## Abstract

Nonlinear metasurfaces have recently brought harmonic generation to subwavelength level, with spectral and polarization control unachievable in bulk crystals. Not only does nonlinear meta-optics enable the investigation of nonlinear physics at the nanoscale, but it also opens promising technological perspectives. To date, however, no full-phase control has been demonstrated on a harmonic field generated with a sufficient efficiency for most practical purposes. In this work, based on the ${\chi ^{(2)}}$ fully tensorial features of AlGaAs metasurfaces, we propose a robust method to generate ${0 - 2}\pi$ phase-encoded second harmonic light for nonlinear wavefront shaping, experimentally demonstrating nonlinear beam steering and all-dielectric metalenses that generate and focus second harmonic beams to sub-wavelength spot sizes. Nonlinear generation with record efficiency and phase control are achieved in nanostructured arrays that are fully compatible with mature III-V semiconductor technology. This breakthrough paves the way for the development of ultrathin, free-space photonic devices for nonlinear imaging, including night vision.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## Corrections

23 March 2021: A typographical correction was made to the author affiliations.

## 1. INTRODUCTION

In recent years, the control over subwavelength light–matter interactions enabled the observation of new linear optical phenomena and the demonstration of promising devices. Two-dimensional (2D) arrangements of metallic or dielectric nanostructures (i.e., optical metasurfaces) allowed for locally shaping the phase of scattered light, establishing a new class of ultrathin devices [1–4] with reported high efficiencies in applications like polarization beam splitting [5], subwavelength focusing [6], and vortex generation in laser cavities [7].

To extend metasurface functionalities and implement nonlinear operations, the scientific community has considered optical metasurfaces for the control of harmonic emitted fields [8,9]. Both fundamental physics and exciting engineering perspectives are presently motivating forefront research, from quantum metasurfaces [10] and topological nonlinear nanophotonics [11] to CMOS-based, upconversion devices for night vision [12].

Light scattering in nonlinear metasurfaces is giving a twist to the 60-year-old field of nonlinear optics, with the role of phase matching being replaced by near-field resonances in non-Hermitian open nanostructures [13,14]. The generation of a harmonic signal has been first demonstrated with metallic nanostructures, where electromagnetic field enhancement by localized plasmon resonances has been exploited for both second- and third-harmonic generation (SHG and THG) [15,16]. Those grounds, combined with experience on phase encoding with metallic meta-grating in the linear regime, led to the first demonstrations of plasmonic nonlinear metasurfaces for THG [17,18] or SHG [19]. They rely on two possible mechanisms: the properties of localized plasmon resonances to control the phase of fundamental frequency (FF) field and thus the nonlinear currents [18] or the extension of the Pancharatnam–Berry phase to the nonlinear regime [17]. However, plasmonic resonators are limited by ohmic losses and weak surface nonlinearities, with maximum frequency conversion efficiencies on the order of ${10^{- 8}}$ before sample damaging in the case of THG. Slightly higher efficiencies have been demonstrated with hybrid metasurfaces for beam steering [20], where frequency conversion and wavefront control are provided by coupled dielectric and plasmonic nanostructures, respectively.

Nanofabrication progress also facilitated the development of subwavelength all-dielectric nanostructures with high refractive index contrast, which can combine bulk optical nonlinearities, transparency, and strong field confinement inside their volume. Pioneering work on isolated silicon nanoresonators resulted in THG efficiency of ${10^{- 7}}$ before saturation [21]. In this regard, the silicon-on-insulator (SOI) platform enabled the demonstration of fully 3D ${\chi ^{(3)}}$ metasurfaces for the generation of TH vortex beams [22], nonlinear imaging [23], and nonlinear holography [24,25], with reported conversion efficiencies on the order of ${10^{- 6}}$ [22–24]. Despite this improvement with respect to nonlinear plasmonics, such THG efficiency still seems to be low for possible applications of wavefront engineering on the harmonic field. Moreover, two-photon absorption (TPA) in Si limits the range of the pump power before the onset of saturation. This is why the much higher nonlinearity of non-centrosymmetric quadratic media [26] has strongly motivated the development of ${\chi ^{(2)}}$ nanophotonics [27–30], even if their full tensor nature makes metasurface design more challenging. Among these materials, the most investigated in nanophotonics to date have been the III-V semiconductors GaAs [27] and AlGaAs [28,29], whose gigantic ${\chi ^{(2)}}({\approx}200{\rm pm/V})$ resulted in SHG efficiencies of ${10^{- 6}}$ in a single nanoantenna [28,29,31], and $5 \times {10^{- 5}}$ in a metasurface [32]. Furthermore, III-V technology provides a large control on alloy composition and energy bandgap, thus enabling TPA-free operation in the near infrared.

