Logically combined photonic crystal - A Fabry Perot optical cavity

We address the logical combination, as opposed to the linear superposition, of two one - dimensional photonic crystals of slightly different periodicities. The original short range translational symmetry is destroyed in these quasi - periodic system. This induces a strong coupling between Bloch modes of different translational wavevectors, and results in a large number of slow modes in such logically combined photonic crystal. In this article, we show by exploiting the beating feature characteristics of the topology of our system, that these slow modes can be effectively described as modes of a Fabry Perot optical cavity made of a homogenous metamaterial with a dispersive refractive index. The homogenized refractive index of the equivalent metamaterial can be obtained from the band structure calculations, using an extended zone scheme. The density of the slow modes in the logically combined photonic crystal is inversely proportional to the group index of the equivalent metamaterial.