Analytical description of band gaps in a ternary metallo-dielectric stack

Metallo Dielectric Photonic Crystals formed by same periodicity metallic inserts in a Dielectric Photonic Crystal show three kind of band gaps, those at the well know dielectric band gap, the ones attributed to the absorption of metal to low frequencies and a new class of metallic bandgaps. Numerical studies have confirmed that while the dielectric band gap width is basically described by the refraction index contrast, the width of the metallic band is described by the thickness of the metal inserts. In this work we carry on the corresponding analytical analysis of both band gaps for this one dimensional ternary dielectric-dielectric-metal structure. The stack that we are proposing is a quarter-wave for the dielectrics and the thickness of the metallic layers is changed as a free parameter. Using standard transfer matrix formalism, we find a closed form of the dispersion relation and from it; we have analytically demonstrated the formation and width of the dielectric band gap and its metallic perturbation, as well as those of the additional metallic band gap.