Applicability of effective medium description to photonic crystals in higher bands: Theory and numerical analysis

We consider conditions under which photonic crystals (PCs) can be homogenized in the higher photonic bands and, in particular, near the point. By homogenization we mean introducing some effective local parameters eff and μeff that describe reflection, refraction, and propagation of electromagnetic waves in the PC adequately. The parameters eff and μeff can be associated with a hypothetical homogeneous effective medium. In particular, if the PC is homogenizable, the dispersion relations and isofrequency lines in the effective medium and in the PC should coincide to some level of approximation. We can view this requirement as a necessary condition of homogenizability. In the vicinity of a point, real isofrequency lines of two-dimensional PCs can be close to mathematical circles, just like in the case of isotropic homogeneous materials. Thus, one may be tempted to conclude that introduction of an effective medium is possible and, at least, the necessary condition of homogenizability holds in this case. We, however, show that this conclusion is incorrect: complex dispersion points must be included into consideration even in the case of strictly nonabsorbing materials. By analyzing the complex dispersion relations and the corresponding isofrequency lines, we have found that two-dimensional PCs with C4 and C6 symmetries are not homogenizable in the higher photonic bands. We also draw a distinction between spurious -point frequencies that are due to Brillouin-zone folding of Bloch bands and “true” -point frequencies that are due to multiple scattering. Understanding of the physically different phenomena that lead to the appearance of spurious and “true” -point frequencies is important for the theory of homogenization.