High-harmonic generation in a metasurface

Abstract

The nonlinear optical effects arising from the interaction between ultrafast lasers and metasurfaces have become a significant focus of photonics research. Among these effects, high-order harmonic generation is regarded as an effective pathway toward realizing extreme ultraviolet light sources and attosecond pulse generation. High-harmonic generation (HHG) based on metasurfaces also offers new strategies for the design of integrated light sources. During this process, the driving conditions for metasurfaces are distinct from those for gas media and bulk materials. Clarifying the physical mechanisms governing the interaction of light with media during HHG, as well as the relationship between these mechanisms and structural parameters, is essential; this understanding also holds great application potential in the research and development of new materials and microscopic dynamics characterization. Here, we review recent advancements in HHG from metasurfaces, focusing on the effects of different resonance modes, structural symmetries, and molecular arrangements in the medium on harmonic yield, polarization control, and the precise manipulation of spatiotemporal distribution. Finally, we discuss improvement strategies for metasurfaces with the aim of more effectively mitigating damping in nonlinear processes, and explore future applications of metasurface-based HHG in frontier fields.

Graphical abstract

Keywords

high-harmonic generation / metasurface / resonance / polarization / crystal

1 Introduction

The human cognition of the world is pervaded by light. From supernova explosions in the vast universe to the moment when cellular life forms are born, light exists throughout [1, 2]. In addition, the various optical effects resulting from the interaction between light and matter are important pathways through which we perceive and understand the world. Since the invention of lasers, research on light has expanded from classical optics to the fields of nonlinear optics and quantum optics. When the electric field intensity of a laser pulse reaches or even exceeds the Coulomb potential of atoms and molecules, the concept of treating a laser field as a perturbation to the motion of electrons constrained by the atomic Coulomb field no longer applies, and a series of complex nonlinear dynamic processes arises [3, 4]. Through the application of chirped pulse amplification theory and mode-locking techniques [5−7], the power density of lasers has been increased to the terawatt level, which has extended the study of light–matter interactions into the domain of strong physical fields [8−10]. Among these studies, high-harmonic generation (HHG) under extreme ultraviolet light sources, ultrafast dynamics of microscopic particles, and biomedical applications have garnered significant attention because of their potential (Fig. 1) [11−16], forming the core of attosecond optics research [17−22].

In addition to intense light serving as an excitation source, the medium interacting with strong light is a key factor in the efficient and stable generation of high-order harmonics (HHs), and there are two main types: gas media and solid media. Among the theoretical models describing HHG, the “three-step model” collision theory, based on nonlinear perturbation theory, provides a good explanation for the process of HHG in gas media [18]. However, in studies of HHG in solid media, Ghimire et al. [23] first discovered that the relationship between the harmonic generation efficiency and the order of nonlinear harmonics differs, and they noted that the high-energy cutoff region of solid-state HHs is linearly correlated with the peak amplitude of the light field, which contradicts the quadratic correlation predicted by the three-step collision model [23]. This discrepancy arises because the propagation of electrons in the laser field after tunnel ionization cannot be neglected. The electron transition dynamics are more complex, meaning that the aforementioned “three-step model” collision theory does not apply to solid media [24, 25]. The mechanism of HHG in solid media is still unclear. The currently accepted processes can be summarized into three steps: i) Under the action of an ultrafast laser pulse, electrons enter the conduction band through multiphoton or tunneling ionization, generating corresponding holes in the valence band. ii) Electrons and holes are accelerated by the laser electric field within their respective energy bands, generating intraband currents in a periodic Bloch oscillation manner, thus contributing intraband harmonics. iii) Interband currents generated by electron transitions from the conduction band to the valence band contribute interband harmonics. The intraband and interband currents are coupled, emitting harmonic signals. During the process of refining the mechanism description, models related to HHG in multiple platforms, such as those involving band structure corrections, the intraband preacceleration process of electrons, the dependence of the HHG cutoff frequency on the incident light field, and modifications to electron–hole collision conditions, have been proposed and validated [9, 26−30]. In practical applications, defect states significantly influence the electronic structure and energy level arrangements in solid media, particularly in semiconducting materials. Defects primarily regulate the characteristics of nonlinear processes by modulating the intensity and dynamic behavior of these responses. In the context of HHG processes, defects can directly affect both the efficiency and threshold of HHG by modifying the internal energy levels or local electric fields within the material [31].

Currently, HHG based on gas media has been well-developed; however, there is still significant room for improvement in terms of the excitation conditions and conversion efficiency of HHs. Compared with gas media, solid media possess the advantages of greater atomic density and periodic structures, providing better control over excitation conditions and harmonic signals [25, 29, 30]. Among the various types of solid media, metasurfaces — synthetic two-dimensional nanostructured materials — offer exceptional capabilities for manipulating electromagnetic waves that cannot be achieved with natural materials. Owing to the unique structure of metasurfaces, their optical properties are determined by the arrangement of nanostructural elements, allowing metasurfaces to overcome some limitations of traditional optical materials. For example, while harmonic generation is typically forbidden in conventional uniform media, metasurfaces with uniform thin films or laterally subwavelength patterns are not subject to this restriction. Additionally, metasurfaces are not constrained by the phase-matching conditions that typically affect nonlinear optical processes. Specifically, the high nonlinear response of dielectric metasurfaces and their ability to locally couple light fields enable high-order light–matter interactions. This makes metasurfaces excellent platforms for the generation and control of HHs [32−47]. This review introduces, from an experimental perspective, the HHG characteristics of metasurfaces based on different resonance modes and structural features and provides an overview of recent advances in metasurface HHG research.

