r^2}({\mathit{d}}_{\mathit{\omega }}\times \mathit{n})(\mathrm{i}\mathit{kr}-1){\mathrm{e}}^{\mathrm{i}\mathit{kr}},\end{array}$$(1) where ${\mathit{d}}_{\mathit{\omega }}$ is the Fourier transform of the electric dipole, $\mathit{r}$ is the radius-vector of the observation point, $\mathit{n}=\mathit{r}/r$ is the unit vector pointing from the dipole to the observer and $k=\mathit{\omega }/c$ is module of the wave vector. In this section, we assume that the magnetic field is not linearly polarized. Then, we can define the plane to which vector $\mathit{H}$ is bound. Let the axis $\mathit{OZ}$ be orthogonal to this plane as shown in Figure 1. Then, the harmonic component of the magnetic field vector will describe an ellipse in the plane $\mathit{XY}$, which we call polarization ellipse of vector ${\mathit{H}}_{\mathit{\omega }}$. Now we know that the source of the field is on $\mathit{OZ}$-axis, but we do not know in which direction of the axis – positive or negative it lies. If the magnetic field is linearly polarized, one cannot define the axis $\mathit{OZ}$ unambiguously. Next, we represent vector ${\mathit{d}}_{\mathit{\omega }}$ by its Cartesian coordinates(2) $$\begin{array}{c}\hfill {\mathit{d}}_{\mathit{\omega }}=(d_{0x}\exp \mathrm{i}{\mathit{\alpha }}_x,d_{0y}\exp \mathrm{i}{\mathit{\alpha }}_y,d_{0z}\exp \mathrm{i}{\mathit{\alpha }}_z),\end{array}$$(2) where ${\mathit{\alpha }}_j$ are the initial phases, $d_{0j}$ are the real positive amplitudes.

Fig. 1 Reference system.

Respectively, Cartesian coordinates of vector ${\mathit{H}}_{\mathit{\omega }}$ are(3) $$\begin{array}{cc}\hfill H_{\mathit{\omega }x}& =\frac{\mathrm{i}k}{r^2}\mathit{\epsilon }{d}_{0y}(\mathrm{i}\mathit{kr}-1)\exp [\mathrm{i}(\mathit{kr}+{\mathit{\alpha }}_y)],\hfill \\ \hfill H_{\mathit{\omega }y}& =-\frac{\mathrm{i}k}{r^2}\mathit{\epsilon }{d}_{0x}(\mathrm{i}\mathit{kr}-1)\exp [\mathrm{i}(\mathit{kr}+{\mathit{\alpha }}_x)],\hfill \end{array}$$(3) where $\mathit{\epsilon }=n_z=\pm 1$.

The size and orientation of the magnetic field polarization ellipse can be defined in terms of polarization parameters; for example, Stokes parameters. Polarization parameters are quadric combinations of components $H_{\mathit{\omega }x}$ and $H_{\mathit{\omega }y}$ and can be easily measured experimentally. We define the Stokes parameters as in [19]:(4) $$\begin{array}{cc}\hfill & s_0=H_{\mathit{\omega }x}{H}_{\mathit{\omega }x}^{\ast }+H_{\mathit{\omega }y}{H}_{\mathit{\omega }y}^{\ast },\hfill \\ \hfill & s_1=H_{\mathit{\omega }x}{H}_{\mathit{\omega }x}^{\ast }-H_{\mathit{\omega }y}{H}_{\mathit{\omega }y}^{\ast },\hfill \\ \hfill & s_2=H_{\mathit{\omega }x}{H}_{\mathit{\omega }y}^{\ast }+H_{\mathit{\omega }y}{H}_{\mathit{\omega }x}^{\ast },\hfill \\ \hfill & s_3=\mathrm{i}(H_{\mathit{\omega }y}{H}_{\mathit{\omega }x}^{\ast }-H_{\mathit{\omega }x}{H}_{\mathit{\omega }y}^{\ast }).\hfill \end{array}$$(4) We would like to emphasize that the field (3(3) $$\begin{array}{cc}\hfill H_{\mathit{\omega }x}& =\frac{\mathrm{i}k}{r^2}\mathit{\epsilon }{d}_{0y}(\mathrm{i}\mathit{kr}-1)\exp [\mathrm{i}(\mathit{kr}+{\mathit{\alpha }}_y)],\hfill \\ \hfill H_{\mathit{\omega }y}& =-\frac{\mathrm{i}k}{r^2}\mathit{\epsilon }{d}_{0x}(\mathrm{i}\mathit{kr}-1)\exp [\mathrm{i}(\mathit{kr}+{\mathit{\alpha }}_x)],\hfill \end{array}$$(3) ) is not the field of plane wave. It means that you cannot calculate Stokes parameters from the electric field, as it is usually done. The electric field is neither equal nor orthogonal to the magnetic one. Evidently, one can represent the Stokes parameters in the form(5) $$\begin{array}{cc}\hfill s_0=& |H_{\mathit{\omega }x}{|}^2+|H_{\mathit{\omega }y}{|}^2,\hfill \\ \hfill s_1=& |H_{\mathit{\omega }x}{|}^2-|H_{\mathit{\omega }y}{|}^2,\hfill \\ \hfill s_2=& 2|H_{\mathit{\omega }x}||H_{\mathit{\omega }y}|\cos ({\mathit{\varphi }}_x-{\mathit{\varphi }}_y),\hfill \\ \hfill s_3=& 2|H_{\mathit{\omega }x}||H_{\mathit{\omega }y}|\sin ({\mathit{\varphi }}_x-{\mathit{\varphi }}_y),\hfill \end{array}$$(5) where ${\mathit{\varphi }}_x-{\mathit{\varphi }}_y$ is the phase difference between the projections of ${\mathit{H}}_{\mathit{\omega }}$ onto axes $\mathit{OX}$ and $\mathit{OY}$.

Substituting expression (3(3) $$\begin{array}{cc}\hfill H_{\mathit{\omega }x}& =\frac{\mathrm{i}k}{r^2}\mathit{\epsilon }{d}_{0y}(\mathrm{i}\mathit{kr}-1)\exp [\mathrm{i}(\mathit{kr}+{\mathit{\alpha }}_y)],\hfill \\ \hfill H_{\mathit{\omega }y}& =-\frac{\mathrm{i}k}{r^2}\mathit{\epsilon }{d}_{0x}(\mathrm{i}\mathit{kr}-1)\exp [\mathrm{i}(\mathit{kr}+{\mathit{\alpha }}_x)],\hfill \end{array}$$(3) ) into (4(4) $$\begin{array}{cc}\hfill & s_0=H_{\mathit{\omega }x}{H}_{\mathit{\omega }x}^{\ast }+H_{\mathit{\omega }y}{H}_{\mathit{\omega }y}^{\ast },\hfill \\ \hfill & s_1=H_{\mathit{\omega }x}{H}_{\mathit{\omega }x}^{\ast }-H_{\mathit{\omega }y}{H}_{\mathit{\omega }y}^{\ast },\hfill \\ \hfill & s_2=H_{\mathit{\omega }x}{H}_{\mathit{\omega }y}^{\ast }+H_{\mathit{\omega }y}{H}_{\mathit{\omega }x}^{\ast },\hfill \\ \hfill & s_3=\mathrm{i}(H_{\mathit{\omega }y}{H}_{\mathit{\omega }x}^{\ast }-H_{\mathit{\omega }x}{H}_{\mathit{\omega }y}^{\ast }).\hfill \end{array}$$(4) ), we get:(6) $$\begin{array}{cc}\hfill & s_0=J(d_{0x}^2+d_{0y}^2),\hfill \\ \hfill & s_1=J(d_{0y}^2-d_{0x}^2),\hfill \\ \hfill & s_2=-2{\mathit{Jd}}_{0x}{d}_{0y}\cos ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y),\hfill \\ \hfill & s_3=2{\mathit{Jd}}_{0x}{d}_{0y}\sin ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y).\hfill \end{array}$$(6) Here,(7) $$\begin{array}{c}\hfill J=\frac{k^2(1+k^2{r}^2)}{r^4}.\end{array}$$(7) Comparing (5(5) $$\begin{array}{cc}\hfill s_0=& |H_{\mathit{\omega }x}{|}^2+|H_{\mathit{\omega }y}{|}^2,\hfill \\ \hfill s_1=& |H_{\mathit{\omega }x}{|}^2-|H_{\mathit{\omega }y}{|}^2,\hfill \\ \hfill s_2=& 2|H_{\mathit{\omega }x}||H_{\mathit{\omega }y}|\cos ({\mathit{\varphi }}_x-{\mathit{\varphi }}_y),\hfill \\ \hfill s_3=& 2|H_{\mathit{\omega }x}||H_{\mathit{\omega }y}|\sin ({\mathit{\varphi }}_x-{\mathit{\varphi }}_y),\hfill \end{array}$$(5) ) and (6(6) $$\begin{array}{cc}\hfill & s_0=J(d_{0x}^2+d_{0y}^2),\hfill \\ \hfill & s_1=J(d_{0y}^2-d_{0x}^2),\hfill \\ \hfill & s_2=-2{\mathit{Jd}}_{0x}{d}_{0y}\cos ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y),\hfill \\ \hfill & s_3=2{\mathit{Jd}}_{0x}{d}_{0y}\sin ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y).\hfill \end{array}$$(6) ), we see that projection of the dipole polarization ellipse onto plane $\mathit{XY}$ is equal to that of magnetic field, but is turned by angle $\mathit{\pi }/2$. It follows also from (6(6) $$\begin{array}{cc}\hfill & s_0=J(d_{0x}^2+d_{0y}^2),\hfill \\ \hfill & s_1=J(d_{0y}^2-d_{0x}^2),\hfill \\ \hfill & s_2=-2{\mathit{Jd}}_{0x}{d}_{0y}\cos ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y),\hfill \\ \hfill & s_3=2{\mathit{Jd}}_{0x}{d}_{0y}\sin ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y).\hfill \end{array}$$(6) ) that if the Stokes parameters are known, then the dipole components can be defined only up to a scaling multiplier. Let us denote ${\mathit{\delta }}_i=\sqrt{I}{d}_{0i}$. Then, the solution of (6(6) $$\begin{array}{cc}\hfill & s_0=J(d_{0x}^2+d_{0y}^2),\hfill \\ \hfill & s_1=J(d_{0y}^2-d_{0x}^2),\hfill \\ \hfill & s_2=-2{\mathit{Jd}}_{0x}{d}_{0y}\cos ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y),\hfill \\ \hfill & s_3=2{\mathit{Jd}}_{0x}{d}_{0y}\sin ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y).\hfill \end{array}$$(6) ) takes the form(8) $$\begin{array}{c}\hfill {\mathit{\delta }}_x=\sqrt{\frac{s_0-s_1}{2}},{\mathit{\delta }}_y=\sqrt{\frac{s_0+s_1}{2}},\\ \hfill \cos ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y)=-\frac{s_2}{\sqrt{s_0^2-s_1^2}},\\ \hfill \sin ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y)=\frac{s_3}{\sqrt{s_0^2-s_1^2}}.\end{array}$$(8) These relations define the dipole polarization ellipse up to a scaling multiplier.

