Artificial versus natural crystals: effective wave impedance of printed photonic bandgap materials

Printed metallo-dielectric photonic bandgap (PBG) materials are analyzed using an analytical approach based on multipole expansions for the scattered fields of individual scatterers and a transfer-matrix method for reconstructing the total scattered fields created by successive lattice planes of the artificial crystal. An effective description of the PBG medium is derived and its correspondence with natural crystals is further advanced through an analysis based on Lorentzian response functions, which characterize natural crystals. The effective wave impedance and bulk reflection coefficient of the medium are provided and their properties inside and outside the bandgaps are examined. The presented treatment for these effective response functions extends far beyond the traditional effective medium theory (EMT) limits.