In photonics research based on metasurfaces, resonance is fundamental. Nonlinear optical processes associated with metasurfaces are always accompanied by different types of resonances, including localized resonance modes within periodic unit structures (surface plasmon resonances, Mie resonances) [48−59], global structural resonance modes (surface plasmon polariton resonances, surface lattice resonances) [60−67], and material resonances (in epsilon-near-zero materials) [50, 68−75]. Notably, the flexibility of the metasurface design allows different resonance modes to simultaneously coexist, and the interactions between them can significantly enhance the nonlinear response (Fano resonances, bound states in the continuum) [37, 42, 76−87]. As an extreme upconversion process, the efficiency of HHG is directly related to the intensity of the local energy field at the interaction site of the incident light. Metasurfaces enhance local fields by orders of magnitude through resonance with the incident optical field. Therefore, different types of resonances are always involved in HHG processes enabled by metasurfaces.

2 Plasma metasurfaces − Surface plasmon resonance

Plasma metasurfaces are typically composed of periodic two-dimensional arrays of metallic micro- and nanostructures and serve as a common platform for interactions between light and matter. However, metals themselves cannot function as HHG media because the electron structure, conductivity, and reflective properties of metals cause strong radiative damping, making them unsuitable for HHG. Nonetheless, metal surfaces contain a high density of free electrons capable of strong coupling with incident light fields. Additionally, plasma metasurfaces made of metals can easily enhance the local electric field by several orders of magnitude. Therefore, metal-based plasma metasurfaces are highly suitable for enhancing HHG processes.

Plasma metasurfaces exhibit two fundamental resonance modes under wavelength-matching conditions: localized surface plasmon resonance (LSPR) and surface plasmon polariton (SPP) resonance. The LSPR mode typically occurs on the surface of metal nanoparticles, whereas the SPP mode arises at metal-dielectric interfaces and propagates along them. The structural characteristics of plasma metasurfaces determine the generation mode of local field enhancement, which may involve the combined effect of one or more modes. The near-field decay length of the SPP mode is tens of times greater than that of the LSPR mode, resulting in a larger effective scale in enhancing HHG in the medium. In the process of enhancing HHG with plasma metasurfaces, these metasurfaces are typically integrated with thin nonlinear dielectric layers (with layer thicknesses usually ranging from a few hundred nanometers to a few micrometers). Based on the sensitivity of the polarization strength of the nanoscale structures in the long-axis direction, the plasma metasurface generates an SPP mode under specific polarization conditions of the incident light field. The electric field intensity within the gaps between adjacent structures is increased by several orders of magnitude [54, 88]. Furthermore, HH emission is produced by the thin dielectric layer. To generate an effective nonlinear electric polarization intensity in the dielectric layer, if the dielectric layer has a regular atomic arrangement, then the relationship between the atomic arrangement and the polarization of the driving field must be considered. The long axis of the plasma structure within a unit period should be aligned as closely as possible to the atomic plane to meet the polarization phase-matching conditions. According to existing reports, in SPP mode, the HHG enhancement of a dielectric integrated with a plasma metasurface is several to more than ten times greater than that of a standalone thin dielectric layer [51].

3 Structural resonance

HHG is a manifestation of the strong interaction between light and matter. As a carrier of HHG, the thin dielectric layer mentioned above strongly interacts with incident light through the plasma metasurface. This interaction during the HHG process results in a local electric field intensity enhancement of several orders of magnitude. The metallic structural units that typically comprise plasma metasurfaces contain high-density free electrons, and the structural parameters of neighboring periodic nanoscale units are highly consistent. This consistency ensures that the local electric field intensity meets the requirements for strong nonlinear processes under the resonance conditions. For HHG based on nonmetallic metasurfaces, the structures within the metasurface unit cell or their axial configurations are more diverse to satisfy the strong light–matter interaction condition. In this case, the resonance modes of the metasurface exhibit structural dependence, with a notable characteristic being that the metasurface has resonance spectral lines with a high quality factor (Q-factor).

3.1 Fano interference

In photonics, interference is a classical manifestation of electromagnetic wave scattering, typically observed as the superposition or suppression of fields. In nonlinear processes based on light–matter interactions, “constructive” interference between different electromagnetic field modes is required to achieve the field enhancement necessary for the medium to generate a nonlinear response. As a classical resonance mode, Fano resonance [89] is caused by the interference between discrete quantum states and a continuum of states. This phenomenon is characterized by an asymmetrical line shape in the absorption spectrum and a high Q-factor [90]. The optical properties of metasurfaces are directly related to their structural parameters. The two resonant states of a metasurface correspond to different microstructures under resonant excitation conditions, and the periodic unit structure is asymmetric. Here, a classical coupled-oscillator model is used to describe Fano interference in a metasurface during HHG [91], with its matrix equation expressed as

(1)\left(\begin{array}{c}{f}_1\\ f_2\end{array}\right)=-{\left(\begin{array}{cc}\omega -{\omega }_1-\mathrm{i}{\gamma }_1& \kappa \\ \kappa & \omega -{\omega }_2-\mathrm{i}{\gamma }_2\end{array}\right)}^{-1}\left(\begin{array}{c}{E}_1\\ E_2\end{array}\right).