Hence, knowing the spectral properties or polarization of the dipole magnetic field, we can define the line connecting dipole with the observer and projection of the dipole polarization ellipse onto the plane orthogonal to that line. Still remains unknown the absolute value of the dipole, one of the two directions to the field source and the distance to the source. Some of these quantities can be found if the spectral components of the electric field are known. In this paper, we discuss another possibility to introduce additional input data. Namely, we suppose that the gradient of spectral amplitude of the magnetic field is known. We start with the most general case – we suppose that all derivatives of ${\mathit{H}}_{\mathit{\omega }}$ with respect to all coordinates are known. In Section 4, we present some solutions of the inverse problem when not all the derivatives are used.

Inverse problem for derivatives of the magnetic field

Let us rewrite coordinates of vector ${\mathit{H}}_{\mathit{\omega }}$ in an arbitrary Cartesian coordinate system $(x_1,x_2,x_3)$ as follows $H_{\mathit{\omega }m}=|H_{\mathit{\omega }m}|\exp \mathrm{i}{\mathit{\phi }}_m$, $m=1,2,3.$ Next, we take derivative over coordinate $x_l$ and divide the result by $H_{\mathit{\omega }m}$. This produces a matrix(9) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{\mathit{lm}}\equiv \frac{1}{H_{\mathit{\omega }m}}\frac{\mathit{\partial }{H}_{\mathit{\omega }m}}{\mathit{\partial }{x}_l}=\frac{1}{|H_{\mathit{\omega }m}|}\frac{\mathit{\partial }|H_{\mathit{\omega }m}|}{\mathit{\partial }{x}_l}+\mathrm{i}\frac{\mathit{\partial }{\mathit{\phi }}_m}{\mathit{\partial }{x}_l}.\end{array}$$(9) As we see, the real part of the matrix element represents relative rate of change of the magnetic field amplitude along the respective axis. Imaginary part of ${\mathrm{\Delta }}_{\mathit{lm}}$ is the phase rate.

Suppose all matrix elements ${\mathrm{\Delta }}_{\mathit{lm}}$ are known. The real part of each element can be measured as ratio of the magnetic field increment to coordinate increment for two near points of space. Respectively, the imaginary part can be measured as relative increment of phase for the same two points.

Now, the inverse problem is defined as follows: there are 9 complex or 18 real numbers ${\mathrm{\Delta }}_{\mathit{lm}}$ (because each complex number is defined by two real numbers). Two vectors have to be found – a real vector $\mathit{r}$ and a complex vector ${\mathit{d}}_{\mathit{\omega }}$. This makes 9 unknown real quantities. Hence, the problem is over-specified, which means that the matrix elements ${\mathrm{\Delta }}_{\mathit{lm}}$ are not independent – there are some constraint equations which we shall ascertain later on.

Let us take derivatives from (1(1) $$\begin{array}{c}\hfill {\mathit{H}}_{\mathit{\omega }}(\mathit{r})=\frac{\mathrm{i}k}{r^2}({\mathit{d}}_{\mathit{\omega }}\times \mathit{n})(\mathrm{i}\mathit{kr}-1){\mathrm{e}}^{\mathrm{i}\mathit{kr}},\end{array}$$(1) ) with respect to coordinates and substitute the results in (9(9) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{\mathit{lm}}\equiv \frac{1}{H_{\mathit{\omega }m}}\frac{\mathit{\partial }{H}_{\mathit{\omega }m}}{\mathit{\partial }{x}_l}=\frac{1}{|H_{\mathit{\omega }m}|}\frac{\mathit{\partial }|H_{\mathit{\omega }m}|}{\mathit{\partial }{x}_l}+\mathrm{i}\frac{\mathit{\partial }{\mathit{\phi }}_m}{\mathit{\partial }{x}_l}.\end{array}$$(9) ). We introduce a dimensionless matrix $D_{\mathit{lm}}={\mathrm{\Delta }}_{\mathit{lm}}/k$. The matrix elements are:(10) $$\begin{array}{c}\hfill D_{\mathit{lm}}=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill p_2{d}_{\mathit{\omega }3}\hfill & \hfill -p_3{d}_{\mathit{\omega }2}\hfill \\ \hfill -p_1{d}_{\mathit{\omega }3}\hfill & \hfill 0\hfill & \hfill p_3{d}_{\mathit{\omega }1}\hfill \\ \hfill p_1{d}_{\mathit{\omega }2}\hfill & \hfill -p_2{d}_{\mathit{\omega }1}\hfill & \hfill 0\hfill \end{array}\right)-a\left(\begin{array}{ccc}\hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill \\ \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill \\ \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill \end{array}\right),\end{array}$$(10) where(11) $$\begin{array}{c}\hfill p_j=\frac{1}{{({\mathit{d}}_{\mathit{\omega }}\times \mathit{\rho })}_j},\mathit{\rho }=k\mathit{r},a=\frac{3+2{\mathit{\rho }}^2-i{\mathit{\rho }}^3}{{\mathit{\rho }}^2(1+{\mathit{\rho }}^2)}.\end{array}$$(11) It follows from (10(10) $$\begin{array}{c}\hfill D_{\mathit{lm}}=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill p_2{d}_{\mathit{\omega }3}\hfill & \hfill -p_3{d}_{\mathit{\omega }2}\hfill \\ \hfill -p_1{d}_{\mathit{\omega }3}\hfill & \hfill 0\hfill & \hfill p_3{d}_{\mathit{\omega }1}\hfill \\ \hfill p_1{d}_{\mathit{\omega }2}\hfill & \hfill -p_2{d}_{\mathit{\omega }1}\hfill & \hfill 0\hfill \end{array}\right)-a\left(\begin{array}{ccc}\hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill \\ \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill \\ \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill \end{array}\right),\end{array}$$(10) ) that the diagonal elements of matrix $D_{\mathit{lm}}$ are dependent only on coordinates and do not depend on dipole components. Denote by $R_m$ and $I_m$ the real and imaginary parts of diagonal elements:(12) $$\begin{array}{c}\hfill R_m=\mathrm{Re}{D}_{\mathit{mm}},I_m=\mathrm{Im}{D}_{\mathit{mm}}.\end{array}$$(12) Let us calculate the quantities(13) $$\begin{array}{cc}\hfill R& ={\left(\sum _m{R}_m^2\right)}^{1/2}=\frac{3+2\mathit{\rho }}{\mathit{\rho }(1+{\mathit{\rho }}^2)},\hfill \\ \hfill I& ={\left(\sum _m{I}_m^2\right)}^{1/2}=\frac{{\mathit{\rho }}^2}{1+{\mathit{\rho }}^2}.\hfill \end{array}$$(13) Equations (13(13) $$\begin{array}{cc}\hfill R& ={\left(\sum _m{R}_m^2\right)}^{1/2}=\frac{3+2\mathit{\rho }}{\mathit{\rho }(1+{\mathit{\rho }}^2)},\hfill \\ \hfill I& ={\left(\sum _m{I}_m^2\right)}^{1/2}=\frac{{\mathit{\rho }}^2}{1+{\mathit{\rho }}^2}.\hfill \end{array}$$(13) ) and (14(14) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{I}{1-I}}.\end{array}$$(14) ) allow to find the distance $r$ between the observer and the dipole, because $R$ and $I$ are known quantities. For example, we get from (14(14) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{I}{1-I}}.\end{array}$$(14) )(14) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{I}{1-I}}.\end{array}$$(14) Coordinates of the unit vector $\mathit{n}$ can be derived from Equations (10(10) $$\begin{array}{c}\hfill D_{\mathit{lm}}=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill p_2{d}_{\mathit{\omega }3}\hfill & \hfill -p_3{d}_{\mathit{\omega }2}\hfill \\ \hfill -p_1{d}_{\mathit{\omega }3}\hfill & \hfill 0\hfill & \hfill p_3{d}_{\mathit{\omega }1}\hfill \\ \hfill p_1{d}_{\mathit{\omega }2}\hfill & \hfill -p_2{d}_{\mathit{\omega }1}\hfill & \hfill 0\hfill \end{array}\right)-a\left(\begin{array}{ccc}\hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill \\ \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill \\ \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill \end{array}\right),\end{array}$$(10) ), (12(12) $$\begin{array}{c}\hfill R_m=\mathrm{Re}{D}_{\mathit{mm}},I_m=\mathrm{Im}{D}_{\mathit{mm}}.\end{array}$$(12) ) and (14(14) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{I}{1-I}}.\end{array}$$(14) ):(15) $$\begin{array}{c}\hfill n_m=-\frac{R_m}{R}=\frac{I_m}{I}.\end{array}$$(15) As a consequence, we obtain relations(16) $$\begin{array}{c}\hfill {\mathit{IR}}_m=-{\mathit{RI}}_m,\end{array}$$(16) which represent two independent equations of constraints for the diagonal elements of matrix $D_{\mathit{lm}}$. The third constraint can be derived by elimination of variable $\mathit{\rho }$ from relations (13(13) $$\begin{array}{cc}\hfill R& ={\left(\sum _m{R}_m^2\right)}^{1/2}=\frac{3+2\mathit{\rho }}{\mathit{\rho }(1+{\mathit{\rho }}^2)},\hfill \\ \hfill I& ={\left(\sum _m{I}_m^2\right)}^{1/2}=\frac{{\mathit{\rho }}^2}{1+{\mathit{\rho }}^2}.\hfill \end{array}$$(13) ) and (14(14) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{I}{1-I}}.\end{array}$$(14) ). This gives(17) $$\begin{array}{c}\hfill R^{2}I={(3-I)}^2(1-I).\end{array}$$(17) Hence, among the three complex diagonal elements $D_{\mathit{lm}}$, only three real numbers are independent.

