The effective permittivity and hyperbolic quality of a one-dimensional metamaterial

We derive the effective permittivity of a one-dimensional metamaterial system based on the Bloch theory and transfer matrix method. The results show to what extent one can use the concept of the effective permittivity in the metamaterial system. It is also demonstrated that in the model of the effective plane wave the derived permittivity is not only temporally dispersive but also spatially dispersive and the validity is confirmed by comparing the band structure with the dispersion relation of the effective medium. Moreover, we investigate the hyperbolic quality and the effect on the (local) density of states of the hyperbolic metamaterial where one of the permittivity tensor elements is negative. This indicates that increasing the composition with negative permittivity such as the filling ratio can enhance the hyperbolic quality and the (local) density of states