Optical metasurfaces are quasi-two-dimensional artificial materials that consist of subwavelength-scale meta-atoms. Thanks to the ultrathin footprints and versatile design degrees of freedom, a variety of metasurfaces have been designed and implemented to achieve novel optical devices or applications such as metalenses, meta-holograms, polarizers, waveplates, spin-to-orbit angular momentum converters, image encryption and polarimeters. By choosing the material constituents and geometries of the meta-atoms, one can easily manipulate the degrees of freedom of light fields, such as amplitude, polarization, phase, and frequency. The ability to exploit frequency as an additional channel relies on nonlinear optical processes, which involve the generation of nonlinear waves at new frequencies. Previous studies in nonlinear optics mainly focus on improving the conversion efficiencies of nonlinear processes, and the manipulation of the generated nonlinear waves is usually realized by linear optical elements. One of the most prominent advantages of nonlinear photonic metasurfaces is their capability to manipulate nonlinear waves while generating them, and therefore people can greatly shrink the devices into a more compact form.

The phase matching condition is of critical importance in traditional nonlinear optical processes based on photonic crystals (Table 1). The quasi-phase matching technique is proposed to improve conversion efficiency when the rigorous phase matching condition is not met (Fig. 1). Nonlinear photonic crystals are a class of artificially engineered structures that can be modulated spatially. They are capable of fulfilling the phase matching condition and realizing nonlinear wavefront shaping simultaneously. As for metasurfaces, because of the subwavelength-scale feature size, the phase matching condition is less rigorous than that in conventional nonlinear crystals. There are many materials and mechanisms that can be chosen to enhance nonlinear responses and to enrich the functionalities of nonlinear photonic metasurfaces (Fig. 2). With the rapid development of nonlinear metasurfaces in recent years, it is time to review the progress in the area. This paper discusses the fundamentals of the effects of symmetries and geometric phases on the nonlinear responses of the metasurfaces and the applications in nonlinear wavefront shaping, quantum information processing, and terahertz wave generation and manipulation based on nonlinear metasurfaces.

The important roles of symmetries and geometric phases in nonlinear photonic metasurfaces are first discussed. While the symmetries of the meta-atoms can decide the allowed and forbidden nonlinear processes, they can also affect the chiral optical responses of the metasurfaces (Fig. 3). The nonlinear geometric phase is dependent on the order of the harmonic generations, the circular polarizations of the fundamental and nonlinear waves, and the spatial orientations of the meta-atoms (Fig. 4). It provides a convenient route to continuously control the phase imparted into the nonlinear waves (Table 2), which underpins the multi-dimensional nonlinear wavefront shaping by metasurfaces.

The applications based on nonlinear metasurfaces are then discussed. The direct applications of nonlinear metasurfaces are wavefront shaping devices (Fig. 5). With the ability to control the phases in nonlinear optical processes such as second harmonic generation, third harmonic generation, sum frequency generation, difference frequency generation, and four-wave mixing, the nonlinear metasurfaces have enabled nonlinear wavefront shaping like focusing, imaging, beam steering, vortex beam generation, holography, and image encryption. By exploiting the quantum entanglement characteristics of spontaneous down conversion processes, one can also use metasurfaces to generate high-dimensional entangled photons (Fig. 6). Several applications such as high-dimensional spatially entangled photon pairs and orbital angular momentum-carrying entangled photon pairs based on plasmonic and dielectric metasurfaces have been experimentally demonstrated. The nonlinear metasurfaces can be used for terahertz wave generation and manipulation as well. Terahertz waves possess unique advantages in applications such as nondestructive measurements and communications, but the development of terahertz technology is impeded by the lack of terahertz sources, detectors, and elements. Nonlinear metasurfaces represent a novel platform for simultaneously generating and manipulating terahertz waves. The concept of geometric phase has been successfully applied to the terahertz wave generation process (Fig. 7), which may lead to more functional devices in the terahertz spectral region.

To push forward the practical applications of nonlinear photonic metasurfaces, the key issue is to improve nonlinear conversion efficiency. All-dielectric metasurfaces can avoid the thermal heating effect that leads to the breakdown of the nanostructures in plasmonic metasurfaces and operate at a high pumping intensity to achieve high conversion efficiency. However, the nonlinear phase control ability of dielectric metasurfaces is very sensitive to the geometries of the nanostructures, which poses challenges to nanofabrication. The nonlinear geometric phases demonstrated on plasmonic metasurfaces provide an elegant way to manipulate the phase of nonlinear optical waves, which may be applied to more material systems. Moreover, the hybrid system of linear metasurfaces combined with traditional nonlinear crystals can provide a route to achieve highly efficient nonlinear wavefront engineering. Novel materials like new crystals or physical mechanisms such as electric field-induced second harmonic generation may also be exploited to improve the efficiency of second harmonic generation.