Further, we find the dipole vector ${\mathit{d}}_{\mathit{\omega }}$. Substituting the found expression for $\mathit{\rho }=\mathit{\rho }\mathit{n}$ into (10(10) $$\begin{array}{c}\hfill D_{\mathit{lm}}=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill p_2{d}_{\mathit{\omega }3}\hfill & \hfill -p_3{d}_{\mathit{\omega }2}\hfill \\ \hfill -p_1{d}_{\mathit{\omega }3}\hfill & \hfill 0\hfill & \hfill p_3{d}_{\mathit{\omega }1}\hfill \\ \hfill p_1{d}_{\mathit{\omega }2}\hfill & \hfill -p_2{d}_{\mathit{\omega }1}\hfill & \hfill 0\hfill \end{array}\right)-a\left(\begin{array}{ccc}\hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill \\ \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill \\ \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill \end{array}\right),\end{array}$$(10) ), we get (as $n_m$ we take $-R_m/R$):(18) $$\begin{array}{cc}\hfill D_{\mathit{lm}}& =\left(\begin{array}{ccc}\hfill 0\hfill & \hfill p_2{d}_{\mathit{\omega }3}\hfill & \hfill -p_3{d}_{\mathit{\omega }2}\hfill \\ \hfill -p_1{d}_{\mathit{\omega }3}\hfill & \hfill 0\hfill & \hfill p_3{d}_{\mathit{\omega }1}\hfill \\ \hfill p_1{d}_{\mathit{\omega }2}\hfill & \hfill -p_2{d}_{\mathit{\omega }1}\hfill & \hfill 0\hfill \end{array}\right)\hfill \\ \hfill & -(R-\mathrm{i}I)\left(\begin{array}{ccc}\hfill R_1\hfill & \hfill R_1\hfill & \hfill R_1\hfill \\ \hfill R_2\hfill & \hfill R_2\hfill & \hfill R_2\hfill \\ \hfill R_3\hfill & \hfill R_3\hfill & \hfill R_3\hfill \end{array}\right),\hfill \end{array}$$(18) where(19) $$\begin{array}{c}\hfill p_j=\frac{R}{\mathit{\rho }{({\mathit{d}}_{\mathit{\omega }}\times \mathit{R})}_j}.\end{array}$$(19) The left part of (19(19) $$\begin{array}{c}\hfill p_j=\frac{R}{\mathit{\rho }{({\mathit{d}}_{\mathit{\omega }}\times \mathit{R})}_j}.\end{array}$$(19) ) is known and the second matrix of the right side is already defined. Undefined remains only vector ${\mathit{d}}_{\mathit{\omega }}$. We introduce notation $F_{\mathit{lm}}$ for the matrix(20) $$\begin{array}{c}\hfill F_{\mathit{lm}}=\mathit{\rho }{D}_{\mathit{lm}}+\mathit{\rho }(R-\mathrm{i}I)\left(\begin{array}{ccc}\hfill R_1\hfill & \hfill R_1\hfill & \hfill R_1\hfill \\ \hfill R_2\hfill & \hfill R_2\hfill & \hfill R_2\hfill \\ \hfill R_3\hfill & \hfill R_3\hfill & \hfill R_3\hfill \end{array}\right).\end{array}$$(20) Then, the vector ${\mathit{d}}_{\mathit{\omega }}$ satisfies the matrix equation(21) $$\begin{array}{c}\hfill \mathit{\rho }\left(\begin{array}{ccc}\hfill 0\hfill & \hfill p_2{d}_{\mathit{\omega }3}\hfill & \hfill -p_3{d}_{\mathit{\omega }2}\hfill \\ \hfill -p_1{d}_{\mathit{\omega }3}\hfill & \hfill 0\hfill & \hfill p_3{d}_{\mathit{\omega }1}\hfill \\ \hfill p_1{d}_{\mathit{\omega }2}\hfill & \hfill -p_2{d}_{\mathit{\omega }1}\hfill & \hfill 0\hfill \end{array}\right)=F_{\mathit{lm}}.\end{array}$$(21) One can easily prove that the equations for diagonal elements are satisfied identically. For the off-diagonal elements, we have six linear homogeneous equations:(22) $$\begin{array}{cc}\hfill & F_{12}{R}_3{d}_{\mathit{\omega }1}+(1-F_{12}{R}_1)d_{\mathit{\omega }3}=0,\hfill \\ \hfill & F_{13}{R}_2{d}_{\mathit{\omega }1}+(1-F_{13}{R}_1)d_{\mathit{\omega }2}=0,\hfill \\ \hfill & F_{21}{R}_3{d}_{\mathit{\omega }2}+(1-F_{21}{R}_2)d_{\mathit{\omega }3}=0,\hfill \\ \hfill & (1-F_{23}{R}_2)d_{\mathit{\omega }1}+F_{23}{R}_1{d}_{\mathit{\omega }2}=0,\hfill \\ \hfill & (1-F_{31}{R}_3)d_{\mathit{\omega }2}+F_{31}{R}_2{d}_{\mathit{\omega }3}=0,\hfill \\ \hfill & (1-F_{32}{R}_3)d_{\mathit{\omega }1}+F_{32}{R}_1{d}_{\mathit{\omega }3}=0.\hfill \end{array}$$(22) It follows directly from (22(22) $$\begin{array}{cc}\hfill & F_{12}{R}_3{d}_{\mathit{\omega }1}+(1-F_{12}{R}_1)d_{\mathit{\omega }3}=0,\hfill \\ \hfill & F_{13}{R}_2{d}_{\mathit{\omega }1}+(1-F_{13}{R}_1)d_{\mathit{\omega }2}=0,\hfill \\ \hfill & F_{21}{R}_3{d}_{\mathit{\omega }2}+(1-F_{21}{R}_2)d_{\mathit{\omega }3}=0,\hfill \\ \hfill & (1-F_{23}{R}_2)d_{\mathit{\omega }1}+F_{23}{R}_1{d}_{\mathit{\omega }2}=0,\hfill \\ \hfill & (1-F_{31}{R}_3)d_{\mathit{\omega }2}+F_{31}{R}_2{d}_{\mathit{\omega }3}=0,\hfill \\ \hfill & (1-F_{32}{R}_3)d_{\mathit{\omega }1}+F_{32}{R}_1{d}_{\mathit{\omega }3}=0.\hfill \end{array}$$(22) ) that $\sum _l{R}_l{F}_{\mathit{lm}}=1$. This defines next three constraints for elements $D_{\mathit{lm}}$(23) $$\begin{array}{cc}\hfill & R_2{F}_{21}+R_3{F}_{31}=1,\hfill \\ \hfill & R_1{F}_{12}+R_3{F}_{32}=1,\hfill \\ \hfill & R_1{F}_{13}+R_2{F}_{23}=1.\hfill \end{array}$$(23) The last relations transform the Equations (23(23) $$\begin{array}{cc}\hfill & R_2{F}_{21}+R_3{F}_{31}=1,\hfill \\ \hfill & R_1{F}_{12}+R_3{F}_{32}=1,\hfill \\ \hfill & R_1{F}_{13}+R_2{F}_{23}=1.\hfill \end{array}$$(23) ) into Equations (24(24) $$\begin{array}{cc}\hfill & F_{21}{d}_{\mathit{\omega }2}+F_{31}{d}_{\mathit{\omega }3}=0,\hfill \\ \hfill & F_{12}{d}_{\mathit{\omega }1}+F_{32}{d}_{\mathit{\omega }3}=0,\hfill \\ \hfill & F_{13}{d}_{\mathit{\omega }1}+F_{23}{d}_{\mathit{\omega }2}=0.\hfill \end{array}$$(24) ) and vice versa. Hence, we can define the vector ${\mathit{d}}_{\mathit{\omega }}$, for example, from (23(23) $$\begin{array}{cc}\hfill & R_2{F}_{21}+R_3{F}_{31}=1,\hfill \\ \hfill & R_1{F}_{12}+R_3{F}_{32}=1,\hfill \\ \hfill & R_1{F}_{13}+R_2{F}_{23}=1.\hfill \end{array}$$(23) ), which by use of (25(25) $$\begin{array}{cc}\hfill & F_{21}^0{d}_{02}-F_{31}^0{d}_{03}=0,\hfill \\ \hfill & F_{12}^0{d}_{01}-F_{32}^0{d}_{03}=0,\hfill \\ \hfill & F_{13}^0{d}_{01}-F_{23}^0{d}_{02}=0,\hfill \\ \hfill & {\mathit{\alpha }}_2-{\mathit{\alpha }}_3={\mathit{\psi }}_{31}-{\mathit{\psi }}_{21}+(2n+1)\mathit{\pi },\hfill \\ \hfill & {\mathit{\alpha }}_1-{\mathit{\alpha }}_3={\mathit{\psi }}_{32}-{\mathit{\psi }}_{12}+(2n+1)\mathit{\pi },\hfill \\ \hfill & {\mathit{\alpha }}_1-{\mathit{\alpha }}_2={\mathit{\psi }}_{23}-{\mathit{\psi }}_{13}+(2n+1)\mathit{\pi },\hfill \end{array}$$(25) ) can be simplified as follows(24) $$\begin{array}{cc}\hfill & F_{21}{d}_{\mathit{\omega }2}+F_{31}{d}_{\mathit{\omega }3}=0,\hfill \\ \hfill & F_{12}{d}_{\mathit{\omega }1}+F_{32}{d}_{\mathit{\omega }3}=0,\hfill \\ \hfill & F_{13}{d}_{\mathit{\omega }1}+F_{23}{d}_{\mathit{\omega }2}=0.\hfill \end{array}$$(24) Here, $F_{\mathit{lm}}$ and $d_{\mathit{\omega }l}$ are complex numbers. Substituting $F_{\mathit{lm}}$ as (see also (2(2) $$\begin{array}{c}\hfill {\mathit{d}}_{\mathit{\omega }}=(d_{0x}\exp \mathrm{i}{\mathit{\alpha }}_x,d_{0y}\exp \mathrm{i}{\mathit{\alpha }}_y,d_{0z}\exp \mathrm{i}{\mathit{\alpha }}_z),\end{array}$$(2) )) $F_{\mathit{lm}}=F_{\mathit{lm}}^0$$\exp \mathrm{i}{\mathit{\psi }}_{\mathit{lm}},$ where $F_{\mathit{lm}}^0$ and ${\mathit{\psi }}_{\mathit{lm}}$ are real numbers, we rearrange the last equations to a set of six real equations(25) $$\begin{array}{cc}\hfill & F_{21}^0{d}_{02}-F_{31}^0{d}_{03}=0,\hfill \\ \hfill & F_{12}^0{d}_{01}-F_{32}^0{d}_{03}=0,\hfill \\ \hfill & F_{13}^0{d}_{01}-F_{23}^0{d}_{02}=0,\hfill \\ \hfill & {\mathit{\alpha }}_2-{\mathit{\alpha }}_3={\mathit{\psi }}_{31}-{\mathit{\psi }}_{21}+(2n+1)\mathit{\pi },\hfill \\ \hfill & {\mathit{\alpha }}_1-{\mathit{\alpha }}_3={\mathit{\psi }}_{32}-{\mathit{\psi }}_{12}+(2n+1)\mathit{\pi },\hfill \\ \hfill & {\mathit{\alpha }}_1-{\mathit{\alpha }}_2={\mathit{\psi }}_{23}-{\mathit{\psi }}_{13}+(2n+1)\mathit{\pi },\hfill \end{array}$$(25) where $n$ is integer. The first three equations define the ratio between the Fourier transform of the dipole components; the next three equations assign the phase difference between the components of vector ${\mathit{d}}_{\mathit{\omega }}$. All six equations define the dipole polarization ellipse up to a scaling multiplier. There is no way to define this multiplier using only the coordinate derivatives of ${\mathit{H}}_{\mathit{\omega }}$, because the initial matrix equation contains only the dipole components ratio. Absolute values of the dipole components or the scaling multiplier can be calculated using the Stokes parameter $s_0$. For example, one can use (7(7) $$\begin{array}{c}\hfill J=\frac{k^2(1+k^2{r}^2)}{r^4}.\end{array}$$(7) ) and exclude the distance between the observer and the dipole by use of (15(15) $$\begin{array}{c}\hfill n_m=-\frac{R_m}{R}=\frac{I_m}{I}.\end{array}$$(15) ).

