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 quickly 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 of each possible constitutive relation that has to pass in order to be physically and mathematically liable. As will be shown, only two out of these nine Padé approximations pass the checklist. The work is meant to be a guideline applicable to decide which constitutive relation actually makes any sense at all. It is an essential ingredient for future research on composite media as any possible constitutive relation to be discussed should pass it.

Figure 1

Illustration of the domain in which the light propagates, the upper-half space is occupied by vacuum and the lower-half space is occupied by a homogenized MM. The surface separating the two half-spaces is denoted $\mathrm{\Gamma }$. The normal $\mathbf{n}$ is outward directed from the homogenized MM.

Figure 2

Propagating modes inside a slab MM. We have two forward modes defined on the the surface ${\mathrm{\Gamma }}^{+}$ and the other backward modes are defined on the surface ${\mathrm{\Gamma }}^{-}$.