Controlling the phase of optical nonlinearity with plasmonic metasurfaces

1 Introduction

Metamaterials are artificial photonic structures that enable unconventional control of light propagation [1], [2], [3]. Electromagnetic properties of the metamaterials can be engineered through the designs of each individual meta-atom. In the last two decades, various exotic optical phenomena such as negative refraction, super-imaging, invisibility, gigantic chirality, and nontrivial topologies have been enabled by metamaterials [1], [2], [3], [4]. In particular, the two-dimensional (2D) counterpart of metamaterials, so-called metasurfaces, have attracted growing attention due to their capability of precise control over the wave front of light with relatively easy fabrication procedures [5], [6], [7], [8].

A metasurface consists of only a single layer of meta-atoms, which can surprisingly control the wave front very precisely despite its deep subwavelength thickness [5], [6], [7], [8]. Over the last few years, many different metasurface designs have been proposed for introducing spatially variant abrupt phase changes across the interface [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33]. Plasmonic nano-antennas are the most commonly used building blocks, which can strongly interact with light with their ultrathin subwavelength thicknesses. The phase of the transmitted or reflected light depends on both the geometry and orientation of the antennas. In the pioneering work by Yu et al. V-shaped antennas were employed to control the phase of cross-polarized scattered light for a linearly polarized incident beam [5]. The phase manipulation without significant variation in amplitude was enabled by carefully controlled resonances along the two orthogonal directions. Since then, various types of metasurfaces have been realized based on different operating mechanisms. Many interesting optical phenomena have been investigated using metasurfaces, such as extraordinary reflection and transmission, imaging by flat lenses, optical holography, optical spin Hall-Effect, and the generation of optical vortex beams [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]. Over the years, the efficiency of the metasurface has been dramatically improved by employing reflective designs or by introducing dielectric resonators [17], [18], [20], [22], [26], [41], [42].

Among the various types of metasurfaces, the so-called geometric phase metasurfaces have gained much popularity due to the simplicity of the phase control by rotating the orientation angle of the constituent meta-atoms of the metasurface [11], [17], [21], [22], [23], [24], [29], [30], [31]. The precise control over the phase using metasurfaces paves the way towards many practical applications. The underlying mechanisms of the phase control is spin-orbit coupling of light enabled by meta-atoms with anisotropic optical responses [11], [15], [21], [43], which converts the circular polarization state of part of the incident light into its opposite polarization state. The inversion of the circular polarization is accompanied by a geometric phase, or Pancharatnam-Berry (P-B) phase, leading to an antenna orientation controlled phase which does not depend on the specific antenna design or wavelength of light [44], [45], [46]. This interesting feature makes its performance broadband and highly robust against fabrication uncertainties.

The concept of metasurfaces has been recently extended to nonlinear optics for manipulating the generation and propagation of nonlinear optical waves [47], [48]. In the past, there has been a large body of works dedicated to the second [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74] and third harmonic generations (SHG and THG) [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87] and four-wave-mixing [88], [89], [90], [91], [92], [93], [94], [95] in plasmonic metasurfaces. Some of the works have focused on the enhancement of nonlinear conversion due to the plasmonic resonances, while some other works have paid particular attention to the effect of symmetry breaking on SHG. Some of these works were nicely summarized in review articles [96], [97], [98], [99]. In this review, we will narrow our scope down to the recent progress of nonlinear photonic metasurfaces for controlling the local nonlinearity phase for harmonic generations. First, we will introduce the concept of nonlinear selection rule for circular polarized fundamental beams. Then, a new approach for precisely controlling the nonlinear optical phase is explained and followed by a discussion of its application in nonlinear holography, imaging, and generation of vortex beams in nonlinear optical processes. Finally, we will briefly summarize and present the outlook of potential future developments of nonlinear metasurfaces.