Alternative solutions

The elements of matrix $D_{\mathit{nm}}$ are not independent (if the source is in fact a point dipole). They are connected by nine real equations which can be written in a form of three real Equation (17(17) $$\begin{array}{c}\hfill R^{2}I={(3-I)}^2(1-I).\end{array}$$(17) ) and (18(18) $$\begin{array}{cc}\hfill D_{\mathit{lm}}& =\left(\begin{array}{ccc}\hfill 0\hfill & \hfill p_2{d}_{\mathit{\omega }3}\hfill & \hfill -p_3{d}_{\mathit{\omega }2}\hfill \\ \hfill -p_1{d}_{\mathit{\omega }3}\hfill & \hfill 0\hfill & \hfill p_3{d}_{\mathit{\omega }1}\hfill \\ \hfill p_1{d}_{\mathit{\omega }2}\hfill & \hfill -p_2{d}_{\mathit{\omega }1}\hfill & \hfill 0\hfill \end{array}\right)\hfill \\ \hfill & -(R-\mathrm{i}I)\left(\begin{array}{ccc}\hfill R_1\hfill & \hfill R_1\hfill & \hfill R_1\hfill \\ \hfill R_2\hfill & \hfill R_2\hfill & \hfill R_2\hfill \\ \hfill R_3\hfill & \hfill R_3\hfill & \hfill R_3\hfill \end{array}\right),\hfill \end{array}$$(18) ), and three complex Equation (25(25) $$\begin{array}{cc}\hfill & F_{21}^0{d}_{02}-F_{31}^0{d}_{03}=0,\hfill \\ \hfill & F_{12}^0{d}_{01}-F_{32}^0{d}_{03}=0,\hfill \\ \hfill & F_{13}^0{d}_{01}-F_{23}^0{d}_{02}=0,\hfill \\ \hfill & {\mathit{\alpha }}_2-{\mathit{\alpha }}_3={\mathit{\psi }}_{31}-{\mathit{\psi }}_{21}+(2n+1)\mathit{\pi },\hfill \\ \hfill & {\mathit{\alpha }}_1-{\mathit{\alpha }}_3={\mathit{\psi }}_{32}-{\mathit{\psi }}_{12}+(2n+1)\mathit{\pi },\hfill \\ \hfill & {\mathit{\alpha }}_1-{\mathit{\alpha }}_2={\mathit{\psi }}_{23}-{\mathit{\psi }}_{13}+(2n+1)\mathit{\pi },\hfill \end{array}$$(25) ). Hence, only 9 real numbers out of 18 real values of matrix $D_{\mathit{nm}}$ are independent. This number of independent variables is theoretically enough to solve the inverse problem, i.e. to find nine real numbers – three components of the radius-vector and three complex numbers ${\mathit{d}}_{\mathit{\omega }}$ – the amplitudes and phases of the dipole. Evidently, the solution of the inverse problem can be presented in different forms, but all of them can be transformed to each other by use of equations of constraint.

Further we show, for example, that the inverse problem can be solved without recourse to the diagonal elements of matrix $D_{\mathit{nm}}$. It is easy to show that (10(10) $$\begin{array}{c}\hfill D_{\mathit{lm}}=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill p_2{d}_{\mathit{\omega }3}\hfill & \hfill -p_3{d}_{\mathit{\omega }2}\hfill \\ \hfill -p_1{d}_{\mathit{\omega }3}\hfill & \hfill 0\hfill & \hfill p_3{d}_{\mathit{\omega }1}\hfill \\ \hfill p_1{d}_{\mathit{\omega }2}\hfill & \hfill -p_2{d}_{\mathit{\omega }1}\hfill & \hfill 0\hfill \end{array}\right)-a\left(\begin{array}{ccc}\hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill \\ \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill \\ \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill \end{array}\right),\end{array}$$(10) ) yields(26) $$\begin{array}{cc}\hfill & D_{21}{\mathit{\rho }}_2+D_{31}{\mathit{\rho }}_3=1+a({\mathit{\rho }}_2^2+{\mathit{\rho }}_3^2),\hfill \\ \hfill & D_{12}{\mathit{\rho }}_1+D_{32}{\mathit{\rho }}_3=1+a({\mathit{\rho }}_1^2+{\mathit{\rho }}_3^2),\hfill \\ \hfill & D_{13}{\mathit{\rho }}_1+D_{23}{\mathit{\rho }}_2=1+a({\mathit{\rho }}_1^2+{\mathit{\rho }}_2^2).\hfill \end{array}$$(26) There are no dipole components in this set of equations. Hence, we can solve the set with respect to the radius-vector $\mathit{r}$. Substituting the solution for $\mathit{r}$ back into (10(10) $$\begin{array}{c}\hfill D_{\mathit{lm}}=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill p_2{d}_{\mathit{\omega }3}\hfill & \hfill -p_3{d}_{\mathit{\omega }2}\hfill \\ \hfill -p_1{d}_{\mathit{\omega }3}\hfill & \hfill 0\hfill & \hfill p_3{d}_{\mathit{\omega }1}\hfill \\ \hfill p_1{d}_{\mathit{\omega }2}\hfill & \hfill -p_2{d}_{\mathit{\omega }1}\hfill & \hfill 0\hfill \end{array}\right)-a\left(\begin{array}{ccc}\hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill \\ \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill \\ \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill \end{array}\right),\end{array}$$(10) ), we can find the vector ${\mathit{d}}_{\mathit{\omega }}$.

Another method, may be practically more useful, is the following. Suppose the vector ${\mathit{H}}_{\mathit{\omega }}$ is not linearly polarized. Let the coordinate system be defined so that $\mathit{OZ}$ is orthogonal to the plane of polarization of vector ${\mathit{H}}_{\mathit{\omega }}$. Then, $x=y=0$ and vector $\mathit{\rho }$ is presented by the only Cartesian component ${\mathit{\rho }}_3$. The sign of this component is not defined so far. In this case, only first two of Equation (28(28) $$\begin{array}{c}\hfill D_{31}{\mathit{\rho }}_3=D_{32}{\mathit{\rho }}_3=\frac{-3-{\mathit{\rho }}^2+\mathrm{i}{\mathit{\rho }}^3}{1+{\mathit{\rho }}^2}.\end{array}$$(28) ) remain:(27) $$\begin{array}{c}\hfill D_{31}{\mathit{\rho }}_3=1+a{\mathit{\rho }}_3^2,D_{32}{\mathit{\rho }}_3=1+a{\mathit{\rho }}_3^2,\end{array}$$(27) because the elements $D_{13}$ and $D_{23}$ are undefined in accordance with (10(10) $$\begin{array}{c}\hfill D_{\mathit{lm}}=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill p_2{d}_{\mathit{\omega }3}\hfill & \hfill -p_3{d}_{\mathit{\omega }2}\hfill \\ \hfill -p_1{d}_{\mathit{\omega }3}\hfill & \hfill 0\hfill & \hfill p_3{d}_{\mathit{\omega }1}\hfill \\ \hfill p_1{d}_{\mathit{\omega }2}\hfill & \hfill -p_2{d}_{\mathit{\omega }1}\hfill & \hfill 0\hfill \end{array}\right)-a\left(\begin{array}{ccc}\hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill & \hfill {\mathit{\rho }}_1\hfill \\ \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill & \hfill {\mathit{\rho }}_2\hfill \\ \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill & \hfill {\mathit{\rho }}_3\hfill \end{array}\right),\end{array}$$(10) ). The last relations yield $D_{31}=D_{32}$. Substituting $a$, we get(28) $$\begin{array}{c}\hfill D_{31}{\mathit{\rho }}_3=D_{32}{\mathit{\rho }}_3=\frac{-3-{\mathit{\rho }}^2+\mathrm{i}{\mathit{\rho }}^3}{1+{\mathit{\rho }}^2}.\end{array}$$(28) Real parts of $D_{31}$ and $D_{32}$ define the rate of magnetic field amplitude as a function of distance. One can see from the last expression that the field amplitude is decreasing with the distance, no matter how the axis $\mathit{OZ}$ is directed. This gives us direction from the observer to dipole which we did not defined until now. Let the axis $\mathit{OZ}$ point from the dipole to observer. Then, ${\mathit{\rho }}_3>0$ and ${\mathit{\rho }}_3=\mathit{\rho }$. Let us find the rates of amplitude and phase by extracting real and imaginary parts from the last equation. For the real part, we obtain relations(29) $$\begin{array}{c}\hfill \frac{1}{k|H_{\mathit{\omega }1}|}\frac{\mathrm{d}|H_{\mathit{\omega }1}|}{\mathrm{d}r}=\frac{1}{k|H_{\mathit{\omega }2}|}\frac{\mathrm{d}|H_{\mathit{\omega }2}|}{\mathrm{d}r}=-\frac{2+{\mathit{\rho }}^2}{\mathit{\rho }(1+{\mathit{\rho }}^2)}.\end{array}$$(29) In the near zone, at the distances much less than the wave length ($r\ll k^{-1},\mathit{\rho }\ll 1$), amplitude of magnetic field drops with the distance as $-2/r$; but in far zone, this rate is half as great. Equation (31(31) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{{\mathit{\phi }}^{\prime }}{1-{\mathit{\phi }}^{\prime }}}.\end{array}$$(31) ) is a cubic equation on $r$. Similarly, the imaginary part of (30(30) $$\begin{array}{c}\hfill {\mathit{\phi }}^{\prime }=\frac{\mathrm{d}{\mathit{\phi }}_1}{k\mathrm{d}r}=\frac{\mathrm{d}{\mathit{\phi }}_2}{k\mathrm{d}r}=\frac{{\mathit{\rho }}^2}{1+{\mathit{\rho }}^2}.\end{array}$$(30) ) gives the rate of phase change ${\mathit{\phi }}^{\prime }$ with the distance(30) $$\begin{array}{c}\hfill {\mathit{\phi }}^{\prime }=\frac{\mathrm{d}{\mathit{\phi }}_1}{k\mathrm{d}r}=\frac{\mathrm{d}{\mathit{\phi }}_2}{k\mathrm{d}r}=\frac{{\mathit{\rho }}^2}{1+{\mathit{\rho }}^2}.\end{array}$$(30) Looking at the Equations (31(31) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{{\mathit{\phi }}^{\prime }}{1-{\mathit{\phi }}^{\prime }}}.\end{array}$$(31) ) and (32(32) $$\begin{array}{c}\hfill \frac{\mathit{\delta }\mathit{\rho }}{\mathit{\rho }}=\mathit{\eta }\frac{\mathit{\delta }R}{R},\mathit{\eta }=\frac{(1+{\mathit{\rho }}^2)(3+2{\mathit{\rho }}^2)}{(1+2{\mathit{\rho }}^2)(3+{\mathit{\rho }}^2)}.\end{array}$$(32) ), we see that at small distances ($r\ll k^{-1}$) the phase rate is much less than the relative amplitude rate. As to large distances, the phase rate is equal to $k$ and does not depend on $r$. Hence, the solution of the last equation with respect to the distance(31) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{{\mathit{\phi }}^{\prime }}{1-{\mathit{\phi }}^{\prime }}}.\end{array}$$(31) cannot be used, because at large distance ${\mathit{\phi }}^{\prime }\to 1$. It would be expected, because in the wave zone, at distances much greater than the wave length, the field is a field of radiation. Hence, the phase displacement at distance of a wave length is equal to $2\mathit{\pi }$ for any $r$.

