All-dielectric nonlinear metasurface: from visible to vacuum ultraviolet

The research content of nonlinear optics is the nonlinear interaction of light and matter with significant intensity¹. Phase mismatch limits the nonlinear conversion efficiency of traditional nonlinear optical crystals. In order to obtain efficient nonlinear optical devices, the research of traditional nonlinear optics is mainly focused on phase matching methods, including birefringence phase matching technology and quasi-phase matching technology^2,3,4,5. However, these two methods have certain limitations. The birefringence phase matching method has a limited adjustment range due to the high requirements on the line width and angle adjustment accuracy of the pump beam. The quasi-phase matching method requires periodic reversal of the crystal orientation of the nonlinear crystal, which makes crystal processing more difficult.

A metasurface refers to a nano-optical antenna array composed of subwavelength structures^6,7,8,9,10, and the scale of its interaction with light is in the nanoscale^11,12, which solves the problem of nonlinear efficiency reduction caused by phase mismatch. Therefore, nonlinear metasurfaces have gradually attracted researchers’ attention. Furthermore, compared with traditional nonlinear optical crystals, the performance of nonlinear metasurfaces is also related to the resonant response of optical antennas. It means that the reasonable selection of metasurface materials and the optimization of antenna structures can customize the multi-resonance effect of metasurfaces and enhance local light fields at a subwavelength spatial scale. The local field enhancement will promote the interaction of light and matter and achieve an enormous nonlinear response. According to the classification of materials, nonlinear metasurfaces can be divided into plasmonic metasurfaces^13,14,15,16 and all-dielectric metasurfaces^{17,18,19,20,21,22}. Plasmonic metasurfaces use surface plasmon resonance to enhance the local light field, producing large nonlinearities at the interface. However, the low damage threshold of metals and the small overlap volume of light field and antennas limit the further improvement of nonlinear conversion efficiency. Therefore, many research groups have studied all-dielectric nonlinear metasurfaces, such as lithium niobate (LiNbO₃), gallium arsenide (GaAs), silicon (Si), zinc oxide (ZnO), and titanium dioxide (TiO₂), etc. Unlike plasmonic metasurfaces, the resonance supported by all-dielectric metasurfaces can confine the light field inside the optical antenna, and the light field and antennas have larger overlap volumes, which can achieve more efficient nonlinear frequency conversion^21,23,24. Besides, it is easy for all-dielectric metasurfaces to control the phase, amplitude, and diffraction pattern of nonlinear light at a subwavelength spatial scale, thereby realizing more complex metasurface functions^20,25,26.

Nonlinear metasurfaces can generate light of new frequencies through frequency doubling, summing, or four-wave mixing (FWM) of pump light. In the beginning, all-dielectric nonlinear metasurfaces were mainly designed to generate harmonics in the visible, while ultraviolet light with a shorter wavelength has a wide range of applications in basic science and industry, including nanolithography, super-resolution imaging, disinfection, photochemistry, and spectroscopy, etc. Especially the ultraviolet light source is an important component of the lithography system, and the resolution of lithography depends on the wavelength of the light source^27,28. The Abbe limit shows that the smaller the wavelength, the higher the spatial resolution. Ultraviolet light has high photon energy, and its radiation can cause ionization and breakage of chemical bonds. Therefore, ultraviolet light can inactivate bacteria and viruses, which makes ultraviolet light widely used in air purification and wastewater treatment^29,30,31. In addition, ultraviolet light has essential applications in the characterization of materials. Near-ultraviolet circular dichroism spectroscopy can be used to characterize protein structure, and deep ultraviolet (DUV) circular dichroism spectroscopy can assist in the development of chiral drugs and assess their quality^32,33. Therefore, many researchers have selected more suitable materials and optimized the geometric structure of all-dielectric metasurfaces to generate nonlinear harmonics with shorter wavelengths and improve the nonlinear efficiency. Through various nonlinear modulation methods, the harmonics in the ultraviolet band can also be used to realize the complex functions of the counterpart in the visible band, such as nonlinear focusing^20,34,35, nonlinear imaging³⁴, nonlinear vortex generation^36,37, nonlinear holography²⁵, nonlinear chirality^38,39,40, and information encryption^26,41 in the ultraviolet band.

In this review, we discuss the latest progress of all-dielectric nonlinear metasurfaces, from visible light to vacuum ultraviolet (VUV). The structure of this review is shown in Fig. 1. In the “Multipolar resonances enhanced nonlinear polarization” section, we show the theoretical framework for studying the nonlinear all-dielectric metasurfaces, including multipole analysis and nonlinear polarization theory. In the “Nonlinear efficiency enhancement” section, we discuss the contribution of various resonant modes to the enhancement of nonlinear efficiency of all-dielectric metasurfaces. In the “Nonlinear modulation” section, we introduce different nonlinear modulation strategies, which can further improve the collection efficiency of harmonics or realize some specific functions. In the “Summary and prospects” section, we summarize and discuss potential research directions.

