Journal of Computational Mathematics Vol.31, No.5, 2013, 439–448. http://www.global-sci.org/jcm doi:10.4208/jcm.1304-m4388 A LOW-FREQUENCY ELECTROMAGNETIC NEAR-FIELD INVERSE PROBLEM FOR A SPHERICAL SCATTERER * Nikolaos L. Tsitsas Department of Informatics, Aristotle University of Thessaloniki, Thessaloniki, Greece Email: ntsitsas@csd.auth.gr Abstract The interior low-frequency electromagnetic dipole excitation of a dielectric sphere is uti- lized as a simplified but realistic model in various biomedical applications. Motivated by these considerations, in this paper, we investigate analytically a near-field inverse scatter- ing problem for the electromagnetic interior dipole excitation of a dielectric sphere. First, we obtain, under the low-frequency assumption, a closed-form approximation of the series of the secondary electric field at the dipole’s location. Then, we utilize this derived approx- imation in the development of a simple inverse medium scattering algorithm determining the sphere’s dielectric permittivity. Finally, we present numerical results for a human head model, which demonstrate the accurate determination of the complex permittivity by the developed algorithm. Mathematics subject classification: 34L25, 78A46, 78A40, 41A60, 33C05. Key words: Near-field inverse problems, Low-frequency region, Dipoles, Hypergeometric functions. 1. Introduction The exact field solutions of direct scattering problems by canonical shapes are often ex- pressed by complicated series of the corresponding eigenfunctions [1,2]. For example, for spher- ical scatterers the fields are expressed by series of products of spherical Bessel and Hankel func- tions. In inverse scattering these series are difficult to manipulate in order to obtain algorithms which extract a specific set of the problem’s parameters. However, under the low-frequency assumption k 0 a ≪ 1(k 0 the free-space wavenumber and a a characteristic dimension of the scatterer) [3]- [6], the field solutions are greatly simplified so that the low-frequency realm offers a better environment for inverse scattering, since the corresponding field quantities are much more workable. In this paper, we investigate analytically a near-field inverse scattering problem concerning the low-frequency interior dipole excitation of a dielectric sphere. The low-frequency assumption permits us to obtain an analytical expression, via hypergeometric functions, of the secondary electric field at the dipole’s location by exact summation of the series representing it. This problem is motivated by potential applications considered in the low-frequency region and mentioned below. Applications of low-frequency internal source excitation of a homogeneous sphere in elec- troencephalography (EEG) have been pointed out in [7]. In particular, the interior excitation of a spherical human head by a low-frequency point-dipole constitutes a suitable EEG model * Received December 21, 2012 / Revised version received February 25, 2013 / Accepted April 16, 2013 / Published online August 27, 2013 /