Electrodynamics of metallic photonic crystals and problem of left-handed materials

An exact analytical theory of the electromagnetic waves in metallic photonic crystals with a small volume fraction of a metal is presented. It is shown that there are waves with a very low cutoff frequency $\omega_0$ and that the permittivity $\epsilon$ is negative below $\omega_0$. We show that if the crystal is embedded into a medium with negative $\mu$, it has 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.