Abstract
We study sufficient conditions for the applicability of the classical parametric approximation in three-wave interactions when the pump intensity is very large compared to signal and idler intensity. To derive such conditions we express the exact classical solutions given by Jacobian elliptic functions in terms of hyperbolic functions. Thereby the first minimum of the pump intensity is correctly described but the periodicity is lost. We derive new approximations for the initial conditions using pump coordinate scaling and find the interval that defines complete pump depletion. We show that the classical parametric approximation with a fixed and sharp pump amplitude and phase can be used for an increasing fraction of this interval if the pump intensity is made to grow. By choosing higher and higher pump intensities the nonlinearity is shifted to the end of that interval. As an instructive example for the application of these findings the generation of two-mode squeezing is briefly considered.
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