Perturbative analysis of electromagnetic homogenization near the Γ-point in higher bands

The behavior of electromagnetic waves in periodic structures near the Γ-point in the first band has been examined thoroughly over the years. In particular, it can be proved rigorously that (i) the dispersion relation is linear and positive (Tsukerman, JOSA B 25, 927, 2008), and (ii) if both the dispersion relation and the impedance of the medium are considered, then the only valid homogeneous description of the medium is obtained by assigning the medium a trivial magnetic permeability µ = 1 (Markel & Schotland, PRE 85, 066603, 2012). For propagating waves with the wave vector q = qn̂, q being a complex wave number and n̂ being a unit vector of direction, one has (for µ = 1) Im (q · q) = Im q² = 2 Re q Im q = (w/c)² Im ε > 0. This relation means that the direction of phase velocity and the direction of spatial decay of a wave always coincide. The more complicated case of waves with more general wave vectors propagating in anisotropic media is considered by Markel & Schotland (J. Opt. 12, 015104, 2010). In any case, all such phenomena are consistent with the linearity and positivity of the slope of all dispersion curves near the Γ-point in the first band. We can refer to this propagation regime as to the homogenization limit.