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I'm presenting a new elegant formulation of the theory of fully nonlinearadiabatic TWM (FNA-TWM) in terms of elliptic function here. Note that thelinear case of SFG and DFG in the undepleted pump approximation described bythe FVH representation has been exploited several years ago. For the sake ofcompleteness, I present the pseudo-FVH representation to describe OPA.Moreover, I'm trying to display an overview of TWM processes and show that boththe linear cases, the linear adiabatic SFG(DFG) and the linear OPA, are onlythe special cases of my theory. Finally I also point out that the geometricimage of the so-called adiabatic basis acts as the geodesic line of thegeneralized Bloch sphere.

New approach to Fully Nonlinear Adiabatic TWM Theory