Simultaneous second- and third-harmonic generation in one-dimensional photonic crystals

Abstract

By use of the envelope-function approach, the equation governing propagation of the TE plane wave in a one-dimensional periodic structure is reduced to a set of coupled-mode dynamical equations for slowly varying amplitudes. We applied this method to layered media possessing both χ(2) and χ(3) nonlinearities, to study the possibility of simultaneous second- and third-harmonic generation. The phenomenon is based on the geometry of the structure and is observed in a wide class of photonic crystals of different natures, provided that the thickness and refractive indices of alternating dielectric layers are appropriately chosen. By imposing various initial distributions of the energy among the individual modes, we studied the evolution of the intensities. We found that the presence of two channels for the energy transfer from the fundamental mode prevents concentration of the total energy in either of the higher modes. In all the cases considered the energy exchange among the modes is observed. We present a particular solution for the energy transfer from the fundamental and the third harmonic to the second harmonic, obtained when the cubic nonlinearity is negligible.