Essential parameters in the formation of photonic band gaps

Abstract

It is generally believed that the spatial periodic variation of dielectric constant, ε, or refractive index, $\sqrt{με}$, can give rise to photonic band gaps (PBG``s). However, we demonstrated that the wave impedance, which depends on $\sqrt{μ/ε}$, is an essential parameter giving PBG rather than the dielectric constant or the refractive index, using microwave experiments of microstrip lines with a periodic array of holes. The role of wave impedance in the formation of PBG``s explains well the effects of magnetic permeability, μ, on PBG``s already reported. We investigated the roles of wave impedance and refractive index in photonic crystals by means of analytical expressions for edges frequencies of PBG in a one-dimensional (1D) photonic crystal with magnetic and dielectric properties. The analytical expressions were derived when the optical thicknesses of layers are the same. The wave impedance governs the formation of PBG``s. Meanwhile, the position of PBG``s and the creation of defect modes are related to the refractive index. We also showed that the contrast of impedance $\sqrt{Cρ}$, neither of mass density ρ nor of velocity $\sqrt{C/ρ}$, in elastic composites plays a crucial role in the formation of acoustic band gaps (ABG``s) by investigating the roles of elastic constant C and mass density in the formation of ABG``s of periodic elastic composites. ABG``s become wider when the mass density and the elastic constant have their maximum values in the same material, but they are getting smaller and eventually disappearing in the opposite case. As the volume fraction increases, the upper (bottom) edge of the stop band increases (decreases) and then decreases (increases) in the composites with the impedance variation, only, so that the mid-gap frequency changes very little and a larger ABG can be created. When the impedance ratio increases, the upper (bottom) edge of the stop band increases (decreases), so that the mid-gap frequency slightly decreases and the...