Nonlinear wavefront control by geometric-phase dielectric metasurfaces: Influence of mode field and rotational symmetry

Nonlinear Pancharatnam-Berry phase metasurfaces facilitate the nontrivial phase modulation for frequency conversion processes by leveraging photon-spin dependent nonlinear geometric-phases. However, plasmonic metasurfaces show some severe limitation for nonlinear frequency conversion due to the intrinsic high ohmic loss and low damage threshold of plasmonic nanostructures. Here, we systematically study the nonlinear geometric-phases associated with the third-harmonic generation process occurring in all-dielectric metasurfaces, which are composed of silicon nanofins with different in-plane rotational symmetries. We find that the wave coupling among different field components of the resonant fundamental field gives rise to the appearance of different nonlinear geometric-phases of the generated third-harmonic signals. The experimental observations of the nonlinear beam steering and nonlinear holography realized in this work by all-dielectric geometric-phase metasurfaces are well explained with our developed theory. Our work offers a new physical picture to understand the nonlinear optical process occurring at nanoscale dielectric resonators and will help in the design of nonlinear metasurfaces with tailored phase properties.


Introduction
Optical metasurfaces offer the ultimate solution for modern diffractive optical elements design by leveraging the artificially tailored nanostructures of subwavelength geometries, which facilitate the flexible control of the phase, amplitude, and polarization of the light with an ultra-compact profile. [1,2] Since the propagation of the light is substantially determined by the local phase distribution of the wavefront, metasurface structures with full-phase transmitted or reflected phase discontinuities are highly desired for realizing numerous optical functionalities. [3][4][5][6][7][8] In particular, Pancharatnam-Berry phase or geometric-phase metasurfaces simplify the optical wavefront engineering by utilizing the linear Pancharatnam-Berry phase, which can be precisely controlled by locally rotating the nanostructures. [9][10][11][12] At the very beginning, plasmonic metasurfaces gained its popularity for the simplicity of obtaining an effective electric and magnetic response with spatially tailored induced surface currents. [13] Besides that, due to the strong local field enhancement near the surface of the plasmonic nanostructures and free of restrictions by the phase-matching condition, plasmonic metasurfaces with boosted nonlinear generation efficiency show 3 great applicability in nonlinear nanophotonics. [14,15] However, the plasmonic materials inevitably introduce large ohmic losses and suffer from low damage thresholds while the all-dielectric materials offer a promising alternative. Benefiting from the multipole Mie resonances supported by the dielectric nanostructures, all-dielectric metasurfaces with greatly decreased dissipation loss and robustness against high power lasers are widely applied to the efficient linear and nonlinear optical wavefront manipulation. [16][17][18][19][20][21] In addition, the sharp Fano resonances supported by the all-dielectric metasurfaces [22][23][24][25] and the excitation of the nonradiative modes in the structures [26][27][28] give rise to the extreme local field confinement and provide stronger light-matter interaction for the efficient nonlinear signal generations.
Recently, the nonlinear optical phase obtained with designed nanostructures has attracted tremendous research interest for the convenience they provide in controlling the nonlinear optical waves. Besides the nonlinear Huygens metasurfaces, [21,29,30] the photon-spin dependent nonlinear geometric-phase opens another avenue to the efficient phase modulation of the photons generated during the frequency conversion process.
The pioneering work that takes advantage of the hybrid metasurfaces based on plasmonic structures and multiple-quantum-wells in semiconductors reveals the possibility of exploiting the spatially varied nonlinear geometric-phase for advanced nonlinear wavefront control with high-efficiency. [31][32][33] In addition, by revisiting the selection rule that holds for the nonlinear crystals, it was revealed that plasmonic nanostructures with specific structural symmetry also show similar selection behavior for the harmonic generation processes. [34][35][36][37][38] In nonlinear geometric-phase metasurfaces, 4 recently reported works mostly focusses on implementing plasmonic nanostructures while studies on all-dielectric nonlinear geometric-phase metasurfaces are still missing.
Here, we theoretically and experimentally study the nonlinear geometric-phases associated with the third-harmonic generation (THG) process occurring in all-dielectric metasurfaces, which are composed of silicon nanofins with different in-plane rotational symmetries. In our previous work, [39] we experimentally exploited the potential of the nonlinear geometric-phase for spin-multiplexed nonlinear holograms using silicon nanofin of two-fold symmetry. However, the underlying physics of the geometric-phase in a nonlinear process using such spatially extended nanofins was only based on some phenomenological assumptions. Here, we analytically develop a complete theory to reveal the underlying physics for the generation of nonlinear geometric-phase and describe the influence of the structural symmetry and local optical mode structure on the possible phase factors. Based on the full-wave simulation and experimental measurements, we test our theoretical analysis by realizing nonlinear beam steering and nonlinear holography based on such metasurfaces.