However, the crucial milestone of efficient SHG with a ${0 - 2}\pi$ phase control in an all-dielectric metasurface has not been reached to date. We ascribe this difficulty to the inherent anisotropy of AlGaAs quadratic susceptibility, which makes metasurface design more challenging.

In this paper, the theoretical grounds for tensorial full phase covering with ${\chi ^{(2)}}$ metasurfaces are provided in Section 2, while Section 3 is dedicated to the demonstration of such SHG wavefront engineering. Specifically, we first validate experimentally that a predominantly on-axis SH can be obtained with a (100) AlGaAs nanochair resonator, as predicted by some of us in [33] (Section 3.A). Then, based on such meta-atoms, we provide the guidelines for phase-encoded SHG in the corresponding metasurfaces (Section 3.B). After testing the latter on the benchmark of SH beam steering (Section 3.C), we demonstrate highly efficient quadratic nonlinear metalenses with different focal distances and subwavelength spot sizes (Section 3.D). Finally, Section 4 includes our numerical (Section 4.A), fabrication (Section 4.B) and experimental (Section 4.C) methods, and in Section 5 we discuss our results, indicating some perspectives for functional metasurfaces in nonlinear imaging.

## 2. THEORY

At variance with the ${\chi ^{(3)}}$ response of silicon, which can still be isotropic with a nonlinear polarization $P_i^{(3)} = {\varepsilon _0}\chi _{\textit{iiii}}^{(3)}E_i^3$ ($i = x,y,z$), the tensor nature of ${\chi ^{(2)}}$ in zincblend crystals ($\bar 43\,\,{\rm m}$ symmetry) results in a nonlinear polarization vector of the type reported hereafter, with ${d_{14}} = {d_{25}} = {d_{36}} = \chi _{\textit{xyz}}^{(2)}/2$, so

The operation of a nonlinear metasurface for SHG with a controlled wavefront first requires the efficient generation of nonlinear current sources in the constituent meta-atoms (i). In each of them, the excitation coefficient ${\alpha _l}$ of the $l$-th resonant mode at SH scales as

where ${Q_l}$, ${Q_{m,n}}$ are the quality factors of the excited mode $l$ at SH, and modes $m$, $n$ at FF respectively, and ${\zeta _{\textit{lmn}}}$ is the overlap integral of the three [13,34]. Thus, the natural choice for obtaining high SHG efficiency is to employ high-Q modes at FF. Indeed, high-Q resonances result in sharp spectral features and high sensitivity to their environment, which in passing makes them ideal for sensing, ultrafast switching, or all-optical modulation [35,36]; albeit less attractive for broadband operation. An additional requirement (ii) is full (i.e., 0 to ${2}\pi$) phase covering of the nonlinear current source,Taking inspiration from phased-array antennas [20], a subwavelength array of independent dielectric resonators provides a suitable solution to locally generate nonlinear currents that act as secondary sources of harmonic fields. However, the unwanted optical coupling between neighbor resonators, which occurs when one minimizes their distance to finely sample the SHG phase, is more difficult to control in case of high-Q modes, which stem from nonlocal interactions. Satisfying both conditions (i) and (ii) with a single element requires a well-balanced trade-off, which can be achieved with low-Q all-dielectric resonators. Moreover, condition (iii) is not available in the case of (100)-epitaxial AlGaAs cylindrical nanoresonators. The reason for this is that (100) crystallographic orientation results in a quadratic susceptibility tensor with nonzero elements $\chi _{\textit{ijk}}^{(2)}$ only for $i \ne j \ne k$. As a consequence, when such cylindrical resonators are excited at normal incidence, the symmetry of the system imposes a null for SHG in the normal direction [29,37], as shown in Fig. 1(a). Tilting the excitation beam [37] or exploiting the constructive interferences of periodic arrays in the far field [32] can offer a workaround solution for some applications. However, to efficiently control the SH wavefront for beam-shaping applications, a main emission close to the normal direction is required under normal incidence of the FF beam.

