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.

Figure 1

The real part of the dimensionless cutoff frequency ${\chi }_0$ as a function of the volume fraction of metal $f$. Solid, dashed, and dotted lines represent solutions of Eqs. (8, 9, 2), respectively. The experimental data of Ref. 3 ($★$) are shown together with the numerical data of Ref. 6 ($▴$, $•$, $\blacksquare$). On the main plot $\alpha =0.024$, $d=12.7\mu \mathrm{m}$ for upper solid, upper dashed lines, and for the point $▴$; $\alpha =0.078$, $d=1.27\mu \mathrm{m}$ for middle solid, middle dashed lines, and for the point $•$; $\alpha =0.246$, $d=0.13\mu \mathrm{m}$ for lower solid, lower dashed lines, and for the point $\blacksquare$. In the inset $\alpha =3.4\times {10}^{-4}$, $d=8.0\mathrm{m}\mathrm{m}$.