  Nonlinear wavefront control is a crucial requirement in realizing nonlinear
optical applications with metasurfaces. Numerous aspects of nonlinear frequency
conversion and wavefront control have been demonstrated for plasmonic
metasurfaces. However, several disadvantages limit their applicability in
nonlinear nanophotonics, including high dissipative loss and low optical damage
threshold. In contrast, it has been shown that metasurfaces made of high-index
dielectrics can provide strong nonlinear responses. Regardless of the recent
progress in nonlinear optical processes using all-dielectric nanostructures and
metasurfaces, much less advancement has been made in realizing a full wavefront
control directly with the generation process. Here, we demonstrate the
nonlinear wavefront control for the third-harmonic generation with a silicon
metasurface. We use a Pancharatnam-Berry phase approach to encode phase
gradients and holographic images on nanostructured silicon metasurfaces. We
experimentally demonstrate the polarization-dependent wavefront control and the
reconstruction of an encoded hologram at the third-harmonic wavelength with
high fidelity. Further, we show that holographic multiplexing is possible by
utilizing the polarization states of the third harmonic generation. Our
approach eases design and fabrication processes and paves the way to an easy to
use toolbox for nonlinear optical wavefront control with all-dielectric
metasurfaces.
