{'fraction_non_alphanumeric': 0.026238532110091743, 'fraction_numerical': 0.014831804281345565, 'mean_word_length': 5.062476826103078, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
All-dielectric χ (2) metasurfaces: recent progress

Optical metasurfaces, i.e. arrays of nanoantennas with sub-wavelength size and separation, enable the manipulation of light-matter interactions in miniaturized optical components with no classical counterparts. Six decades after the first observation of the second harmonic generation (SHG) in bulk crystals, these devices are expected to break new ground in the field of nonlinear optics, shifting the focus from the phase matching approach achieved within long propagation dis-tances to that of near-field resonances interplay in leaky nanocavities. Here we review the recent progress in SHG with all-dielectric metasurfaces. We discuss the most used technological platforms which underpinned such advances and analyze different SHG control approaches. We finally compare their performances with other well-established technolo-gies, with the hope to delineate the current state-of-the-art and figure out a few scenarios in which these devices might soon offer unprecedented opportunities.

The possibility to control, trap and shrink light down to the sub-wavelength scale 1 is one of the ultimate objectives of nanophotonics. At larger scales than light wavelength, the properties of the electromagnetic field can be harnessed via the electric permittivity and the magnetic permeability of the materials which host it. As such, the range of possible implementations in the visible and near-IR spectrum is restricted by the narrow window of accessible permittivities with natural dielectrics (typically ). In order to overcome such limitations, both metals and dielectrics can be fabricated with a nano-structuration at the sub-wavelength scale, resulting in optical components with effective properties beyond those of bulk crystals, hence the name meta-optics. For example, single nanoantennas can act as optical transducers 2,3 , confine light in deep sub-wavelength volumes 4,5 or control spontaneous emission 6 from single molecules 7 and quantum-dots 8 , while nanopatterned 2D or 3D arrays can manipulate light propagation 9−11 and group velocity 12 . Such arrays correspond to metasurfaces, metamaterials and photonic crystals. The term metasurface, as a whole, includes both blazed binary gratings or high-contrast transmitarrays, designed to outperform conventional diffractive optical elements, and periodic arrangements of metallic or dielectric nanoresonators. In recent years, the latter have drawn the attention of the scientific community, especially due to the phenomenological richness of highly multi-mode Mie-type resonators. This unique feature emerged as an intriguing solution to induce non-trivial interference effects, which lead, among others, to the observation of Fano resonances 13−15 , unidirectional scattering 16−18 , anapole states 19 , and bound states in the continuum 20−23 (BICs). Although metals proved attractive to obtain resonances with extremely small mode volumes 5 , in some applications their absorption losses can be a problem 24 , and dielectric nanostructures 25 have progressively set in as a bettersuited solution. This is the case, among others, of biosensing 26−28 , quantum optics 29−31 and nonlinear devices 32 . Strong field confinement in photonic structures is an important ingredient for nonlinear optical generation to create sources of entangled photons 33 , tunable cavities 34 , and ultra-fast modulators 35−37 . In this context, nonlinear metasurfaces made of purely dielectric resonators appear as a promising alternative to already existing platforms, like nonlinear waveguides. This domain is evolving rapidly, and several reviews have already been published on the topic 38−41 .
The present work aims at reviewing the recent advancements in the field of nonlinear all-dielectric metasurfaces operating in the visible/near-infrared (IR) window, with a particular focus on second harmonic generation (SHG). The manuscript is organized as follows: in Section SHG in all-dielectric nanoantennas we introduce the topic of nonlinear generation in Mie resonators, pointing out the main figures of merit for a high efficiency in non-Hermitian systems. Then, in Section SHG in all-dielectric metasurfaces, we overview the main approaches adopted in recent years to control or boost harmonic generation in metasurfaces. Finally, the conclusion in Section Conclusion and outlook provides a comparison with other existing technologies, outlining the current state-of-the-art and the possible perspectives in the near future.