In Eq. (1), f₁ and f₂ represent two radiation states with driving frequency ω, which are coupled to the incident field. E₁ and E₂ are the amplitudes of the model, ω₁ and ω₂ are the resonance frequencies, and γ₁ and γ₂ are the damping constants of the two oscillators. κ represents the coupling parameter between the two oscillators. By normalizing the absolute value of κ with respect to γ₁ and γ₂ and comparing their magnitudes and the changes in the natural frequencies of the oscillators, theoretical models such as Fano resonance, electromagnetically induced transparency (EIT), and Borrmann effect models can be obtained [as shown in Fig. 3(a)]. Specifically, in the Fano resonance model, resonance occurs between two oscillators with significantly different damping factors, producing both a narrow spectral line and a broad spectral line. The coupling constant κ is much smaller than the larger damping factor γ₁. When one of the oscillators (oscillator 1, which has a larger damping factor) is driven, this corresponds to the case in which f₁ ≠ 0 and f₂ = 0. The shape of the resonance spectral line is determined by the detuning between the two oscillator frequencies, ω₂ − ω₁ (the ratio of the odds of electronic transitions to the discrete state and the continuous state, which is defined as q).

In practical applications, by introducing spatial symmetry breaking into the nanoscale structures within the unit cell of a metasurface, the coherent superposition of two radiation states (typically manifested as the superradiant dipole mode and the subradiant quadrupole mode) can generate Fano resonance [33]. Symmetry breaking in nanostructures can take various forms, such as “etching” a single structure or forming a dimer by combining different structures. When the spectral detuning condition between two radiation resonance modes is satisfied, Fano resonance occurs. The Q factor of the resonance spectral line can be increased by optimizing the structural parameters to achieve efficient coupling with the incident light field. Owing to the symmetry breaking of the structure, the far-field radiation modes can be controlled by adjusting the structural parameters. For the HHG process, this structural dependence allows the metasurface the overlapping position of the resonances of two radiation states to be flexibly controlled through the metasurface, thereby improving the spectral emission efficiency.

As mentioned above, the detuning of the spectra of the two oscillators determines the shape of the Fano resonance spectral line. Consider a special case in which f₁ ≠ 0 and f₂ = 0. When the resonance frequencies of the two oscillators with strong and weak damping exactly match or the spectral detuning is very small (i.e., ω₁ = ω₂), the parameter q = 0. This scenario describes a situation in which only oscillator 1 can couple with the incident light, whereas oscillator 2 barely couples with the incident light, commonly referred to as the “bright” mode and “dark” mode. The interaction between the two oscillators causes a slight shift in their natural frequencies. When the condition ω − ω₁ ≤ ω₁ is met, the frequency is close to resonance. Typically, the “bright” mode, which has a larger damping factor, manifests as a broad stopband in the transmission spectrum. However, through coupling with the “dark” mode (which has the opposite phase), the “bright” mode can return to its original resonant state. Compared with the Fano resonance case in which q ≠ 0, this coupling achieves a resonance compensation effect, resulting in a narrow transparency window whose spectral line resembles the characteristic EIT profile. In this case, EIT can be viewed as a special scenario of Fano resonance in which there is no spectral detuning between the “bright” and “dark” modes. Ultimately, this manifests as constructive interference that enhances the field.

For metasurfaces, this EIT-like resonance typically occurs in dimer structures [78, 92]. In practical applications, both the “bright” mode and “dark” mode structures of a metasurface can be excited, but the damping factors need to be significantly different. This results in the “dark” mode structure having weak coupling to the incident light. Therefore, the “dark” mode structure cannot be directly excited by the incident light; its energy coupling is driven by the “bright” mode structure, which represents a passive coupling mode. The intensity of the EIT-like resonance is highly sensitive to the structural parameters of the metasurface. Typically, when the frequency meets the resonance condition, the EIT resonance intensity is determined by the spacing between dimers, and shows low sensitivity to the external environment [93]. Based on this, compared to plasmonic metasurfaces, dielectric metasurfaces exhibit higher-quality-factor resonance lines owing to lower Ohmic losses. Reis and colleagues [36] achieved EIT resonance using a silicon metasurface. Compared with that of a pure silicon medium, the HHG emission is enhanced by two orders of magnitude. In this process, the location of the structural “hotspot” dynamically changes, and its decay time is slower. While maintaining a high intensity, this effectively prolongs the residence time of the “hotspot”. In fact, this characteristic is present in all EIT-like processes based on metasurfaces, providing a reference for achieving highly efficient HHG.

3.2 Bound states in the continuum

In photonics, a bound state in the continuum (BIC) refers to a localized state within the continuous spectrum whose energy level lies in the forbidden region of the continuum spectrum. It is an eigenstate of the system, and its generation mechanism can be described by band theory and multilevel matrix superposition [94, 95]. Compared with traditional photonic states, the far-field radiation of metasurfaces with BIC structures is completely suppress while maintaining a high-quality factor. This allows these metasurfaces to sustain strong coupling with the incident light field, significantly enhancing the interaction between light and matter. The BIC structure exhibits characteristics similar to those of the anapole state, which favors the conditions required for achieving HHG in dielectric materials [81]. The application of field enhancement through BIC metasurfaces spans multiple research areas; however, here, we focus only on the effects of these metasurfaces on nonlinear optical processes. BICs can be excited in and generated by various types of dielectric metasurfaces, such as those made from dielectric materials, perovskites.

