1. Introduction

Metasurfaces are ultrathin planar metamaterials made of periodic subwavelength metallic or dielectric nanostructures, also called meta-atoms [1,2]. The most intriguing feature of metasurfaces is that one can select the geometry and chemical composition of meta-atoms to control the phase, amplitude or the polarization state of light waves propagating through, reflected or scattered from them [3–13]. Unlike conventional optical components, such as lenses, wave-plates and prisms, which rely on gradual phase changes accumulated along the optical path to control and manipulate light, metasurfaces diminish the dependence on the propagation effect by introducing spatial inhomogeneity in optical properties over the thin interface. Thus, metamaterials provide a new degree of freedom of controlling the wavefront of light [1,2]. So far, metasurfaces with abrupt and controllable phase changes have been utilized to realize beam steering [3,4], vortex beams [3,5], ultrathin focusing lenses [6,7], flat waveplates [8,9], optical spin-orbit interactions [10,11], and photonic spin Hall effect [12,13].

Recently, the concept of phased metasurfaces has been extended to the nonlinear regime to enable both coherent generation and manipulation [14–21]. For example, a π phase shift can be imposed on the local second harmonic generation (SHG) nonlinearity by simply inverting the orientation of the metallic split ring resonator (SRR) or complementary SRR (CSRR) [14,15]. Based on this mechanism, binary nonlinear metasurfaces consisting of SRRs or CSRRs have been designed to implement the engineered diffraction, intense focusing, polarization control, and the generation of Airy and vortex beams of the output SHG signals [14–17]. For more complex beam manipulation in nonlinear processes, a continuous and spatially varying phase manipulation is highly desirable. The recent realization of the geometric Berry phase in nonlinear metasurfaces enabled a continuous control over the nonlinear phase change from 0 to 2π by simply rotating the meta-atoms with respect to the laboratory frame [18]. Although the concept of the nonlinear Berry phase provides a simple route for better manipulation of the beam profile of the nonlinear radiation and realization of nonlinear holography [19], it requires circularly polarized fundamental wave (FW) and only generates circularly polarized harmonic generation. More recently, it was demonstrated that under linearly polarized illumination, a full-range nonlinear phase of four-wave mixing (FWM) and third harmonic generation (THG) signals could be obtained in the metasurfaces made of rectangular nanoapertures and V-shaped nanoantennas, respectively, by changing the size and shape of the meta-atoms [20,21]. So far, however, literature addressing the continuous phase engineering of the SHG radiation from the nonlinear metasurfaces illuminated by linearly polarized light is rather limited.

In this study, we demonstrate that by finely adjusting the structural parameters of the meta-atoms made of the CSRRs, the resultant spectral shift between the resonances in the CSRRs and the fundamental wave can introduce a local nonlinear phase with continuous tuning range exceeding π into the SHG processes for linearly polarized excitation. The nonlinear phase tuning range can be further extended to cover the entire 2π range by reversing the orientation of the CSRRs, and consequently imposing an additional π phase shift on the SHG signals [15]. The complete phase control over the nonlinear emission enables us to design flat nonlinear optical components that combine the coherent generation and manipulation of harmonic waves. As a demonstration of this concept, we show that the CSRRs-based metasurfaces with properly designed phase gradient can be utilized to bend and tightly focus the SHG signals.

2. Results and discussions

Figure 1(a) schematically depicts the gold CSRR element serving as the building block of the nonlinear metasurface, in which the local nonlinear effect originates mainly from a microscopic symmetry break at the surface of the CSRR [22]. In this work, CSRRs are chosen because they not only exhibit high quadratic nonlinearities that arise from the coupling of the exciting field and the generated nonlinear surface currents to bright radiating plasmonic modes of the CSRRs [22,23], but also have transmittance maxima at both the excitation and SHG wavelengths [15,22]. Each CSRR element is characterized by a set of structural parameters l (the arm length), w (the arm width), g (the width of the central metal patch) and t (the thickness of the gold film), where g and t are fixed to 50 nm and 20 nm, respectively, while l and w are allowed to be varied. The linear optical properties of the regular arrays of CSRRs sitting on a fused silica (amorphous SiO₂) substrate are first investigated using a commercial finite-element method (FEM) software package COMSOL Multiphysics. In the calculations, periodic boundary conditions are employed to mimic periodicities (a = 400 nm) along the x- and y-directions. All the sharp corners are rounded with a radius of curvature of 5 nm to avoid possible numerical artifacts. The refractive index of the fused silica substrate is assumed to be 1.46, and the permittivity of gold is taken from the experimental data of Johnson and Christy [24]. Figure 1(b) shows the transmission spectra of the metasurfaces illuminated by the y-polarized plane-wave propagating along the z-direction (normal incidence) for a fixed arm width (w = 75 nm) while different arm lengths (l). As expected, the transmittance through the CSRRs are interchanged with the reflectance of the corresponding SRRs [23], which ensures transmittance maxima at the fundamental resonance for the CSRRs rather than transmittance minima for the SRRs. It is also seen that the plasmonic resonances of the CSRRs can be spectrally tuned by changing the arm length. For example, as the arm length varies from l = 160 nm to l = 100 nm, the fundamental resonance peak blue-shifts drastically from λ ≈1400 nm to λ ≈1100 nm.

