Homogenization of quasi-1d metamaterials and the problem of extended bandwidth

We derive approximate analytical expressions for the effective permittivity tensor of two-phase metamaterials whose geometry is close to one-dimensional (quasi-one-dimensional metamaterials). Specifically, we consider the metamaterial made of parallel slabs with width given by a linear or parabolic function. Using our approach, the design of epsilon-near-zero, ultra-low and high refractive index metallodielectric metamaterials with extended bandwidth has been demonstrated. In addition, generalizations to the three-dimensional case and some limitations of the presented technique are briefly considered.