Scaling law of second harmonic generation in one-dimensional photonic systems

Summary form only given: A theoretical study of second harmonic generation in one-dimensional photonic crystals and planar microcavities is made by means of nonlinear transfer matrix method. The purpose of this work is to analyse the various mechanisms which lead to a high conversion efficiency in fully periodic photonic crystals or when a structural defect is introduced. In a one-dimensional periodic structure, great enhancement in second harmonic conversion efficiency can be obtained, when resonant coupling of the pump and harmonic fields with photonic modes near the band-edges, as well as phase matching, are achieved. Even though numerous theoretical and experimental works have been done, the scaling law of second harmonic conversion efficiency with the number N of periods in these systems has not been completely understood. It seems that there is no consensus about the maximum reachable scaling and in which structures it can be realized. Here, we consider two different one-dimensional photonic crystals: a dielectric Bragg reflector constituted by the repetition of two layers, and a non-conventional one characterized by a period that is constituted by four layers. We demonstrate that in the first case it is possible to achieve a maximum scaling of the second harmonic conversion efficiency prop N⁶. On the other hand, by considering more complex structures with four layers in a period, a larger scaling prop N⁸ can be obtained by satisfying phase matching and double resonance conditions for the pump and harmonic field, at the lower band-edges of the first and third order photonic gap, respectively. Another possible route for increasing second harmonic generation is to embed the nonlinear source in a Fabry-Perot cavity, which can be obtained by inserting a structural defect in a one-dimensional periodic system. By exploiting the resonance of the pump and/ or harmonic field with a cavity mode, it is possible to enhance pump field intensity and/- r to optimize the extraction efficiency of the harmonic field. In this case we observe an exponential scaling of second-harmonic signal with the number N of mirror periods, which is maximum when double resonance and phase matching conditions are simultaneously fulfilled