SHG in all-dielectric nanoantennas
Historically, nonlinear generation at the nanoscale was firstly detected in metallic nanoresonators 42−47 , due to the easier fabrication technology and more established characterization of these structures. The excitation of localized surface plasmon resonances (LSPRs) in metallic d λ/n d ≈ λ/n nanoparticles provides a strong field enhancement close to the metal surface, where the centrosymmetry of the material is broken and both third-and second-order nonlinear phenomena can be revealed. It took almost ten years before the first demonstration of harmonic generation in silicon nanodisks 48,49 , but these pioneering studies immediately revealed two main advantages of dielectric over metallic nanostructures: negligible ohmic losses, and a remarkable field enhancement inside the resonator volume, which enables to exploit not only surface nonlinearity but also the stronger bulk nonlinearity in the whole volume. In dielectrics, Mie-type resonances originate when the diameter of the nanoparticle is reduced so as to become comparable with the effective wavelength in the medium. At variance with their plasmonic counterparts, the first mode to appear (when ) is a magnetic dipole resonance, and in turn the electromagnetic field can be concentrated in the core of the nanoparticles even for small sizes. n χ (2) In this framework, the nanofabrication processes of III-V semiconductors guided-wave optics enabled the development of optical nanoantennas with noncentrosymmetric crystal structure and thus second-order nonlinear response. The most widely used materials are reported in Table 1, which provides a few factors that can explain the pros and cons of a given platform with respect to the others: 1) a high-refractive index or, more precisely, the possibility to achieve a high refractive-index contrast between the nanoresonator and its substrate ① , which in turn determines how tightly the field can be confined; 2) a strong quadratic bulk susceptibility ; and 3) a broad transparency spectral window. These points motivated a series of experimental investigations on SHG in GaAs 50 , Al 0.18 Ga 0.82 As 51,52 and GaP 53 nanoresonators, with a maximum conversion efficiency 51,52 , and more recently on lithium niobate 54,55 and multi-quantum-well structures 56 . Currently, transition metal dichalcogenide 57 (TMDC) platforms are also being explored, but their performances are presently limited by fabrication technology and a smaller nonlinear overlap provided by these monolayers than nanoresonators, as it will be more clearly shown in the following.
Far from being a mere outcome of technological development, the micro-to-nano transition subtends a real paradigm change for optics, with analogies and striking differences between these two worlds. At variance with microcavities 63,64 , nanoantennas are open systems with non-negligible coupling to free radiation 65 , and the related nonlinear processes can be described as follows, within the formalism of non-Hermitian physics 66,67 (see ω ω m = Ω m − iΓ m /2 • An external pump at fundamental frequency (FF) transfers a part of its energy to the resonator, by exciting its leaky eigenmodes with complex eigenfrequencies ; κ in • The coupling coefficient , which describes the injection efficiency into such modes, depends on their spectral and spatial overlap with the FF field; • The eigenmode at FF couples with higher-order modes at the second harmonic (SH) through the quadratic susceptibility . The coupling efficiency depends on the spatial overlap of these modes, mediated by the nonlinear tensor; 2ω • The field at is radiated with a time constant , which is inversely proportional to the mode quality factor ; the coupling efficiency with the different radiative channels are described by the coefficient m l Assuming for sake of simplicity that only two modes are involved, which we label with at FF and at SH respectively, we can quantify the SHG efficiency as 67,68 where, with respect to the formulations in ref. 67,68 , we included the spectral overlap inside the coupling coefficient. The mode spatial overlap in (1) determines the selection rules for nonlinear generation in nanoresonators, which can be derived analytically for the particular case of a spherical resonator in a uniform medium 69 , but generally have be computed numerically 67 . In the configuration of Fig. 1, the nanoresonator behaves like an antenna: it receives an external excitation, it stores the energy in the near field converting the FF signal to a localized current distribution at SH, and the latter radiates to the far field. In order to operate properly, an antenna has to maximize the radiated energy and the spectral bandwidth (low Q-factor), as well as the coupling with free space, i.e. impedance matching (high ). Indeed, the first experimental investigations in this domain exploited low-order Mie-type resonances in dielectric nanoparticles, in many cases a magnetic dipole 48,52,70 , with low quality factors (of the order of 10) as for most dipolar antennas 71 . These features are clearly different from those of guided optics: nanoantennas operate in free space and can be integrated in optical cavities; they can provide a larger in-and out-coupling efficiency than the evanescent coupling adopted in many guided systems, and they typically exhibit a broader spectral and spatial acceptance.