Uniqueness and stability of the solutions

Uniqueness of the obtained solutions follows directly from the used methods. Namely, we have found all solutions of a set of algebraic equations. In some cases, the solution is obtained in different forms like, for example, expressions for $r$ given by Equations (13(13) $$\begin{array}{cc}\hfill R& ={\left(\sum _m{R}_m^2\right)}^{1/2}=\frac{3+2\mathit{\rho }}{\mathit{\rho }(1+{\mathit{\rho }}^2)},\hfill \\ \hfill I& ={\left(\sum _m{I}_m^2\right)}^{1/2}=\frac{{\mathit{\rho }}^2}{1+{\mathit{\rho }}^2}.\hfill \end{array}$$(13) ) and (14(14) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{I}{1-I}}.\end{array}$$(14) ). The first equation allows to find $r$ from the real part of diagonal elements of matrix $D_{\mathit{lm}}$, while the second one uses the imaginary part of this matrix. It means that in the first case, $r$ is calculated by the use of measurements of the gradient of magnetic field amplitude, while in the second case calculations use measurements of the phase gradient $\mathit{\partial }{\mathit{\phi }}_m/\mathit{\partial }{x}_m$ (see Equation (9(9) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{\mathit{lm}}\equiv \frac{1}{H_{\mathit{\omega }m}}\frac{\mathit{\partial }{H}_{\mathit{\omega }m}}{\mathit{\partial }{x}_l}=\frac{1}{|H_{\mathit{\omega }m}|}\frac{\mathit{\partial }|H_{\mathit{\omega }m}|}{\mathit{\partial }{x}_l}+\mathrm{i}\frac{\mathit{\partial }{\mathit{\phi }}_m}{\mathit{\partial }{x}_l}.\end{array}$$(9) )). The real and imaginary parts of the field gradient are not independent because the nine partial derivatives (9(9) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{\mathit{lm}}\equiv \frac{1}{H_{\mathit{\omega }m}}\frac{\mathit{\partial }{H}_{\mathit{\omega }m}}{\mathit{\partial }{x}_l}=\frac{1}{|H_{\mathit{\omega }m}|}\frac{\mathit{\partial }|H_{\mathit{\omega }m}|}{\mathit{\partial }{x}_l}+\mathrm{i}\frac{\mathit{\partial }{\mathit{\phi }}_m}{\mathit{\partial }{x}_l}.\end{array}$$(9) ) are defined only by three components of the dipole (2(2) $$\begin{array}{c}\hfill {\mathit{d}}_{\mathit{\omega }}=(d_{0x}\exp \mathrm{i}{\mathit{\alpha }}_x,d_{0y}\exp \mathrm{i}{\mathit{\alpha }}_y,d_{0z}\exp \mathrm{i}{\mathit{\alpha }}_z),\end{array}$$(2) ). The real and imaginary parts of the diagonal elements are bound up with Equation (18(18) $$\begin{array}{cc}\hfill D_{\mathit{lm}}& =\left(\begin{array}{ccc}\hfill 0\hfill & \hfill p_2{d}_{\mathit{\omega }3}\hfill & \hfill -p_3{d}_{\mathit{\omega }2}\hfill \\ \hfill -p_1{d}_{\mathit{\omega }3}\hfill & \hfill 0\hfill & \hfill p_3{d}_{\mathit{\omega }1}\hfill \\ \hfill p_1{d}_{\mathit{\omega }2}\hfill & \hfill -p_2{d}_{\mathit{\omega }1}\hfill & \hfill 0\hfill \end{array}\right)\hfill \\ \hfill & -(R-\mathrm{i}I)\left(\begin{array}{ccc}\hfill R_1\hfill & \hfill R_1\hfill & \hfill R_1\hfill \\ \hfill R_2\hfill & \hfill R_2\hfill & \hfill R_2\hfill \\ \hfill R_3\hfill & \hfill R_3\hfill & \hfill R_3\hfill \end{array}\right),\hfill \end{array}$$(18) ). Hence, there can be different forms of the solution, but all of them should give the same result in numerical calculations. If in some real experimental situation, solutions for $r$ derived from Equations (13(13) $$\begin{array}{cc}\hfill R& ={\left(\sum _m{R}_m^2\right)}^{1/2}=\frac{3+2\mathit{\rho }}{\mathit{\rho }(1+{\mathit{\rho }}^2)},\hfill \\ \hfill I& ={\left(\sum _m{I}_m^2\right)}^{1/2}=\frac{{\mathit{\rho }}^2}{1+{\mathit{\rho }}^2}.\hfill \end{array}$$(13) ) and (14(14) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{I}{1-I}}.\end{array}$$(14) ) are different, then it can indicate that the field measured is not a field of a dipole. The same reasoning is applicable to any other results of this paper, because as mentioned above, the initial set of equations is overdetermined. One do not need to know all the derivatives of the field in order to solve the inversion problem.

Let us estimate the stability of the obtained solutions. We study how errors in the measurements of the magnetic field translate into errors in the reconstructions. We start with analysis of expressions for $r$ (13(13) $$\begin{array}{cc}\hfill R& ={\left(\sum _m{R}_m^2\right)}^{1/2}=\frac{3+2\mathit{\rho }}{\mathit{\rho }(1+{\mathit{\rho }}^2)},\hfill \\ \hfill I& ={\left(\sum _m{I}_m^2\right)}^{1/2}=\frac{{\mathit{\rho }}^2}{1+{\mathit{\rho }}^2}.\hfill \end{array}$$(13) ) and (14(14) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{I}{1-I}}.\end{array}$$(14) ), then we proceed to stability of the solution for $\mathit{n}$ and finally we discuss stability of the solution for the Fourier transform of the dipole vector.

Let the reconstructed parameter $y$ be a function of measurements $x$: $y=f(x)$. We think of small reconstruction error $\mathit{\delta }y$ as the module of the linear part of change in $y$ corresponding to the small change $\mathit{\delta }x$ in $x$. So, we write$$\begin{array}{c}\hfill \mathit{\delta }y=\left|\frac{\mathrm{d}f(x)}{\mathrm{d}x}\mathit{\delta }x\right|.\end{array}$$Taking derivative from Formula (13(13) $$\begin{array}{cc}\hfill R& ={\left(\sum _m{R}_m^2\right)}^{1/2}=\frac{3+2\mathit{\rho }}{\mathit{\rho }(1+{\mathit{\rho }}^2)},\hfill \\ \hfill I& ={\left(\sum _m{I}_m^2\right)}^{1/2}=\frac{{\mathit{\rho }}^2}{1+{\mathit{\rho }}^2}.\hfill \end{array}$$(13) ), we obtain the relative error for $\mathit{\rho }$(32) $$\begin{array}{c}\hfill \frac{\mathit{\delta }\mathit{\rho }}{\mathit{\rho }}=\mathit{\eta }\frac{\mathit{\delta }R}{R},\mathit{\eta }=\frac{(1+{\mathit{\rho }}^2)(3+2{\mathit{\rho }}^2)}{(1+2{\mathit{\rho }}^2)(3+{\mathit{\rho }}^2)}.\end{array}$$(32) It is easy to check that $\mathit{\eta }\le 1$. Next, we express $\mathit{\delta }R/R$ in terms of $\mathit{\delta }{R}_m$:$$\frac{\mathit{\delta }R}{R}=\frac{R_1\mathit{\delta }{R}_1+R_2\mathit{\delta }{R}_2+R_3\mathit{\delta }{R}_3}{R_1^2+R_2^2+R_3^2}.$$Let $R_g$ be the greatest among $|R_1|,|R_2|,|R_3|$ and we assume that $\mathit{\delta }{R}_1=\mathit{\delta }{R}_2=\mathit{\delta }{R}_3=\mathit{\delta }{R}_g$. Then,(33) $$\begin{array}{c}\hfill \frac{\mathit{\delta }R}{R}\le 3\frac{\mathit{\delta }{R}_g}{|R_g|}.\end{array}$$(33) According to definition (9(9) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{\mathit{lm}}\equiv \frac{1}{H_{\mathit{\omega }m}}\frac{\mathit{\partial }{H}_{\mathit{\omega }m}}{\mathit{\partial }{x}_l}=\frac{1}{|H_{\mathit{\omega }m}|}\frac{\mathit{\partial }|H_{\mathit{\omega }m}|}{\mathit{\partial }{x}_l}+\mathrm{i}\frac{\mathit{\partial }{\mathit{\phi }}_m}{\mathit{\partial }{x}_l}.\end{array}$$(9) )$$R_g=\frac{\mathit{\partial }|H_g|}{k|H_g|\mathit{\partial }{x}_g}\approx \frac{\mathrm{\Delta }|H_g|}{k|H_g|\mathrm{\Delta }{x}_g},$$where $\mathrm{\Delta }|H_g|=|H_g{|}_2-|H_g{|}_1$ and $\mathrm{\Delta }{x}_g={(x_g)}_2-{(x_g)}_1$ are the finite differences measured in order to calculate the gradient of the magnetic field. The relative error in $R_g$ is calculated then as(34) $$\begin{array}{c}\hfill \frac{\mathit{\delta }{R}_g}{|R_g|}=\frac{\mathit{\delta }(\mathrm{\Delta }|H_g|)}{\mathrm{\Delta }|H_g|}+\frac{\mathit{\delta }|H_g|}{|H_g|}+\frac{\mathit{\delta }|\mathrm{\Delta }{x}_g|}{|\mathrm{\Delta }{x}_g|}.\end{array}$$(34) Combining this with the expressions (34(34) $$\begin{array}{c}\hfill \frac{\mathit{\delta }{R}_g}{|R_g|}=\frac{\mathit{\delta }(\mathrm{\Delta }|H_g|)}{\mathrm{\Delta }|H_g|}+\frac{\mathit{\delta }|H_g|}{|H_g|}+\frac{\mathit{\delta }|\mathrm{\Delta }{x}_g|}{|\mathrm{\Delta }{x}_g|}.\end{array}$$(34) ) and (35(35) $$\begin{array}{c}\hfill \frac{\mathit{\delta }\mathit{\rho }}{\mathit{\rho }}\le 3\mathit{\eta }(\frac{\mathit{\delta }(\mathrm{\Delta }|H_g|)}{\mathrm{\Delta }|H_g|}+\frac{\mathit{\delta }|H_g|}{|H_g|}+\frac{\mathit{\delta }|\mathrm{\Delta }{x}_g|}{|\mathrm{\Delta }{x}_g|}).\end{array}$$(35) ), we get the estimation of relative error in $\mathit{\rho }$(35) $$\begin{array}{c}\hfill \frac{\mathit{\delta }\mathit{\rho }}{\mathit{\rho }}\le 3\mathit{\eta }(\frac{\mathit{\delta }(\mathrm{\Delta }|H_g|)}{\mathrm{\Delta }|H_g|}+\frac{\mathit{\delta }|H_g|}{|H_g|}+\frac{\mathit{\delta }|\mathrm{\Delta }{x}_g|}{|\mathrm{\Delta }{x}_g|}).\end{array}$$(35) Hence, the relative error in $\mathit{\rho }$ of formula (13(13) $$\begin{array}{cc}\hfill R& ={\left(\sum _m{R}_m^2\right)}^{1/2}=\frac{3+2\mathit{\rho }}{\mathit{\rho }(1+{\mathit{\rho }}^2)},\hfill \\ \hfill I& ={\left(\sum _m{I}_m^2\right)}^{1/2}=\frac{{\mathit{\rho }}^2}{1+{\mathit{\rho }}^2}.\hfill \end{array}$$(13) ) is linearly dependent on the sum of relative errors in: measurements of the magnetic field difference in two near points of space; measurement of the absolute value of the magnetic field; and measurement of the distance between the two near points. The inverse problem in this case is said to be a well-posed problem.

