Causality relations in the homogenization of metamaterials

We investigate the causality relations in homogenized metamaterial arrays and determine the reasons why metamaterial effective (bulk) constitutive parameters obtained using classic point-dipole approximations may violate basic causality conditions represented by the Kramers-Kronig relations. We show that noncausality is inherently introduced by the use of point-dipole approximations within Maxwell-Garnett homogenization procedures and that these artifacts become particularly significant for more densely packed arrays. In contrast we show that properly defined and exactly computed bulk constitutive parameters of periodic metamaterial arrays always satisfy causality for a fixed spatial frequency in a rigorous homogenization framework. Finally we show how the Maxwell-Garnett approach can be modified to effectively remove the noncausal effects.