Electromagnetic Interrogation of Dielectric Materials 1 2 Introduction and Problem Formulation 2.1 Motivation and the General Inverse Problem

We investigate time domain based electromagnetic inverse problems. This is done in the context of general polarization models (which include the familiar Debye and Lorentz models as special cases) for dielectric materials. A one-dimensional model based on time-windowed pulsed inputs is formulated and well-posedness results are given for the simulation problem. Numerical investigations of problems involving estimation of dielectric parameters in a slab geometry are carried out. We also present results for use of secondary reeections to estimate geometry of the slab. A survey of the mathematical literature reveals considerable interest in the iden-tiication of material parameters describing electromagnetic phenomenon. For our purposes, we categorize the materials and the models employed to describe them as either dispersive or non-dispersive, where dispersive materials are those in which electromagnetic waves of diierent frequencies have diierent phase velocities. When modeled in the frequency domain, this is manifested as parameters which depend explicitly on frequency. In time domain models, the same phenomenon can be captured with constitutive laws in which the electric and/or the magnetic polarizations are expressed in terms of the convolution of the history of the electric and magnetic elds. The equivalence of the two in the case of electric polarization dispersion is shown by Jackson Jac, pp.306] Forward problems in this eld are dominated by one dimensional scattering problems where planar electromagnetic waves impinge on dielectric slabs. In a series of four papers KK, KK2, KK3, KK4] Kristenson and Krueger examine this problem with a wave splitting technique and the derivation of scattering operators which satisfy imbedding equations. The reconstruction of the functions representing the physical parameters proceeds from the imbedding equations and a scheme is presented which is shown to be robust in the presence of noise. The physical model covers stratiied media , meaning that the material is inhomogeneous in the direction of the propagation of the waves. A similar approach is applied to a slightly diierent model by Weston Wes] who considers a dissipative wave equation equivalent to the the problem of planar waves in stratiied media. These results are extended by Krueger Kru, Kru2, Kru3] to cover media in multiple slabs, thus containing multiple discontunities in the material parameters. Corones and Sun CS] use the same method of wave splitting and 1 invariant imbedding to reconstruct coeecients in a one dimensional wave equation with a source term. In another paper, He and Strr om HS] also consider …