Now we make the some analysis with respect to the solution (15(15) $$\begin{array}{c}\hfill n_m=-\frac{R_m}{R}=\frac{I_m}{I}.\end{array}$$(15) ). Taking derivative from Equation (15(15) $$\begin{array}{c}\hfill n_m=-\frac{R_m}{R}=\frac{I_m}{I}.\end{array}$$(15) ), we get$$\frac{\mathit{\delta }\mathit{\rho }}{\mathit{\rho }}=\frac{1}{2(1-I)}\frac{\mathit{\delta }I}{I}.$$The denominator of this equation shows that the relative error in $\mathit{\rho }$ grows infinitely as $I\to 1$ or, as it is seen from (14(14) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{I}{1-I}}.\end{array}$$(14) ), as $\mathit{\rho }\to \infty$. Substituting $\mathit{\delta }I/I$ from Equation (9(9) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{\mathit{lm}}\equiv \frac{1}{H_{\mathit{\omega }m}}\frac{\mathit{\partial }{H}_{\mathit{\omega }m}}{\mathit{\partial }{x}_l}=\frac{1}{|H_{\mathit{\omega }m}|}\frac{\mathit{\partial }|H_{\mathit{\omega }m}|}{\mathit{\partial }{x}_l}+\mathrm{i}\frac{\mathit{\partial }{\mathit{\phi }}_m}{\mathit{\partial }{x}_l}.\end{array}$$(9) ) and (14(14) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{I}{1-I}}.\end{array}$$(14) ), we obtain(36) $$\begin{array}{c}\hfill \frac{\mathit{\delta }\mathit{\rho }}{\mathit{\rho }}\le \frac{3}{2}(1+{\mathit{\rho }}^2)(\left|\frac{\mathit{\delta }(\mathrm{\Delta }{\mathit{\phi }}_g)}{\mathrm{\Delta }{\mathit{\phi }}_g}\right|+\left|\frac{\mathit{\delta }(\mathrm{\Delta }{x}_g)}{\mathrm{\Delta }{x}_g}\right|),\end{array}$$(36) $g$ is the subscript of the greatest among $|\mathrm{\Delta }{\mathit{\phi }}_i/\mathrm{\Delta }{x}_i|$. Again, the error in reconstructed data depends linear on the errors in measurement. But this time, the constant of proportionality $1+{\mathit{\rho }}^2$ grows with the distance $r$. Figure 5 shows an example of such behaviour. Therefore, at great distances $r\gg \mathit{\lambda }$ expression (13(13) $$\begin{array}{cc}\hfill R& ={\left(\sum _m{R}_m^2\right)}^{1/2}=\frac{3+2\mathit{\rho }}{\mathit{\rho }(1+{\mathit{\rho }}^2)},\hfill \\ \hfill I& ={\left(\sum _m{I}_m^2\right)}^{1/2}=\frac{{\mathit{\rho }}^2}{1+{\mathit{\rho }}^2}.\hfill \end{array}$$(13) ) gives better results than Equation (14(14) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{I}{1-I}}.\end{array}$$(14) ) or (15(15) $$\begin{array}{c}\hfill n_m=-\frac{R_m}{R}=\frac{I_m}{I}.\end{array}$$(15) ). The same conclusion is applicable to the pair of solutions (31(31) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{{\mathit{\phi }}^{\prime }}{1-{\mathit{\phi }}^{\prime }}}.\end{array}$$(31) ) and (33(33) $$\begin{array}{c}\hfill \frac{\mathit{\delta }R}{R}\le 3\frac{\mathit{\delta }{R}_g}{|R_g|}.\end{array}$$(33) ).

Let us estimate errors in the unit vector of direction (16(16) $$\begin{array}{c}\hfill {\mathit{IR}}_m=-{\mathit{RI}}_m,\end{array}$$(16) ). Using a vector form $\mathit{R}=(R_1,R_2,R_3)$, we obtain next expression for the differential of $\mathit{n}$$$\mathrm{d}\mathit{n}=\frac{((\mathit{R}\times \mathrm{d}\mathit{R})\times \mathit{R})}{R^3},|\mathrm{d}\mathit{n}|=\frac{\mathrm{d}R}{R}\sin \mathit{\varphi },$$where $\mathit{\varphi }$ is the angle between vectors $\mathit{R}$ and $\mathrm{d}\mathit{R}$. As $\mathrm{d}{n}_m\le |\mathrm{d}\mathit{n}|$, we estimate the relative error in direction from the observer to the source of the field as$$\mathit{\delta }{n}_m=\frac{\mathit{\delta }{R}_g}{|R_g|}$$with $\mathit{\delta }{R}_g/|R_g|$ defined by Equation (36(36) $$\begin{array}{c}\hfill \frac{\mathit{\delta }\mathit{\rho }}{\mathit{\rho }}\le \frac{3}{2}(1+{\mathit{\rho }}^2)(\left|\frac{\mathit{\delta }(\mathrm{\Delta }{\mathit{\phi }}_g)}{\mathrm{\Delta }{\mathit{\phi }}_g}\right|+\left|\frac{\mathit{\delta }(\mathrm{\Delta }{x}_g)}{\mathrm{\Delta }{x}_g}\right|),\end{array}$$(36) ). Following the reasoning in paragraph next to Equation (37(37) $$\begin{array}{c}\hfill T=\left(\begin{array}{ccccc}\hfill \cos \mathit{\beta }\hfill & \hfill \hfill & \hfill -\sin \mathit{\beta }\cos \mathit{\alpha }\hfill & \hfill & \sin \mathit{\beta }\sin \mathit{\alpha }\hfill \\ \hfill -\sin \mathit{\beta }\hfill & \hfill \hfill & \hfill \cos \mathit{\beta }\cos \mathit{\alpha }\hfill & \hfill & -\cos \mathit{\beta }\sin \mathit{\alpha }\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill \sin \mathit{\alpha }\hfill & \hfill & \sin \mathit{\beta }\hfill \end{array}\right)\end{array}$$(37) ), we conclude that the solution for $\mathit{n}$ is stable.

Evidently, the same analysis is applicable to calculation of the dipole polarization from Equation (27(27) $$\begin{array}{c}\hfill D_{31}{\mathit{\rho }}_3=1+a{\mathit{\rho }}_3^2,D_{32}{\mathit{\rho }}_3=1+a{\mathit{\rho }}_3^2,\end{array}$$(27) ) or calculations of the Fourier components of the dipole. This results also in the linear dependence between the errors in measurements and errors in calculations of the sought quantities.

Hence, we can establish that the inversion problem discussed in this paper is well-posed and the obtained solutions are unique.

It is interesting to see how well the obtained solutions do work at very high and very low frequencies. As one can see, the frequency does not appear explicitly in the final formulas. It is included only in the reduced distance $\mathit{\rho }=\mathit{\omega }r/c$. The errors of calculations depend only on relative distance $r/\mathit{\lambda }$. Therefore, for fixed distance, working with low frequency means working close to the source of the field. There are no problems in calculation of the distance in case $r\ll \mathit{\lambda }$ by use of gradient of magnetic field. Because, the gradient of the field and the field itself are relatively great and can be easily measured, even in the case of stationary field. But when using for calculation, the phase gradient can cause large errors, because at small distances or by slowly varying field the phase gradient is small and tends to zero as $\mathit{\rho }\to 0$ or $\mathit{\lambda }\to \infty$ (see Equation (32(32) $$\begin{array}{c}\hfill \frac{\mathit{\delta }\mathit{\rho }}{\mathit{\rho }}=\mathit{\eta }\frac{\mathit{\delta }R}{R},\mathit{\eta }=\frac{(1+{\mathit{\rho }}^2)(3+2{\mathit{\rho }}^2)}{(1+2{\mathit{\rho }}^2)(3+{\mathit{\rho }}^2)}.\end{array}$$(32) )).

At high frequencies, problems can be encountered with measurements of the derivatives of the magnetic field. These measurements must be made in a small area which is much less than the wavelength. But in case of short wavelength, the measurement accuracy can be restricted, for example, by dimensions of the used device or probe. The relative error $\mathit{\delta }(\mathrm{\Delta }|H_g|)/\mathrm{\Delta }|H_g|$ in Equation (37(37) $$\begin{array}{c}\hfill T=\left(\begin{array}{ccccc}\hfill \cos \mathit{\beta }\hfill & \hfill \hfill & \hfill -\sin \mathit{\beta }\cos \mathit{\alpha }\hfill & \hfill & \sin \mathit{\beta }\sin \mathit{\alpha }\hfill \\ \hfill -\sin \mathit{\beta }\hfill & \hfill \hfill & \hfill \cos \mathit{\beta }\cos \mathit{\alpha }\hfill & \hfill & -\cos \mathit{\beta }\sin \mathit{\alpha }\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill \sin \mathit{\alpha }\hfill & \hfill & \sin \mathit{\beta }\hfill \end{array}\right)\end{array}$$(37) ) can be intolerable great in this case.

Significant errors can arise in case of weak magnetic field. There are at least two origins of errors in this case: (i) noise – it must be sufficiently less then the measured field; (ii) error $\mathit{\delta }H/H$ (second item in Equation (37(37) $$\begin{array}{c}\hfill T=\left(\begin{array}{ccccc}\hfill \cos \mathit{\beta }\hfill & \hfill \hfill & \hfill -\sin \mathit{\beta }\cos \mathit{\alpha }\hfill & \hfill & \sin \mathit{\beta }\sin \mathit{\alpha }\hfill \\ \hfill -\sin \mathit{\beta }\hfill & \hfill \hfill & \hfill \cos \mathit{\beta }\cos \mathit{\alpha }\hfill & \hfill & -\cos \mathit{\beta }\sin \mathit{\alpha }\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill \sin \mathit{\alpha }\hfill & \hfill & \sin \mathit{\beta }\hfill \end{array}\right)\end{array}$$(37) )) in measurement of weak field can become rather great.

An example

Here, we test the derived formulas in a numerical experiment. We define some numerical values of the Fourier components of a dipole at some point of space. Then, we calculate the magnetic field produced by the dipole at another point of space. We use the calculated values of the field as ‘measured’ vales and apply the developed method for location of the dipole position. Next, we add a random disturbance to the calculated field values and calculate the position of the dipole again in order to analyse the stability of the solution.

Let the monochromatic component ${\mathit{d}}_{\mathit{\omega }}{e}^{-\mathrm{i}\mathit{\omega }t}$ of a dipole describe an ellipse in the coordinate plane $\mathit{XY}$ as shown in Figure 2. The dipole components are defined as follows:$${\mathit{d}}_{\mathit{\omega }}=(d_{\mathit{\omega }x},d_{\mathit{\omega }y},0)=(d_x,\mathrm{i}{d}_y,0),$$where $d_x$ and $d_y$ are real numbers.

Fig. 2 Reference systems.

