The inverse problem of reconstructing time-harmonic minimum energy current distributions in a spheroidal volume from given data of far-field radiation is addressed. Following the procedure outlined by Marengo and Devaney , we formulate, upon deriving a spherical harmonics expansion of the electromagnetic field radiated by a current inside a prolate spheroid, the inverse problem in terms of linear operator theory. Owing to the lack of orthogonality of spheroidal vector wave functions, every eigenfunction will couple with several spherical radiation modes at a time, making the solution rather involved. Simplification is achieved in the special case of rotationally symmetric fields, for which numerical examples are given. As an application, the use of minimum energy currents for identifying distributions of nonradiating current in a spheroidal volume is pointed out.
 In a recent communication by Marengo and Devaney , the inverse problem of electromagnetics, more specifically that of reconstructing the time-harmonic minimum energy (ME, or minimum L2 norm solution) current distribution in a spherical source volume from the radiating field was solved by means of linear operator formalism. This solution and its scalar counterpart [Marengo et al., 2000] are remarkable benchmarks, not only because they allow spherical radiation modes to be connected to three dimensional (free-field) volume distributions in lieu of the classical multipole point sources at the expansion centre, but also because ME solutions enable a unique decomposition of any localized source (current) distribution into a radiating (the ME current) and a nonradiating (NR) source distribution [Marengo and Ziolkowski, 2000]. Since NR currents have a large import on the input impedance and bandwidth of a radiating element [Marengo and Ziolkowski, 2000], the interest to identify such current distributions is manifest.
 Because a typical linear dipole, by its employment of space, is more efficiently approximated by a prolate spheroid than by a sphere, an attempt will be made in the present paper to generalize the linear inversion procedure of Marengo and Devaney  from spherical to prolate spheroidal source volumes. A connection will be developed between spherical harmonic radiation modes and the moments of a current with respect to spheroidal vector eigenfunctions. Since the spheroidal vector wave functions are generally not orthogonal [Flammer, 1957; Li et al., 2002], unlike the spherical ones, calculation of the amplitudes of the spheroidal modes inside the source volume becomes more involved. A considerable simplification is achieved by only considering fields which are rotationally symmetric around the axis of the spheroid.
 The plan for the paper at hand is the following: The electromagnetic field generated by the current J(r) confined in a spheroid is expressed in terms of the Green function in spheroidal coordinates. This is achieved by developing analytical expressions for the radial field components, thus enabling a spherical harmonics expansion of the radiated field. Having stated the forward problem in terms of a linear operator involving spheroidal vector wave functions, the minimum energy source is expressed in terms of the adjoint operator. Numerical aspects are discussed along with examples.
2. Field Due to a Volume Current
 Time-harmonic electromagnetic fields E, H, with time dependence ejωt (henceforth suppressed) satisfy the Helmholtz wave equations
Outside the source region (where r · J ≡ 0) these wave equations for the radial field components have the solutions
respectively, where V is the volume in which the current is contained and
is the unbounded Green function.
 Let the surface enclosing the radiator be
where the z-axis is the revolving axis, see Figure 1, and c and b denote the lengths of the polar and equatorial radii. The case c < b is a flattened or oblate spheroid while c > b is an elongated or prolate spheroid, and we will concentrate on the latter. To reproduce the oblate case only simple replacements are needed, as explained later.
 Hence, we employ prolate spheroidal coordinates (f; ξ, η, ϕ), which are related to the Cartesian ones by the transformation [Flammer, 1957, chapter 2; Li et al., 2002, chapter 2]
f = denoting the semifocal distance (for the oblate, f = ). The prolate coordinates are defined in the intervals: η ∈ [−1, 1], ϕ ∈ [0, 2π] and ξ ∈ [1, ∞]. Surfaces of constant ξ are spheroids, and the surface enclosing the source (14) is ξ0 = c/f.
 In the prolate spheroidal system of coordinates, G can be written in terms of the Lamé products [Flammer, 1957, chapter 2; Li et al., 2002, p. 67]
Here, Smn is the angular spheroidal function, composed of a weighted sum of associated Legendre functions Pnm(.)
while Rmn(1,4) represent radial spheroidal functions of the first or fourth kind, respectively. In terms of spherical Bessel functions jn(.) and Hankel functions hn(2)(.), they are
The expansion coefficients dp∣m∣n for (18) and (19) may be determined recursively using the eigenvalues (separation constants) of the differential equations for Smn and Rmn [Flammer, 1957, chapter 3]. In definition (19) the expansion coefficients have been normalized as
which has the effect that Smn → Pn∣m∣(η) when η → 1. Furthermore, in (17)
is a normalization factor stemming from the orthogonality of the Legendre functions. It is introduced to harmonize with the Green function in the spherical case, where a similar factor also appears.