The nonlinear materials that generate harmonics from visible to ultraviolet bands include LiNbO₃, GaAs, Si, ZnO, TiO₂, etc. Under the irradiation of strong pump light, these materials undergo nonlinear polarization and generate nonlinear harmonics. However, the nonlinear efficiency of unpatterned thin films is low, and resonance needs to be introduced to enhance the local field intensity and promote nonlinear conversion. Due to the flexibility of metasurface design, researchers can manipulate certain characteristics of nonlinear light to achieve specific functions, such as radiation shaping, which can improve the nonlinear collection efficiency of light; polarization, amplitude, and phase modulation can achieve nonlinear imaging, nonlinear vortex light generation, nonlinear multi-path holography, etc. The images of dielectric materials (from left to right) are reproduced with permission^{17,18,19,20,21}. Copyright 2020, American Chemical Society, Copyright 2018, American Chemical Society, Copyright 2014, American Chemical Society, Copyright 2022, American Chemical Society, Copyright 2019, American Chemical Society. The images of “Efficiency enhancement” (from left to right) are reproduced with permission^21,23,24. Copyright 2015, American Chemical Society, Copyright 2024, American Chemical Society, Copyright 2019, American Chemical Society. The images of “Nonlinear modulation” (from left to right) are reproduced with permission^20,25,26. Copyright 2018, American Chemical Society, Copyright 2019, American Chemical Society, Copyright 2022, American Association for the Advancement of Science (AAAS).

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All-dielectric metasurfaces use nanoantennas with different shapes and permittivities to support different resonant modes. The theory of electromagnetic resonance multipolar expansion is an essential means to study their resonant modes. The local field enhancement generated by resonance can enhance the nonlinear polarization of the dielectric antenna. Any system’s optical response mode can be divided into three categories: electric mode, magnetic mode, and toroidal mode^42,43,44,45. The response of light can be regarded as the joint contribution of infinite-order modes, and the contribution of high-order modes is small and can even be ignored in many cases.

Multipolar expansion

The external electromagnetic field will stimulate the current density ({boldsymbol{j}}) or charge density ({boldsymbol{rho }}) inside dielectric nanoantennas, and the far-field radiation can be regarded as the sum of the radiation generated by different multipoles^46,47. In Cartesian coordinates, the multipole response of a nanoantenna can be described by Eqs. (1)–(6):

where ({boldsymbol{P}}) represents the electric dipole (ED) moment, ({boldsymbol{M}}) represents the magnetic dipole (MD) moment, ({boldsymbol{T}}) represents the toroidal dipole (TD) moment, ({Q}_{alpha ,beta }^{(E)}) represents the electric quadrupole (EQ) moment, ({Q}_{alpha ,beta }^{(M)}) represents the magnetic quadrupole (MQ) moment, ({Q}_{alpha ,beta }^{(T)}) represents the toroidal quadrupole (TQ) moment, (c) is the speed of light in vacuum, (omega) is the angular frequency, and ({boldsymbol{r}}) is the displacement vector from the origin to any point in space. ({alpha }^{{prime} },,{beta }^{{prime} }equiv x,y,z) represent the Cartesian coordinate components. After the multipole moments have been obtained, the total scattering intensity to the far field can be calculated using Eq. (7):

where the first term signifies the scattering energy of the ED in the far field, the second term is the scattering energy of the MD, the third term represents the interference between the ED and the TD, the fourth term means the scattering energy of the TD, and the fifth and sixth terms are the scattering energies of the EQ and the MQ, respectively. (o(frac{1}{{c}^{5}})) means the contribution of higher-order multipoles, such as TQs and electric and magnetic octopoles, which is very small. Among them, ({{boldsymbol{P}}}^{{boldsymbol{* }}}) represents the complex conjugate of the ED ({boldsymbol{P}}). The ED, MD, EQ, MQ, TD, and TQ are shown in Fig. 2 “Multipolar expansion”. According to the above equations, it can be seen that the scattering energy of each multipole can be calculated based on the charge and current intensity of each multipole generated by the incident wave. Then, the scattering energy of each multipole is analyzed and compared, and the contribution of each multipole at the resonance can be known.