However, as it is clear from Eq. (1), maximizing SHG requires high-Q resonances. Ideally, one would envisage suppressing all the radiation channels apart from those used to inject and extract power, in order to both efficiently pump the resonator and increase the quality factor of the nonlinearly interacting modes. In recent years, several strategies have been proposed for boosting nonlinear generation by optimizing optical confinement 72,73 , through anapole states 53,74−77 or high-Q BIC modes 68,78,79 , and controlling SH radiation pattern 80−85 or polarization properties 52,86,87 .
These theoretical and experimental studies allowed to extensively investigate nonlinear optical effects in open cavities and engineer their emission properties. However, for many practical purposes, it is clearly more advantageous to dispose of these elementary units, called in the following meta-atoms, in large arrays. According to the specific end, different solutions have been explored and they will be analyzed in detail in the next section.
To provide a fair comparison in terms of frequencyconversion performances, a unique figure of merit has to be defined. Hereafter, we will refer to the power-independent conversion efficiency . However, in some cases, the conversion ratio will be reported along with the peak pump power , to provide a helpful indication of the maximum SHG power before the onset of saturation or other undesired effects. All the reported powers refer to peak values of temporal pulses.

SHG in all-dielectric metasurfaces
Arranging nanoantennas in 2D arrays, i.e. metasurfaces, not only increases the active surface and in turn the nonlinearly generated power, but it also enables a mutual interaction among the meta-atoms as an additional degree of freedom to control nonlinear emission. The field perceived by each nanoantenna in the array is the sum of the incident and the scattered waves by all the neighboring elements. According to the ratio of these two contributions, one of the two following regimes prevails: • Far-field interaction. For large inter-particle distances, but still in the sub-diffraction limit, mutual interactions act as a minor perturbation. The effect of the scattered field by the nanoparticles in the array is to slightly enhance the electromagnetic field confinement and the quality factor of the resonant modes, yet the main radiation properties of isolated meta-atoms stay unaltered.
• Near-field interaction. In compact arrays, optical coupling between close neighbors can strongly perturb the resonant behavior of isolated nanoparticles and promote the onset of high-Q delocalized modes, which are particularly interesting for nonlinear generation.
It is also worth mentioning that for larger periods, comparable with the incident wavelength, the in-plane first diffraction order is no longer evanescent (Rayleigh anomaly) and additional peaks, due to surface lattice resonances, appear in the extinction spectrum 88,89 . This case will not be analyzed in detail in the present work, and we therefore refer the interested reader to thematic reviews 90−92 .
Let us also remark in passing that optical nanocavities and metasurfaces have been often suggested as an intriguing solution to avoid the well-known need of phase matching in microcavities. However, it is worth noting that such statement is not entirely appropriate for two reasons: 1) since phase matching is a condition aimed at maximizing the in-phase oscillation of overlapped FF ζ lmm ζ lmm and SH fields, it is not a problem to avoid but an opportunity to exploit; 2) its concept stems from the overlap integral which, in translationally or rotationally invariant cavities, can be decomposed in a transverse and a propagative contribution. In nanocavities this decomposition is no longer feasible, but still has an amplitude and phase which has to be taken into account to maximize SHG efficiency 67 , especially in the case of delocalized collective resonances.