In this framework, two different pathways have been recently proposed to break the system symmetry: (1) moving to a different AlGaAs crystallographic orientation like (111) or (110) to change the selection rules imposed by the ${\chi ^{(2)}}$ tensor [38]; or (2) resorting to axially asymmetric nanostructures to excite resonant SH modes with a highly directive radiation pattern close to the normal [33]. Here, because of the robust technological protocols developed for (100) AlGaAs, we opt for the latter solution. Figure 1 depicts the starting point of the present work: the design of a set of nanochair-shaped resonators to control the phase of on-axis SHG.

## 3. DESIGN AND EXPERIMENTAL RESULTS

#### A. On-Axis SHG from (100) AlGaAs Metasurfaces

The unit cell of our (100) AlGaAs metasurface is designed by developing the concept proposed and numerically investigated in [33]. As sketched in Fig. 2(a), it consists of a cylinder with elliptical basis, where one fourth of the volume has been etched away, and the two halves with heights $h$ and $h/2$ enable SHG along the normal to a sapphire substrate. The latter allows working in a transmission geometry. In this paper, we will restrict our attention to this experimental configuration, which is the most attractive for wavefront-shaping applications. The FF beam, with a wavelength 1550 nm, impinges from the substrate along $z$ and is linearly polarized along $x$. By properly choosing the basis semiaxes $a$ and $b$, it is possible to maximize the generation of a cross-polarized SH field in the forward direction. To operate in a transparency regime at FF and SH, ${{\rm Al}_{0.18}}{{\rm Ga}_{0.82}}{\rm As}$ was chosen for the resonator material because this aluminum molar fraction allows us to increase pump power without incurring TPA-driven SHG saturation [28].

In the following, we will focus on square periodic arrays of many resonators of this type. However, we would like to stress that the behavior of such metasurfaces strictly depends on the properties of the single resonators. When the period is large enough, the properties of the individual structures are mostly preserved in the 2D grating, and the far-field response can be approximated as the product of single-emitter radiation and the array factor [32,39]. For smaller periods, probing the array properties also allows the evaluation of the possible contribution of near-field interactions between neighboring meta-atoms. After careful design and experimental feedback, we are able to maximize SHG close to normal. A detailed comparison between the properties of isolated structures and periodic arrays is reported in Fig. S1 of Supplement 1.

To probe the SHG efficiency and directivity, we first experimentally compared two uniform metasurfaces: one based on the nanochair meta-atoms, as shown in Figs. 2(a) and 2(b), and the other one composed of their cylindrical counterparts. The measurements were performed with a horizontal microscope setup, as described in Section 4 of Supplement 1. The incoming collimated beam with waist ${w_0} \sim 100\;{\unicode{x00B5}{\rm m}}$ ensured a uniform excitation of the entire arrays. The SHG signal was collected in forward direction by a large numerical aperture objective (${\rm NA} = 0.8$) and its radiation pattern was studied through back focal-plane imaging. Figure 2(c) compares the SHG in $k$ space from the nanochair metasurface to a reference array of cylindrical nanoresonators with radius $r = 210\;{\rm nm}$. At variance with the latter, the nanochairs exhibit a main lobe along $z$. Figure 2(d) highlights this feature in a polar coordinate along the $xz$ plane. Integration over the azimuthal angle $\varphi$ vs. the emission angle $\theta$ proves that $60\%$ of the SHG power is emitted for $\theta \lt 8^\circ$. For an input time-average power $\bar P_{\rm FF}=7.4\,\,{\rm mW}$ impinging on the metasurface, a SH power $\bar P_{\rm SH}=96\,\,{\rm nW}$ was collected, as shown in Supplement 1, Fig. S12. Normalization with respect to the laser peak power allows the extrapolation of an intrinsic conversion efficiency ${\eta _{{\rm SHG}}} \sim 9.3 \times {10^{- 7}}{{\rm W}^{- 1}}$ for a single resonator. The reader can refer to Section 4 and Supplement 1 for details on the laser source and the SHG efficiency calculation.