Nonlinear polarization excited in the dielectric geometric-phase metasurfaces
For plasmonic geometric-phase metasurfaces, the harmonic generation process can be phenomenally understood as the result of the excitation of the effective nonlinear surface currents which substantially carry the spin-dependent nonlinear geometricphases. [32] Moreover, the nonlinear response of the plasmonic metasurfaces is usually modeled as an equivalent surface with effective tensor nonlinear susceptibilities where 5 the wave coupling and phase matching along the propagating direction is generally neglected due to the short propagation distances. [32,33] However, the nonlinear optical signals generated from dielectric nanostructures are built up by the overall radiation of the induced volume nonlinear polarization. Due to the large aspect ratio and the high refractive index of the dielectric structures, the propagation of the light confined inside the structures makes the wave coupling effect conspicuous. Hence, the nonlinear optical process occurring in dielectric nanostructures will inevitably involve the wave coupling among different field components of the resonant fundamental wave (FW), and the radiated nonlinear waves can carry additional geometric-phases beyond what has been predicted in their plasmonic counterpart.
A conceptual image of the used metasurface is shown in Figure 1, which illustrates the effect of anomalous third-harmonic (TH) beam steering exhibited by the nonlinear metasurface with designed spatially linear phase modulation. When a circularly polarized FW normally impinges onto such a metasurface, the generated LCP and RCP TH signals, which are modulated with the spin-dependent nonlinear geometric-phases, will be deflected into different spatial directions. In our work, we always assume planar circularly polarized FW propagating along the normal of the metasurface, which is collinear with the z-axis of the laboratory frame, illuminates the metasurface from the substrate, see Figure 1. For the THG processes occurring in the dielectric metasurface, the wave coupling between different components of the confined FW field can be described through the effective third-order susceptibility tensor and l refer to the cartesian coordinates x, y, and z, respectively. Because the nonlinear 6 geometric-phase is carried by the circularly polarized TH light waves, it is more convenient to study the nonlinear response in the circular polarization basis. The effective third-order susceptibility tensor in the circular basis can be obtained by a coordinate transformation, which finally gives ( ) Here, α, β, γ, and δ refer to the circular coordinates L, R, and z, and i   is the element of the transformation matrix between the circular basis and the cartesian basis. When rotating the in-plane orientation of the dielectric nanostructure, the rotated nonlinear polarization will differ from the unrotated nonlinear polarization in the laboratory frame by an additional phase, which gives rise to the nonlinear geometric-phase of the radiated TH signals. As shown in Figure 2(a), when the silicon nanofin is counterclockwise rotated by an angle θ, the effective third-order susceptibility tensor of the nanostructure in the circular basis can be written as , here the prime refers to the local frame, and ( ) ' R    is the element of the rotation matrix between the local and laboratory frame in the circular basis. Therefore, the nonlinear optical process that involves the wave coupling of specific resonant FW field components correspondingly carries the geometric-phase contributed by the local and laboratory frame rotation transformation, i.e., Considering only the left-handed (LCP) and right-handed (LCP) TH signals that arise from the forward THG processes, we analytically obtain the corresponding volume third-order nonlinear polarizations as:   It is well known that the selection rule verified by the plasmonic nanoantenna with mfold in-plane rotational symmetry gives the allowed harmonic generation orders by 1 n lm = when the metasurface sample is pumped by a circularly polarized FW. [34] Here, l is an arbitrary integer and '+' and '−' signs refer to the co-polarized and crosspolarized harmonic signals, respectively. Therefore, in this work, we primarily focus on the THG processes occurring in the silicon nanofins with one-fold (C1), two-fold (C2) and four-fold (C4) in-plane rotational symmetry and how the structural symmetry influences the spin and corresponding geometric-phases carried by the TH signals.