Quasi-independent nanoantennas
In the radio-frequency (RF) domain, antenna arrays are used to control the far-field properties of emitted signals. In that case, local sources enable to separately drive each element of the array whose behavior is totally independent. In the case of metasurfaces for nonlinear optics the scenario is a bit different. The driving term usually stems from free radiation and non-Hermitian cavities are always coupled due to their leaky nature, even if their distance is large compared to the wavelength 93 . This especially holds true for low-order Mie resonances like electric (ED) and magnetic (MD) dipoles, insofar as they strongly scatter light in the far field. However, for arrays of Si nanoparticles with a period large enough to apply coupled-dipole approximation, it was shown that the scattering properties at wavelengths larger than the interparticle distance are mostly determined by the scattering properties of individual constituents 89 . Figure 2(a) compares the near-IR scattering spectrum of an isolated GaAs nanocylinder, computed with perfectly matched layers at the borders of the simulation domain, and the transmission spectrum of arrayed elements with sub-λ distance. Decomposing the field inside a nanocylinder of the array into vector spherical wavefunctions, it appears that the dips in the transmission spectrum of the array originate from Mie-type resonances of isolated elements, rather than non-local Bragg resonances typical of the photonic crystal regime 94 . The in-plane scattering of neighboring structures essentially results in an electromagnetic-field enhancement inside the nanoparticle volume, see Fig. 2(b), and a partial blue-shift of Mie-type resonances at large wavelengths, whereas additional lattice resonances appear when the wavelength is comparable with the period.
These features were formerly exploited to enhance the nonlinear generation from nanocylinders relying on the properties of isolated elements, for the design of metasurfaces working in the sub-λ regime at FF and featuring diffraction lobes at harmonic frequencies 48 25 , which both enhances the field inside the resonator volume and keeps its size small. The choice of the period is usually aimed at maximizing the density of resonators in the array, while keeping the role of non-local interactions weak.
I far tot N Within this approach, in analogy with antennas arrays in the RF domain, the total SH far-field intensity emitted from the metasurface can be written as the sum of quasi-independent emitters: where is the array factor and is the emission from a single element. The role of the array is therefore to modulate the radiation pattern of the isolated emitter. While for a given metasurface is set, one can modify the nonlinearly excited modes and by changing the pump configuration (e.g. its polarization) 67,103 . In turn, depending on the state of the pump, the nonlinear emission can be funneled into different diffraction lobes, see Fig. 3(a, b). However, in many cases of practical interest it is more convenient to control the SH polarization while staying in the same diffraction order (e.g. the zero-diffraction order for on-axis applications). In these circumstances the lattice constant can be optimized to maximize the overlap between and and partially redirect the nonlinear generation, see Fig. 3(c, d).
The low-Q antenna regime lends also itself to nonlinear beam shaping applications, as it can combine a broad spectral bandwidth with a large coupling to free space. At variance with linear metasurfaces, where resonances are usually detrimental for wavefront shaping 104 , here a strong local field enhancement and thus resonances are necessary for an efficient nonlinear conversion, as highlighted in the previous section. In recent years, two main mechanisms have been shown to encode the harmonic field phase: 1) tuning of the interaction between FF and SH modes by changing the size of the meta-atoms; 2) extension of Pancharatnam-Berry phase concept to the nonlinear regime 105 .
[0, 2π] The former was inspired by a series of works on ultrathin plasmonic metasurfaces 110,111 and later extended to third harmonic generation 112,113 . More recently, it was investigated in detail to design highly directional emitters implementing the first Kerker condition at harmonic frequencies, and thus to reproduce the concept of Huygens' metasurfaces in the nonlinear regime. Wang et al. 106 predicted that balancing the excitation of modes with opposite symmetry is possible to extract a set of resonators with unidirectional THG covering the whole phase range. They experimentally validated this approach with a Si metasurface steering the THG beam into the first diffraction order, with a record 92% efficiency, see Fig.  4(a). The possibility to upconvert an optical signal and control its wavefront within a sub-micrometer-thick metasurface is not only attractive for scalability and integrability purposes, but it also enables to manufacture optical devices that would not be otherwise conceivable with bulk elements. This idea allowed e.g. to study the spatial correlation between an object and its upconverted  image, when observed through a nonlinear meta-lens, and generalize the lens equation to the nonlinear regime 107,114 , see Fig. 4(b).
