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The application of the Pancharatnam-Berry (PB) phase approach to the design of nonlinear metasurfaces has recently enabled subdiffractive phase control over the generated nonlinear fields, embedding phased array features in ultrathin structures. Here, we rigorously model, analyze, and design highly efficient nonlinear metasurfaces with advanced functionalities, including the generation of pencil beams steered in arbitrary directions in space, as well as vortex beams with polarization-dependent angular momentum, and we extend the PB approach to various nonlinear processes. To this purpose, we develop an accurate and efficient theoretical framework-inspired by the linear phase array theory-based on the effective nonlinear susceptibility method, thus avoiding the use of time-consuming numerical simulations. Our findings allowexploiting the flat nonlinear optics paradigm, enabling exciting applications based on subwavelength field control over flat and large-scale structures with giant nonlinear responsesclos

Advanced control of nonlinear beams with Pancharatnam-Berry metasurfaces