Dielectric metasurfaces are typically fabricated from high-refractive-index materials such as silicon (Si), gallium arsenide (GaAs), and silicon nitride (Si₃N₄) using nano/microfabrication techniques, and they can support BIC modes. Dielectric-based BIC metasurfaces have been designed for various subwavelength photonic devices [86, 96−101]. Compared with plasmonic metasurfaces, dielectric metasurfaces offer lower energy losses, higher nonlinear responses, and strong localized optical field characteristics. In energy coupling processes, the fundamental resonance mode of dielectric metasurfaces is Mie resonance. BICs arise from the strong coupling between different resonance modes, enhancing the intensity of light–matter interaction within the structure. For example, the destructive interference of the two eigenmodes of the toroidal dipole in a dimer structure of dielectric disks along the y- and z-axes can achieve a BIC [80, 96]. This reveals the connection between the toroidal dipole and the BIC, with both eigenmodes exhibiting an infinite-Q-factor characteristic, significantly enhancing the coupling of the toroidal multipole to energy [102]. Additionally, Cong et al. [103] reported that under normally incident light polarized along the x- and y-axes, breaking of the in-plane C₂ symmetry causes the transmission spectrum of the structure to exhibit Fano-like and EIT-like line shapes, respectively. This discovery highlights that structural symmetry can serve as a new degree of freedom for tuning BICs [104, 105], which has attracted widespread research interest. Kupriianov et al. [80] designed highly efficient metasurfaces with controlled high-Q resonances by breaking the symmetry of a 2D planar lattice out of the plane. In the microwave range, they achieved selective excitation of quasi-BICs by varying the relative heights of the nanopillars, which altered the position of the “hotspot”. This symmetry-breaking design enabled a tunable Q-factor of the directional frequency of the BIC [80]. Zograf et al. [37] designed a silicon metasurface with symmetry-broken quasi-BIC resonances. The BIC resonance at the fundamental frequency resulted in a high energy concentration within the metasurface, enabling HHG from the 3rd to 11th orders. The resulting harmonic emission spanned the near-infrared to visible spectrum [37].

Technological development in the field of photonics have always been supported by optical materials, with high-performance optical materials being the key to achieving practical applications. Among these materials, perovskites have been widely used in fields such as solar cells and novel photonic devices due to their excellent characteristics, including a high refractive index, a high nonlinear coefficient, a tunable bandgap, high-efficiency photoluminescence, and low cost [114−118]. Additionally, perovskite thin films exhibit remarkable nonlinear properties, as their optical nonlinearity is several orders of magnitude greater than the nonlinear permeability of traditional semiconductor materials [119, 120]. To further enhance the nonlinear properties of perovskites, especially their higher-order nonlinear responses, forming metasurfaces with perovskites as nonlinear media is an effective approach. Perovskite materials themselves possess high luminescence quantum efficiencies, tunable bandgaps, and good solution-processability, which facilitates excitation of BICs in the metasurfaces constructed from them compared with metasurfaces composed of metal or dielectric materials. Furthermore, by loading spin-related geometric phases in BIC mode [121], perovskite metasurfaces can exhibit spin–valley locking, which can be used to increase the directional collimation emission efficiency of light-emitting devices. Tonkaev et al. [46] were the first to observe fifth-order harmonic emission from a perovskite metasurface made from the MAPbBr₃ halide. The grating design of the structure led to excitation of the BIC resonance and guided mode resonance (GMR). By tuning the frequency degeneracy of the BIC and GMR through suppression of the even-order Fourier components of the grating dielectric function, fifth-order harmonic emission was observed from the resonant structure [46]. Compared with the thin-film state, the fifth-order harmonic emission from the structured perovskite metasurface was enhanced by two orders of magnitude.

3.3 Material resonance − Epsilon-near-zero materials

Epsilon-near-zero (ENZ) materials refer to a class of materials whose real part of the dielectric constant is zero within a specific spectral range [70, 122−124]. These materials exhibit reduced reflection and absorption compared to conventional materials. Owing to their excellent optical properties, ENZ materials are considered potential materials for next-generation optoelectronic devices. The subwavelength structures and near-zero characteristics of ENZ materials reduce the phase matching limitations in nonlinear processes, resulting in a strong nonlinear response [68, 75, 125, 126]. For thin-film structures consisting of metal-transparent conductive oxide materials, at specific polarization angles, the electric field is confined within the ENZ dielectric layer, leading to an increase in the local electric field by several orders of magnitude. This enhancement can directly excite HHG in the dielectric layer, with HHG being accompanied by a noticeable spectral redshift and linewidth broadening. This phenomenon is considered to be the result of the combined effects of light-induced electron heating and the time-varying properties of the dielectric material [73]. Currently, various theoretical models are used to describe the enhancement mechanisms of nonlinear responses in ENZ materials. Their strong nonlinear response is associated with enhancement of nonlinear magnetization and the local electric field, which is a strong coupling mechanism within the material itself, often referred to as material resonance. Given the outstanding nonlinear optical characteristics of ENZ materials, their nonlinear responses can be further enhanced by combining them with patterned nanostructures to form ENZ metasurfaces.

In practical applications, two-dimensional plasma nanoantennas are typically integrated onto the surface of ENZ materials to form ENZ metasurface structures. This increases the coupling of the ENZ metasurface with incident light, leading to an increase in the nonlinear response of ENZ materials by several orders of magnitude [69, 127−130]. Additionally, structured plasma nanowire antennas play a role in controlling the light field, which can influence the nonlinear emission of ENZ materials. These types of metasurface structures are usually described by the dielectric constant dispersion of free electrons, based on the Drude model, and show a strong dependence of the energy coupling intensity on the structural parameters:

(2)ϵ\left(\omega \right)={ϵ}_{\mathrm{\infty }}-{\displaystyle \frac{{\omega }_{\mathrm{P}}^2}{{\omega }^2+\mathrm{i}\gamma \omega }}.