#### B. Control of SH Phase

The prevailingly on-axis SHG in the dielectric resonators of Fig. 2(a) makes this structure an ideal building block for nonlinear beam-shaping applications. Its radiation diagram stems from the interplay of several resonances. Within each nanoantenna, the nonlinearly interacting modes at FF and SH can be reshaped in such a way that the latter has an on-demand phase, while preserving a central emission lobe with a roughly constant amplitude. In practice, a lookup table (LUT) can be computed, which associates a given SH phase to the values of semiaxes $a$ and $b$ of the elliptical basis. Numerically, this was implemented by computing first the FF electric near field with fully vectorial FEM simulations in COMSOL Multiphysics under periodic boundary conditions. Then, nonlinear local currents were calculated in the undepleted pump regime, and finally near- to far-field transformations [40] were adopted to get the SH far-field phase, amplitude at normal direction, and main emission lobe direction from the unit cell.

Importantly, these computations were performed on periodic structures, but all our work strictly relies on the properties of the isolated resonators in Fig. 1(a). We minimize the optical interaction of the antennas in the near field because the scattering properties of the metasurface would otherwise stem from collective effects, and the response would depend on the choice of the unit cell position in the metasurface and not from the LUT predictions. Conversely, if the antennas inside the array are optically uncoupled, the presence of new neighboring resonators with different dimensions does not perturb the properties of the unit cell. This makes the LUT computation robust to any phase profile implementation, instead of requiring any further numerical optimization that strongly depends on the specific application [41]. To ensure this condition, we first computed the properties of isolated structures, then we minimized the unit cell size $\Lambda$, keeping as much as possible unaltered the far-field features. The periodic simulations serve in this sense to monitor the slightest deviation from the resonator response inside the array and optimize geometric parameters. A detailed analysis of LUT dependence on unit cell size is provided in Section 1 of Supplement 1. An optimum value is found for $\Lambda = {900}\;{\rm nm}$, below such value the role of collective modes is no longer negligible. Therefore, the choice of the periodicity implies a trade-off between the requirement of optically uncoupled nanoantennas and a sub-wavelength sampling distance.

Here we accept to work just above the subwavelength regime at SH, coping with the presence of one diffraction order at large angles, which imposes lower bounds for the phase sampling distance and a limit for the diffraction efficiency. As the SH radiation pattern of the unit cell is highly directional with a single lobe close to the normal, as shown in Supplement 1, Fig. S3, most of the power is directed to the zero-diffracted order and the first-diffracted order acts as a perturbation to the final result. A more quantitative analysis can be found in Section 2 of Supplement 1. The results of LUT calculations for a fixed periodicity ${\Lambda} = 900\;{\rm nm}$ and pump wavelength ${\lambda _{{\rm FF}}} = 1550\;{\rm nm}$ are reported in Fig. 3(a). Interestingly, the symmetry of the system offers a further degree of freedom to extend the LUT and more easily satisfy all the imposed constraints: Upon a $\pi$ rotation of the “chair” about the $z$ axis, as shown in Fig. 3(b), the $z$ component of the electric field inside the resonator at FF is $\pi$-shifted while the $x$ component stays unaltered, finally resulting in a $\pi$-shifted nonlinear polarization vector $P_y^{{\rm SH}} = {\varepsilon _0}\chi _{\textit{yxz}}^{(2)}{E_x}{E_z}$. The algorithm for the identification of a set of building blocks seeks resonators with the highest possible equal-amplitude SHG efficiency, a main emission lobe close to the normal, and equally spaced phases in the $[{0,\;2\pi}]$ range. A detailed description of the extracted nanochair geometries can be found in Section 2 of Supplement 1. Please note that LUT properties in Fig. 3 depend on the polarization state of the FF beam, as expected for any material exhibiting an anisotropic nonlinear susceptibility.

