Road map for experimental characterization of metamaterials Print E-mail

Recent advances in electromagnetics of complex materials and technological developments have enabled the design and fabrication of numerous new types of micro- and nanostructured artificial electromagnetic media. New electromagnetic phenomena observed in the novel inhomogeneous structures lead to new electromagnetic properties, which could enable novel industrial applications. These properties may extend beyond those of natural materials or even be absolutely new.

To understand the properties of these new materials and evaluate the applicability of the structure to target applications one should first determine if a particular structure can be described by a few parameters characterizing material or the structure demonstrates so-called mesoscopic properties, i.e. properties that change from sample to sample, depending on its size, shape, and environment.

This "road map" gives some guidance on how one can experimentally determine what kind of structure we deal with and, if this is indeed a new material, how to assess the electromagnetic properties of composite material samples. This usually means measurement of effective material parameters, like permittivity and permeability tensors (sometimes also electromagnetic coupling tensor), which define the response of an effectively continuous material to the excitation by electromagnetic waves.

We assume that the sample of the composite material under characterization is shaped as a planar layer: a slab. The challenge is to determine experimentally if this material is an artificial dielectric or artificial magnetic or bianisotropic material, or it is a photonic crystal or something else. If possible, we should retrieve its material parameters from measurement data.


Experimental steps Short explanations
Step 1.
Is the sample a piece of an effectively homogeneous material or a (complicated mesoscopic) structure?
Measurement of diffuse scattering from the sample shaped as a planar layer. The sample is illuminated by a wave beam with nearly flat phase front incident at a certain angle. The beam width is small enough so that the edges of the layer under characterization are not illuminated. This incident wave beam models the plane wave incident on the infinitely extended layer. Amplitude of fields, reflected, transmitted and scattered at all directions, is measured. Measurements at a given frequency can be repeated for several
incidence angles. These measurements can be done at several frequencies.
The aim is to determine the level of diffusion scattering for frequencies and angles of incidence which are of our interest. If we observe that (for a given frequencyÂ Â  and incidence angle) nearly all reflected energy propagates in the specular direction, according to the usual reflection law, then we deal with an effectively homogeneous sample (for this frequency and this incidence angle). If we observe side lobes or high level of diffusion scattering, it means that the sample is not an
effectively homogeneous material, and we deal with a more complex system. This can be a diffraction grating or a photonic crystal or whatever.
If the level of diffusion scattering is small, we can passÂ  to
Step 2. Are the effects of spatial dispersion [1] weak or strong?
Measurement of the plane-wave reflection R andÂ  transmission T coefficients of the sample at the given frequency 1) in a uniform medium andÂ  2) stacked with a conventional material sample, for example, aÂ  dielectric layer (which is referred to as the test layer). The incident field is the same wave beam as in Step 1. First, we find from R and T coefficients the transmission matrix [2,3] of the sample under test and the transmission matrix of the test layer separately. Next, we bring them together, measure the transmission matrix of the whole stack and compare it with the product of the two previously measured matrices. If the sample under characterization is effectively homogeneous, the transfer matrix of the stack of two
Â layers (the composite layer and the test layer) should be equal to the product of the transfer matrices of the two slabs.Â  The exceptions from this rule refer to the case when the presence of the dielectric layer enhances the manifestation of the structure heterogeneity (for the wave in free spaceÂ  the structure behaves as an effectively continuous, but for the wave in the dielectric, as a discrete one). Then one has to take another dielectric layer with a smaller refraction index.
If we find out that the effective response of the sample does notÂ  change when we bring the sample close to a test sample (with the known properties), then the spatial dispersion is weak enough and we can pass to
Step 3. Search for the minimal set of local constitutive parameters
Study the polarization characteristics of reflected and transmitted waves at several incidence angles, using, for example, ellipsometry set-ups. Illumination from both sides is necessary to assess reciprocity and bi-anisotropy of the sample. We determine the type of the material (isotropic, anisotropic, bianisotropic, chiral, gyrotropic) analyzing the polarization ellipses of the reflected and transmitted waves.
Step 4. Retrieval of local constitutive parameters
Using RT-retrieval (e.g., modern variants of the NRW technique [4,5] and its extensions) or ellipsometry (note,Â  however, that ellipsometry methods are well developed only for non-magnetic
layers) determine the bulk effective material parameters. For RT-retrieval, measurements of both amplitude and phase of the reflection and transmission coefficients are necessary. If the material under characterization is prepared on a wafer or another substrate whose thickness is larger than that of the material layer under test, the substrate of the same thickness should be studied
separately in order to measure the S-matrix of the material layer
under test.
At this stage the numerical values of constitutive parameters are determined as functions of the frequency.
Step 5. Analysis of the retrieved parameters and interpretation of the results ("sanity check")
Analyse the results. Elaborate methods of measurement of new (alternative) parameters, if that is necessary, and then make these new measurements. Repeat this analysis stage... until the results
are satisfactory.
To analyze whether the obtained set of constitutive parameters makes physical sense and if these parameters can be indeed used to describe the material of the sample. One should check the following key issues:
  • Energy conservation is not violated (the imaginary parts of the permittivity or permeability do not change signs and the real part of the effective impedance is positive) [1].
  • Constitutive parameters do not depend on the thickness or shape of the sample.
  • Causality principle [1] is not violated - that is, the model does not allow any response before the corresponding stimulus has been applied. This can be confirmed from the analysis of the frequency dependence of the effective permittivity and permeability. For causal materials the real and imaginary parts of the material parameters satisfy the Kramers-Kronig relations [1], and in the low-loss regions and well below resonances the permittivity and permeability grow with increasing frequency) [1].
  • The retrieved parameters do not depend on the incidence angle and the sample thickness (otherwise they characterize not the material, but only this sample for this particular excitation).


If the above procedure successfully "converges", this means that the sample under test can be characterized by conventional (volumetric) effective parameters: permittivity, permeability, chirality parameter, etc. These parameters can be used in device design, because the parameters indeed describe the properties of the material, which will not change if the size or shape or position of the sample will be different than in the measurement set-up.

If this is not the case, the conclusion is that the retrieved material parameters have a limited validity region and, most likely, unconventional physical meaning. However, in studies of metamaterials compromises often have to be made, as in many instances there are no better ways to describe the "material" response than using these "material parameters". They can successfully provide some qualitative description of physical phenomena in the sample, but care should be taken in understanding the limitations for their use.

Additional explanations for experts in electormagnetic characterization of complex materials can be found here.

For practical measurement setup see here.


[1] L.D. Landau and E.M. Lifshitz, Electrodynamics of continuous media, Pergamon Press, Oxford, 1960.
[2] D.M. Pozar, Microwave Engineering, Wiley, 1996.
[3] R.E. Collin, Foundations for Microwave Engineering, Mc-Graw Hill, New York, 1966 (first edition); 1992 (second edition).
[4] A.M. Nicolson and G.F. Ross, IEEE Trans. Instrum. Meas., 17 (1968) 395.
[5] W.W. Weir, Proc. IEEE, 62 (1974) 33.