In Eq. (2), ε_∞ represents the high-frequency dielectric constant, ω_P is the plasma frequency, and γ is the damping coefficient. In structural design, the resonance wavelength of the plasma nanowire antenna needs to coincide with the wavelength corresponding to the ENZ material to produce the LSPR and ENZ dual resonance effect, thereby maximizing the coupling strength. Furthermore, the thickness of the ENZ layer and the spatial separation between the nanowire antenna and the ENZ layer also directly influence the coupling between the structures. Manukyan et al. [127] studied the impact of these structural parameters on the coupling strength between the nanowire antenna and the ENZ layer, highlighting the need to consider the thickness of the ENZ layer, the scale of the plasma resonance, and energy loss factors. This finding is significant for the structural design of nonlinear ENZ metasurfaces. Additionally, Alam et al. [69] demonstrated that the presence of ENZ materials causes a metasurface to exhibit ultrafast changes in the refractive index in response to the light intensity and that ENZ metasurface structures are not constrained by phase matching. This behavior offers the potential to increase the transmission efficiency of ultrafast nonlinear processes such as optical harmonic generation [69]. ENZ metasurfaces can serve as material platforms for the development of highly integrated micro/nano optoelectronic devices and provide ultrafast all-optical means for efficient control of light fields at the micro/nano-scale. Recently, Yan et al. [131] reported that the weak coupling mechanism within ENZ media exhibits continuous resonant states analogous to those observed in Fano resonance. By incorporating discrete resonant structures into ENZ media, they achieved the first observation of Fano resonance in such media. The spectral characteristics of this resonance were found to be directly correlated with the magnetic permeability of the ENZ media [131], offering a novel approach to modulating the resonant modes. Given the inherently low energy loss characteristic of ENZ media, introducing new resonant modes to achieve high-Q resonances theoretically enhances the efficiency of higher-order nonlinear processes.

In addition to the aforementioned resonance modes, high-Q resonances in metasurfaces also include surface lattice resonance (SLR) and GMR. SLR arises from the coupling between the LSPR of periodically arranged metal nanostructures and the in-plane diffraction waves of a metal array. Compared with individual metal structures, SLR results in stronger field enhancement and higher-quality-factor resonance lines [65, 132−136]. This phenomenon has been used in research on highly efficient nonlinear metasurfaces and ultrasensitive optical sensing. Essentially, SLR still belongs to the category of plasmonic resonance, but since the radiation losses in metals can be suppressed through array diffraction coupling, theoretically, metasurfaces based on SLR also offer enhancement effects for HHG. GMR metasurface structures mainly consist of diffraction gratings and planar waveguides. When light is diffracted by a grating, and the diffraction angle matches the propagation constant of the waveguide, energy is strongly coupled into the waveguide mode, causing resonance [137−139]. The physical mechanism behind this is the redistribution of optical energy caused by the coupling between the externally propagating diffracted light and the modulated leaky waveguide modes. GMR features a highly diffraction efficiency and narrow bandwidth resonance lines. In metasurface structures, GMR is often accompanied by the emergence of BICs, which can further enhance the structural performance. This phenomenon has been widely used in the design of subwavelength optical devices. In GMR, energy transmission occurs within the structure, where the structure strongly “binds” the energy and exhibits multifrequency resonance characteristics. Therefore, GMR metasurfaces can be used to precisely control HH frequencies. Currently, there are limited reports on the applications of SLR and pure GMR modes in higher-order nonlinear processes [40, 46], but such metasurfaces themselves can serve as highly efficient platforms for HHG.

4 Regulation of HHG via crystal metasurfaces

In the aforementioned HHG process based on metasurfaces, resonance-induced local field enhancement can meet the energy requirements for electron tunneling within a nonlinear medium. However, for crystalline media, HHG exhibits additional correlations with the medium. The characteristic periodic arrangement of particles (molecules, atoms, ions) in three-dimensional space in crystalline media (excluding quasicrystal structures here) imposes selection rules on the HHG efficiency and polarization control, which are related to the excitation light and the microscopic structure of the medium. Based on these features, the localization properties of fields induced by metasurfaces, combined with the diverse target materials of crystalline media, can be utilized to regulate HHG.

4.1 Nonlinear selection rules of crystal metasurfaces

The complete nonlinear process generated by a metasurface includes in situ nonlinear emission enabled by local field enhancement, and the nonlinear polarization generated by the pump light also radiates into the far field. The latter requires consideration of the influence of the excitation conditions on the nonlinear interaction tensor of the metasurface [140]. For crystal media, the polarization direction of the fundamental wave and the symmetry of the nonlinear tensor determine the spatial modes of the field. This necessitates that the nonlinear tensor components of the metasurface match the input and output beams, and the spatial mode overlap of the input and output beams with the metasurface at each frequency should be optimized. For example, in certain modes, far-field nonlinear emission can be controlled by rotating dipoles [141−143]. Furthermore, for crystalline and other nonplasma media, field enhancement at harmonic frequencies often manifests as high-order multipole modes. These multipole modes can be induced by the medium layer or substrate under the influence of nanostructured antennas, as the field enhancement effect of metasurface structures is often not confined to the interface. The polarization modes of these modes are not exactly the same, which may inhibit directional emission of nonlinear signals. Theoretically, to maximize the energy coupling efficiency, all radiation channels other than those for power injection and extraction should be suppressed [144, 145].

