Towards more general constitutive relations for metamaterials: a checklist for consistent formulations

When the period of unit-cells constituting metamaterials is no longer much smaller than the wavelength but only smaller, local material laws fail to describe the propagation of light in such composite media when considered at the effective level. Instead, nonlocal material laws are required. They have to be derived by approximating a general response function of the electric field in the metamaterial at the effective level that is accurate but cannot be handled practically. But how to perform this approximation is not obvious at all. Indeed many approximations can be perceived and one should be able to decide as quick as possible which of these possible material laws are mathematically and physically meaningful at all. Here, at the example of a second order Padé approximation of the general response function of the electric field, we present a checklist each possible constitutive relation has to pass in order to be physically and mathematically liable. As will be shown, only one out of these nine Padé approximations passes the checklist. The work is meant to be a guideline applicable to decide which constitutive relation makes actually sense at all. ... mehrIt is an essential ingredient for future research on composite media as any possible constitutive relation to be discussed should pass it.