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 epsilon 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.