Second-harmonic phase-matching based on zero refractive index materials

We provide a two-dimensional honeycomb photonic crystal, exhibiting accidental triply-degenerate points for each polarization at two frequencies with 2 times relationships. Using k ⃗ p ⃗ method and group theory, we demonstrate the zero refractive index phenomenon at two different polarizations. Without considering the additional loss, the numerical simulation shows that the honeycomb crystal exhibits greatly enhanced second harmonic generation (SHG) conversion efficiency over the traditional nonlinear materials. Our study has established the prerequisites for the photonic structure to achieve the phase-matching SHG naturally.