The extension of this first resonant approach to the SH case was not straightforward due to the selection rules imposed by the tensor of most widely used noncentrosymmetric materials, which lead to a null on-axis SHG from axially symmetric nanoparticles, impractical for many applications. A workaround solution to this issue was recently demonstrated thanks to AlGaAs-onsapphire meta-atoms, with a broken symmetry that helps to rephase the two SH radiation lobes at large angles and favors normal emission 108 . Such concept was validated on SH beam steering and focusing, see Fig. 4(c), with a normalized conversion efficiency . On the other hand, nonlinear geometric-phase control in centrosymmetric metasurfaces relies on the following idea: when a nanostructure with -fold rotational symmetry is excited by a pump beam with circular polarization state , the -th harmonic dipole moment acquires a phase , where is the rotation angle of the nanostructure and the ' ' and '-' signs correspond to the same and opposite circular polarization with respect to the pump beam, respectively 105 . The symmetry order imposes selection rules on harmonic generation as only the orders , with an integer, are allowed. This principle was validated by a series of explorations on C2 and C3 symmetric metallic nanostructures for THG and SHG, respectively 105,114−116 . More recently, the same mechanism was exploited to encode the phase in Si nano-blocks and demonstrate TH holography in alldielectric metasurfaces with a factor-40 generation enhancement with respect to a bare Si film 117 . Bar-David et al. 109 also showed that centrosymmetry breaking close to the surfaces can be exploited in Si, as done in plasmonic nanoparticles, to induce an effective second-order nonlinearity. Resorting to C1-symmetric meta-atoms, see Fig. 4(d), it was therefore possible to manipulate SH phase in dielectric metasurfaces, yet with a generation efficiency 4 orders of magnitude lower than in metals, which can be attributed to a much smaller field enhancement at the surfaces compared to the case of LSPR. Compared to the previous approach, geometric phase control offers unique advantages as it directly ensures an equal emission amplitude from the different meta-atoms and it requires the optimization of a single structure.
The above-mentioned shaping techniques rely solely on the geometry of the resonators. A different solution, recently demonstrated by Dasgupta et al. 118 , relies on the strong spin-orbit coupling, due to degenerate valleys in TMDC monolayers, to generate optical vortex at the SH with different topological charges. The valley-dependent spin selection rules, in combination with Mie-resonances, might offer a further degree of control on the properties of the generated beam in analogue way as reported for hybrid Au-WS 2 metasurfaces 119 .

Collective resonances in nonlinear metasurfaces
Mie resonators are highly multi-mode open cavities. Their eigenmodes, being non strictly orthogonal within the resonator volume 120 , can either interfere giving rise to Fano resonances 13,15 or strongly couple to each other, originating Friedrich-Wintgen BICs 21,22 . Exploiting this approach, Koshelev et al. 68 reported an experimental record quality factor in an isolated AlGaAs resonator with a maximum SHG efficiency . To go beyond this value, one can either increase the resonator size to excite high-Q whispering gallery modes as in micro-disks or exploit interference between multiple nanoantennas. The latter option was first demonstrated by Yang et al. 121 to boost THG in Si metasurfaces. The coupling between a bright and broad ED mode and a dark MD with smaller damping results in a sharp Fano resonance with an experimental in the transmittance spectrum. The same structure was exploited to enhance high harmonic generation 122 and design lithium niobate metasurfaces for SHG 123 . In the latter case, a sharp Fano resonance ( ) was theoretically predicted to enhance SH conversion up to a value for a pump intensity GM/cm 2 .