Multipolar expansion is the basis for analyzing the resonant mode of nanoresonators. The resonant mode of a resonator has at least one polar resonance mode at work. Different polar resonances of a resonator or resonances of different nanoresonators can interact to generate resonant modes with higher Q values, such as BIC and Fano resonances, to enhance nonlinear polarization in nonlinear processes including SHG, SFG, DFG, THG, FWM, etc. The images of “Multipolar expansion” are reproduced with the permission⁴⁴. Copyright 2014, American Physical Society. The images of “Multipolar interaction” are reproduced with permission^48,51,56,61. Copyright 2014, Springer Nature, Copyright 2015, John Wiley and Sons, Copyright 2014, Springer Nature, Copyright 2020, AAAS.

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Multipolar interaction

The resonant mode of a nanoparticle can be analyzed by multipolar expansion, which can obtain the information of the dominant electromagnetic multipole. However, in many cases, multiple electromagnetic pole resonance modes coexist, and they will interact with each other to form a new field enhancement mode. Generally speaking, the TD and the ED have similar far-field radiation distribution characteristics. Therefore, when the two dipole moments satisfy ({boldsymbol{P}}={boldsymbol{-}}{ik}{boldsymbol{T}}), the destructive interference occurs and the radiation-free anapole mode is generated^43,48,49,50. The radiation-free mode has a strong local field enhancement effect in the near field. Similarly, the Kerker effect^{51,52,53,54,55} is also generated by the interaction of multipoles: the ED and the MD of the same order show different phase symmetry in the forward and the backward radiation. For example, the ED radiation may show even parity, while the MD radiation shows odd parity. Therefore, the forward radiations of the electric field are in phase, and the backward radiations are out of phase. When the ED and MD modes overlap in space, the backscattering is suppressed, and the directionality of the forward scattering is enhanced. The anapole mode and the Kerker effect are also shown in Fig. 2.

In addition to the interaction between multipoles, the coupling between resonant modes will also occur. The Fano resonance arises from the coherent interference between a discrete quantum state and a continuum band of state^18,56. As shown in Fig. 2, the ED (bright mode), supported by the strip resonator, and the MD (dark mode), supported by the ring resonator, interfere with each other to form a Fano resonant system. The resonant mode’s quality factor (Q factor) is used to analyze the interaction strength between light and matter^57,58. Therefore, an optical quasi-bound state in the continuum (q-BIC) with high-Q factors has become an important research point^59,60. A single nanostructure can support different resonant modes at different wavelengths. However, if the geometric parameters of the nanostructure are adjusted, the resonant wavelengths change. When the two resonant wavelengths are close enough, a strong interaction will occur between the two resonant modes. The strong interaction will form the q-BIC and significantly increase the Q factor of the resonance^61,62. The Fano resonance and the q-BIC effect are shown in the “Multipolar interaction” of Fig. 2.

Dielectric polarization

Due to resonance effects such as electromagnetic multipoles, the local electromagnetic field of the dielectric metasurface is enhanced. Under strong light interacting with the matter, the relationship between the electric polarization vector and the electric field vector is nonlinear. The polarization intensity induced by the medium can be expanded into a power series of the intensity of the electric field⁶³:

where ({chi }^{({rm{n}})}) is the n-order electric susceptibility (n = 1, 2, 3, …), which is a tensor with the order of n + 1 and has a total of 3ⁿ⁺¹ tensor elements, and ({boldsymbol{E}}) is the intensity of the local electric field. The expressions of the linear polarization tensor and the second-order polarization tensor are shown in Eqs. (9) and (10),

The polarization intensity P includes two parts: linear and nonlinear components. The nonlinear light is related to nonlinear polarization intensity P_NL, and its expression is shown in Eq. (11):

Figure 2 “Nonlinear polarization” describes several nonlinear effects, including second harmonic generation (SHG)^64,65, sum frequency generation (SFG)^66,67, difference frequency generation (DFG)⁶⁸, third harmonic generation (THG)^69,70 and FWM^71,72. The corresponding nonlinear polarization intensities are as shown in Eqs. (12)–(17):

Combined with Maxwell’s equations, the intensity of the electric field of the nonlinear light propagating in the dielectric medium can be solved by the nonlinear polarization vector, as shown in Eq. (18):

where ({boldsymbol{k}}) and ({{boldsymbol{k}}}^{{boldsymbol{{prime} }}}) are the wave vectors of the electric field and polarization field, respectively. The time domain wave equation in nonlinear media is shown in Eq. (19):

where ({boldsymbol{r}}) is the position vector.

