Electrodynamics of Metallic Photonic Crystals and the Problem of Left-Handed Materials

An analytical theory of low frequency electromagnetic waves in metallic photonic crystals with a small volume fraction of a metal is presented. The evidence for such waves has been obtained recently by experiments and computations. The cutoff frequency of these waves, ${\omega }_0$, is studied. An analytical expression for the permittivity $ϵ$ is obtained and shown to be negative below ${\omega }_0$. If the crystal is embedded into a medium with a negative $\mu$, there are no propagating modes at any frequency. Thus, such a compound system is not a left-handed material (LHM). The recent experimental results on the LHM are discussed.