 

Fig. 1 (a) Schematic of a uniform CSRR array with a periodicity of a = 400 nm. The height of the CSRR is t = 20 nm. The width of the central metal patch is g = 50 nm. Normally incident light propagates along the z-axis and is polarized along the y-axis. (b) Calculated linear transmission spectra of the CSRR arrays with a fixed arm width of w = 75 nm but different arm lengths. For clarity, each curve is shifted vertically by a constant of 0.7 relative to the previous one. The red and blue dashed lines indicate the fundamental and SHG wavelengths, respectively. (c) The normalized SHG conversion efficiency from the CSRRs with w = 75 nm and l = 160 nm. (d) The nonlinear phase shift and amplitude in the array of CSRRs with w = 75 nm but different arm lengths. (e) and (f) Calculated SH phase-shift and amplitude for CSRRs with different l and w.

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In addition, the nonlinear phase shift and efficiency of the SHG from the CSRRs can also be calculated using FEM numerical simulation based on a hydrodynamic model of the free electron dynamics [22,25], which has been used previously to handle SHG process in metallic structures such as SRRs, CSRRs, and T-shaped nanoparticles [22,25]. Figure 1(c) shows the normalized SHG conversion efficiency from the CSRRs with an arm width of w = 75 nm and an arm length of l = 160 nm. The maximum efficiency is found to occur at λ ≈1400 nm [Fig. 1(c)], which corresponds to the fundamental resonance of the CSRRs [Fig. 1(b)]. Figure 1(d) shows the calculated SH phase shift and amplitude (red line) at λ_SH = 700 nm (λ_FW = 1400 nm) for the CSRRs with a fixed arm width (w = 75 nm) and different arm lengths. It is found that although the SH phase shift exhibits a wide (~300-degree) phase coverage by solely varying the arm length from l = 185 nm to l = 105 nm [blue line in Fig. 1(d)], the SH amplitude is drastically varied from 0 to 1 [red line in Fig. 1(d)]. In order to obtain a complete phase control over the SHG and simultaneously large SH amplitudes, both the arm length and arm width of the CSRRs need to be varied in the calculations. Figures 1(e) and 1(f) show the maps of the calculated phase shift and amplitude of the SHG at a given FW wavelength λ_FW = 1400 nm, respectively, as functions of the arm length and width, which actually provide guidance for choosing the CSRRs with appropriate structural parameters to create a metasurface with a given phase gradient. For example, to obtain a π/4 phase gradient, the phase shift contours −45°, 0°, 45°, and 90° are first determined and outlined by dashed lines in Fig. 1(e). Then, the contour lines are superimposed to the colour map of the SHG amplitude in Fig. 1(f). Along each contour line, the phase shift of the SHG could remain the same, while its amplitude can be changed by selecting the CSRRs with different combinations of the arm length and width. Two typical sets of four CSRRs are chosen along the path-1 and path-2, as indicated by solid circles and squares in Fig. 1(f), respectively. There are only small differences between the SHG amplitudes of the four CSRRs in the path-1, while SHG amplitude differences between the four CSRRs in the path-2 are relatively large. By simply mirroring the CSRR with respect to its base, one creates an inverted CSRR whose locally generated SHG radiation has an additional π phase shift [10], which equivalently extends the nonlinear phase coverage shown in Fig. 1(e) by π.

As schematically shown in Figs. 2(a) and 2(b), two eight-resonator unit cells with the same incremental phase shift of Δφ = π/4 (along the + x direction) are thus created from the initial four CSRRs located in the path-1 or path-2 and the corresponding four inverted-CSRRs. The far-field SHG emission of the metasurfaces constructed from the above two super-cells excited by a normally incident fundamental wave (λ_FW = 1400 nm) polarized along the y-direction are calculated and shown in Figs. 2(c) and 2(d). Although the SHG amplitude ratio of the CSRRs along the path-1 (1.1:1.3:1.2:1) and path-2 (1.1:1.5:1.4:1) is different, the SHG radiation patterns of the metasurfaces constructed from these two sets of CSRRs are found to be almost the same. In each case, the SHG could be radiated into the backward hemisphere (90° < θ < 270°, corresponding to the substrate side) and the forward hemisphere (0° < θ < 90° and 270° < θ < 360°, corresponding to the air side) at the emission angles of 192.4° and 347.4° (or equivalently −12.6°), respectively. Since the origins of the two strong side lobes are the same, we only focus on the transmitted SHG signals.

