## 1. Introduction

[2] Composite materials can offer low weight while maintaining high mechanical strength, electrical conductivity and other properties. They also offer possibilities for tuning material parameters by varying the microstructure of the composite, for instance by varying the volume fraction of one or several inclusion materials. The control of electromagnetic material properties is essential in many applications, for instance power transformers, antennas, radomes, integrated circuits etc. In this paper, we address several issues associated with the experimental characterization of composite materials.

[3] Electromagnetic material properties are often measured in a waveguide setting, where the Nicolson-Ross-Weir (NRW) method is used to extract material parameters [*Nicolson and Ross*, 1970; *Weir*, 1974]. The waveguide environment improves the signal-to-noise ratio compared to a free space measurement, but introduces a more complicated wave propagation. In particular, for anisotropic materials very little is known about the influence of the metal walls of the waveguide on the shape of the modes, and they can usually only be computed numerically. One exceptional case is given by *Damaskos et al.* [1984], where the anisotropy axes of the material are aligned with the walls of a rectangular waveguide, in which case the NRW method can be used with some minor modifications. In this paper, we show that the NRW method can still be motivated even in more general cases such as anisotropic materials and periodic structures, but considerable care is necessary before interpreting the results as material data.

[4] Classical homogenization theory has well known results for calculating the homogenized permittivity and permeability matrices **ε**^{hom} and *μ*^{hom} of a composite material with a specified microstructure **ε**(** r**) and

**(**

*μ***), with**

*r***denoting the position vector, see for instance the book by**

*r**Milton*[2002] and references therein. If the microstructure of a composite material has some sort of anisotropy, such as in a layered structure or aligned fiber inclusions, the result is often a homogenized material with anisotropic properties. Even though it is possible in theory to set up an optimization procedure to calculate all material parameters if sufficiently many measurement setups are constructed, in practice the measurement accuracy is greatly enhanced if a priori knowledge of the anisotropy is included from the start. For instance, the method proposed by

*Damaskos et al.*[1984] requires the anisotropy axes to be aligned with the waveguide walls. We give several illustrations on how a priori information is essential in order to be able to determine material properties.

[5] Even though the homogenization method can be used to compute theoretical material parameters, the microstructure is often only partially known, and we need to verify the calculations by measurements. When we characterize the material through a scattering experiment using electromagnetic waves, the primary outputs of the experiment are the wave parameters wave number *k* and wave impedance *Z*, which have to be translated to material parameters ε and *μ* through a theoretical model. Even for an isotropic material it may happen that a combination of *k* and *Z* (both located in the proper complex half plane as required by passivity) can generate unphysical material parameters ε and *μ* (where the imaginary part of one parameter has the wrong sign). This typically occurs when the thickness of the material under test corresponds to one half wavelength. This is a well known problem, and several strategies have been suggested to deal with it [*Baker-Jarvis et al.*, 1990, 1992; *Boughriet et al.*, 1997; *X. Chen et al.*, 2004; *Barroso and de Paula*, 2010]. Using the Cramér-Rao lower bound [*Kay*, 1993; *Gustafsson and Nordebo*, 2006] we show that the instability is not due to the NRW method itself, but can be attributed to the signal quality in reflection data. We demonstrate that at this resonance both material parameters ε and *μ* are ill-determined, but of the wave parameters only *Z* is ill-determined, and the wave number *k* can be retrieved in a stable way from the nonmagnetic variant of the NRW method.

[6] This paper is organized as follows. Some background information on fixed-frequency material models and homogenization is summarized in sections 2 and 3, together with new results using Cramér-Rao bounds for estimating the accuracy of determination of isotropic material properties. The distinction between material parameters as the output from homogenization and wave parameters as the output from experiments is emphasized, and practical issues regarding wave scattering experiments are reviewed in section 4. In this section we also present a new numerical model which can be used to obtain synthetic scattering data, where the practical difficulties can be controlled and varied. The synthetic scattering data are used in section 5 to study the situation of an anisotropic material in a rectangular waveguide, having arbitrary axis of anisotropy. Some conclusions are given in section 6.