2 Selection rule for harmonic generations

The symmetry of crystalline structure plays a very important role for harmonic generation processes in a crystal. For example, a necessary condition for even orders of harmonic generation is breaking of inversion symmetry [47], [48]. This also applies to the symmetry of artificial lattice with an engineered crystal structure, such as a plasmonic lattice [49], [50], [56], [57], [58], [59], [66], [79], [98]. Recent research has shown that the macroscopic symmetry of metasurfaces also leads to specific selection rules of nonlinear process in a similar way as their microscopic counterpart [100], [101]. As shown in Figure 1A–C, there have been extensive investigations on SHG in plasmonic metasurfaces consisting of building blocks with broken inversion symmetry, such as split ring resonators (SRRs) [49], [57], [65], [66] and L-shaped meta-atoms [50], [56]. Despite the tremendous interest in SHG in plasmonic metasurfaces, not much attention has been paid to the symmetry-governed harmonic processes for a circularly polarized fundamental wave (FW). The circular polarization state representing the spin of photons plays an important role in optics, in particular in the emerging field of spin-orbit coupling of light. A few decades ago, it was discovered that harmonic generation for circularly polarized FWs obeys a selection rule that relates the order of the harmonic process, the rotational symmetry of the crystals, and the circular polarization states of the generated harmonic signals [100], [101]. Recently it was found that the same selection rule can also be applied to plasmonic metasurfaces. This selection rule can be rigorously derived through a simple coordinate transformation process, which is detailed below.

Figure 1:

Symmetry consideration in nonlinear plasmonic metasurfaces.

(A) SHG from gold SRRs; (B) SHG from V-shaped plasmonic metasurface; (C) for plasmonic meta-atom with C3 rotational symmetry, the polarization direction of SHG wave exhibits the effect of nonlinear optical activity; (D) for plasmonic meta-atom with C4 rotational symmetry, helicity of THG is opposite to that of FW. A is adapted from Ref. [49], B from Ref. [58], C from Ref. [59], and D from Ref. [79].

Because of their ultrathin thicknesses, plasmonic meta-atoms that constitute the metasurfaces can be treated as a monolayer array of nonlinear dipole moments. Here we suppose the metasurface is in the X-Y plane. For an incident fundamental light of circular polarization σ along +z direction onto the metasurface, its electric field is expressed as E→σ=E˜0e^σ=E˜0(e^x+iσe^y)/2, where σ=±1 represents left- or right-circular polarizations (LCP or RCP). Each plasmonic meta-atom represents a local nonlinear dipole moment. As discussed in Ref. [81], for the mth harmonic generation process, the nonlinear polarization of the plasmonic element can be expressed as

where α↔m is the mth harmonic nonlinear polarizability tensor of the plasmonic structure. For an ultrathin plasmonic structure under illumination of fundamental beam at normal incidence, the excited nonlinear dipole moment is an in-plane vector, which can be always decomposed into two in-plane counterrotating dipole moments: p→mω,σ that has the same rotation direction as the incident circular polarization and p→mω,−σ that is opposite to that of the incident fundamental beam. Under a coordinate rotation of 2π/n, where n is the order of rotational symmetry of the structure, both the incident light and the nonlinear dipole moments acquire a spin-dependent phase due to the rotation spin coupling,

where E→Tσ,p→Tmω,σ, and p→Tmω,−σ are the electric field of the incident FW and the two nonlinear circular dipole moments in the new coordinate, respectively. Despite the coordinate rotation, the nonlinear polarizability α_m should remain the same because of the n-fold rotational symmetry of the structure. Hence, the nonlinear dipole moments are given by

By substituting Eq. (2) into Eq. (3), we have

By comparing Eq. (4) with Eq. (1), we can readily obtain the following selection rule for the nonlinear harmonic generation:

Equation (5) states that the allowed orders of nonlinear harmonic generation are given by m=(nl+1) for the same circular polarization as the incident wave, and m=(nl−1) for the opposite circular polarization, where l is an arbitrary integer number. Note that in a bulk crystal, the selection rule cannot be applied to the C1 and C2 rotational symmetry because the FW cannot be maintained to be circularly polarized during its propagation inside the crystal. However, this is not an issue for a 2D plasmonic metasurfaces due to the deep subwavelength interaction length.

