Propagation of waves in metallic photonic crystals at low frequencies and some theoretical aspects 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 of the existence of such waves has been found recently via experiments and computations. We have obtained an exact dispersion equation for omega (k) and studied the cutoff frequency omega_0 = omega(0) as a function of parameters of the photonic crystal. An analytical expression for the permittivity epsilon is calculated. It is shown, 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.