4.2 Crystal-orientation selection rules for HHG

The crystal planes of single-crystal media possess specific point group symmetries, and the HHG yield varies under different linear polarizations of light, exhibiting crystal-orientation-dependent characteristics. This behavior is attributed to differences in the Brillouin zones corresponding to linearly polarized light [146]. You et al. [147] investigated the dependence of HHG on the crystal orientation of MgO single crystals and demonstrated anisotropic HHG through real-space electron trajectories. They reported a significant increase in HHG when electrons traversed atomic positions across the crystal lattice. Additionally, the crystal orientation affects the polarization and temporal structure of HHs [147]. Langer et al. [148] studied the relationship between HHG and crystal orientation in bulk GaSe single crystals, demonstrating how the crystal symmetry governs the polarization and carrier-envelope phase (CEP) temporal structure of HHs. This finding provides a method to control the polarization angle between even-order and odd-order harmonics [148]. Interestingly, by adjusting the crystal orientation and selecting specific polarizations, a harmonic frequency comb can be switched between ν and 2ν, offering new perspectives for solid-state attosecond photonics.

In the aforementioned HHG processes based on different types of media, polarized incident light is used to match the electrical polarization strength or electronic wave vector of the structure [149, 150]. Recently, circularly polarized HHs have shown potential application in probing solid-state magnetism and chiral-sensitive light–matter interactions [149, 151, 152]. Unlike linearly polarized HHG, direct excitation of HHG in gaseous media using circularly polarized light is not feasible, as the transitioning electrons cannot collide with the parent ions. This progress typically requires additional control over the light field. Owing to the periodic structure and high-density atomic arrangement in crystal media, adjacent atomic orbitals exhibit significant overlap. HHG can be directly excited using circularly polarized light under anisotropy conditions (constrained by angular momentum conservation).

In metasurface structures, crystalline media are transformed into layered structures with thicknesses ranging from several hundred nanometers to a few micrometers. For single-crystal media with different crystal orientations, the internal atomic arrangement is regular. To increase the HH yield, structured plasmonic nanoantennas are typically integrated into thin crystalline media to increase the HHG of the crystal. In this process, the polarized electric field generated by the metasurface is sensitive to the long axis of the structure. Alignment of the long axis of the nanoantennas parallel to the crystal orientation ensures the generation of nonlinear polarization [51].

Additionally, by designing nanoantennas with orthogonal long axes to couple with the polarization components of circularly polarized excitation light, “hot–spots” with relative phases can be formed on the structure surface. The phase control of the light field by the nanoantennas is inherited in the nonlinear process. Jalil et al. [52] utilized nanoantennas with mutually perpendicular long axes to control the polarization and phase of HHs, enhancing circularly polarized HHG in Si single crystals. The relative phase emitted by each antenna was nπ/2 (where n is the harmonic order), and the linear high harmonics emitted by the two orthogonal antennas overlapped in the far field to produce circularly polarized odd-order harmonics [52]. Additionally, the polarization state of HHs is directly related to the rotational symmetry of the internal atomic structure of thin crystals [153, 154]. During harmonic generation with circularly polarized fundamental beams, the rotational symmetry of the atomic structure influences the phase of the nonlinear dipole moment. Specifically, under a time-periodic potential field, the Hamiltonian of atoms or nuclei exhibits periodicity, and its corresponding dynamic symmetry operators remain invariant. Alon et al. [157] investigated the three-dimensional multi-electron Hamiltonian framework under an N-fold dynamical symmetry in non-degenerate Floquet systems, within the theoretical framework proposed by Sambe and Howland [155, 156]. They derived that the dynamical symmetry (DS) operator remains invariant under the combined influence of a fixed potential with N-fold dynamical symmetry and a circularly polarized, time-dependent electric field. This effectively reduces the system’s dynamics to a one-dimensional Hamiltonian description. Additionally, they established selection rules for high-order scattering processes based on dynamical symmetry principles [157]:

(3){\mathit{P}}_{\mathrm{N}}=\left(\phi \to \phi +{\displaystyle \frac{2\pi }{N}},t\to t+{\displaystyle \frac{2\pi }{N\omega }}\right).

Apply the above rules to the thin crystal medium of the C_N axis. Under the influence of a periodic light field, the Hamiltonian of a three-dimensional multielectron system with N-fold dynamic symmetry remains constant within the time-periodic potential field. For circularly polarized fundamental waves propagating along an N-fold rotational symmetry axis in the medium, the nonlinear susceptibility remains invariant after a 2π/N rotational symmetry operation, whereas the nonlinear dipole moment acquires a spin-dependent geometric phase shift. The corresponding polarization intensity is

(4)\begin{array}{rl}& {\mathit{P}}_{+}^{\left(m\right)}\left(\omega \right)={\mathit{P}}_{}^{\left(m\right)}\left(\omega \right){\mathrm{e}}^{\mathrm{i}\left(\phi +\frac{2\pi }{N}\right)}{\mathrm{e}}^{-\mathrm{i}m\omega \left(t+\frac{2\pi }{N\omega }\right)},\\ & {\mathit{P}}_{-}^{\left(m\right)}\left(\omega \right)={\mathit{P}}_{}^{\left(m\right)}\left(\omega \right){\mathrm{e}}^{-\mathrm{i}\left(\phi +\frac{2\pi }{N}\right)}{\mathrm{e}}^{-\mathrm{i}m\omega \left(t+\frac{2\pi }{N\omega }\right)}.\end{array}

