Random microstructure metamaterials

Metamaterials are usually defined as artificial materials designed to have values of relative permittivity and, in the case of magnetic materials, relative permeability not found in nature. Some authors restrict the definition to include only those materials which have negative values but this is rather unuseful since it would require that a metamaterial must be a non-metamaterial at certain frequencies (it is not possible to obtain a material with negative permittivity or permeability at all frequencies – e.g. see  negrefr). With this point in mind one could define a metamaterial as a material for which there exists a frequency for which either the relative permittivity or permeability is negative, but this is true for very many lossless heterogeneous composite materials when the particles in the composite become comparable with a wavelength. Furthermore, such a definition fails properly to take the effects of loss into account leading to values of the real part of the permittivity or permeability that may be less than unity but not negative.  With these problems in mind, we will keep a rather more relaxed view of the definition and drop the requirement for negativity.  Metamaterials, in this more general sense, have been around for a long time.

The design of metamaterials in current literature has largely been restricted to materials with perfectly or near-perfectly ordered microstructure. There are two good reasons why this is the case. Firstly, analysis (either numerical or analytic) is much more tractable. Secondly, perfect order (but see below) is an almost necessary requirement for low loss in highly dispersive structures and such dispersion is a necessary requirement for the existence of negative values for relative permittivity and permeability. Any randomness in the microstructure gives rise to inhomogeneous scattering losses and even small non-zero imaginary parts (characterising loss) of the relative permittivity or permeability make such materials unsuitable for many of the applications for which negative values are deemed useful.

The study of randomness in the microstructure of artificial materials or metamaterials, and how this impacts on their bulk characteristics, is thus an important area of research. We have conducted work in this area (so far unpublished in the open literature) in a number of catagories:

• The analysis of multi-layered structures with a microstructure characterised by an effective permittivity and permeability with a random perturbation defined by a variance and autocorrelation function on a Poisson or Gaussian distribution. This is an extension of work by Lachiany  and others.
• The development of approximate and exact techniques to construct representative scattering from finite structures containing these materials.
• The development of methods to determine the statistical parameters necessary to compute such scattering, given a heterogeneous two-phase material with known geometry.

There are, however, some types of random microstructures which are not characterised by a Poisson or Gaussian distribution and have quite different properties. The hyperuniform and stealthy hyperuniform distributions are particularly interesting and may allow the construction of random metamaterials which do not generate scattering loss. See research topic on Hyperuniform materials.

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