Q → ∞
The symmetry of modes in periodic structures plays a central role to confine photons for longer times and thereby boost nonlinear generation. As was shown for photonic-crystal slabs 128 or lamellar gratings 129 , the presence of guided resonances or asymmetric Bloch modes, which are symmetry protected at normal incidence, can result in sharp peaks in the transmission spectrum with a typical Fano line shape, as soon as the in-plane k-vector is different from zero. When the symmetry of the system is perfectly preserved, i.e. in the case of a normally impinging plane wave on a symmetric array, modes with opposite symmetry are completely decoupled promoting the creation of BIC modes infinitely confined in the structure 130 ( ). Please note that, strictly speaking, symmetry protected BIC modes exist only in infinite Gigli C et al. Opto-Electron Adv 5, 210093 (2022) https://doi.org/10.29026/oea.2022.210093 structures, and the presence of an edge would destroy this condition. By reciprocity, these modes cannot be excited from free space and they are not of practical interest. However, by slightly breaking the symmetry of the system, one can observe the presence of quasi-BIC modes with large quality factors. This result can be achieved either by slightly tilting the incident beam or by carefully designing asymmetric meta-atoms. Several configurations are analyzed and compared in ref. 131 , and they have been the initial guess for the design of many nonlinear metasurfaces, see Fig. 5. Relying on the broken symmetry resonators proposed in ref. 132 , Vabishchevich et al. demonstrated a GaAs metasurface with a maximum field intensity enhancement inside the resonators volume and , see Fig. 5(a), resulting in threetimes higher SHG conversion efficiency than in nanodisk metasurfaces with a standard MD resonance 50 . In the Fano picture, such symmetry breaking can be seen as the introduction of a coupling term between bright and broad resonances with dark and narrow ones, which would be otherwise uncoupled by symmetry. The close link between quasi-BIC modes and the appearance of Fano resonances was adopted to design several metaatoms with broken symmetry to enhance nonlinear effects. In ref. 133 , the authors resort to spindle-shaped Si nanoparticles to implement a metasurface that features a resonance with and 300 times enhanced THG with respect to a uniform slab. Similarly, ref. 125 reports a maximum TH conversion ratio for relying Q Q r Q nr 1/Q = 1/Q nr + 1/Q r Q r = Q nr on meta-atoms made of two Si nanorods with different sizes, see Fig. 5(b). Importantly, in the same work the authors demonstrate the central role played by the nonradiative losses due to surface roughness, structural disorder and finite size of the array. The total quality factor exhibits a radiative ( ) and a non-radiative ( ) contribution according to , and the maximum conversion efficiency is achieved at critical coupling ( ).
The same meta-atoms have also been used to boost the nonlinear response of TMD monolayers. In ref. 66 the authors measured a SHG enhancement of ≈1140 times with respect to the bulk Si film, in a layer of tungsten disulfide (WS 2 ) deposited on a Si metasurface exhibiting a quasi-BIC with at critical coupling. A collective resonance with Q-factor ≈ 550 from a similar Si metasurface was also exploited in ref. 134 to boost the nonlinearity of an MoS 2 monolayer, detecting a maximum conversion efficiency .
A larger non-radiative was experimentally achieved with Si nanoblocks 126 featuring a symmetry breaking along one in-plane direction, see Fig. 5(c). However, due to their available broadband short-pulse source, the authors could only obtain a conversion efficiency exploiting a resonance with a smaller . The last symmetry breaking mechanism shown in Fig.  5(d) relies on zigzag arrays of nanocylinders with elliptical basis. The high-Q resonance due to the quasi-BIC mode and the strong field enhancement close to the res-  1.0 S180 S175 S170 S165 0.  ) and a SHG efficiency in continuous-wave regime, which constitutes one of the most promising result in the near-IR with metasurfaces to date.

