Homogenization of metamaterials with spatial dispersion

A study is made of the problem of metamaterial homogenization, which is the attempt to represent an artificially fabricated inhomogeneous periodic structure as a homogeneous medium with an electromagnetic response described by a number of constitutive parameters (permittivity, permability, etc.) In particular, the importance of spatial dispersion in metamaterials and the need to characterize metamaterials with wavevector dependent constitutive parameters is explained an examined. A brief survey of important previous attempts at metamaterial homogenization is presented. This is followed by a discussion of spatial dispersion in metamaterial crystals. The importance of spatial dispersion in metamaterials is justified and some manifestations of spatial dispersion described. In particular the little known phenomenon of bianisotropy in centrosymmetric crystals due to spatial dispersion is explained. Also, the effects of spatial dispersion on physical quantities such as energy flux and dissipation are identified. We then describe a new method for solving for the free eigenmodes of a metamaterial crystal with a complex wavevector eigenvalue simulation. Next, two different theoretical attempts by the author at metamaterial homogenization are described, both accompanied by tests of the calculated constitutive parameters and critical examination of the strengths and weaknesses of each approach. Finally, strong evidence of the presence and importance of spatial dispersion in metamaterials is presented.