#### C. SH Beam Steering

First, we propose the steering of SH light, as a proof of principle for a full ${0 - 2}\pi$ SH phase control from an all-dielectric metasurface. While requiring moderate computational costs and offering useful figures of merit to quantitatively compare modeling and experiments, this constitutes a well-suited benchmark to probe the efficiency of the method before moving on to more complicate phase profiles. Two different metasurfaces have been modeled and fabricated: the former is composed of identical meta-atoms with the same parameters as in Fig. 2(b); the latter includes a supercell of $N = 8$ resonators, which implements a sawtooth phase profile corresponding to a SH steering angle ${\theta _B} = {\sin ^{- 1}}({{\lambda _{{\rm SH}}}/N{\Lambda}}) \approx 6.2^\circ$. Both supercells have been modeled with periodic boundary conditions. The $y$ component of the SH electric near field of the eight meta-atoms, reported in Fig. 4(a), predicts the generation of a wavefront with the designed steering angle ${\theta _B}$. The propagation of a continuous wavefront from the final supercell was verified and is reported in Supplement 1, Fig. S7. The LUT properties are preserved in the supercell, albeit at the expenses of an upper limit of 52% in the forward diffraction efficiency, which is mainly dictated by the size of the unit cell. (For a more comprehensive comparison between LUT predictions and the results of final device, please refer to Section 2 of Supplement 1.) Figure 4(b) reports a SEM picture of the fabricated metasurfaces (see details in the Methods section), and Fig. 4(c) shows the far-field Poynting vector in the $k$ space. SH emission from the supercell has been corrected with the array factor to consider the final size of the metasurface and compare it with experiments. In the former case, all the emitted power is funneled into the zero-diffracted order, while the latter highlights the steering mechanism. Finally, a more quantitative comparison between the numerical model and the experimental result is provided in Fig. 4(d), in polar coordinates: a total of 47% of the SH collected within ${\rm NA }= 0.8$ is directed into the first order, but this value is expected to largely increase with technological accuracy. Funneling of part of the SH power into undesired higher diffraction orders is ascribed to proximity effects in the lithographic process, which introduce a deviation with respect to the nominal values of $a$ and $b$. Diffraction efficiency is maximum for the design wavelength ${\lambda _{{\rm FF}}} = 1550$ nm and remains larger than 40% for ${\lambda _{{\rm FF}}} = 1540$ to 1600 nm, as shown in Supplement 1, Fig. S14. Although numerical predictions in Fig. 4(d) show additional contributions at large angles, due to the limited size of unit cell, the highly directional radiation pattern of meta-atoms funnel most of the SHG into the central lobe, as shown in Supplement 1, Fig. S3. In addition, we expect that optimizing the nanoresonator geometry, the unit-cell size could be reduced and diffraction performances further enhanced.

#### D. Nonlinear Metalenses

Integrated photonics would greatly benefit from the realization of an efficient nonlinear flat lens, not only for its compactness but also for an easier phase control with respect to bulk optics, which relies on the phase-matching condition. The opportunity to generate and focus nonlinear beams to a diffraction-limited spot size with a single metalens would therefore herald new perspectives for nonlinear imaging and nonlinear microscopy. A metasurface focusing the SHG signal at a distance $f$ was designed by imposing an hyperboloidal phase profile to SH emission ${\varphi _{{\rm SH}}}({x,y}) = ({2\pi /{\lambda _{{\rm SH}}}})({\sqrt {{f^2} + ({{x^2} + {y^2}})} - | f |})$.

