  In a nonlinear three-wave mixing process, the interacting waves can
accumulate an adiabatic geometric phase (AGP) if the nonlinear coefficient of
the crystal is modulated in a proper manner along the nonlinear crystal. This
concept was studied so far only for the case in which the pump wave is much
stronger than the two other waves, hence can be assumed to be constant. Here we
extend this analysis for the fully nonlinear process, in which all three waves
can be depleted and we show that the sign and magnitude of the AGP can be
controlled by the period, phase and duty cycle of the nonlinear modulation
pattern. In this fully nonlinear interaction, all the states of the system can
be mapped onto a closed surface. Specifically, we study a process in which the
eigenstate of the system follows a circular rotation on the surface. Our
analysis reveals that the AGP equals to the difference between the total phase
accumulated along the circular trajectory and that along its vertical
projection, which is universal for the undepleted (linear) and depleted
(nonlinear) cases. Moreover, the analysis reveals that the AGPs in the
processes of sum-frequency generation and difference-frequency generation have
opposite chirality. Finally, we utilize the AGP in the fully nonlinear case for
splitting the beam into different diffraction orders in the far field.