Partial integration and the use of Gauss' theorem gives
The surface integral vanishes because J = 0 on the boundary of the current carrying region, and hence
In a similar manner
The two sets of vector eigenfunctions thus defined; Zmn★ and ∇′ × Zmn★/k, are identical with the ones used by Kokkorakis and Roumeliotis  in the investigation of resonances of spheroidal cavities.
3. Spherical Wave Expansion of the Radiation Field
 An electromagnetic field outside a spherical volume source can be expanded as [Stratton, 1941]
where Z0 = and
are the vector spherical harmonics, (r, θ, ϕ) being the usual spherical coordinates.
 In the far zone hn(2)(kr) ∼ jn+1e−jkr/(kr) and ∇ → −jkur, so that (26) may be written
If the field distribution f(ur) is known (measured or calculated), the mode coefficients amn, bmn may be obtained from
by application of orthogonality properties of exponentials and the Legendre functions, respectively.
 For determining the spherical wave expansion one can show that the r-weighted radial components r · E and r · H are sufficient [Papas, 1988]. Because the M-vector by definition is transverse to r, only N contributes. In the far-field, the desired component of the Nmn-mode can be written explicitly
by virtue of orthogonality of the modes in the ϕ-direction. Analogously, putting r · (26) equal to (25) we get
We notice that, in the far field, the spheroidal η-coordinate takes the same values as cos θ, where θ is the azimuthal angle of the spherical system. Therefore, orthogonality of Legendre functions is applicable: multiplying both sides by Pℓ∣m∣(η) and integrating η from −1 to 1 gives, after some simplification and renaming of indices,
As a check for this, letting f → 0, all other dn−∣m∣∣m∣ℓ disappear except that for ℓ = n, which approaches unity, hence producing the formulas
can be formulated by inserting (38) and (39) in (32). The dyadic operator
maps the source region scalarly to the far-field sphere. In order to establish the equation for the inverse problem, a pair of inner products must be introduced. They are defined in the Hermitean fashion
where X is the space of L2 vectors within the spheroid ξ ≤ ξ0 and
Y being the space of transverse L2 vectors of direction (θ, ϕ). Now, we may postulate a unique mapping †: Y → X, called the adjoint of : X → Y and satisfying
Developing the inner product and using the formulae (33) and (34)
But in view of (46) this is equal to 〈†f∣J〉X★. Hence, we identify
where the transformed data vector [†]−1f = , following the terminology of Marengo and Devaney , is called the ‘filtered’ data. Existence of the inverse operator depends on the quality of the data, more specifically, its physical relevance and its degree of noise.
 Supposing that our data is free from noise and physically relevant so that exists and can be written in terms of spherical harmonics as
then it must also be possible to calculate the quantity † = f. The accomplishment of this operation is, however, made somewhat complicated by the fact that the two sets of spheroidal vector wave functions; Zmn and ∇ × Zmn/k, defined in (24) and (25), unlike the spherical eigenfunctions, are generally neither orthogonal by themselves, nor to each other. In extenso they are given by
omitting, for brevity, indices and arguments of Rmn(1)(kf; ξ) and Smn(kf; η). A considerable simplification is achieved when rotational symmetry is assumed (m = 0), as shall be demonstrated below.
 Upon forming the equation † = f we get
where αmℓm′ℓ′, βmℓm′ℓ′ and χmℓm′ℓ′ are TE/TE, TM/TM and TE/TM-mode interaction coefficients, respectively,
wherein the factor f3(ξ2 − η2)dξ dη dϕ is the volume element in the prolate spheroidal coordinates.
 The minimum energy source
can be evaluated when the ‘filtered’ coefficients ãmn, mn have been obtained from the system of equations (involving a finite but sufficient number of modes) formed by equations (53) and (54). The solution is somewhat elusive, but symbolically, it can be written
where , and represent the mode coupling matrices, , the data vectors and ã, the ‘filtered’ data vectors. To illustrate the solution in more detail, we consider the case of rotational symmetry, when m = 0 and the two sets of vector wave functions (51) and (52) are orthogonal by direction. Hence, the TE/TM-cross-coupling matrix = 0 and we have
The ‘filtered’ coefficients, ã0n and 0n may now be solved trough inversion of and .
