## 1. Introduction

[2] In a recent communication by *Marengo and Devaney* [1999], the inverse problem of electromagnetics, more specifically that of reconstructing the time-harmonic minimum energy (ME, or minimum *L*^{2} 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.

[3] 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* [1999] 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.

[4] The uses of spheroidal vector wave functions in electromagnetic theory have hitherto included, inter alia, light scattering by spheroids [*Asano and Yamamoto*, 1975; *Li et al.*, 2002], radiation of spheroidal antennas [e.g., *Wait*, 1969], and the resonances of spheroidal cavities [*Kokkorakis and Roumeliotis*, 1998]. More recently, theoretical limitations of the radiation quality factor of spheroidal modes was studied by *Sten* [2003].

[5] 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.