Reconstruction of dielectric profile from total electric field data

An inverse scattering problem of determining the electrical property of an unknown dielectric object is considered. The electromagnetic wave generated by an impressed electric current with a short duration time illuminates the dielectric object. The measured data of the total field, the sum of the incident and scattered fields, over an observation surface S is used to reconstruct the electrical property distributions of the object without explicit information on the incident field. Using the tangential component of the electric field over the observation surface S, we can set up a boundary value problem which is equivalent to the original direct scattering problem in the interior region Ωi bounded by the surface S. Suppose that the total field data is measured over the observation surface S till the time t = T by which almost all the total energy leaves the interior region Ωi through the surface S and the field in the interior region becomes negligibly small. In this interior equivalent problem, the electrical property distribution must be the same as in the original scattering problem, i.e., the scattering object placed in the interior region must be the same as the original one. If a different object is placed in the interior region, reflection occurs on the surface in order for the total field to satisfy the boundary value. The reflected wave keeps bouncing in the interior region and remains even after the end time T of measurement. Noting this fact, we introduce the following functional of the electrical property distribution p(r): Q(p)= 1/2 ∫(Ωi) [|Eeq(p; r, T)|2 and |ηHeq(p; r, T)|2]dr where η is the intrinsic impedance of free space, and Eeq(p; r, T) and Heq(p; r, T) are the electric and magnetic fields at the time t = T of the boundary value problem with the electrical property distribution p. If the distribution p is identical to the original one, the equivalent electric and magnetic fields are null at the time t = T. Therefore, the inverse problem is reduced to finding a minimizer of the functional Q(p). We apply a conjugate gradient method to the minimization problem. Several numerical simulations show the validity of the proposed method.