Effective permittivity and permeability functions of metalo-dielectric photonic band gap materials

Composite periodic structures, also named photonic band gap (PBG) materials, have attracted a lot of attention due to their controllable dispersive properties that allow a wide range of applications in novel antenna structures and frequency selective surfaces. Up to now, the analysis of PBG structures has been performed by numerical methods. Therefore optimizing the performance of a particular PBG design usually requires a case-by-case, trial and error method which is both CPU-time consuming and physically obscure. In order to bypass these limitations and isolate the response of the medium from the device in use, an effective description for PBG crystals is needed far beyond the quasistatic limit of traditional effective medium theories. Such an effective description for a metalo-dielectric photonic band gap (PBG) material with disk inclusions has been developed analytically. In this paper, we generalize this approach by presenting the effective permittivity and permeability functions for inclusions of a general canonical shape. This is useful in optimising the band gap formation as a function of the shape of the implants. We focus on flat inclusions of infinitesimal thickness that create thin planar geometries which are relevant for microwave devices.