The above selection rules in the SHG and THG have been independently observed in plasmonic metasurfaces by two groups. In Ref. [59], the SHG from a thin gold film with a hexagonal lattice of circular apertures was investigated. The lattice had a C3 rotational symmetry, i.e. n=3. Based on the selection rule, for a circularly polarized incident beam at normal incidence, the allowed orders of harmonic generation with the same circular polarization as the incident beam, given by m=(nl+1), are 1, 4, 7…, while that with the opposite circular polarization, given by m=(nl−1), are 2, 5, 8…. Hence, for SHG (m=2), only nonlinear signal with the opposite circular polarization can be generated. This was confirmed by the experimental result shown in Figure 1C. In a separate work Chen et al. [79] constructed a metasurface consisting of a square lattice array of gold nano-crosses (Figure 1D), in which both the individual building blocks and the lattice exhibit C4 rotational symmetry. Polarization-resolved THG for a circularly polarized FW was measured, which showed that only THG wave of opposite circular polarization as that of the incident wave could be observed. This observation was nicely explained by the nonlinear selection rules [59], [79], [100], [101], [102], [103].

3 Controlling the local phase of harmonic generations

Controlling the local phase of nonlinear polarizability plays a very important role in nonlinear optics as it can help compensate the phase mismatch between the fundamental and harmonic generation signals in a nonlinear crystal. This can dramatically enhance the nonlinear conversion efficiency in a bulk crystal [47], [48]. The most commonly employed method for spatially engineering the phase of the nonlinearity (particularly for SHG) is called electric poling [104], [105], [106], which involves the application of a strong static electric field to a ferroelectric crystal via patterned electrodes on the crystal surface. For a field strength above the so-called coercive field strength, domain reversal can occur, leading to a change of sign of the second order polarizability. Poling has been mainly applied periodically along a single direction for achieving the so-called quasi-phase matching condition.

Although electric poling has already been successfully employed in practical applications, it has certain limitations. Firstly, electric poling only produces binary phase states (0 and π) for the nonlinear polarizability, which may lead to undesired nonlinear optical processes. In addition, the domain size of the poled ferroelectric materials is usually much larger than the wavelength of light, which can result in multiple diffraction orders in the harmonic signal. Recently, the development of nonlinear metasurfaces provides a new platform to overcome these limitations through a continuous control over the local nonlinear polarizability down to the subwavelength scale [81].

The concept of poling for achieving a binary phase in the nonlinear material polarization has been recently applied to the design of nonlinear metasurfaces using “SRR” as the building blocks with sub-wavelength pixel size [66]. There, the mirror symmetry of the SRR was used to reverse the sign of the effective nonlinear susceptibility χ⁽²⁾. By simply flipping the orientation of the SRR, a π phase shift of the local SHG radiations can be induced (shown in Figure 2A). Using this concept, Segal et al. demonstrated SHG with well-controlled diffraction angle and achieved a focusing effect for SHG signals. However, for more complex beam manipulation in a nonlinear process a continuous and spatially variant phase manipulation is more desirable.

Figure 2:

Phase control on nonlinear metasurfaces.

(A) Nonlinear metasurface consisting of SRR meta-atoms. The arrangement of the meta-atoms in supercells with opposite orientations introduces π shift of nonlinearity phase for SHG. (B) THG from a nonlinear metasurface based on geometric P-B phase meta-atoms with C2 and C4 symmetry. The continuous phase gradient for the THG signal results in a nonlinear beam deflection. (C) Metal-quantum well hybrid metasurface consists of SRR meta-atoms exhibiting a P-B phase for SHG. A is adapted from Ref. [66], B from Ref. [81], and C from Ref. [70].

Very recently, the concept of the linear P-B phase has been extended to nonlinear harmonic generations of arbitrary orders, paving the way towards the design of nonlinear metasurfaces with continuously controllable phase of the local effective nonlinear polarizability [81]. As shown in Figure 2B and C, the metasurface consists of an array of plasmonic meta-atoms of the same shape but spatially variant orientations. Similar to the proof of nonlinear selection rule, the nonlinear P-B phases can be derived by a coordinate transformation process. For a circularly polarized FW propagating along the rotational axis of a meta-atom, the local effective nonlinear dipole moment can be expressed as p→θnω=α↔θ(E→σ)n, where α_θ is the nth harmonic nonlinear polarizability tensor of the meta-atom with orientation angle θ, and E→σ is the electric field of FW, with σ=±1 representing the LCP or RCP, respectively. Since the only parameter that distinguishes different meta-atoms is their orientation angle θ, it is natural to assume that α_θ is a function of θ. To simplify the derivation, we define a local coordinate (referred to as local frame) which is attached to each meta-atom. Hence, the local coordinate (x′, y′) axes are rotated by an angle of θ with respect to the laboratory frame (x, y). Similar to Eq. (2), the incident light acquires a spin-dependent phase due to the rotation spin coupling [81],

where the index “L” denotes the meta-atom’s local coordinate frame. Since the local coordinate is fixed, the nth harmonic nonlinear polarizability of the meta-atom’s local frame is simply α(0). Thus, the nth harmonic nonlinear dipole moment in the local frame is given by

