Derivation and characterization of dispersion of defect modes in photonic band gap from stacks of positive and negative index materials

An expression is derived to predict the incident angle dependence of the frequency of defect modes inside the zero-n¯ gap of one-dimensional photonic crystals stacking with positive and negative index materials. From this expression, the dependence of the dispersion type of the defect mode on both the refractive index and the wave impedance of the component layers of the photonic crystal are given. For situations with a given value of permeability, it is found that, as the adjacent layers of the defect are low refractive index materials, only positive or negative type of angular dispersion exists; as the adjacent layers of the defect are high refractive index materials, all three types of angular dispersion, including the near-zero dispersion, appear with the refractive index of the defect changes. The dispersion relation diagrams of such photonic crystals are presented, the simulated results agree with the theoretical prediction.