Simple analytical expression for the effect of initial interaction in Cerenkov second-harmonic generation

We analyze the nonlinear coupled-mode equations describing the Cerenkov second-harmonic generation in a nonlinear dielectric waveguide. An approximate analytic solution in a closed form is newly obtained for the pump depletion and nonlinear phase-shift of the fundamental field. Our solution, valid over the whole length, shows that the conversion efficiency is reduced by the feature of the initial process of the nonlinear interaction. It gives a simple expression for the efficiency with better accuracy for short device length. We have compared our solution with the previous analytic solutions derived by others. The validity of our solution is also checked by the nonlinear beam propagation method. Furthermore, the solution can be expressed in terms of a leaky mode phase-mismatch, and it shows explicitly that a resonant pump depletion and nonlinear phase-shift of the fundamental field occurs when the phase-matching to the lowest order leaky mode is accomplished. Extending our analysis method, we have also performed higher order numerical analysis for the nonlinear coupled-mode equations.