Restriction of the phase matching condition on refractive index modulation of 1D photonic crystals in second harmonic generation

Phase and group velocity matching equations of second harmonic generation are reconsidered for 1D photonic crystals to analyze an induced structure restriction. A multiplicity of solutions is found through a numerical computation, which separates the restricted structure parameters into infinite units. The local and global characteristics of the restriction on refractive indices and material lengths are discussed in detail as well. The analysis shows that increasing either the dispersion of the nonlinear material or the thickness of a single layer will require a larger refractive index contrast.