Exact iterative solution of simultaneous second-harmonic and third-harmonic generation in nonlinear photonic crystals

A computational study on simultaneous second-harmonic and third-harmonic generation and enhancement through a χ(2) three-wave mixing process in one-dimensional nonlinear photonic crystals is presented. The mathematical model partly overcomes the shortcoming of some existing models based on the undepleted pump approximation, which is derived from a nonlinear system of Maxwell equations. We introduce not only a variational approach that combined the finite element method and the fixed-point iteration to study the nonlinear frequency conversion but also a continuation technique depends on the weak formulation of the incremental fields to ensure the convergence of the iterative procedure when the nonlinearity is very strong. Two designed photonic crystals matched the phase by utilizing and balancing the interplay of material dispersion and the geometrical dispersion are reported, as well as a tested photonic crystal to show the validity and efficiency of our approaches. Numerical experiments indicate that the conversion efficiencies of SH and TH generation can be significantly enhanced when the frequencies of fundamental and harmonic waves are tuned at the photonic band edges or are located to the defect states.