## Realizability of metamaterials with prescribed electric permittivity and magnetic permeability tensors

#### Graeme MiltonUniversity of UtahMathematics

We show that any pair of real symmetric tensors epsilon and mu can be realized as the
effective electric permittivity and effective magnetic permeability of a metamaterial
at a given fixed frequency. The construction starts with two extremely low loss
metamaterials, with arbitarily small microstructure, whose existance is ensured by
the work of Bouchitt´e and Bourel and Bouchitt´e and Schweizer, one having at the
given frequency a permittivity tensor with exactly one negative eigenvalue, and a
positive permeability tensor, and the other having a positive permittivity tensor,
and a permeability tensor having exactly one negative eigenvalue. To achieve the
desired effective properties these materials are laminated together in a hierarchical
multiple rank laminate structure, with widely separated length scales, and varying
directions of lamination, but with the largest length scale still much shorter than
the wavelengths and attenuation lengths in the macroscopic effective medium.

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