Nonlinear response of a geometric-phase metasurface composed by silicon nanofins with C2 in-plane rotational symmetry
To simplify our analysis, we first study the properties of the TH signals generated from silicon nanofins with C2 in-plane rotational symmetry. It is well known that Neumann's principle determines the dipole allowed nonlinear optical processes for structures with specific symmetry. Therefore, the effective volume nonlinear polarizations excited 9 inside the silicon nanofins with C1 and C2 in-plane rotational symmetries are analytically given by Equation 1. Next, we conduct the full-wave simulations to facilitate the numerical study of the nonlinear response of the silicon nanofin array. The nanofin is made of amorphous silicon (α-Si) and placed on the low-refractive-index silicon dioxide (SiO2) substrate, and the surrounding medium of the above half-space is assumed to be air. In our work, the silicon nanofin is designed to be of a subwavelength scale to ease the wavefront phase encoding of the TH signals. As shown in Figure   Based on the silicon nanofin shown in Figure 2(a), we further simulate the anomalous refractions of the TH signals generated from the metasurface with a gradient nonlinear geometric-phase change along the x-direction. By gradually rotating the silicon nanofin along the x-direction, a spin-dependent surface phase gradient is obtained that diffracts the TH beams at different angles. In our simulation, the FW wavelength is selected as 1300 nm, and the gradient-phase metasurface period is composed of 36 units.

Nonlinear response of the geometric-phase metasurface composed of silicon nanofins with C1 and C4 in-plane rotational symmetry
For further demonstration of the selection rule, we study the geometric-phases associated with the TH waves generated from the silicon nanofins with C1 and C4 inplane rotational symmetry. Figure 3(a) illustrates the used C1 silicon nanofins. In our design, the C1 silicon nanofin has a U-shape cross-section, and its geometric parameters are selected as w = 150 nm, l = 300 nm, d 1 = 80 nm and d 2 = 60 nm. The height and lattice constant of such silicon nanofin are optimized as h = 550 nm and p = 400 nm to maintain higher THG conversion efficiency. By locally rotating the silicon nanofin from 0° to 360° with an angular rotation step of 5°, the THG conversion efficiency will change due to the variation of the coupling among the neighboring units, see Figure 3 (b). The larger fluctuation of THG conversion efficiency indicates stronger coupling among silicon nanofins occurs in our C1 design, and bigger distortion on the nonlinear geometric-phases is expected. In our simulation, the FW wavelength is selected as 1240 nm to ensure that the TH wavelength is bigger than the lattice constant, which suppresses the higher-order diffraction and increases the uniformity of the TH far-field. Based on Neumann's principle, for silicon nanofins of C4 in-plane rotational symmetry, its effective volume nonlinear polarizations are given by: Here, the TH waves radiated from the locally rotated array silicon nanofin with C4 inplane rotational symmetry would carry the geometric-phase of ( ) exp 4i  .   In Figure 4(e) and (f), the full-wave simulation shows the field distribution of the crosspolarized and co-polarized TH signals radiated from the C4 silicon nanofins with a gradient local geometric-phase modulation. In our full-wave simulation, the FW wavelength is selected as 1350 nm to increase the linearity between the local rotation angle and the geometric-phase, and the designed gradient geometric-phase metasurface is composed by 20 silicon nanofins within one period. According to our calculation, we obtain an anomalous refraction angle of 3.36°, which is close to the theoretical values of 3.31° calculated by generalized Snell's law. In addition, the TH waves radiated from the C4 structure could also be equivalently understood as the superposition of the TH waves radiated from two overlapped C2 nanofins with a 90° intersection angle. In this case, the co-polarized TH wave obtained in the far-field is the superposition of two co-polarized TH waves differing by 180° phase delay and hence interfere destructively.
In contrast, the cross-polarized TH wave is the superposition of two cross-polarized TH waves in phase. Therefore, only cross-polarized TH waves can be observed in the farfield due to the constructive interference that occurs for the cross-polarized TH waves 15 radiated from the nanofins.

