The effective properties of a composite material can be computed from knowledge of the component materials and the geometry of the microstructure. To verify these results at a certain frequency, wave scattering experiments can be performed, where typically reflection and transmission data are recorded and subsequently analyzed in a theoretical scattering model to extract material data. In this paper, we show that the wave scattering experiment is more closely related to the wave parameters (wave number and wave impedance), than to the material parameters (permittivity and permeability). We focus on anisotropic materials, demonstrate a method to extract the effective permittivity of such materials, and show the importance of knowledge of the anisotropy direction in order to interpret the data.
 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.
 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. , 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.
 Classical homogenization theory has well known results for calculating the homogenized permittivity and permeability matrices εhom and μhom of a composite material with a specified microstructure ε(r) and μ(r), with r denoting the position vector, see for instance the book by Milton  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.  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.
 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.
 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.
2. Material Models
 Our analysis is conducted in the frequency domain (time convention ejωt), where the most general local, linear, causal, time translation invariant model is
The requirements of passivity and causality restrict the matrices ε, ξ, ζ, and μ. In the isotropic case, where ξ = ζ = 0, ε = εI, and μ = μI, the scalars ε and μ have negative imaginary part and are analytic functions of ω in the lower complex half plane. In this paper, we mostly consider models where ξ = ζ = 0 and μ = μ0I, that is, a nonmagnetic model where the permittivity matrix may be anisotropic and complex valued.
 We often think of materials as having intrinsic properties that do not depend on the shape or size of the material object. For metamaterials [Capolino, 2009a, 2009b], the microstructure is often comparable to wavelength, which implies that the large scale parameters depend on direction of propagation and frequency, ε = ε(k, ω) [Landau et al., 1984; Sjöberg et al., 2005; Sjöberg, 2005]. This implies spatial dispersion
which means that in order to compute the value of D at a certain point r in space, we need to include values of E in a neighborhood of r. Thus, the material model is not local, and the boundary of the material needs detailed attention, which is not easily reconciled with most methods to compute scattering. With spatial dispersion present, the boundary may have a great influence on the problem [Simovski, 2009a, 2009b]. In order to extract the bulk properties of such structures, it is necessary to compensate for the surface effects. When only the wave number of the metamaterial is to be measured, this can be done by the method presented in section 5.
 The above mentioned problems for spatially dispersive structures mean that when we talk about a material that can be described by simple material parameters ε and μ, we must usually be in a parameter region where a/λ → 0, where a is a typical length of the microscopic structure (for instance the length of a unit cell of a periodic microstructure).
2.1. Isotropic Model
 Isotropic materials can be characterized either using material parameters and μ, or wave parameters k (wave number) and Z (wave impedance). The wave parameters are defined from the description of a plane wave (in this case propagating in the positive z direction), E = E0e−jkz and H = H0e−jkz with the constant amplitudes related by E0 = ZH0 × . By passivity, the parameters , μ, and k are restricted to the lower complex half plane (negative imaginary part), and Z is restricted to the right half plane (positive real part). Thus, the parameters (jωε, jωμ, jk, Z) are restricted to the right half complex plane, and are related to each other as jk = and Z = , which are well defined operations with the standard square root having a branch cut along the negative real axis, enabling straight-forward numerical implementation. However, in practice these relations are frequently contaminated by measurement uncertainties, such as noise and systematic errors.
 Reflection and transmission coefficients r and t are restricted to the unit circle in the complex plane. The NRW method can be used to extract k and Z from r and t, both being in the correct half planes. But taking arbitrary points jk′ and Z′ in the right half plane, does not always give new points jωε′ = jk′/Z′ and jωμ′ = jk′Z′ in the right half plane. A trivial example is (ignoring units and setting ω = 1)
We need to consider the mapping of a product and a quotient between two points in the right half complex plane, and determine when the results are also in the right half. The points k′ and Z′ must then be contained in the region determined by |arg(jk) ± arg(Z)| < π/2, which is more restricted than the right half plane, see Figure 1. L. F. Chen et al.  have a pragmatic approach to this problem: regions where this condition is not satisfied are simply omitted from the measurement results.
2.2. Cramér-Rao Lower Bound
 It is possible to estimate the accuracy of the determination of material data using the Cramér-Rao lower bound [Kay, 1993, chapter 3]. In principle, this provides a lower bound for the variance of the estimated parameters in terms of the measured data, regardless of the method used for the estimation. Although it is an idealized bound, depending on assumptions like unbiased estimator (meaning all systematic errors must be eliminated) and Gaussian noise, it can still provide a reasonable estimate of the error in situations where the necessary differentials of the probability distribution function can be explicitly calculated.