Equations (12)–(18) show that a large nonlinear electric susceptibility is also required to induce strong nonlinear polarization besides the strong local electromagnetic field. Furthermore, the absorption of nonlinear metasurface at the pump wavelength will affect the distribution of the electromagnetic field, decrease the Q factor and hinder the enhancement of nonlinear efficiency. The absorption at the nonlinear wavelength will directly reduce the nonlinear harmonic generation. They are properties of the nonlinear material, so the nonlinear metasurface selection is also crucial. The even-order harmonic generation, such as SHG, requires materials with non-centrosymmetric structures, such as GaAs⁷³, gallium aluminum arsenide (AlGaAs)^74,75, gallium phosphide (GaP)⁷⁶, and LiNbO₃¹⁷. However, GaAs and AlGaAs have significant absorption in the visible band, affecting harmonics’ conversion efficiency⁷⁷. GaP has a larger band gap, which helps to solve the problem^76,78. LiNbO3 also has a large band gap, but it is more difficult to fabricate. In addition to these single-crystal materials, two polycrystalline materials, barium titanate (BaTiO₃) and ZnO, are also used to generate second harmonics. The SHG efficiency of polycrystalline materials does not directly correspond to the components of the second-order polarizability tensor but to the average second-order polarizability, which leads to a certain degree of reduction in SHG emission. However, by rationally designing the structure of the microresonator, a stronger nonlinear efficiency can also be obtained^79,80. The odd harmonic generation, such as THG, is not limited by the crystal structure centrosymmetry. Commonly used materials for THG in the VIS-UV band are Si⁸¹, germanium (Ge)⁸², molybdenum disulfide (MoS₂)⁸³, arsenic sulfide (As₂S₃)⁸⁴, and TiO₂²¹. Semiconductor nonlinear materials have large nonlinear polarizability⁸⁵ and fast response speed, but the absorption loss of the material is relatively high. MoS₂⁸³ maintains a high nonlinear polarizability over a wider band, but temperature significantly affects its performance. Arsenic trisulfide and titanium dioxide have relatively low nonlinear polarizabilities^86,87, but their band gaps are relatively high⁸⁸, and their absorption in the visible band is relatively weaker. Table 1 summarizes the materials, corresponding nonlinear processes, and linear and nonlinear characteristics of all-dielectric nonlinear metasurfaces.

Spontaneous parametric down-conversion (SPDC), high-order harmonic generation (HHG), THG, and SHG are the most widely studied nonlinear processes in nonlinear all-dielectric metasurfaces. One of the most critical metrics that needs to be considered is nonlinear efficiency. The harmonic wavelengths generated by these nonlinear processes cover the band from visible to ultraviolet light. However, the four mentioned nonlinear processes’ nonlinear efficiency is low for bare dielectric film. An efficient strategy is incorporating resonances, such as Mie, Fano, and BIC resonance, into nonlinear metasurfaces to enhance the nonlinear efficiency.

SPDC^89,90,91 is also a second-order nonlinear process that can convert pump photons into a pair of entangled signal and idle photons with lower frequencies. It can generate nonlinear light with wavelengths longer than the pump wavelength. The introduction of the above resonances can greatly increase the generation rate of spontaneous and entangled photon pairs. Therefore, all-dielectric nonlinear metasurfaces are promising for making quantum light sources, which is crucial for obtaining high-efficiency and high-dimensional quantum optical meta-devices^92,93,94,95. Santiago-Cruz et al.⁹⁶ prepared a LiNbO₃ metasurface that supports Mie resonance to enhance the SPDC to generate two-photon pairs. The two-photon pair generation rate is 5.4 Hz, 20 times higher than unpatterned LiNbO₃ films. Among the multipolar resonances discussed in the “Multipolar resonances enhanced nonlinear polarization” section, q-BIC modes consistently exhibit extremely high-Q factors. This characteristic allows them to confine light for extended durations, leading to significant field enhancements. Santiago-Cruz et al.⁹⁷ also utilized GaAs nonlinear metasurfaces operating at ~720 nm to excite the SPDC. By breaking the symmetry of the metasurfaces, both ED-q-BIC and MD-q-BIC resonances were simultaneously induced. As illustrated in Fig. 3a, these q-BIC resonances can generate degenerate or degenerate photon pairs over a wide spectral range, with generation efficiency three orders of magnitude higher than unpatterned GaAs thin films. The high entangling of photon pairs is also a widely concerned issue in the quantum field^98,99. Nonlinear metasurfaces can combine quantum state manipulation with generating many correlated quantum pairs to construct the required quantum states. Zhang et al. demonstrated the potential of nonlinear metasurfaces to generate spatially entangled photon pairs. They covered a layer of silicon grating on a LiNbO₃ film. This structure can excite guided-mode resonance, which forms a BIC under the protection of rotational symmetry. When the incident light is oblique, the rotational symmetry is destroyed, and a rotational symmetry-protected BIC can be generated, forming two high-Q resonances on both sides of the Γ point (λ = 1570 nm). Topological frequency splitting makes the idler and signal photon pairs highly entangled in space to meet the lateral phase matching condition. Compared with the unpatterned film, this metasurface’s entangled photon generation rate is increased by 450 times¹⁰⁰.

