  The geometric phase can be acquired for the idler or signal wave through
rotation of the quasi-phase-matching (QPM) parameters in the $\chi^{(2)}$
nonlinear process using sum-frequency generation (SFG) or difference-frequency
generation (DFG). In the case of a weak idler and signal coupled by a strong
pump, the system is linear, and the geometric phase can be well predicted via
the available knowledge. However, once the intensities of the signal and idler
become comparable to that of the pump, the system becomes nonlinear, and the
geometric phase accumulated via the circular rotation of QPM parameters
deviates from the prediction at the linear limit. This deviation is a
nonmonotonic function and strongly depends on the intensities of the input
waves, which brings great uncertainty in the precise creation or manipulation
of the geometric phase in the pump-dependent process. In this paper, we study
two rotation schemes, full-wedge rotation, and half-wedge rotation, of the QPM
parameters in the process of fully nonlinear three-wave mixing. In the
full-wedge rotation scheme, the geometric phase is independent of pump
depletion due to the nonadiabatic variation of the phase factor of the QPM
parameters. In contrast, in the half-wedge rotation scheme, in which the three
QPM parameters vary adiabatically, the geometric phase linearly depends on the
pump depletion level. The results show that the geometric phase can be
accurately controlled in the nonlinear frequency conversion. Finally, a design
scheme of all-optically controlled second-harmonic beam shaping is presented.
This finding effectively suppresses the uncertainty in creating the geometric
phase in the depleted nonlinear frequency conversion process and provides an
avenue toward constant control of the geometric phase in nonlinear optics
applications and quantum information processing.