Due to the ultrathin thickness of the meta-atoms, it is safe to assume that the nonlinear dipole moment lies in the X-Y plane, which can be further decomposed into two in-plane counter-rotating dipole moments as

We finally make a transformation from the local frame back to the laboratory frame, with the two rotating dipole moments expressed as

Hence, the nonlinear polarizabilities of the individual meta-atom are given by

Equation (10) shows that the nth nonlinear polarizability of a meta-atom contains two circular components with different angle-dependent phases; the one with the same circular polarization as the FW has a phase of (n−1)σθ, and the other with the opposite circular polarization has a phase of (n+1)σθ. However, based on the rotational symmetry of the meta-atom, one or both of the circular components of the nonlinear polarizability may not be present, according to the aforementioned nonlinear selection rules [59], [81], [100], [101], [102], [103].

The continuous phase control over the nonlinearity was recently experimentally demonstrated. In Ref. [81], phase control over THG wave was demonstrated by using metasurfaces consisting of plasmonic meta-atoms with two- and four-fold rotational symmetries (Figure 2B). For a meta-atom with two-fold rotational symmetry (C2), THG signals with both the same and opposite circular polarizations as the FW can be generated with a spin-dependent phase of 2σθ and 4σθ, respectively (Figure 3A). On the other hand, a meta-atom with three- and four-fold rotational symmetry (C4) prohibits the generation of the SHG and THG signal with the same polarization state as the incident polarization. Hence, only a single SHG and THG signal with opposite circular polarizations to that of the incident FW is generated with a geometric phase of 3σθ and 4σθ, respectively (Figure 3B and C).

Figure 3:

Nonlinear geometric P-B phase elements.

(A–C) Nonlinear phase elements with two-, three-, and four-fold rotational symmetry (C2–C4). Based on the rotational symmetry, only particular nonlinear processes and circular polarization states are allowed. (A) SHG from a C2 symmetric structure is forbidden; the P-B phases of 2θ and 4θ for the THG are observed. (B) For a C3 meta-atom, the P-B phase of SHG with opposite helicity compared to the FW is 3θ. (C) SHG from C4 meta-atoms is forbidden, and the P-B phase for the THG with opposite circular polarization is equal to 4θ. θ is the orientation angle of the meta-atom. A–C are adapted from Ref. [98].

To precisely measure the third-order nonlinearity phase, a nonlinear phase meta-grating was designed, which consisted of supercells composed of two sets of antennas of different orientation angles. As the periodicity of the supercell was greater than the third harmonic wavelength, for an incident FW, there existed at least the first diffraction orders for the THG signal. The relative intensities of the 0th and the 1st orders are determined by the nonlinearity phase difference between the two sets of antennas making up the supercell. For the two sets of gratings forming an angle of 45°, the corresponding nonlinearity phase between them was 180°, which resulted in a completely destructive interference in the 0th order.

Thus, by exploiting the rotational symmetry of the meta-atoms (such as C4 symmetry for THG and C3 symmetry for SHG), only a single circularly polarized harmonic generation signal exists with well-defined nonlinearity phase. Importantly, for rotational symmetry order greater than two, each meta-atom has isotropic electromagnetic responses in the in-plane directions, which is very similar to a circular disk. In this case, the linear optical response of each antenna does not depend on its orientation, and therefore the linear optical response is decoupled from its nonlinear optical responses. For example, by assembling meta-atoms of C3 or C4 rotational symmetries with spatially varying orientations in a 2D lattice or 3D lattice, a nonlinear metasurface or metamaterial can be formed which shows homogeneous linear properties but locally an inhomogeneous nonlinear polarizability distribution for a circularly polarized FW.