a GaAs metasurface generating complex quantum states based on BIC. b The subwavelength AlGaAs resonator enhancing high-order harmonics in the visible band using BIC. c Silicon metasurface enhancing the visible-band third-order harmonic based on the Fano resonance. d The ZnO metasurface enhancing VUV generation based on magnetic resonance. a Reproduced with permission⁹⁷. Copyright 2022, AAAS. b Reproduced with permission¹⁰⁶. Copyright 2016, American Chemical Society. c Reproduced with permission¹¹². Copyright 2023, AAAS. d Reproduced with permission⁸⁰. Copyright 2019, American Chemical Society.

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Complementary to the process that converts short-wavelength photons into long-wavelength photons, generating short-wavelength light is a crucial focus of this review, which can be achieved through harmonic generation^{101,102,103,104}. The higher the harmonic order, the shorter of the wavelength of the harmonic. Therefore, many scientists utilized the HHG to obtain the nonlinear harmonic with a shorter wavelength. For instance, Liu et al.¹⁰⁵ also used a silicon metasurface that supports high-Q Fano resonance to enhance the generation of high-order harmonics. At an excitation intensity of 0.071 TW cm⁻², they detected high-order harmonic signals of 5–11 orders, among which the 5-order harmonic (λ = 464 nm) signal intensity was enhanced by more than 30 times compared with that of unstructured silicon films. BIC can also greatly enhance the generation of high-order harmonics. Zalogina et al.¹⁰⁶ utilized AlGaAs metasurfaces to enhance the high-harmonic generation process^106,107,108 using Friedrich-Wintgen BICs^62,109,110 at an incident wavelength of 3.75 μm. Their theoretical analysis, shown in Fig. 3b, demonstrates the generation of a 9-order harmonic wave around 420 nm, and the intensity of the 9-order harmonic is by 6 orders of magnitude larger than that of the unstructured sample. To generate high-order harmonics with smaller wavelengths, J. K. An added a reflective substrate at the bottom of the ZnO disk metasurface. The reflective substrate can enhance the Mie resonance of the metasurface. Simulation results show that the metasurface generates strong 9-order harmonics (172 nm) at a resonant wavelength of 1550 nm, and the nonlinear efficiency is 10 orders of magnitude higher than that of ZnO thin film¹¹¹. However, it should be acknowledged that generating short wavelengths through high-harmonic processes may not be the most optimal approach due to the rapidly declining intensity of high-harmonic waves.

The third-order and second-order harmonics have larger intensity than that of high-order harmonics. To enhance the harmonic intensity harmonic in the ultraviolet (UV) region, scientists researched on THG or SHG using the pumping light with the wavelength in the visible band. Shorokhov et al. employed a quadrumer metasurface composed of hydrogenated amorphous silicon (a-Si: H) clusters to enhance THG via high-quality collective modes associated with magnetic Fano resonance, as depicted in Fig. 3c. This approach resulted in a third harmonic wave at 450 nm, achieving up to a 100-fold enhancement compared to a bare a-Si: H film¹¹². To solve the problem of Q value decrease caused by strong mode leakage, Kim added a layer of reflective substrate to the ZnO metasurface. The microunit consists of a pair of asymmetrically arranged cuboids, and the metasurface supports q-BIC with a Q value of 243 under the condition of high refractive index contrast between the substrate and microunits. The pump light with a wavelength of 1060 nm can generate the third harmonic of 353 nm with a nonlinear efficiency of 0.11%¹¹³. Abdelraouf et al.²⁴ designed the silicon metasurface with trapezoidal nanoresonators. By introducing symmetry defects, a q-BIC with a Q factor of 180 can be formed at a wavelength of 840 nm. Experimental measurements show that when the pump light power is 15 GW cm⁻², the third harmonic power at a wavelength of 280 nm is 14.5 nW, achieving the highest nonlinear harmonic power in the DUV band. Their nonlinear enhancement factor is 48 times that of the other method compared. Semmlinger et al. designed a TiO₂ metasurface that supports anapole resonance at the wavelength of 555 nm to enhance the generation of the third harmonic (λ_THG = 185 nm). Compared with unpatterned TiO₂, the generation efficiency of the third harmonic is increased by 180 times²¹.

