Mechanism of formation of band gaps in one-dimensional photonic quasi-crystals

A 1D photonic quasi-crystal in the form of a system of layers with different spatial periods dζ and dn of the impedance and the refractive index, respectively, is analyzed. The dependence of the positions of band gaps formed in photonic crystals on the commensurability of the impedance and refractive-index periods is studied. It is shown that the band gaps formed in such systems are determined by the superperiod, which is the least common multiple of the admittance and refractive-index periods, rather than the refractive-index period (the Bragg condition) or the impedance period. This fact means that, in the case of incommensurability, a band gap is formed at a wave number of zero.