Relaxed dispersion constraints and metamaterial effective parameters with physical meaning for all frequencies

Metamaterial effective parameters may exhibit freedom from typical dispersion constraints. For instance, the emergence of a magnetic response in arrays of split ring resonators for long wavelengths cannot be attained in a passive continuous system obeying the Kramers-Kronig relations. We characterize such freedom by identifying the three possible asymptotes which effective parameters can approach when analytically continued. Apart from their dispersion freedom, we also demonstrate that the effective parameters may be redefined in such a way that they have a certain physical meaning for all frequencies. There exist several possible definitions for the effective permittivity and permeability whereby this is achieved, thereby giving several possible frequency variations for high frequencies, while nevertheless converging to the same dispersion for long wavelengths.