For SHG, the low-refractive-index nonlinear material used for harmonic excitation in the ultraviolet band will also lead to leakage of the enhancement mode to the substrate. Therefore, Kim placed the reflective substrate between the silica substrate and the dielectric array, which further enhanced the Mie resonance of the array and generated a 400 nm second-order harmonic with a nonlinear enhancement of 280¹¹⁴. Besides the reflective layer, the group added a LiNbO₃ film between the LiNbO₃ array and the silica substrate. The far field can excite guided-mode resonance in the LiNbO₃ film, thereby obtaining the q-BIC. The second harmonic wavelength is 387.3 nm, and the nonlinear efficiency of transmission and reflection is 0.14% and 0.095%, respectively, higher than the above metasurface with reflective layer¹¹⁵. Semmlinger et al.⁸⁰ proposed a ZnO-based metasurface, as shown in Fig. 3d, which exhibits an MD resonance at a wavelength of 394 nm. This setup can generate VUV light at 197 nm by leveraging the second-order nonlinearity of ZnO material. The corresponding SHG efficiency is nearly a thousand times higher than that of a bare ZnO film, and this efficiency can be further enhanced by increasing the incident angle.

The resonance used to enhance the nonlinear generation, the nonlinear process, the metasurface material, and the enhancement factor are summarized in Table 2.

Various nonlinear manipulation methods have emerged because of the flexibility of the nonlinear metasurface design. In addition to introducing the resonant modes mentioned above, the diffraction pattern modulation strategy can also achieve nonlinear efficiency enhancement. The nonlinear light generated by nonlinear metasurfaces will diffract to different diffraction orders^116,117, so the energy is relatively dispersed. The modulation of the nonlinear radiation pattern can make the nonlinear harmonic radiation of the metasurface directional, which is helpful for the collection of nonlinear light energy. Ghirardini et al. introduced an in-plane asymmetric holographic grating to redirect the second harmonic (λ_SHG = 785 nm) propagating in directions other than the normal direction of the AlGaAs cylindrical metasurface to the normal direction¹¹⁸. The research group also used an axially asymmetric AlGaAs nanostructure-chair resonator to excite a nonlinear mode with a normal radiation direction¹¹⁹. The interaction between the resonances at the fundamental (λ_FF = 1550 nm) and the harmonic (λ_SHG = 775 nm) frequencies makes the asymmetric resonator have a relatively stronger central emission lobe than the cylindrical resonator, as shown in Fig. 4a. The research on nonlinear diffraction pattern modulation is concentrated mainly in the visible light band, using the laser with a wavelength of about 1550 nm as the pump light to generate the second harmonic or the third harmonic, so the harmonic wavelength is between 500 and 800 nm^{120,121,122,123,124}.

a The nano-chair metasurface modulating the nonlinear radiation pattern. b The silicon metasurface able to achieve ultra-fast nonlinear modulation. c The metasurface able to achieve asymmetric transmission of nonlinear light based on amplitude modulation. d The nonlinear meta-lens generating focused VUV light based on geometric phase modulation. a Reproduced with permission¹¹⁹. Copyright 2021, Optical Society of America. b Reproduced with permission¹²⁷. Copyright 2015, American Chemical Society. c Reproduced with permission¹³². Copyright 2022, Springer Nature. d Reproduced with permission²⁰. Copyright 2019, AAAS.

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Nonlinear modulation also includes the ultrafast modulation of the light in the time domain utilizing several nonlinear effects, such as free carrier generation, thermal effect, and Kerr effect. For all-dielectric metasurface constructed by silicon, the additional free-carrier effect caused by two-photon absorption can lead to electron-hole pair recombination rates of hundreds of picoseconds to several nanoseconds, which seriously limits the optical modulation speed^125,126. Shcherbakov et al.¹²⁷ introduced the MD resonance to suppress the carrier effect significantly. They achieved a transmittance modulation time of 65 fs for light with a wavelength of 780 nm, as shown in Fig. 4b. However, the limitation of the indirect electronic band gap of the silicon greatly hinders the improvement of the maximum modulation depth. The research group used GaAs, a material with a direct electronic band gap, to make a metasurface to overcome the problem. The nanoantenna supported an MD resonance at 800 nm and achieved a reflectivity modulation time of 350 fs¹²⁸. The research on ultrafast optical control using all-dielectric metasurfaces is mainly focused on the visible and near-infrared bands^129,130, and there are few studies on ultrafast optical modulation in the shorter wavelength band.