HHG satisfies the selection rule m = nN ± 1, where m is the harmonic order, n is any integer, and N ≥ 3 represents the number of rotational symmetry axes of coplanar atoms within the planar periodic structure. The “+” and “−” signs correspond to polarization states (left- or right-handed circular polarization) that are the same as or opposite to that of the fundamental wave. Owing to the translational invariance of crystals, N can only take values of 1, 2, 3, 4, or 6 (excluding the nonlinear effects of quasicrystalline structures). Moreover, the characteristics of harmonic signal emission during the half-cycle of the fundamental wave in monochromatic fields result in the absence of even-order harmonic signals, which implies that for certain values of N, specific orders of circularly polarized HH signals are absent. For example, when N = 3, the 9th harmonic signal with circular polarization cannot be generated; when N = 4, neighboring harmonic orders exhibit different polarization states. According to the above selection rules, the polarization state of HHs is directly related to the harmonic order, and by configuring the “receiving port” and the polarization state of the fundamental wave, HHs can be selectively generated [158]. An intriguing aspect is that polarization control of HHs by metasurfaces occurs in situ, without the need for any additional light field modulation. This finding provides a novel approach for carrier-envelope control in the extreme ultraviolet (EUV) regime and for attosecond-scale localized field manipulation. Furthermore, when applying the above selection rules to study the HH characteristics of molecules with an N-fold symmetry axis in a circularly polarized field, the target molecules must be oriented perpendicular to the direction of light propagation (DS constraint), however this restriction can be avoided in thin crystals possessing a C_N axis. Theoretically, the selection rules described above are solely determined by the specific point group symmetry of the crystal plane in the medium, which can also be applied to other nonlinear media, and the conclusions are universal.

5 Conclusion and outlook

In the field of photonics, light–matter frequency conversion is an important aspect of all kinds of nonlinear optical phenomena caused by the interactions between strong light and matter. Among these phenomena, the use of low-frequency light to excite materials to generate high-energy photons has been a focal point of research. In particular, the generation of high-quality periodic EUV attosecond pulses not only validates and refines the theories of strong-field optics and quantum optics but also serves as the foundation for attosecond optical experiments [18, 21, 22]. There are two possible pathways for generating attosecond pulses based on nonlinear optical theory: cascaded Raman scattering and HHG. According to existing reports, HHG is a more efficient method (Raman scattering has a low efficiency, and its spectral phase is difficult to measure) [22, 159−163]. As a high-order nonlinear process, HHG cannot be described by the physical picture of second or third harmonics. Vampa et al. [163−165] theoretically categorized the mechanisms responsible for solid-state HHG into two distinct models: intraband and interband processes. In the intraband model, high-order harmonics arise from the drift acceleration of charge carriers (electrons and holes) along the energy band under the influence of a strong laser field, leading to Bloch oscillations [166]. In the interband model, the mechanism involves laser-induced electron transitions induced by the strong laser field between different energy bands, generating a polarization current. This process resembles the three-step model of HHG observed in gases. Under a strong laser field, electrons transition from the valence band to the conduction band. Subsequently, these electrons and holes move at their respective drift velocities through the conduction and valence bands, preserving the integrity of the electronic band structure. Influenced by lattice scattering and the intense laser field, electrons transition back from the conduction band to the valence band, where they recombine with holes to emit energy in the form of photons. This mechanism effectively couples the electronic states, enabling the release of high-energy photons via coherent emission. Since electrons are ionized by a strong laser field, the electron motion is a complicated problem that requires elaborate computational operations and increases complexity of the progress. Although the “three-step model” explains the electron motion, its use limited by computational resources, and the model requires simplifications based on weak-field, spatial uniformity, and relaxation time energy assumptions, and the above-threshold ionization (ATI) theory used to describe strong-field electron ionization and the phenomenon of non-sequential double ionization (NSDI) are closely related to the “three-step model”, leading to limitations in the theoretical model. For electron many-body interactions or nonuniform systems, the above theoretical models cannot be perfectly adapted. Currently, although the experimental explanation of HHG in solid media is still primarily based on the “three-step model”, the description of the mechanism for solid-state HHG is not yet complete.

The optical properties of metasurfaces strongly depend on the enhancement of the local electromagnetic field, which arises through resonant mechanisms. Because the thickness of the metasurface or the integrated medium is much smaller than the excitation light wavelength, nonlinear processes are not constrained by phase-matching conditions. The individual microelements of a metasurface, as a structure-sensitive material, can possess arbitrary characteristics, and the controlled surface electric field exhibits geometric features related to the structure, enabling flexible tuning of light–matter interactions. The diversity of “structural designs” allows metasurfaces to couple energy through various resonance mechanisms. This “structural design” refers not only to the geometric features of the structure but also to the selection of the medium layer in composite metasurfaces, as the way layered structures interact with electromagnetic waves is still determined by their dielectric constants [83, 137, 167−169].

Metallic surface materials have a high concentration of free electrons, and plasmonic metasurfaces made from noble metals exhibit strong coupling to incident light fields, often outperforming dielectric metasurfaces. Through LSPR, enhancement of the local electric field by several orders of magnitude is easy, making these metasurfaces commonly used as harmonic “amplifiers”. However, the dielectric constant of metals has a significant imaginary component, indicating both strong energy coupling and high losses. This limits the energy utilization efficiency, and reducing the intrinsic losses of metals is a key aspect for achieving high-efficiency HHG. To address this issue, recent work by Tsai et al. [170] revealed that by changing the height of metallic nanoparticle arrays, the resonance mode can be shifted from a localized, high-loss LSPR to a nonlocal, low-loss SPP. This change in the localization of the resonance mode increased the quality factor of the structure by two orders of magnitude, significantly reducing material losses. Tsai et al. [170] cleverly extended the local resonance of individual units to the collective resonance of a nanoparticle array by merely altering the structural height. This “local” to “global” transformation provides new insights for the design of metasurfaces. In addition, compared with dielectric metasurfaces, metal metasurfaces face the issue of lower damage thresholds. For metallic metasurfaces, during the HHG process, the power density of the excitation light typically does not exceed the order of 10¹⁰ W/cm², which limits the HH emission intensity of metal composite structure metasurfaces. Recently, in studies of HHG on metal-nitride-based metasurfaces, the damage threshold of the metasurfaces exceeded the terawatt range (10¹² W/cm²), which opens up the possibility for enhancing extreme EUV generation in vacuum using plasmonic nanostructures [171].