We designed two different lenses with a diameter of 70 µm and target focal distances of $175\;{\unicode{x00B5}{\rm m}}$ and $70\;{\unicode{x00B5}{\rm m}}$, corresponding to a numerical aperture ${\rm NA} \approx \;0.2$ and 0.5, respectively. The total set of resonators was extracted from the data in Fig. 3(a), by minimizing the error from the target function ${\varphi _{{\rm SH}}}({x,y})$ and maximizing the emission amplitude. A SEM picture of the first metalens is reported in Fig. 5(a). To experimentally characterize its focusing behavior, the SH was captured at different image planes along the ${z}$ axis, with a spatial resolution of 1 µm. The stacked images were post-processed to extract the profile of the emitted beam and the result is reported in Figs. 5(b) and 5(c). The inset of this figure shows the experimental acquisitions at the metasurface plane ($z = 0$) and at the focal plane ($z = f$). A 3D reconstruction of the beam can be found in Supplement 1, Fig. S16. We furtherly retrieved the beam waist at each position to reconstruct the waist profile $w(z)$, which is shown with red dots in Figs. 5(b) and 5(c), implementing a numerical knife-edge analysis detailed in Section 4 of Supplement 1. In the paraxial approximation, assuming a Gaussian shape for the SH beam, the waist profile can be fitted by $w(z) = {w_0}\sqrt {1 + {{({z/{z_R}})}^2}}$, with ${z_R}$ the Rayleigh range, enabling the extraction of the waist radius at focal point ${w_0}$. Approximating the Airy disk pattern at the focal plane with a Gaussian profile, the size of a diffraction-limited spot focused by a lens is ${w_0} \sim 0.42{\lambda _{{\rm SH}}}/\rm NA$. For the two designed lenses, this corresponds to ${w_0} = 1.63\;{\unicode{x00B5}{\rm m}}$ and $0.65\;{\unicode{x00B5}{\rm m}}$, but we measured ${w_0} = 1.75\;{\unicode{x00B5}{\rm m}}$ and $0.65\;{\unicode{x00B5}{\rm m}}$, with an experimental estimated uncertainty of ${0}{\rm .07\,\,\unicode{x00B5}{\rm m}}$ (see Supplement 1, Fig. S15), at focal distances $f = 185\;{\unicode{x00B5}{\rm m}}$ and $75\;{\unicode{x00B5}{\rm m}}$, respectively. Furthermore, for technological convenience, we fabricated square metalenses [see Fig. 5(a)]; thus, the above values of NA are slightly underestimated. Finally, considering the difficulty to model the experimental nonidealities due to fabrication tolerances, whose minimization is beyond the scope of this work, we can conclude that the measured performances are in good agreement with the theoretical expectations.

## 4. METHODS

#### A. Numerical Simulations

The scattering properties of isolated or arrayed resonators at FF and SH were modeled with FEM simulations in COMSOL Multiphysics. For isolated resonators simulations, perfectly matching conditions were imposed at the edges, while Floquet-Bloch boundary conditions were adopted for periodic structures. The sapphire substrate was assumed semi-infinite with a refractive index ${n_s} = 1.75$. The refractive index of AlGaAs was extracted from [42], and for the nonlinear susceptibility a dispersion-less value ${\chi ^{(2)}} = 200\;{\rm pm}/{\rm V}$ in the near infrared was assumed [43]. The pump beam was modeled as a plane wave at constant wavelength ${\lambda _{{\rm FF}}} = 1550\;{\rm nm}$, intensity ${I_0} = 1\;{\rm GW}/{{\rm cm}^2}$, and Poynting vector aligned along the nanocylinder axis and linearly polarized along $x$ in Fig. 2(a), corresponding to the [100] AlGaAs crystallographic axis. The nonlinear current inside the nanocylinder was calculated in the undepleted pump regime. The electric near field of the unit cell at SH was computed in COMSOL Multiphysics and then projected in the far-field domain in the air semi-infinite plane through near- to far-field transformations [40]. This enabled the extraction of the phase and amplitude of the cross-polarized SH field $E_y^{{\rm SH}}$ in the zero-diffracted order and the main lobe angular distance from the normal. For the simulation of the beam-steering device in Fig. 4(a), a supercell made by eight elements was considered with a total dimension of $7200\;{\rm nm} \times 900\;{\rm nm}$ in the $xy$ plane. Periodic conditions were imposed at the boundaries and the far field response was modulated by the array factor of a $70\;{\unicode{x00B5}{\rm m}} \times 70\;{\unicode{x00B5}{\rm m}}$ 2D grating to reproduce the experimental conditions.

