Abstract
In this study, an analytical solution is proposed to solve the general problems of three-wave mixing with pump depletion. The so-called quasi-phase-matching (QPM) for χ(2) nonlinearities, including sum frequency generation (SFG), second harmonic generation (SHG) and difference frequency generation (DFG), is amended when the depletion of the pumping light power is considered. The optimum domain width and the corresponding conversion efficiency are discussed. The optimum domain widths deviate from the coherence length lc=π/Δk, but the deviation is so small that it can be ignored according to the actual situation. The applicable scope of undepleted pump approximation (UPA) is estimated by a proposed relative tolerance. Analysis indicates that the relative tolerance showing the difference of conversion efficiency between the UPA and the exact solution is solely determined by the conversion efficiency. Based on the relative tolerance curves for SFG, SHG, and DFG, two models are developed: one can be used to describe the SFG and SHG processes, whereas the other can be used for the DFG process. These findings can be used to provide direct guidance for practical application, and can also make the estimation of practical samples more convenient.
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