Changing the colour of light is one of the most fundamental processes of nonlinear optics and can in principle be achieved by mixing light beams in nonlinear crystals. However, such processes are considered unrealistic in small nano-crystals due to the negligible conversion efficiency, related to their short length. Nevertheless, for more than three decades  researchers have been actively looking for ways of increasing the efficiency of nonlinear frequency conversion in ultra-thin surfaces. Plasmonic (metallic) nanostructures were considered as a possible solution, due to their strong field enhancement, however, up to now, there has been limited progress, mainly due to their dissipative losses and low mode volume. A major breakthrough in increasing the efficiency was enabled by the use of high-refractive-index resonant dielectric nanoantennas  to demonstrate third harmonic generation (THG) with an efficiency several orders of magnitude higher than what is possible in plasmonics . Even further frequency conversion enhancement can be achieved in the case of second harmonic generation (SHG) in AlGaAs nanoantennas due to their large quadratic nonlinear susceptibility, reaching conversion efficiencies of ~10−4 [4-6]. These record-higher efficiencies open a wide range of possible applications, including nonlinear imaging and holography. In this work, we review the recent progress of nonlinear frequency mixing in all-dielectric nanoantennas and metasurfaces and explain the underlying physics behind the enhancement of the nonlinear processes in high-refractive-index nano-crystals, including AlGaAs. Importantly, we demonstrate the ability to design the radiation pattern of SHG emission from the nanocrystals, to create complex beam radiation shapes with high conversion efficiency, including nonlinear images.
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