Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method.

The transfer matrix method has been widely used to calculate the scattering of electromagnetic waves. In this paper, we develop the conventional transfer matrix method to analyze the problem of second harmonic generation in a one-dimensional multilayer nonlinear optical structure. In the designed nonlinear photonic crystal structure, the linear and nonlinear optical parameters are both periodically modulated. We have taken into account the multiple reflection and interference effects of both the linear and nonlinear optical waves during the construction of the transfer matrix for each composite layer. Application of this method to multilayer nonlinear photonic crystal structures with different refractive indices indicates that the proposed method is an exact approach and can simulate the generation of the second harmonic field precisely. In an optimum structure, the second harmonic generation efficiency can be several orders of magnitude larger than in a conventional quasi-phase-matched nonlinear structure with the same sample length. The reason is that, due to the presence of photonic band gap edges, the density of states of the electromagnetic fields is large, the group velocity is small, and the local field is enhanced. All three factors contribute to significant enhancement of the nonlinear optical interactions.