The field of the dipole is measured at a point with position vector $\mathit{r}$. The magnetic field of the dipole changes in the plane orthogonal to the vector $\mathit{r}$. The observer adopts a coordinate system with axis $Z^{\prime }$ orthogonal to the plane in which the magnetic field varies, and with arbitrary directed axes $X^{\prime }$ and $Y^{\prime }$. In order to calculate the magnetic field at point $\mathit{r}$, we denote the angle between the axis $Y^{\prime }$ and the plane $\mathit{YZ}$ by $\mathit{\beta }$. Then, the transition matrix from the coordinate system $(x,y,z)$ to the system $(x^{\prime },y^{\prime },z^{\prime })$ takes the form(37) $$\begin{array}{c}\hfill T=\left(\begin{array}{ccccc}\hfill \cos \mathit{\beta }\hfill & \hfill \hfill & \hfill -\sin \mathit{\beta }\cos \mathit{\alpha }\hfill & \hfill & \sin \mathit{\beta }\sin \mathit{\alpha }\hfill \\ \hfill -\sin \mathit{\beta }\hfill & \hfill \hfill & \hfill \cos \mathit{\beta }\cos \mathit{\alpha }\hfill & \hfill & -\cos \mathit{\beta }\sin \mathit{\alpha }\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill \sin \mathit{\alpha }\hfill & \hfill & \sin \mathit{\beta }\hfill \end{array}\right)\end{array}$$(37) Using Equation (1(1) $$\begin{array}{c}\hfill {\mathit{H}}_{\mathit{\omega }}(\mathit{r})=\frac{\mathrm{i}k}{r^2}({\mathit{d}}_{\mathit{\omega }}\times \mathit{n})(\mathrm{i}\mathit{kr}-1){\mathrm{e}}^{\mathrm{i}\mathit{kr}},\end{array}$$(1) ), we find the components of the magnetic field vector in the primed coordinate system(38) $$\begin{array}{c}\hfill H_{\mathit{\omega }{x}^{\prime }}={\mathit{Ad}}_{\mathit{\omega }{y}^{\prime }},H_{\mathit{\omega }{y}^{\prime }}=-{\mathit{Ad}}_{\mathit{\omega }{x}^{\prime }},H_{\mathit{\omega }{z}^{\prime }}=0,A=\frac{\mathrm{i}k}{r^2}(\mathrm{i}\mathit{kr}-1){\mathrm{e}}^{\mathrm{i}\mathit{kr}}.\end{array}$$(38) Here, $d_{\mathit{\omega }{j}^{\prime }}$, $j^{\prime }=x^{\prime },y^{\prime }$ are the dipole components in the primed reference system, which are defined by equations$$\begin{array}{c}\hfill d_{\mathit{\omega }{j}^{\prime }}=T_{\mathit{jl}}{d}_{\mathit{\omega }l},\end{array}$$where $T_{\mathit{jl}}$ are the elements of transition matrix $T$.

In order to calculate the measurable values – Fourier amplitude and phase – we extract the real and imaginary parts of $H_{\mathit{\omega }{j}^{\prime }}$. This gives(39) $$\begin{array}{c}\hfill H_{\mathit{\omega }{x}^{\prime }}=\frac{k}{r^2}[{\mathit{Sd}}_x\sin \mathit{\beta }+{\mathit{Cd}}_y\cos \mathit{\beta }\cos \mathit{\alpha }-\mathrm{i}({\mathit{Cd}}_x\sin \mathit{\beta }-{\mathit{Sd}}_y\cos \mathit{\beta }\cos \mathit{\alpha })],\\ \hfill H_{\mathit{\omega }{y}^{\prime }}=\frac{k}{r^2}[{\mathit{Sd}}_x\cos \mathit{\beta }-{\mathit{Cd}}_y\sin \mathit{\beta }\cos \mathit{\alpha }-\mathrm{i}({\mathit{Cd}}_x\cos \mathit{\beta }+{\mathit{Sd}}_y\sin \mathit{\beta }\cos \mathit{\alpha })],\\ \hfill S=\sin \mathit{\rho }-\mathit{\rho }\cos \mathit{\rho },C=\cos \mathit{\rho }+\mathit{\rho }\sin \mathit{\rho },\mathit{\rho }=\mathit{kr}.\end{array}$$(39) We can simplify these expressions supposing that the observer chooses the axes $X^{\prime }$ and $Y^{\prime }$ in a way that the phase difference between $H_{\mathit{\omega }{x}^{\prime }}$ and $H_{\mathit{\omega }{y}^{\prime }}$ is $\mathit{\pi }/2$. One can see from the last equations that it happens if $\mathit{\beta }=0$ or $\mathit{\pi }/2$. Further, we consider the case $\mathit{\beta }=0$. Then, Equations (41(41) $$\begin{array}{c}\hfill |H_{\mathit{\omega }{x}^{\prime }}|=\frac{k^3}{{\mathit{\rho }}^2}{d}_y\cos \mathit{\alpha }\sqrt{1+{\mathit{\rho }}^2},|H_{\mathit{\omega }{y}^{\prime }}|=\frac{k^3}{{\mathit{\rho }}^2}{d}_x\sqrt{1+{\mathit{\rho }}^2},\end{array}$$(41) ) take the form(40) $$\begin{array}{cc}\hfill H_{\mathit{\omega }{x}^{\prime }}& =\frac{k}{r^2}{d}_y\cos \mathit{\alpha }(C+\mathrm{i}S),\hfill \\ \hfill H_{\mathit{\omega }{y}^{\prime }}& =\frac{k}{r^2}{d}_x(S-\mathrm{i}C).\hfill \end{array}$$(40) The real values which are measured in practice are: the amplitudes(41) $$\begin{array}{c}\hfill |H_{\mathit{\omega }{x}^{\prime }}|=\frac{k^3}{{\mathit{\rho }}^2}{d}_y\cos \mathit{\alpha }\sqrt{1+{\mathit{\rho }}^2},|H_{\mathit{\omega }{y}^{\prime }}|=\frac{k^3}{{\mathit{\rho }}^2}{d}_x\sqrt{1+{\mathit{\rho }}^2},\end{array}$$(41) and the phases(42) $$\begin{array}{c}\hfill {\mathit{\phi }}_{x^{\prime }}=\arctan \frac{S}{C}\pm \mathit{\pi }n,{\mathit{\phi }}_{y^{\prime }}=-\arctan \frac{C}{S}\pm \mathit{\pi }n,\end{array}$$(42) $n$ are the integer numbers.

Using Equations (42(42) $$\begin{array}{c}\hfill {\mathit{\phi }}_{x^{\prime }}=\arctan \frac{S}{C}\pm \mathit{\pi }n,{\mathit{\phi }}_{y^{\prime }}=-\arctan \frac{C}{S}\pm \mathit{\pi }n,\end{array}$$(42) ) and (43(43) $$\begin{array}{c}\hfill \frac{\mathrm{d}f(\mathit{\rho })}{\mathrm{d}\mathit{\rho }}=\frac{f(\mathit{\rho }+\mathrm{\Delta }\mathit{\rho })-f(\mathit{\rho }-\mathrm{\Delta }\mathit{\rho })}{2\mathrm{\Delta }\mathit{\rho }},\end{array}$$(43) ), we calculate amplitudes and phases at three close points along the axis $Z^{\prime }$: at $z^{\prime }=\mathit{\rho }$ and at $z^{\prime }=\mathit{\rho }\pm \mathrm{\Delta }\mathit{\rho }$. Then, we calculate numerically the derivatives as(43) $$\begin{array}{c}\hfill \frac{\mathrm{d}f(\mathit{\rho })}{\mathrm{d}\mathit{\rho }}=\frac{f(\mathit{\rho }+\mathrm{\Delta }\mathit{\rho })-f(\mathit{\rho }-\mathrm{\Delta }\mathit{\rho })}{2\mathrm{\Delta }\mathit{\rho }},\end{array}$$(43) where(44) $$\begin{array}{c}\hfill f=|H_{\mathit{\omega }{x}^{\prime }}|,|H_{\mathit{\omega }{y}^{\prime }}|,{\mathit{\phi }}_{x^{\prime }},{\mathit{\phi }}_{y^{\prime }}.\end{array}$$(44) We use these derivatives to calculate the distance between the observer and the source of the field according to formulas (31(31) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{{\mathit{\phi }}^{\prime }}{1-{\mathit{\phi }}^{\prime }}}.\end{array}$$(31) ) and (33(33) $$\begin{array}{c}\hfill \frac{\mathit{\delta }R}{R}\le 3\frac{\mathit{\delta }{R}_g}{|R_g|}.\end{array}$$(33) ) and to find relative error of these calculations. Calculated values of the derivatives of $|H_{\mathit{\omega }1}|$ and $|H_{\mathit{\omega }2}|$ in Equation (31(31) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{{\mathit{\phi }}^{\prime }}{1-{\mathit{\phi }}^{\prime }}}.\end{array}$$(31) ) and derivatives of ${\mathit{\varphi }}_1$ and ${\mathit{\varphi }}_2$ Equation (32(32) $$\begin{array}{c}\hfill \frac{\mathit{\delta }\mathit{\rho }}{\mathit{\rho }}=\mathit{\eta }\frac{\mathit{\delta }R}{R},\mathit{\eta }=\frac{(1+{\mathit{\rho }}^2)(3+2{\mathit{\rho }}^2)}{(1+2{\mathit{\rho }}^2)(3+{\mathit{\rho }}^2)}.\end{array}$$(32) ) are equal. The relative error $\mathit{\epsilon }$ in calculation of $\mathit{\rho }$ is defined as follows: $\mathit{\epsilon }=(\mathit{\rho }-{\mathit{\rho }}_0)/{\mathit{\rho }}_0$, where ${\mathit{\rho }}_0$ is the ‘exact’ value of $\mathit{\rho }$ used by direct calculation of the magnetic field, and $\mathit{\rho }$ is the result of solution of the inverse problem. The dependence $\mathit{\epsilon }(\mathit{\rho })$ is shown in Figure 3. In calculation of derivatives, we put $\mathrm{\Delta }\mathit{\rho }=0.05$. In other words, the field is ‘measured’ at the points which are $\mathrm{\Delta }r=0.05\mathit{\lambda }/2\mathit{\pi }$ apart.

Fig. 3 Relative errors of the distance calculation. 1 – error in the case when the distance is calculated from the formula (33(33) $$\begin{array}{c}\hfill \frac{\mathit{\delta }R}{R}\le 3\frac{\mathit{\delta }{R}_g}{|R_g|}.\end{array}$$(33) ); 2 – error in the case when the distance is calculated from the formula (31(31) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{{\mathit{\phi }}^{\prime }}{1-{\mathit{\phi }}^{\prime }}}.\end{array}$$(31) ).

Fig. 4 Relative error which occurs if the distance is calculated by use of Equation (31(31) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{{\mathit{\phi }}^{\prime }}{1-{\mathit{\phi }}^{\prime }}}.\end{array}$$(31) ). A 1% noise is added.

Fig. 5 Relative error which occurs if the distance is calculated by use of Equation (33(33) $$\begin{array}{c}\hfill \frac{\mathit{\delta }R}{R}\le 3\frac{\mathit{\delta }{R}_g}{|R_g|}.\end{array}$$(33) ). A 1% noise is added.