Compared with plasmonic metasurfaces, dielectric metasurfaces exhibit greater nonlinear responses and lower energy losses. The nonlinear generation in dielectric metasurfaces strongly depends on resonance, with Mie resonance typically dominating lower-order nonlinear processes such as third-harmonic generation (THG). Owing to the high energy requirements of HHG, optical excitation is used to induce resonances such as Fano resonances, BICs, multilevel Mie resonances, and guided-mode resonances in the metasurfaces. These high-Q resonances enhance the subwavelength confinement of local fields in the nanostructures, enabling strong coupling of incident energy. High-Q resonance metasurface structures are often composed of nanostructured antennas combined with other material layers, such as perovskites, or ENZ materials. In addition to enhancing HHG, the optical properties of these materials are inherited by the metasurface, allowing it to modulate the light field in both space and time, thereby expanding the range of applications for metasurfaces. With the application of novel materials and the demand for functionalization and integration of metasurfaces, the dielectric composition and geometric features of individual metasurfaces are becoming increasingly diverse. These metasurfaces exhibit hybrid electric-dipole and magnetic-dipole resonance modes [172], such as BIC. This field interaction not only enhances the nonlinear response capabilities of metasurfaces but also offers new possibilities for regulating and integrating complex functionalities to suit various application scenarios. In subsequent developments, this is also an important aspect of designing functionalized metasurfaces.

Crystals possess specific spatial symmetries, and for different crystal orientations, the number of independent components of the dielectric tensor in the principal axis coordinate system varies. This anisotropic characteristic results in spatial differences in the nonlinear polarization strength within crystals. For crystal metasurfaces, directional nonlinear far-field emission requires spatial mode overlap at different frequencies. In practical operation, this mode overlap manifests as the overlap of resonance directions, which can be optimized by adjusting the polarization direction of the fundamental wave. Furthermore, in crystal metasurfaces, the crystal orientation of the material provides an additional degree of freedom for selectively controlling circularly polarized HHG. This selection rule does not require additional manipulation of the incident light field and depends solely on the number of rotational symmetry axes of coplanar atoms within the unit cell, which is also applicable to other solid media. Since this process occurs in situ on the metasurface, it avoids the dispersion of HH signals that would result from the addition of extra optical components. This offers a new approach for carrier control in the deep-ultraviolet region and for attosecond-scale localized field manipulation.

HHG exhibits extensive application potential across multiple cutting-edge fields. This potential manifests as precise control over “small-scale events”, and HHG based on metasurfaces enhances flexibility and directivity. In the biomedical field, the electric field distribution controlled by metasurfaces enables HH to target the nonlinear responses of specific molecules or cells for selective detection, thereby guiding localized photodynamic effects. This capability is crucial for applications such as biological imaging, drug tracking, and treatment evaluation. In optical information technology, the subwavelength structural features of metasurfaces facilitate high-speed data transmission and high-density optical storage at the microscale. Moreover, the strong correlations between harmonics during the HHG process exhibit quantum-level entanglement characteristics, which are essential for constructing quantum communication protocols. In computer vision, HH can enhance image recognition performance via nonlinear optical effects, improving the accuracy of machine learning algorithms in understanding complex scenes. Among the many application prospects, with the use of HHG as an EUV light source being a key aspect. The realization of EUV light sources based on gas media HHG has been extensively studied. In the future, generating high-repetition-rate EUV light sources based on HHG will be an important research direction. These sources will also be a crucial tool for measuring material structures and achieving higher-precision time- and space-resolved imaging of microscopic particles. However, the generation of high-repetition-rate HHG based on gas media still faces challenges, as driving HHG in gas media requires high single-pulse energy. This places stringent requirements on both the repetition rate and single-pulse energy of the driving light source. Compared with gases and bulk media, metasurfaces have the advantage of enhancing local optical fields, which reduces the single-pulse energy requirements for HHG. This enables the generation of high-repetition-rate EUV light based on metasurfaces. Moreover, the compact structure and on-chip integration of metasurfaces endow them with significant potential for use as platform-based EUV light sources. In summary, the study of HHG based on metasurfaces is not only a significant breakthrough in the field of nonlinear optics, but the interdisciplinary research. This work provides a new choice for improving the performance of optical devices, and drives the advancement of fields such as biomedicine, optical information science, and quantum technology, and offers theoretical support for the design of novel materials. The distinctive properties of metasurfaces render them an ideal platform for HHG, and this research orientation will unlock new possibilities for the development of frontier domains.

Here, we review the resonance mechanisms and control of HHG in metasurfaces, highlighting the advantages of metasurface-based HHG and its promising applications. Although HHG based on metasurfaces is still limited by the relatively low conversion efficiency, low harmonic order and damage threshold of the materials, these challenges related to damping are being addressed with the advancement of micro/nanofabrication technologies, the application of new materials, and structural optimization through algorithms [173]. We believe that metasurfaces will provide strong support for the future development of ultrafast optics.

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