## 1. Introduction

[2] With the recent remarkable progress of computational resources, computer-aided design has become a valuable component in developing systems of microwave power engineering. Knowledge of complex permittivity (ɛ = ɛ′ − *i*ɛ″) of materials involved in an application is critical for creating an adequate model and thus for successful system design. However, the dielectric constant ɛ′ and the loss factor ɛ″ are not always available. The lack of data regarding realistic materials motivates further development of robust and practical techniques of determining complex permittivity.

[3] Since ɛ′ and ɛ″ are not directly measured, but calculated given the data on some measurable characteristics, a related numerical simulator may be made involved in determination of material parameters through a numerical solution of a corresponding inverse problem. This approach has been taken in a number of techniques using the finite element method [*Coccioli et al.*, 1999; *Thakur and Holmes*, 2001; *Olmi et al.*, 2002; *Santra and Limaye*, 2005], the finite-difference time-domain (FDTD) method [*Wäppling-Raaholt and Risman*, 2003] and the finite integration technique [*Requena-Pérez et al.*, 2006] for modeling of the entire experimental fixtures. Further exploring this trend, *Eves et al.* [2004] and *Yakovlev et al.* [2005] have recently developed the novel neural-network-based FDTD-backed technique capable of efficiently determining the dielectric constant and the loss factor of materials placed in a transmission-line-type cavity. In this case, the experimental part is reduced to measuring the reflection and transmission coefficients of the systems. The technique is demonstrated to be versatile, robust, frequency- and cavity-independent, and applicable to the samples and fixtures of arbitrary configuration. However, while it has been shown by *Eves et al.* [2007] that the reconstructed ɛ′ and ɛ″ can be easily validated and are proved to be accurate, uniqueness of this reconstruction remains to be an assumption. The latter circumstance may become an issue when using this or other modeling-based technique for determining effective complex permittivity of such increasingly important materials as nano-composites and metal powder, typically characterized by very high values of ɛ′. This paper presents the first results of the original study aiming to show that determination of complex permittivity of a body in a waveguide is unique when ɛ′ and ɛ″ are reconstructed from the related reflection and transmission coefficients.

[4] More specifically, a goal of our study is to develop solution techniques elaborated by *Shestopalov and Sirenko* [1989] and *Shestopalov and Shestopalov* [1996] for the direct and inverse boundary value problems (BVPs) for Maxwell's and Helmholtz equations associated with the wave propagation in the waveguides with dielectric inclusions. Such problems arise also in mathematical models of the wave propagation and diffraction in inhomogeneous media [*Colton and Kress*, 1998]. In our approach, the BVPs and eigenvalue problems are formulated in unbounded domains and with partial radiation conditions at infinity that contain the spectral parameter; the method of solution employ integral equations (IEs) constructed using Green's function of the domain occupied by a regular guide. The methods of reconstructing the shape of the scatterer or its permittivity are developed by *Colton and Kress* [1998] mainly for the cases when the obstacles are supposed to be perfectly conducting or dielectric bodies in two- or three-dimensional space. The recent paper by *Shestopalov and Lozhechko* [2003] suggests the technique for cylindrical scatterers whose (two-dimensional) cross sections are formed by domains with infinite noncompact boundaries. The uniqueness for such problems stated in the whole space or in the half-space is proved when the data in the inverse problem of finding the shape of the scatterer or permittivity of the inclusion (1) consist of the far-field patterns of the scattered field given for the plane wave irradiating the obstacle from all directions, and (2) are available for all frequencies varying in a certain interval.

[5] However, when a dielectric body is situated in a waveguide, similar results concerning the unique solvability and efficient solution techniques for the inverse scattering problems of reconstructing permittivity of the scatterer are not available. This fact becomes a driving force of our effort in developing a new approach to the solution of both direct and inverse scattering problems in waveguides. The present paper is devoted to the proof of uniqueness for a parallel-plane waveguide.