#### B. Sample Fabrication

The sample was obtained by a planar structure, grown by molecular beam epitaxy, consisting of a high quality 400 nm thick ${{\rm Al}_{0.18}}{{\rm Ga}_{0.82}}{\rm As}$ layer deposited on a [100] GaAs substrate. A 500 nm thick ${{\rm Al}_{0.8}}{{\rm Ga}_{0.2}\rm As}$ sacrificial layer was inserted before the GaAs epitaxial growth. The sample was then glued on a sapphire host substrate with a flip-chip process. The growth substrate and the sacrificial layer were then removed by mechanical and selective chemical etching, leaving only the 400 nm thick GaAs as a mirror-flat surface. Two consecutive e-beam lithography and inductively coupled plasma (ICP) etching processes were performed to fabricate $70\;{\unicode{x00B5}{\rm m}} \times \;70\;{\unicode{x00B5}{\rm m}}$ metasurfaces with a period of $900\;{\rm nm}$. The nanochair resonators in Fig. 2(a) are patterned with a double e-beam lithography. We used ma-N 2403 assisted by TI Prime adhesion promoter as a negative resist. To avoid electron charging, Electra92 was spun on the resist. An accelerating voltage of $20\;{\rm kV}$ with an exposure dose of $120\;{\unicode{x00B5}\rm C}/{{\rm cm}^2}$ were set for the lithography. A first, ICP reactive-ion etching (ICP–RIE) dry etching with ${{\rm SiCl}_4}$ transferred the first half of cylinder pattern to the AlGaAs layer for $200\;{\rm nm}$ thickness, followed by a second aligned lithography and ICP etching to realize the final nanochair resonators. A step-by-step description and additional SEM images can be found in Section 3 of Supplement 1.

#### C. Nonlinear Optical Characterization

The core of the experimental setup is a home-built horizontal microscope. The excitation pulsed beam is generated by an optical parametric amplifier (Mango, Amplitude/APE) pumped by a mode-locked ytterbium-doped fiber laser (Satsuma, Amplitude). The pulse duration was ${\tau _p} \sim 160\;{\rm fs}$ with a repetition rate of $1\;{\rm MHz}$. The input polarization was controlled by a half-wave plate and a polarizer, to ensure the alignment with the [100] crystallographic axis of AlGaAs. The fundamental beam was focused by a $f = 400\;{\rm mm}$ lens on the back focal plane of $10 \times$ microscope objective (${\rm NA} = 0.2$) ensuring a collimated excitation beam with waist ${w_0} \sim 100\;{\unicode{x00B5}{\rm m}}$ [44], as characterized by knife edge measurement (see Fig. S9 of Supplement 1). The generated SH was collected with a high-NA objective (LMPLFLN 100X, ${\rm NA }= 0.8,\;{\rm Olympus}$) cleaned by a low-pass filter at $850\;{\rm nm}$ and measured by a high-QE CCD camera (Trius SX825, Starlight Xpress). Back focal plane imaging was implemented through a $200\;{\rm mm}$ Bertrand lens along the collection path.