 The Cramér-Rao lower bound for fixed frequency material parameters is given by first computing the Fisher information matrix (assuming uncorrelated Gaussian noise in reflection and transmission coefficients r and t)
where σr is the noise level for the reflection coefficient, and σt is the noise level for the transmission coefficient. The derivatives are computed from a theoretical model where the reflection and transmission coefficients are those for a homogeneous isotropic slab, which can be found in many textbooks, for instance [Born and Wolf, 1999]. In this model, r and t are holomorphic functions of ε and μ which justifies using the complex derivate above. If this condition is not met, differentiation has to be done with respect to real and imaginary parts separately. Details on the computations can be found in the work of Sjöberg and Larsson .
where [−1]εε indicates the permittivity diagonal element of the inverse of the Fisher information matrix. Corresponding bounds can be computed for the estimation of wave number k and wave impedance Z.
 An experimental demonstration of the bounds is given in Figure 2. Reflection and transmission data for an epoxy slab with thickness d = 9.61 mm are measured 50 times in an X-band rectangular waveguide, and material data are calculated with the Nicolson-Ross-Weir algorithm [Nicolson and Ross, 1970; Weir, 1974]. Using the full NRW algorithm to extract both ε and μ, it is concluded that the material is nonmagnetic and the permittivity is practically constant in the frequency band. Using the nonmagnetic NRW algorithm [Boughriet et al., 1997] and taking the mean value over all measurements and frequencies, the permittivity is concluded to be ε/ε0 = 2.8 − 0.02j. This is used together with μ = μ0 and the mean noise figures estimated from measured data, σr2 = −83 dB and σt2 = −79 dB, to draw the dashed curves in Figure 2. The variances of the material and wave parameters are computed independently for each frequency point from the output of the full NRW algorithm.
 The frequency variation of the bounds in Figure 2 depends on the experimental setup rather than the material properties. All parameters are difficult to determine at the cutoff frequency 6.6 GHz, which depends on the width of the waveguide. At 10.1 GHz the slab thickness corresponds to one half wavelength. Here, both material parameters ε and μ are difficult to estimate, but of the wave parameters only the impedance Z is difficult to determine and the wave number k remains stable.
 We now return to the problem of composite materials, consisting of a mixture of different component materials. To predict the effective properties of a composite material, it is a common practice to analyze the model problem of a periodic structure where the unit cell is small compared to the wavelength. In the limit of a vanishingly small unit cell, the resulting problems are static with periodic boundary conditions, called the local problems:
Usually, these equations are solved with zero sources, J = 0 and ρ = 0, and the fields are generated by postulating the average of two fields, for instance
These conditions together with ∇ × E = 0 imply the representation E = E0 − ∇ϕ, where ϕ is a periodic scalar potential, determined by the elliptic equation ∇ · [ · (E0 − ∇ϕ)] = 0. Similar results apply to H. Having solved the equations above for given E0 and H0, the homogenized parameters εhom and μhom are defined by averages
The equations can also be solved in a dual setting, postulating the average 〈D〉 = D0, introducing a periodic vector potential representation D = D0 + ∇ × F, and using E(r) = ε(r)−1 · D(r) (with corresponding formulations for the magnetic fields). The direct and dual formulations can be used to construct bounds on the homogenized parameters as in the works of Sjöberg [2009a], Wiener , and Milton . For instance, the Wiener bounds are
In general, the homogenized parameters εhom and μhom are not scalars but matrices, and the inequalities should be understood in terms of all quadratic forms over the matrices. If the parameters are complex, the bounds become regions in the complex plane bounded by circles [see Milton, 2002]. The anisotropy can be due to, for instance, the lamination direction of a stratified structure. For a structure consisting of isotropic sheets laminated in the z direction, the homogenized permittivity is given by
that is, it has the arithmetic mean of the component properties in the plane of the sheets, and the harmonic mean in the normal direction of the sheets. This is a uniaxial material, typical of many manufacturing processes.
 The homogenization procedure described in this section can be used to calculate or estimate the material parameters of any material where the microstructure is very small compared to the wavelength. However, when we want to verify these calculations by measurements on real materials at frequencies relevant for applications where wave propagation occurs, we typically measure wave parameters like wave number k and wave impedance Z, rather than permittivity and permeability directly. To translate k and Z to material parameters requires a theoretical model of the scattering problem. As we saw at the end of section 2, the comparison with real measurement data may present some challenges even for isotropic media.