The nonlinear chiral metasurface can be obtained by destroying spatial mirror symmetries of nanoparticles. It can generate different responses to circularly polarized light in two spin states, thereby realizing polarization modulation of nonlinear harmonics. It is significant for nonlinear chiral sensing and nonlinear metasurface holography. Kang et al. designed a nonlinear metasurface based on the optical Kerr effect by adding asymmetric grooves on both sides of Si strips of the Si grating, so it supports guided-mode resonance to enhance the Kerr effect for an incident wavelength of 1500 nm. The metasurface has different transmittance for left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) light. Therefore, the polarization state of the transmitted light can be actively modulated by changing the incident intensity³⁸. Moreover, nonlinear metasurfaces can also show polarization selectivity in frequency conversion processes. Koshelev et al. prepared a nonlinear metasurface composed of L-shaped silicon nanoparticles supporting Mie resonance and multiple q-BIC resonances. Therefore, it can produce strong third-order harmonics between 413 and 500 nm at the pump wavelength from 1240 to 1500 nm. In addition, at resonant wavelength, the metasurface has a nonlinear circular dichroism (CD) value close to ±1³⁹. Kim et al. extended the harmonic wavelength of the nonlinear chiral metasurface to the ultraviolet band. They added a reflective substrate under the Z-shaped LiNbO₃ nanoparticle array to enhance the Mie resonance of the low-refractive-index material. The metasurface can produce hexapole resonance under the incidence of 792 nm RCP light, while the hexapole resonance is weak under LCP light. Therefore, the nonlinear conversion efficiency of RCP pumping light is 16 times higher than that of the LCP pumping light, and the wavelength of the second-order harmonic is 396 nm⁴⁰.

The amplitude modulation and phase modulation can adjust the arrangement of nanoantennas on the metasurface to customize the complex functions of the nonlinear metasurface¹³¹, such as nonlinear meta-lens, optical encryption, and nonlinear beam shaping. The harmonic wavelengths cover a wide wavelength range from visible light to VUV light. Kruk et al.¹³² used Si and SiN to prepare axially asymmetric nonlinear metasurfaces. Four basic nanocolumns were selected by changing the major and minor axes of elliptical nanocolumns. They can apply different amplitude modulations to the third harmonic (λ_THG = 492 nm) transmitting forward and backward. Therefore, they can encode two different transmission patterns for the forward and backward directions, as shown in Fig. 4c.

Furthermore, the geometric, resonance, and propagation phases can also be used to modulate nonlinear wavefronts to realize nonlinear imaging, vortex, and holography. Schlickriede et al. studied imaging objects through the dielectric nonlinear meta-lens. The Mie-type nanoresonators were used to generate nonlinear propagation phase modulations. The object was illuminated with infrared light (λ_FF = 1550 nm), and the image was captured at a visible wavelength (λ_THG = 517 nm). By reviewing classical imaging theory, they proposed and verified the generalized Gaussian lens equation applicable to nonlinear imaging³⁴. Hail et al. proposed a Si nonlinear metasurface supporting local high-order Mie resonance with high Q value to achieve nonlinear phase modulation while maintaining high nonlinear conversion efficiency. Changing the parameters of nanoresonators can modulate the resonant phase and realize the focusing of the third-order harmonic with the wavelength from 413 to 453 nm³⁵. For nonlinear vortex generation, Mao et al. used the pump light with the wavelength of 1266 nm to generate second harmonic vortex beams with different orbital angular momentum based on the relationship between the rotation angle of SiN nanofins and the second-order harmonic phase. The harmonic wavelength is 633 nm³⁶. Besides the geometric modulation, Wang used the resonant phase modulation method to control the THG phase. The selected Si nano units have a similar amplitude modulation to the THG. However, the phase modulation covers a full 2π. Therefore, the third-order harmonic vortex beam with a wavelength of 538 nm can be generated by arranging the cylinders on the metasurface according to the phase of the vortex beam³⁷. As for nonlinear holography, Gao et al. introduced a new mechanism for nonlinear holography using silicon metasurfaces with C-shaped meta-atoms. These nanoantennas enhance the fundamental resonance, and the THG signal is redirected to the air gap region through high-order resonance. The resonant phase will change by changing the inner radius, outer radius, and opening angle of the C-shaped meta-atom. They experimentally generated efficient cyan and blue THG holograms²⁵ by arranging meta-atoms with different resonant phases. Reineke used geometric phase modulation to realize nonlinear holography. He used the different relationships between the Si nanoparticle rotation angle and the geometric phase modulations for the LCP and RCP third-order harmonic to accomplish holographic multiplexing. The harmonic wavelength was 400–450 nm²⁶. Liu et al. studied the geometric phase modulation effect of Si nanoparticles of different rotational symmetry types, and realized polarization-controlled multi-channel holographic multiplexing for the 400 nm third-order harmonic⁴¹. The above studies all manipulated the wavefront of harmonics in the visible band, and phase modulation can also focus UV light into a very small area. The large-energy UV light in an extremely small area is significant in many fields. For example, laser processing can accurately remove materials and process complex objects under strict tolerance limits. In biomedicine, it can precisely treat diseased areas and reduce the damage to healthy tissues. For lithography, it can also overcome the problems caused by the low sensitivity of the resist and achieve ultra-high resolution processing. To obtain high-energy VUV light concentrated in a small area, Tseng et al. designed a meta-lens composed of a 150 nm thick C3 symmetric ZnO nanoresonator, based on geometric phase modulation to generate and focus VUV, the second harmonic (λ_SHG = 197 nm). The focused light spot has a diameter of only 1.7 μm. Its power density enhancement is increased 21 times compared with the meta-lens surface²⁰, as shown in Fig. 4d.