 A special case of interest is the spherical limit f → 0, when dn−∣m∣∣m∣ℓ → 1 for n = ℓ and zero for n ≠ ℓ. Then (58) approaches
while the coupling matrices and become highly singular and the relations between the ‘filtered’ coefficients ãmn, mn and the data amn, bmn can be stated as
 At first it is appropriate to adduce, concerning the nature of the elements Ann′, Bnn′ of the coupling matrices and , that there will be a drastic decrease in their values for n or n′ larger than kfξ0, which implies that the influence of modes of the order higher than kfξ0 becomes, by way of the respective inverse matrices, strongly pronounced. The sensitivity of the method to small errors in the data as well as in the matrix elements themselves is thereby emphasized, thus setting a practical limit to the achievable accuracy and resolution of the spatial structure of the source [Marengo and Devaney, 1999]. Quite a similar delicacy in the numerical calculation was encountered when the radiation Q was estimated for spheroidal radiating modes, especially when considering eigenfunctions whose mode index n considerably exceeded the electrical size of the source [Sten, 2003].
 To illustrate, let us consider a spheroidal source volume whose semifocal distance is f = 1/k m, k being 2π m−1, corresponding to a wavelength of 1 m. The shape of the spheroid is fixed by the parameter ξ0 = 3, corresponding to an ellipticity (=axial ratio ) of 2/3 ≈ 0.94, which is almost spherical. Let us here concentrate only on the -matrix, pertaining to TM or electric dipole modes, the behavior of the -matrix being essentially similar.
 In the following calculations, modes up to n = 5 were considered, although in view of the size of the spheroid the two lowest modes, TM01 and TM03, are practically the most important ones. The part of the inverse coupling matrix which pertains to these modes is
For the evaluation of the integrals of (55) and (56) a 500 times 500 discretization of the integration range was effected (the integration over ϕ readily gives 2π, as a consequence of the rotational symmetry). The whole operation was accomplished on PC-Matlab in a few minutes, and on the whole, the most time-consuming part of the operation is the evaluation of the spheroidal wave functions, Smn and Rmn. If more modes are taken into account the evaluation inevitably slows down due to the denser discretization required.
 Now, assuming that the data vector consists of only one nonzero element, say b01 = 1, then the ‘filtered’ coefficients become 01 ≈ 11.9 · 10−7, 03 ≈ −0.55 · 10−7. In comparison, equations (16) and (17) of Marengo and Devaney  for a spherical source of the same size give as 01/b01 approximately 9.24 · 10−7, which does not differ much from the spheroidal case. Correspondingly, if the only nonzero element of the data vector is b03 = 1, the ‘filtered’ coefficients become 01 ≈ −0.14 · 10−6, 03 ≈ 4.71 · 10−6. Equations (16) and (17) of Marengo and Devaney  now give 03/b03 ≈ 3.91 · 10−6. The relevant eigenfunctions appear in Figure 2.
 Next, let the source volume be characterized by the coordinate ξ0 = 1.05, corresponding to an ellipticity of /21 ≈ 0.3, the semifocal distance being f = 20/(7k) and k = 2π m−1. By so doing, the electrical size of the spheroid remains kfξ0 = 3, and we may restrict ourselves to the lowest few modes, as above. The coupling matrix now becomes
where the cross-coupling (off-diagonal) terms are significantly larger than in the previous case.
 Now, if the only nonzero data element is b01 = 1, the ‘filtered’ coefficients are 01 ≈ 15.2 · 10−6, 03 ≈ −7.3 · 10−6 while if the only nonzero data element is b03 = 1, the result is 01 ≈ −1.87 · 10−5, 03 ≈ 9.85 · 10−5. These numbers evidently differ significantly from the nearly-spherical case above, but so are also the eigenfunctions totally different in this case, as illustrated in Figure 3.
6. Final Remarks
 A method has been given for reconstructing the minimum energy current source inside a specified prolate spheroidal volume from a given radiation field, expressed in terms of modes of spherical wave functions. As a corollary of the nonuniqueness of the solution for the inverse problem at hand, the difference between the present minimum energy solution and any true current distribution, generating an exactly similar radiation pattern and occupying the same volume, represents a nonradiating source.
 For reasons of simplicity and transparency of exposition, an explicit solution was provided only in the case of rotational symmetry. However, matrix equations for carrying out the expansion for more general fields were given symbolically, leading to a considerably more involved calculation routine. Finally, numerical examples illustrating the evaluation of the amplitude coefficients for the vector eigenfunctions were given. In these examples a simple but time-consuming two-dimensional integration scheme was employed, which, in view of future applications, should be made more efficient.
 As mentioned in Section 2.1, the oblate case can be treated in much a similar way. Instead of the prolate spheroidal wave functions Smn and Rmn, the corresponding oblate functions should be employed. Furthermore, due to a difference in the coordinate system, factors involving ξ2 − 1 will be replaced by ξ2 + 1, while those involving ξ2 − η2 will be replaced by ξ2 + η2. Also, the lower endpoint of integration over ξ in (55)–(57) shall be changed to ξ = 0.
 The author is grateful for the suggestions by Dr. Marengo and an anonymous reviewer.