While a single harmonic signal with well-defined phase is usually preferred, multiple nonlinear signals with different phases can sometimes be useful for information multiplexing in holography. In such cases, meta-atoms with the lowest rotational symmetry, i.e. C1 symmetry, such as SRRs, can be used. For example, SRRs can generate a phase shift of σθ and 3σθ for SHG signals of different circular polarizations [70], [71] (Figure 2C). In Figure 3, by taking into account the nonlinear selection rules, we summarize the angle-dependent nonlinearity phase for different orders of harmonic generation and for nano-antennas of various rotational symmetries. The nonlinear Berry phase provides a powerful yet simple route to controlling the wavefront of nonlinear optical radiation without affecting the uniformity of the linear optical response.

4 Device applications of nonlinear metasurfaces

4.1 Nonlinear metasurface holography

In the linear optical regime, due to their superior capability of phase manipulation, metasurfaces have been used for various practical applications, such as flat lenses [11], [26] and optical holography [13], [14], [19], [20], [21], [22], [23], [24], [31], [107]. Extending the techniques of linear optical wavefront engineering to the nonlinear optical regime leads to a demonstration of various optical functionalities, such as nonlinear holography, nonlinear imaging, and nonlinear spin-orbit interaction of light. Combining the concept of multiplexed phase control for nonlinear signals of different polarization states with holography optimization techniques, multiple holographic images (Figure 4A and B) can be generated that are multiplexed by the polarization state of the FW, the nonlinear signals, or both [71], [86].

Figure 4:

Nonlinear optical holography and encryption.

(A) Nonlinear metasurface holography. The local P-B phase of SHG wave comes from the orientation of the SRR meta-atoms and circular polarization of FW. This technique allows encoding different images into one metasurface using two opposite polarization states, which correspond to different P-B phases. (B) Nonlinear holography using THG from V-shaped meta-atoms. (C) Nonlinear metasurface for information encryption. If the metasurface is illuminated with unpolarized white light, no image can be observed. Under pumping of circularly polarized FW at the wavelength of 1240 nm, an image with the characters “META” appears at the SHG wavelength (620 nm); for a direct illumination with unpolarized light at a wavelength of 620 nm, no information on the “META” characters is observed. A is adapted from Ref. [86], B from Ref. [71], and C from Ref. [73].

Recently, it was shown that a nonlinear metasurface consisting of an array of SRRs with spatially variant orientations could be used to generate three completely independent holographic images in the transmitted waves of the fundamental and the second harmonic wavelengths [71]. For a circularly polarized FW with a spin σ incident onto a SRR metasurface with an orientation angle of φ, the transmitted beam with spin −σ at the fundamental wavelength acquires a geometric P-B phase of 2σφ. Meanwhile, the SHG signals of both spins can be generated due to the C1 rotational symmetry of the SRR. Both the SHG signals with spin σ and −σ can be generated, which acquire a nonlinear Berry phase of σφ and 3σφ, respectively [71]. In this case, the linear and nonlinear geometric phases, corresponding to optical signals of different frequencies and different spins, are correlated to each other. Nevertheless, three independent holographic images can be designed by utilizing a single metasurface, which was designed using an improved Fidoc algorithm. Experimentally, three independent holographic images of the letters “R” and “L” were observed with frequency/spin combinations of (2ω, −σ) and (2ω, σ), respectively (Figure 4A). In another work, Almeida et al. [86] constructed a double-layer nonlinear metasurface holography, with each layer operating for FW with one particular linear polarization only. The cross talk between the two layers was minimized due to the orthogonality between the polarization states of light that the two layers could interact with. Experimentally, two independent images were observed at the SHG wavelength, which were multiplexed by using the linear polarization states of the FW. The polarization and/or wavelength-dependent nonlinear metasurface holography presented above may have applications in multi-dimensional optical data storages and optical encryptions.

4.3 Nonlinear metasurface lens

The first nonlinear metasurface lens in the form of a Fresnel zone plate was demonstrated by Segal et al. [66]. The nonlinear zone plate consisted of annular zones, with adjacent zones composed of SRRs of opposite directions, leading to binary nonlinear phases of 0 and π. This was a direct analogue of poling along an in-plane direction. For an incident Gaussian FW, focused SHG signal in the transmission direction was observed. A focal spot of 7 μm with an enhancement of 73-fold in intensity relative to the SHG at the metasurface was observed. Nonlinear metasurface lens was also demonstrated for four-wave mixing (FWM) by Almeida et al. [94]. The metasurface consisted of an array of rectangular apertures in a gold film. The phase of the nonlinear signal generated by each aperture was controlled by its width and length. Based on this, a series of nonlinear metasurfaces were designed and fabricated, which focused the FWM signal at different focal distances ranging from 5 to 30 μm.

