The real part of the dielectric function of a polar crystal [39, 43] with the presence of dissipation can be written as [[epsilon].sub.[infinity]]+ X([omega]), where X([omega]) = ([[omega].sub.0] - [[epsilon].sub.[infinity]]) [[omega].sup.2.sub.T]([[omega].sup.2.sub.T] - [[omega].sup.2])/[([[omega].sup.2.sub.T] - [[omega].sup.2]).sup.2] + [[gamma].sup.2][[omega].sup.2] [43].

McGurn, "Photonic band structures of two-dimensional systems fabricated from rods of a cubic polar crystal," Phys.

These include direct PWE formulations for PCs constructed with lossless metals [37], metals with dissipations [38], polar crystals [39, 40], and superconducting composite [41].

Specifically we showed that the roots of a simple cubic equation accurately yield the eigenfrequencies of a 2D PC constructed with polar crystals. The nature of the analytical roots of the cubic equation explains the appearance of dispersionless bands in the photonic band structure of 2D PCs with polar crystals [39, 40, 43].

In Table 1, we have tabulated the nonzero matrix coefficients of [??]([omega]) for dielectric functions of a two-level quantum dot [49], polar crystals [39,43], and Drude-like materials.

We will consider the rods to be made of polar crystals. Previously, polar crystals have been used in the investigations for the enhancement of the photonic band gap and investigation of coexisting polaritonic band gap in PCs [39,40,54,55].

Let's analyze the asymptotic behavior of the scalar polynomial for the case of PCs made of polar crystals. The cubic-scalar polynomial [in [[omega].sup.2]] for polar crystals with nonzero dissipation (gamma) is P([omega]) = [[omega].sup.6] - [[omega].sup.4][[omega].sup.2.sub.n] + [[omega].sup.2.sub.T](2 + [epsilon][[theta].sub.n]) - [[gamma].sup.2]] - [[omega].sup.2][[[omega].sup.2.sub.n]([[gamma].sup.2] - 2[[omega].sup.2.sub.T]) - [[omega].sup.4.sub.T](1 + [epsilon][[theta].sub.n])] - [[omega].sup.2.sub.n][[omega].sup.4.sub.T].

Indeed, by numerical example we showed that for the particular case of a two-dimensional photonic crystal made of polar crystals, scalar polynomial [which is a result of Equation (8) with only one mode of the backbone photonic crystal] yields accurate eigenfrequencies.