  We study the frequency dependence of the effective electromagnetic parameters
of left-handed and related meta-materials of the split ring resonator and wire
type. We show that the reduced translational symmetry (periodic structure)
inherent to these meta-materials influences their effective electromagnetic
response. To anticipate this periodicity, we formulate a periodic effective
medium model which enables us to distinguish the resonant behavior of
electromagnetic parameters from effects of the periodicity of the structure. We
use this model for the analysis of numerical data for the transmission and
reflection of periodic arrays of split ring resonators, thin metallic wires,
cut wires as well as the left-handed structures. The present method enables us
to identify the origin of the previously observed resonance/anti-resonance
coupling as well as the occurrence of negative imaginary parts in the effective
permittivities and permeabilities of those materials. Our analysis shows that
the periodicity of the structure can be neglected only for the wavelength of
the electromagnetic wave larger than 30 space periods of the investigated
structure.