Relative great errors at small distance, especially those represented by curve 2, are caused alone by numerical differentiation. At small distances, when $r$ is comparable with the wavelength $\mathit{\lambda }$, the phase, and especially the magnetic field amplitude grow rapidly. The used method (44(44) $$\begin{array}{c}\hfill f=|H_{\mathit{\omega }{x}^{\prime }}|,|H_{\mathit{\omega }{y}^{\prime }}|,{\mathit{\phi }}_{x^{\prime }},{\mathit{\phi }}_{y^{\prime }}.\end{array}$$(44) ) for numerical differentiation is too rough in this case. Evidently, these errors can be made as small as desired by use of appropriate numerical methods of calculations.

In order to analyse the stability of solutions with regard to the noise, we have added to the ‘measured’ data listed in (45(44) $$\begin{array}{c}\hfill f=|H_{\mathit{\omega }{x}^{\prime }}|,|H_{\mathit{\omega }{y}^{\prime }}|,{\mathit{\phi }}_{x^{\prime }},{\mathit{\phi }}_{y^{\prime }}.\end{array}$$(44) ) a 1% random noise defined by formula $f_{\mathrm{noise}}=f(1+0.02R)$, where $R$ is a random variable of interval $[-0.5,0.5]$. The result is shown in Figures 4 and 5.

As one can see, the solution (31(31) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{{\mathit{\phi }}^{\prime }}{1-{\mathit{\phi }}^{\prime }}}.\end{array}$$(31) ) is rather stable – the relative error does not exceed 5% above the distance $\mathit{\rho }=1$ ($r>\mathit{\lambda }/2\mathit{\pi }$). But the plot in Figure 5, which represents the solution given by formula (33(33) $$\begin{array}{c}\hfill \frac{\mathit{\delta }R}{R}\le 3\frac{\mathit{\delta }{R}_g}{|R_g|}.\end{array}$$(33) ), demonstrates the unacceptable errors at large distances. These results of numerical modelling agree with the discussions following the Equations (37(37) $$\begin{array}{c}\hfill T=\left(\begin{array}{ccccc}\hfill \cos \mathit{\beta }\hfill & \hfill \hfill & \hfill -\sin \mathit{\beta }\cos \mathit{\alpha }\hfill & \hfill & \sin \mathit{\beta }\sin \mathit{\alpha }\hfill \\ \hfill -\sin \mathit{\beta }\hfill & \hfill \hfill & \hfill \cos \mathit{\beta }\cos \mathit{\alpha }\hfill & \hfill & -\cos \mathit{\beta }\sin \mathit{\alpha }\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill \sin \mathit{\alpha }\hfill & \hfill & \sin \mathit{\beta }\hfill \end{array}\right)\end{array}$$(37) ) and (38(38) $$\begin{array}{c}\hfill H_{\mathit{\omega }{x}^{\prime }}={\mathit{Ad}}_{\mathit{\omega }{y}^{\prime }},H_{\mathit{\omega }{y}^{\prime }}=-{\mathit{Ad}}_{\mathit{\omega }{x}^{\prime }},H_{\mathit{\omega }{z}^{\prime }}=0,A=\frac{\mathrm{i}k}{r^2}(\mathrm{i}\mathit{kr}-1){\mathrm{e}}^{\mathrm{i}\mathit{kr}}.\end{array}$$(38) ).

Conclusions and discussion

The inverse problem for a dipole magnetic field represented by its Fourier transform can be solved in different ways, dependent on input data. In general, unknown quantities are three coordinates of the field source and three complex components of the dipole. This gives nine unknown real quantities. Hence, at least nine independent real numbers should be known to make the solution possible.

If the magnetic field is not linearly polarized, information on its spectral properties (Fourier transform ${\mathit{H}}_{\mathit{\omega }}$ or Stokes parameters) makes it possible to define two opposite directions from the observer to dipole as a perpendicular to the plane in which the magnetic field vector is varying. Projection of the dipole polarization ellipse onto the plane orthogonal to this line is defined by (8(8) $$\begin{array}{c}\hfill {\mathit{\delta }}_x=\sqrt{\frac{s_0-s_1}{2}},{\mathit{\delta }}_y=\sqrt{\frac{s_0+s_1}{2}},\\ \hfill \cos ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y)=-\frac{s_2}{\sqrt{s_0^2-s_1^2}},\\ \hfill \sin ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y)=\frac{s_3}{\sqrt{s_0^2-s_1^2}}.\end{array}$$(8) ) up to a scaling multiplier. But, the information on the spectral properties of the magnetic field is insufficient for calculation of distance between the dipole and observer. As additional input parameters derivatives of the magnetic field with respect to coordinates at the observation point can be used.

In Section 3, solution is given for the case when all the nine derivatives from the magnetic field are known. The inverse problem in this case has unambiguous solution. Distance to the dipole can be calculated by formula (13(13) $$\begin{array}{cc}\hfill R& ={\left(\sum _m{R}_m^2\right)}^{1/2}=\frac{3+2\mathit{\rho }}{\mathit{\rho }(1+{\mathit{\rho }}^2)},\hfill \\ \hfill I& ={\left(\sum _m{I}_m^2\right)}^{1/2}=\frac{{\mathit{\rho }}^2}{1+{\mathit{\rho }}^2}.\hfill \end{array}$$(13) ), or (14(14) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{I}{1-I}}.\end{array}$$(14) ) or (15(15) $$\begin{array}{c}\hfill n_m=-\frac{R_m}{R}=\frac{I_m}{I}.\end{array}$$(15) ). Direction from the dipole to the observer is defined by the unit vector (16(16) $$\begin{array}{c}\hfill {\mathit{IR}}_m=-{\mathit{RI}}_m,\end{array}$$(16) ), and the dipole polarization vector by (27(27) $$\begin{array}{c}\hfill D_{31}{\mathit{\rho }}_3=1+a{\mathit{\rho }}_3^2,D_{32}{\mathit{\rho }}_3=1+a{\mathit{\rho }}_3^2,\end{array}$$(27) ). Absolute value of the dipole amplitude is given by Equations (6(6) $$\begin{array}{cc}\hfill & s_0=J(d_{0x}^2+d_{0y}^2),\hfill \\ \hfill & s_1=J(d_{0y}^2-d_{0x}^2),\hfill \\ \hfill & s_2=-2{\mathit{Jd}}_{0x}{d}_{0y}\cos ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y),\hfill \\ \hfill & s_3=2{\mathit{Jd}}_{0x}{d}_{0y}\sin ({\mathit{\alpha }}_x-{\mathit{\alpha }}_y).\hfill \end{array}$$(6) ) and (7(7) $$\begin{array}{c}\hfill J=\frac{k^2(1+k^2{r}^2)}{r^4}.\end{array}$$(7) ), because the distance $r$ is already known at this stage.

There are nine complex derivatives $D_{\mathit{lm}}$ defined only by two vectors – a real one $\mathit{r}$ and a complex vector ${\mathit{d}}_{\mathit{\omega }}$. Hence, it can be no more than nine independent elements of matrix $D_{\mathit{lm}}$. There are nine real constraint equations for elements of this matrix which can be written down, for example, in the form (17(17) $$\begin{array}{c}\hfill R^{2}I={(3-I)}^2(1-I).\end{array}$$(17) ), (18(18) $$\begin{array}{cc}\hfill D_{\mathit{lm}}& =\left(\begin{array}{ccc}\hfill 0\hfill & \hfill p_2{d}_{\mathit{\omega }3}\hfill & \hfill -p_3{d}_{\mathit{\omega }2}\hfill \\ \hfill -p_1{d}_{\mathit{\omega }3}\hfill & \hfill 0\hfill & \hfill p_3{d}_{\mathit{\omega }1}\hfill \\ \hfill p_1{d}_{\mathit{\omega }2}\hfill & \hfill -p_2{d}_{\mathit{\omega }1}\hfill & \hfill 0\hfill \end{array}\right)\hfill \\ \hfill & -(R-\mathrm{i}I)\left(\begin{array}{ccc}\hfill R_1\hfill & \hfill R_1\hfill & \hfill R_1\hfill \\ \hfill R_2\hfill & \hfill R_2\hfill & \hfill R_2\hfill \\ \hfill R_3\hfill & \hfill R_3\hfill & \hfill R_3\hfill \end{array}\right),\hfill \end{array}$$(18) ) and (25(25) $$\begin{array}{cc}\hfill & F_{21}^0{d}_{02}-F_{31}^0{d}_{03}=0,\hfill \\ \hfill & F_{12}^0{d}_{01}-F_{32}^0{d}_{03}=0,\hfill \\ \hfill & F_{13}^0{d}_{01}-F_{23}^0{d}_{02}=0,\hfill \\ \hfill & {\mathit{\alpha }}_2-{\mathit{\alpha }}_3={\mathit{\psi }}_{31}-{\mathit{\psi }}_{21}+(2n+1)\mathit{\pi },\hfill \\ \hfill & {\mathit{\alpha }}_1-{\mathit{\alpha }}_3={\mathit{\psi }}_{32}-{\mathit{\psi }}_{12}+(2n+1)\mathit{\pi },\hfill \\ \hfill & {\mathit{\alpha }}_1-{\mathit{\alpha }}_2={\mathit{\psi }}_{23}-{\mathit{\psi }}_{13}+(2n+1)\mathit{\pi },\hfill \end{array}$$(25) ).

In Section 4, an example of inverse problem solution for a case when not all derivatives of the magnetic field are known is presented. It is shown that the solution of the inverse problem exists, if only non-diagonal elements of the matrix $D_{\mathit{lm}}$ are known.The dipole coordinates in this case are defined by (28(28) $$\begin{array}{c}\hfill D_{31}{\mathit{\rho }}_3=D_{32}{\mathit{\rho }}_3=\frac{-3-{\mathit{\rho }}^2+\mathrm{i}{\mathit{\rho }}^3}{1+{\mathit{\rho }}^2}.\end{array}$$(28) ), and an ellipse of polarization of the dipole is described by Equation (27(27) $$\begin{array}{c}\hfill D_{31}{\mathit{\rho }}_3=1+a{\mathit{\rho }}_3^2,D_{32}{\mathit{\rho }}_3=1+a{\mathit{\rho }}_3^2,\end{array}$$(27) ). If polarization of the magnetic field is not linear and we are interested only in position of the field source, then it is sufficient to know the derivative of only one Cartesian coordinate of the magnetic field vector, which should be calculated along direction, orthogonal to the plane of magnetic field polarization. In this case, the distance to a source is defined by Equation (31(31) $$\begin{array}{c}\hfill r=\frac{1}{k}\sqrt{\frac{{\mathit{\phi }}^{\prime }}{1-{\mathit{\phi }}^{\prime }}}.\end{array}$$(31) ) or (33(33) $$\begin{array}{c}\hfill \frac{\mathit{\delta }R}{R}\le 3\frac{\mathit{\delta }{R}_g}{|R_g|}.\end{array}$$(33) ), and the direction to a field source is defined as a perpendicular to the plane of polarization of vector ${\mathit{H}}_{\mathit{\omega }}$.

Uniqueness and stability of the obtained solution are discussed in Section 5. It is shown that the obtained solution of the inverse problem is unique and stable. But, the relative errors depend on which data are used for calculation of sought quantity. A numerical example in Section 6 demonstrates the theory developed in the previous sections.

Acknowledgments

The authors would like to thank Dr V.F. Gordeev and Dr I. Zhukova for valuable discussions. This work has been supported by the grant for LRSS, project No 224.2012.2 and by the Ministry of Education and Science of Russian Federation, project 14.B37.21.0774.