This paper comprehensively reviews the latest developments in all-dielectric nonlinear metasurfaces, which cover the harmonic wavelengths from visible light to VUV. The basis for the excitation of nonlinear effects in all-dielectric metasurfaces is the strong interaction between light and matter. The nonlinear polarization of dielectrics can be enhanced by introducing resonant modes to enhance the local light field. In addition, far-field radiation shaping is an effective strategy to enhance the ability to collect nonlinear signals. Nonlinear amplitude, polarization, and phase control can collect nonlinear signals and customize complex far-field transmission, which is beneficial for nonlinear imaging, sensing, etc. Ultrafast modulation of light can be achieved by utilizing the nonlinear effects of the Kerr effect, thermal nonlinear effect, and free-carrier effect.

Although many researchers have conducted in-depth research on enhancing and controlling nonlinear light in the visible light band, research on nonlinear metasurfaces in the VUV region remains limited. This limitation is primarily due to the challenge of fabricating nanostructures with small line widths. While inducing high-harmonic generation from a metasurface operated at longer wavelengths is a viable alternative, it is often difficult to isolate the desired harmonic order from the multitude of generated harmonic waves. Moreover, the structure supporting high-Q resonance tends to be poorly robust, so the processing error greatly affects the nonlinear efficiency. Although some scientists researched the fabrication robustness of symmetry-protection BIC metasurface and found the optimal structure, the asymmetric ellipses¹³³, there have been no universal principles suitable for all types of BIC metasurface design to obtain robustness up to now. Therefore, enhancing VUV harmonic efficiency to promote practical applications such as in photolithography¹³⁴ and high-energy physics¹³⁵ remains an unresolved issue.

We suggest that researchers can explore solutions from three perspectives. First, the application of integrated-resonant units (IRUs)^136,137 could be beneficial. Research in linear metasurfaces has demonstrated that integrating different resonance modes in a building block can enhance performance, such as efficiency¹³⁸, operational bandwidth¹³⁹, and multifunctionality¹⁴⁰. We believe this approach can potentially transition from the linear to the nonlinear domain, thereby improving the response of VUV metasurfaces. The schematic diagram of the application of IRU metasurface to nonlinear generation is shown in Fig. 5a. Second, the nonlocal effect appears promising as it can integrate efficiency enhancement with phase manipulation. One can achieve a balance between a high-Q factor and flexible phase control by precisely tuning such collective response, including near-field interactions between adjacent meta-atoms and spatially extended modes arising from periodic effects. Recent studies have demonstrated that nonlocal metasurfaces can be effectively utilized for dark-field emission¹⁴¹, high-Q wavefront shaping¹⁴², and nonlinear generation¹⁴³. The concept of nonlocal metasurface enhancing nonlinear generation is shown in Fig. 5b. Additionally, enhancing the nonlinear power density is crucial. Researchers have primarily utilized basic parabolic focusing profiles to boost nonlinear power density. However, insights from linear metasurfaces suggest that focusing efficiency can be further improved by optimizing the layout using inverse design and other artificial intelligence (AI) techniques^144,145. Therefore, the AI algorithms can assist to obtain the desired nonlinear output of metasurface by parameter designing and layout optimization, as shown in Fig. 5c. Through these strategies, it is possible to advance the performance of short-wavelength nonlinear metasurfaces significantly.

a The concept of the IRU metasurface for nonlinear generation. b The concept of the nonlocal metasurface supporting the high-Q resonance which can significantly enhance the nonlinear conversion efficiency. c The concept of the application of AI algorithms to design all-dielectric metasurface to obtain the desired nonlinear metasurface.

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