Conclusion and outlook
Most of the aforementioned studies aimed at optimizing the electromagnetic field confinement inside the metasurfaces, with the primary goal of enhancing nonlinear conversion efficiency. Let us therefore compare their performances with those of other more established photonic platforms aiming at the same target through different physical phenomena.  Figure 6 reports the SHG efficiencies of several published works in the last two decades, grouped according to different types of devices and materials. Guided systems like optical waveguides 62,135 , ring 136−138 or disk 139−144 resonators provide tight electromagnetic field confinement in nonlinear materials for long propagation distances, leading in turn to huge conversion efficiencies (up to ). However, the disks featuring the best performances have a large footprint (of the order of the mm 2 i.e. not fully compatible with high integration), while conversion efficiencies drop to for smaller LiNbO 3 and GaAs microdisks.   Phased-matched guided systems enable high generation efficiency, but they are typically not well suited when a large coupling with free radiation is required or when light confinement has to be squeezed to nanomet-ric volumes. In this perspective, photonic-crystal waveguides 145 and nanocavities 146 gained an increasing interest in the community. The creation of defect states in photonic-crystal slabs (e.g. L3 cavities) enabled to confine the normally impinging light in ultra-small mode volumes and enhance SHG in III-V devices 147−151 .
However, in photonic crystals as well as in metasurfaces, a maximized generation efficiency stems from a well-balanced compromise between field enhancement and mode volume. Indeed, the exploitation of spatially extended resonances can increase the effective interaction volume between harmonic fields and boost nonlinear conversion. Based on the concept of symmetry-protected BIC introduced in Section SHG in all-dielectric metasurfaces, Minkov et al. 152 designed a doubly resonant photonic crystal slab heterostructure providing a fully confined defect mode inside the photonic bandgap at FF and a resonant quasi-BIC mode inside the light cone at SH. Wang et al. 153 experimentally demonstrated that, when pumped at normal direction, such structure generates a radially polarized SH with a conversion efficiency . This good performance arises from the fulfillment of a doubly resonant condition and the optimization of the nonlinear overlap integral 152 . The comparison reported in Fig. 6 brings us to a first relevant conclusion: if one considers conversion efficiency as the sole target, to date guided systems provide the best performances by orders of magnitude over all other platforms. On the other hand, both photonic crystals and metasurfaces offer unprecedented opportunities for integrability 154 , on-axis applications 153 , ultra-fast switching 35,37,155 , and emission control 156 . Metasurfaces appear inherently more robust to the breaking of translational invariance induced by the presence of the edges, and as such more effective to achieve strong resonant enhancement also for small footprints 126 . χ (2) In conclusion, we have reviewed the recent progress in the rapidly evolving domain of all-dielectric metasurfaces. At least two possible outlooks emerged for these devices.
On one side, resorting to coupled nanoantennas and collective resonances seems the wisest strategy to maximize nonlinear generation. However, also the metasurfaces exhibiting the largest quality factors are orders of magnitude less efficient than the other platforms in Fig.  6. This outcome stems from the fact that to date most of the studies adopting this approach, and discussed in Section SHG in all-dielectric metasurfaces, focused on the creation of a single high-Q resonance around FF. Instead, we expect that such gap with the other technologies may be filled with a careful design providing a good balance between free-space coupling and mode quality factors, while implementing a doubly resonant condition and optimizing the nonlinear overlap integral, as described in Eq. (1).
On the other side, low-Q antennas represent a true change of paradigm with respect to both guided structures and photonic crystals. Their presently lower SHG efficiency is largely counterbalanced by intriguing possibilities ranging from the dynamic tunability of individual meta-atom emission 82,155,157,158 , pulse shaping 159 , broadband parametric devices 96 , nonlinear imaging 107 , wavefront shaping 108 and meta-holography 112,160,161 . Their rapid development is currently underpinned by an ongoing progress in nanofabrication, new promising nonlinear materials like TMDCs, and both analytical 69,162 and numerical 67 methods to model nonlinear generation in leaky cavities. The improvement of such mathematical tools seems especially important for non-intuitive design and optimization of highly multi-mode nanoresonators 163 .
Based on the impressive achievements of this new branch of nonlinear optics, we expect a new class of nonlinear photonic meta-devices to arise in the forthcoming years, for high-speed switching, entangled photon sources, supercontinuum generation and nonlinear imaging.