Nonlinear photonic crystal: effects of negative refractive indices and dispersion in the resonant region

We investigate second-harmonic generation of photons in nonlinear photonic crystal composed of N periods of bilayers with dispersive complex magnetic permeability and dielectric permittivity. Analytical expressions for the reflected and transmitted fundamental fields (FFs) and the second-harmonic (SH) fields are obtained. We find interesting features in the SH fields versus pump frequency and relative thickness of the bilayers for the same real values of permittivity and permeability but with different signs. Negative permeability is a substitute for periodical poling. A stronger SH signal is found when one of the layers has negative refractive index. For certain parameters, the backward and forward SH signals can be generated over a broad frequency range. Around the resonant region with strong dispersions, the reflected and transmitted signals depend on the magnetic and dielectric resonances of each layer. The transmitted FF field shows a narrow transparency window that can be controlled by the thickness of the layers. Also, a negative index around the resonant region reduces the SH peak. These results provide insights for designing efficient nonlinear photonics devices with dielectric-magnetic materials.