Experimental verification of the nonlinear beam steering realized by silicon gradient geometric-phase metasurfaces
We proceed to experimentally validate the nonlinear phase manipulation by utilizing the spin-dependent nonlinear geometric-phases associated with the silicon nanofin metasurface. The dielectric metasurface was fabricated on a silica substrate by standard nanofabrication processes of deposition, patterning, lift-off, and etching. Figure 5(a) shows a scanning electron microscopy (SEM) image of the fabricated silicon nanofin gradient geometric-phase metasurface sample composed of C2 silicon nanofins, where one supercell is composed by 20 gradually rotated silicon nanofins with an angular rotation step of 9°. In our design, the geometry parameters of the nanofin are the same as illustrated in Figure 2(a). The experimental setup is schematically shown in Figure   5(b). The FW beam was generated by an optical parameter oscillator (Coherent, Chameleon compact OPO) and passed through a linear polarizer and a quarter waveplate to obtain the desired circularly polarized FW input. Then the FW was focused onto the sample and the TH signal was collected with an objective (Nikon 40x, NA 0.6). The back focal plane was imaged by two lenses to an sCMOS camera (Andor, Zyla 4.2). The polarization state of the collected TH signal was analyzed with another quarter-wave plate and a linear polarizer. Short-pass filters were used to suppress the FW background. Figure 5(c) shows the measured diffraction patterns of the LCP and RCP TH signals when the gradient-phase metasurface sample is illuminated with 50 mW circularly polarized FW at 1200 nm. Take the LCP FW illumination scenario as an example, the measured diffraction angles for the LCP and RCP TH signals are 3.05°±0.01° and 6.09°±0.01° , respectively, which is close to the design values of 3.01° and 6.01° , correspondingly. Figure 5(d) is the intensity profile of the center cutline of the measured TH diffraction pattern. It can be clearly seen here that for both co-polarized and cross-polarized TH signals, we could always observe the zeroth order, which can be explained by the nonlinear polarizations We also fabricate and test the gradient geometric-phase metasurfaces made of C1 and C4 silicon nanofins as shown previously in Figure 3


(cross-polarized). According to the simulation results shown in Figure 4(b), the cross-polarized total FW field confined inside the C4 silicon structure is weaker than the co-polarized total FW field, therefore, the intensity difference between the zeroth-order diffraction of co-and cross-polarized TH signals can be well understood. In addition, the asymmetric line shape of the TH intensity profile when switching the FW spin might be attributed to the low fabrication quality of the C4 sample (see the inset SEM image).

Nonlinear holography based on silicon geometric-phase metasurfaces
A promising application of the nonlinear metasurfaces is the holographic image reconstruction at a different wavelength than the FW (see Figure 7(a)). To demonstrate such potential by the silicon metasurface for the TH signal, we designed and encoded a phase-only hologram into the orientations of the nanofins. Figure 7 The geometric-phases associated with the TH waves we discuss here are based on a silicon nanofin array which involves the contribution of the coupling among the neighboring units. However, numerical simulations show that for an individual silicon nanofin with C1, C2 or C4 in-plane rotational symmetry without the coupling effects, the TH waves carry the spin-dependent geometric-phase modulation expected by the nonlinear geometric phase (Supporting Information). The influence of the coupling effects among the neighboring units on the deviation of the nonlinear phase can be understood as follows: firstly, the FW field confined inside the silicon nanofin will be reshaped when counting the coupling effects; secondly, the scattering of the side lobe of the TH field radiated from the dielectric nanofins will distort the TH far-field and forms the high order diffractions. In addition, our developed theory can also be generalized and expanded to the second-harmonic generation (SHG) and other higherorder harmonic generation processes.

Conclusion
In conclusion, we theoretically and experimentally study the nonlinear geometricphases associated with the THG process occurring in all-dielectric silicon metasurfaces.
The THG process involves the wave coupling among the different components of the FW field excited inside the dielectric nanoresonators and thereby gives rise to numerous nonlinear optical processes carrying different geometric-phases. We show that the dielectric nanofin structures with C1, C2, and C4 in-plane rotational symmetry generate circularly polarized TH signals in forward direction carrying the geometric-phases as predicted by the selection rule for nonlinear processes. Furthermore, we experimentally demonstrate nonlinear k-space holography by using all-dielectric geometric-phase metasurfaces composed by the silicon nanofins with C1, C2, and C4 in-plane rotational symmetry. The experiment results agree with our theoretical model. Our work offers a simple physical picture for the understanding of the nonlinear optical process occurring in all-dielectric geometric-phase metasurface and shows great application potential in nonlinear nanophotonics.

