Effective refractive index analysis of optical Kerr nonlinearity in photonic bandgap structures

A numerical dispersion relation is employed to analyze the linear and nonlinear optical effective refractive indices of 1-D finite periodic photonic bandgap structures. A Bragg reflector (BR) and a photonic crystal microcavity (PCMC) are examined by assuming that the high indexed layer of the two constituent layers possesses a nonlinear optical response. For the BR, the singularity of refractive index, appearing at bandgap edges in a Bloch index description, is removed. In the case of the PCMC, the optical responses at a defect mode and bandgap edges are properly described, thanks to the use of the numerical dispersion relation. This also allows us to quantitatively compare the Kerr nonlinearity observed at the defect mode and the bandgap edges. The efficiencies of the BR bandgap edge and the PCMC defect mode in achieving a given transmission change are compared by calculating the required nonlinear optical refractive index change. The PCMC defect mode is found to be 1.5 times and twice more efficient than the BR bandgap edge in 20 and 10 dB transmission modulation, respectively.