4. Wave Scattering
 Typically, when characterizing electromagnetic material properties by wave scattering, reflection and transmission coefficients are recorded by a network analyzer, and subsequently translated to wave parameters (k and Z) or material parameters (ε and μ) through a theoretical model. Most methods are developed for isotropic media, primarily the NRW method [Nicolson and Ross, 1970; Weir, 1974], but they can sometimes also be applied to anisotropic media if sufficient a priori information on the anisotropy axes is available.
 A typical measurement setup is to measure reflection and transmission coefficients for a planar slab. The sample can be positioned in a coaxial fixture, hollow waveguide, or suspended in free space. In this paper, we study an X-band rectangular waveguide setup as in Figure 3, but the results are easily transferred to other setups. Even though the theoretical analysis to obtain material parameters via the NRW method is straightforward, there are several practical problems to overcome:
 1. For the coaxial fixture and hollow waveguide, relatively small samples with very precise geometry are required. For free space measurements, larger samples are needed where it may be difficult to maintain uniform properties throughout the slab.
 2. Imperfections such as small airgaps and uncertainty in position need to be modeled and compensated for. Even small geometric deviations from the theoretical shape may generate coupling to higher order modes and store reactive power, which is not easily distinguished from material properties. Several of these compensations are described by Baker-Jarvis et al.  and Larsson et al. .
 3. A well known problem is the half-wave resonance. For samples with low loss, the reflection coefficient becomes zero when the thickness of the sample corresponds to half a wavelength. This typically generates unphysical output from the NRW algorithm, which is unstable in this parameter region.
 In order to study the impact of these and other problems, and for evaluation of different inversion algorithms, we have designed a numerical model using the commercial program Comsol Multiphysics (http://www.comsol.com). This program uses finite elements to discretize a given geometry and can solve electromagnetic scattering problems with complex anisotropic permittivity and permeability. To study our waveguide problem, we have set up a numerical X-band rectangular waveguide geometry (cross section width 22.9 mm and height 10.2 mm). Two waveguide ports are connected by a 5 cm hollow waveguide each to a section of variable length, which contains the material under test. The S-parameters can then be obtained by solving the scattering problem for each frequency.
 As a representative example of the calculations of the synthetic S-parameters in section 5.2, a frequency sweep using an adaptive solver with about 20 000 elements in the initial mesh (about 34 000 in the refined mesh), and 100 frequency points for a fixed geometry takes about 30 min on an ordinary dual-core computer. The accuracy of the numerical model was estimated by refining the initial mesh to about 50 000 elements (about 91 000 in the refined mesh) and comparing the output S-parameters. The resulting squared deviation was found to be on the order of −54 dB at the low end of the frequency band (7 GHz) and −30 dB at the high end (15 GHz). The squared deviation between output material parameters varied between −52 dB and −36 dB for 5 cm slabs, and −40 dB to −28 dB for 0.5 cm slabs.
5. Extraction of Anisotropic Properties
 As demonstrated by (15), a composite material with asymmetric microstructure can be anisotropic, even though it consists of isotropic component materials. In general we must assume a full matrix,
which can be reduced to a diagonal matrix for biaxial materials, by choosing a coordinate system along the axes of anisotropy. For gyrotropic materials such as a magnetized ferrite, the diagonalization requires a complex valued change of variables, which corresponds to circular polarization rather than spatial coordinate directions.
 In order to measure the anisotropic parameters, methods developed for isotropic media can be used together with a variation of the setup, such as change of polarizations, angles of incidence etc. The number of measurements needed can be greatly reduced if a priori information on the axes of anisotropy is available. When this is not possible, the results need to be interpreted with care, which we demonstrate in the case of an anisotropic material with arbitrary axis of anisotropy.
5.1. Determination of Wave Number
 We first demonstrate that it is possible to determine the wave number in a slab. The method was previously shown by Janezic and Jargon  in the context of isotropic materials, but it is applicable also to anisotropic materials and structures with spatial dispersion, as long as the interaction between the ends of the structure is due to one well defined mode. Consider a scattering geometry as in Figure 4. For a linear material, the scattered waves are given by the scattering matrix
This can be rearranged as a transfer matrix, relating the waves on either side to each other:
Formulas expressing the T-parameters in terms of the S-parameters are given by Frickey . The T-matrix can be cascaded, such that a sequence of scatterers have a total transfer matrix [T] = [T1][T2] ⋯ s [Tn]. In particular, if the interfaces of the slab are so far apart that they interact only through one mode, the transfer matrix for the slab can be factorized as
where the matrices [Tleft] and [Tright] describe the matching of modes at the interfaces of the material. In general, this includes the excitation of many modes both in the material and in the surrounding waveguide. Under the assumption that the interfaces of the slab are connected by only one mode, all other modes must be strongly damped and can be seen as storing reactive power near the interfaces. The matrix [Tm(d)] is then the only one depending on the thickness of the slab, and has eigenvalues e∓jβd, where β is the wave number of the mode inside the slab. Generalization to several propagating modes is possible [Sjöberg, 2009b], but requires a setup where the number of propagating modes in the surrounding equals the number of modes connecting the slab interfaces.