Experimental Section
Full-wave simulation: To figure out the transmitted phase and THG conversion efficiency of the silicon nanofin array, here we conducted the full-wave simulations by using the finite element method in the frequency domain. First, we calculated the total 22 FW field spatial distribution inside the nanofin. Next, the free space TH field was calculated based on the nonlinear polarization defined by the pre-calculated total FW field. In our simulation, the amplitude of the FW electric field was set as 10 8 V/m, and the third-order nonlinear susceptibilities ( )  were assumed to be 2.45×10 −19 m 2 V −2 . [18] In addition, the third-order nonlinear susceptibilities ( ) were assumed to be one-third of ( ) 3 xxxx  , where i and j refer to the cartesian coordinates x, y, and z. The complex refractive index of the amorphous silicon (α-Si) we utilized in the simulation was adopted from the experimental measurement values of our laboratory fabricated α-Si film. And the refractive index of low-refractiveindex silicon dioxide (SiO2) was adopted from data measured by Maliston. [40] Periodic boundary condition was applied to calculate the far-field property of the infinite silicon nanofin array and the gradient geometric-phase metasurface.
Hologram design: The phase-only hologram is designed by using the Gerchberg-Saxton (GS) algorithm. As a kind of iterative phase retrieval algorithm, the GS algorithm does not calculate the wavefront in the hologram plane directly but constructs an iterative loop between the object plane and the hologram plane via a propagating function, such as the Fourier transform. During the process of iteration, the phase profile of the hologram is optimized with amplitude replacement applied on the object plane and the amplitude normalization applied on the hologram plane. In the nonlinear harmonic generation case, the phase information is calculated through such a method and is converted to the azimuthal angle distribution afterward, according to the nonlinear geometric-phase principle.
Sample fabrication: The all-dielectric silicon metasurfaces were fabricated on a glass substrate following the processes of deposition, patterning, lift-off, and etching. First, through plasma-enhanced chemical vapor deposition (PECVD), we prepared a 620-nmthick amorphous silicon (α-Si) film. Following this, a poly-methyl methacrylate resist layer was spin-coated onto the a-Si film and baked on a hot plate at 170 °C for 2 min to remove the solvent. Next, the desired structures were patterned by using standard electron beam lithography. Subsequently, the sample was developed in 1:3 MIBK:IPA solution and then washed with IPA before being coated with a 20-nm-thick chromium layer by electron beam evaporation. Afterward, a lift-off process in hot acetone was performed. Finally, by using inductively coupled plasma reactive ion etching (ICP-RIE), the desired structures were transferred from chromium to silicon.
Optical Characterization: The nonlinear response of the metasurface is measured by an optical setup. The metasurface is illuminated by a circularly polarized femtosecond laser beam between 1200 nm and 1400 nm. The laser source is a Ti:sapphire femtosecond laser pumped optical parametric oscillator (OPO) with a typical pulse length of 200 fs, a repetition rate of 80 MHz and a typical output power of 300 mW.
The incident laser beam was focused onto the metasurface (f = 50 mm, beam width: ~50 µm) in order to enhance the nonlinear response of the metasurface. The generated third harmonic light from the metasurface is collected by a microscope objective (40x magnification, NA = 0.6), whose back focal plane is imaged onto an sCMOS camera (Andor Zyla 4.2). To image the back focal plane onto the camera, we used two lenses with focal distances of = 150 mm . To distinguish between different circular polarization states, we used a combination of a quarter-wave plate and a linear polarizer.
Optical filters are used to suppress the fundamental light. The diffraction spots and the holograms reconstruct at the back focal plane of the microscope objective, which allows for a unified measurement setup for the gradient and hologram metasurfaces.  Selecting the LCP FW wavelength as 1350 nm, the field distribution of (b) different circularly polarized components of the FW and TH field excited in the silicon nanofin array whose local rotational angle θ is 0°. The nonlinear geometric-phase of (c) the RCP TH signal and (d) the THG conversion efficiency of the RCP (red circle) and LCP (blue circle) TH waves when rotating the silicon nanofins from 0° to 90° with an angular rotation step of 5°. The simulated free space TH field distribution of (e) the anomalous refracted RCP TH waves and (f) surface bounded LCP TH waves. All fields are normalized with input FW electric field intensity.