In the above demonstrations, the quality of focusing was limited by either the binary phase or inaccurate control over the nonlinearity phase. Very recently, nonlinear P-B phase was employed for the design of high quality nonlinear metasurface lenses that were capable of focusing generated SHG signal into a diffraction-limited spot. As shown by the scanning electron microsopy image in Figure 5A, the nonlinear metalens consisted of an array of meta-atoms of C3 rotational symmetry, with the orientation angle of each meta-atom satisfying

Figure 5:

Metasurface for nonlinear imaging and wavefront engineering.

(A) Working principle of the nonlinear metalens. Illustration of the imaging concept: An L-shaped aperture is imaged on a screen with the help of the nonlinear metalens consisting of C3 nanoantennas (right). (B) SHG beam with OAM was generated using SRR metasurface with fork-type singularity. (C) Schematic of spin controlled generation of OAM of SHG by using nonlinear photonic metasurface. For an FW with two spin states (LCP and RCP) normally incident onto the nonlinear photonic metasurface with C3 symmetry, generation of SHG waves with opposite handiness to that of FW is allowed. By encoding the phase singularity into the metasurfaces, the SHG vortex beams with spin controlled topological charges are obtained. A is adapted from Ref. [108], B from Ref. [72], and C from Ref. [73], [74].

(11)θ(r)= k0(f2+r2−|f|)/3

where r is the distance from the axis of the metalens, f is the focal length of the lens, and k0=2πλ is the free-space wave vector of the SHG wave. The nonlinear metalens had a diameter of 300 μm, and the C3 nano-antennas were positioned on concentric rings with a spacing of 500 nm. For a metalens designed at f=500 μm, the measured SHG intensity distribution at different transverse planes in the focal region is shown in Figure 5A, where a diffraction-limited spot is observed. In the experiment, an L-shaped aperture carved into a metal film served as the object, which was positioned at a certain distance in front of the nonlinear metalens. A circularly polarized FW was used to illuminate the “L” aperture. The transmitted wave interacted with the nonlinear metalens, where the SHG waves were generated and imaged in the transmission direction due to the nonlinearity phase distribution across the metalens (Figure 5A). More interestingly, due to the spin-dependent nature of the nonlinear P-B phase, the nonlinear metalens could function either as a convex lens or a concave lens, resulting in formation of real or virtual images depending on the circular polarization states of the FW.

5 Conclusions/perspective

In this review, we presented the recent development of nonlinear metasurfaces for locally controlling the nonlinearity phase of the harmonic generations. We have focused on harmonic generations with circularly incident FW, where the rotational symmetry of the meta-atoms plays a very important role. We first discussed the nonlinear selection rule that dictates the allowable harmonic generation orders for a particular rotational symmetry of the meta-atom. We then extended the concept of P-B phase from linear optics to nonlinear optics, which features a simple relationship between the orientation angle of the meta-atom and the local nonlinearity phase. We discussed a number of applications with nonlinear metasurfaces, such as nonlinear holography, vortex beam generation, and nonlinear metalenses. Nonlinear P-B phase provides us unprecedented level of control over the wavefront of the harmonic waves. However, the conversion efficiency of nonlinear metasurfaces is still very low. Therefore, new nonlinear materials and systems, such as semiconductors with intersubband transition, need to be exploited in the future to enhance the efficiency of nonlinear optical conversions. In addition, by extending the concept of P-B phase from 2D metasurfaces to 3D metamaterials, the local control over the nonlinear optical phase may help design novel optical devices satisfying the phase matching condition for harmonic generations. Then it is possible to develop highly efficient nonlinear photonic devices, which have great potential for applications in optical sensing, imaging, and communications. Moreover, the study of the spin-orbit interaction with nonlinear optical metasurface provides a versatile platform for high-dimensional classical and quantum optical communications. With the further development of the functionalities and efficiencies of nonlinear metasurfaces, we would like to expect an increasing amount of applications and new research fields based on the platform of nonlinear optical metasurface devices.