## 5. DISCUSSION AND CONCLUSION

Shaping the nonlinear emitted beam from 2D arrays of nanostructures with submicrometer thickness dramatically extends the possibility granted by linear flat metasurfaces, offering a true paradigm shift for nonlinear optics [9]. Such metasurfaces have recently provided new routes for nonlinear imaging [23] and might be employed in integrated photonic sources for infrared image upconversion [45]. However, the imperative requirement to achieve this goal is a platform with high compatibility with photonic technology and the identification of reliable set of building blocks, which can be easily used for the implementation of any 2D phase profile at harmonic frequencies. In this work, making use of one LUT relying on (100) AlGaAs nanochair resonators, we demonstrated SHG beam steering and Fresnel lenses metasurfaces based on the properties of isolated resonators. They offer a high integrability combined with a large ${\chi ^{(2)}}$ nonlinearity. In our experiment, due to a contingent upper limitation for our available laser source power, we were forced to use an input FF intensity as low as ${I} = {140}\;{{\rm MW/cm}^2}$; i.e., 50 times lower than in [29]. Even for such a modest excitation, we believe, to the best of our knowledge, that our SHG efficiency of $1.3 \times {10^{- 5}}$ is already six orders-of-magnitude higher than the SHG record in plasmonic metasurfaces [19], three orders-of-magnitude higher than THG in plasmonic metasurfaces [18], and one order-of-magnitude higher than the record value for THG in silicon metasurfaces [22–24]. Yet, it is noteworthy that TPA in Si limits the maximum pump intensity to about ${5}\;{{\rm GW/cm}^2}$ before saturation of THG due to rising free-carrier absorption [21]. Conversely, ${{\rm Al}_{0.18}}{{\rm Ga}_{0.82}}{\rm As}$ enables work in a transparency regime for $\lambda \; \gt \;{740}\;{\rm nm}$, and systematic measurements on isolated resonators show no traces of SHG saturation, even for pump intensities as high as ${7}\;{{\rm GW/cm}^2}$ [29]. Therefore, we believe a reasonable two orders-of-magnitude increase of pump intensity (not available with our source) would boost SHG efficiency of our ${{\rm Al}_{0.18}}{{\rm Ga}_{0.82}}{\rm As}$ metasurfaces up to ${{10}^{- 3}}$, a definitely promising level for practical implementations. While a more detailed comparison of our result with state-of-the-art nonlinear beam shaping can be found in Section 5 of Supplement 1, note that our normalized efficiency ${\eta _{{\rm SHG}}} = 2.8 \times {10^{- 10}}\,\,{{\rm W}^{- 1}}$ is, to the best of our knowledge, half-a-million times higher than the value reported in [19] (${\eta _{{\rm SHG}}} = 6 \times {10^{- 16}}\,\,{{\rm W}^{- 1}}$).

In addition to the demonstration of a nonlinear metalens focusing the SH beam to a spot size of $0.84{\lambda _{{\rm SH}}}$, we have reported on what we believe, to the best of our knowledge, to be the first implementation of nonlinear meta-optics with ${0 - 2}\pi$ phase control with a conversion efficiency that is compatible with real-world applications. With a pump at telecom wavelength, this was made possible by exploiting the huge and fully tensorial ${\chi ^{(2)}}$ nonlinearity of TPA-free (100) oriented AlGaAs-on-sapphire metasurfaces, which are now available for ambitious applications like a flat-optics camera for night vision, based on frequency upconversion and detection by a CMOS focal plane [12]. Current performances are limited by both design (exploration of a restricted number of geometrical parameters) and fabrication (proximity effects and deviations between the size of nominal and fabricated nanostructures). In the future, the design might be improved with machine-learning approaches [46] and fabrication by more advanced lithographic equipment.

## Funding

Agence Nationale de la Recherche (ANR-18CE24-0026).

## Acknowledgment

The authors thank Y. Genuist for the MBE growth of the AlGaAs layers and M. A. Vincenti for fruitful discussions. Author Carlo Gigli thanks S. Suffit, P. Filloux, and C. Belacel for technical advices during fabrication process, as well as A. Borne and V. Vinel for fruitful discussions on experimental characterization. Authors Giuseppe Leo and Jean-Michel Gérard acknowledge financial support by ANR through the NOMOS project.

## Disclosures

The authors declare no conflicts of interest.

## Supplemental document

See Supplement 1 for supporting content.

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