 By combining two samples of thicknesses d1 and d2, we may form the matrix
which can be computed directly from measured S-parameters. This is a similarity transform of the matrix [Tm(d2 − d1)], which has eigenvalues . Since a similarity transform does not alter the eigenvalues of a matrix, we can then compute the propagation factors as the eigenvalues of measurement data [T(d1)]−1[T(d2)]. To determine the wave number β, take the logarithm of the eigenvalue and apply a phase unwrapping to compensate for the 2π periodicity of the exponential function ej. Having computed the wave number β, it is often represented as the effective permittivity εeff = (β2 + kc2)/k02, where kc = π/a is the cutoff wave number, a being the width of the waveguide, and k0 = ω is the free space wave number. The effective permittivity can be seen as the equivalent permittivity of an isotropic material, having the same wave number β in the waveguide as the material under test.
 This method of determining the wave number β or, equivalently, the effective permittivity εeff, can also be applied to metamaterials where the microstructure is comparable to wavelength. Typically, the metamaterial would be a periodic structure, and the two lengths of the sample would correspond to a different number of periods.
 The above procedure requires two samples of different lengths. If the sample has symmetry such that its transfer matrix can be factorized as
the same procedure of extracting the effective permittivity from the eigenvalues can be applied using only one sample. A careful analysis of this single sample, single mode procedure, reveals that it is equivalent to extracting the permittivity via the NRW method assuming a nonmagnetic material.
5.2. A Priori Information Is Important
 The method described in the previous section provides a stable way of determining partial information about the material. However, we must emphasize that what is determined is the effective permittivity, that is, the permittivity an equivalent isotropic material would have in order to cause the same propagation delay per unit length of the material. But if we know the sample is anisotropic, we need to use a priori information on the anisotropy axes in order to extract the permittivity matrix.
 A demonstration of the pitfalls is given by considering scattering by an anisotropic slab in a rectangular waveguide. Let the permittivity be uniaxial,
where the direction of the anisotropy axis is varied between 0 and 90 degrees in the cross section plane of the waveguide according to Figure 5. The S-parameters are computed using the numerical model described at the end of section 4, and the resulting effective permittivity determined by the method in the previous section is shown in Figure 5 (bottom). It is seen that without a priori information on the orientation of the anisotropy, this material characterization method can produce any effective permittivity between the extreme values. When the anisotropy axis is aligned with the waveguide walls (ϕ = 0 and ϕ = 90 degrees), the effective permittivity is an accurate determination of the corresponding permittivity matrix element, as was shown by Damaskos et al. .
 Another careful judgment when setting up the experiment is the requirement that the interfaces of the slab interact only through one mode. In Figure 5, a relatively long sample with d = 5 cm was used, whereas in Figure 6 a much thinner sample with d = 0.5 cm was used. It is seen that for high frequencies the effective permittivity for the thin sample has some resonances, which should not be confused with material properties. These are not the usual half-wavelength resonances in the NRW technique (which are not an issue with the present method), but rather the influence of higher order mode propagation in the slab. The longer sample has higher discrimination of coupling between the slab interfaces via higher order modes, but this depends not only on sample length; using lower losses, σ = 0.01 S/m, causes similar resonances also for the long slab (not shown). In order to further suppress the higher order modes in the sample, it can be placed in a thinner waveguide with higher cutoff frequency as in the work of Sjöberg [2009b]. The improved accuracy for longer samples coincides with the findings by Baker-Jarvis et al. . In the present case, we emphasize that we are only determining partial information on the material (its wave number in a waveguide setting), and additional variation of parameters is necessary in order to fully characterize the material.
 We have discussed several issues of importance for the characterization of composite materials in waveguides. The accuracy by which material parameters can be measured has been illustrated using the Cramér-Rao bound. The theoretical predictions of homogenization theory gives homogenized material parameters hom and μhom, whereas the wave scattering technique primarily gives wave parameters k and Z. Although these are in principle equivalent at least for an isotropic material, noise and model imperfections require great care in order to extract relevant information. In particular, we have demonstrated how a priori knowledge of anisotropy directions and sufficiently long samples can be crucial in order to properly design an experiment from which to extract material properties.
 The financial support of the Swedish Foundation for Strategic Research is gratefully acknowledged.