 

Fig. 2 (a) and (b): Schematic view of eight-resonator super-unit cells with a phase gradient of π/4. One super-unit is created from the CSRRs in the path-1, the other is from the CSRRs in the path-2. (c) and (d) Far-field SHG intensity from the metasurfaces consisting of the super-unit cells presented in (a) and (b).

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The radiation at a specified emission angle of θ from the phased array can be analytically expressed as

In the following, we experimentally demonstrate that the CSRRs-based metasurfaces with properly designed nonlinear phase gradient can be utilized to bend the SHG beam. As schematically shown in Fig. 3(a), femtosecond pulses with a duration of 100 fs and a repetition rate of 1kHz are derived from an optical parametric amplifier. The FW is polarized along the CSRR arm and focused to a ~30 μm-diameter spot onto the samples from the substrate side under normal incidence. The average power of the FW is ~0.2 mW. The transmitted optical signals pass through a short pass filter to block the pump light and then are collected by an objective (50 × , 0.7 N. A.). With tube lenses L1 and L2, the back focal plane of the objective, which is a Fourier image of the radiation from the sample, is imaged onto a cooled back-illuminated electron multiplying charged coupled device (EMCCD) camera. Figures 3(b)-3(d) show the scanning electron microscopy (SEM) images of the nonlinear metasurfaces consisting of one-CSRR unit cell (uniform array with a phase gradient of Δφ = 0), eight-CSRR super-unit cell with Δφ = π/4, and six-CSRR super-unit cell with Δφ = π/3, respectively. All the CSRR arrays with a footprint of 50 μm × 50 μm are fabricated by focused ion beam milling in a 20-nm-thick gold film supported on a fused silica substrate. Figure 3(e) shows the k-space image of SHG emission from the uniform CSRR array pumped by λ_FW = 1400 nm. Due to the periodicity of a = 400 nm is smaller than the wavelength of the SHG beam (λ_SHG = 700 nm), only a zeroth-order SHG diffraction is observed at an emission angle of θ = 0° [Fig. 3(e)]. The k-space images of SHG emission from the nonlinear metasurfaces consisting of eight-CSRR and six-SRR super-unit cells are shown in Figs. 3(f) and 3(g), respectively. In each case, only one spot appears at a negative diffraction angle, which agrees well with the calculated result shown in Fig. 2(c). The measured emission angles are also found to be dependent both on the periodicity (p) and the FW wavelength (λ_FW) [Figs. 3(f) and 3(g)]. Furthermore, all these measured angles are consistent with the first negative order SHG diffraction angles calculated according to θ_-1 = −sin⁻¹(λ/p), which confirm that the CSRRs-based metasurfaces with the proper choice of the phase gradient allow us to control the beam steering of the SHG emission.

 

Fig. 3 (a) Schematic illustration of the experimental setup, where the CDD-camera records either Fourier images or real images of the CSRR-based samples. (b)-(d) SEM images of the nonlinear metasurfaces consisting of one-CSRR unit cell with Δφ = 0, eight-CSRR super-unit cell with Δφ = π/4, and six-CSRR super-unit cell with Δφ = π/3, respectively. Scale bars represent 500 nm. (e)-(g) Measured k-space images of SHG from the samples shown in (b)-(d), respectively. The uniform array in (e) is pumped by FW with a wavelength of 1400 nm. The phase-modulated arrays in (f) and (g) are pumped by FWs with three different wavelengths of 1300nm, 1400 nm, and 1500 nm.

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To further illustrate the flexibility of the continuous phase control over the local second-order nonlinearity, we designed a CSRR-based metasurface that is able to focus SHG beam. In the design of a nonlinear metalens with a desired focal length f, the arm length and width of the CSRRs located in a concentric ring with a given radius of r are chosen from Figs. 1(e) and 1(f) to impose a particular phase shift $\phi (r)\text{=2}\pi \sqrt{f^2+r^2}/{\lambda }_{SHG}$ onto the SHG signals. Figure 4(a) shows the SEM image of the fabricated nonlinear metalens with f = 40 μm. The simulation based on a beam propagation technique [15] for the metalens upon illumination by normally incident FW (λ_FW = 1400 nm) with its polarization along the CSRR arm is shown in Fig. 4(b). In order to experimentally characterize the focusing effect of the fabricated nonlinear metalens, the EMCCD-camera is placed around the focal plane of the tube lens L1, where the real image of the sample appears, as schematically shown in Fig. 3(a). The measured result is shown in Fig. 4(c) and found to be well consistent with the simulation, presenting the intense focusing of the SHG at the desired focus length. In addition, as indicated by dashed lines in Fig. 4(c), the cross-sectional images of the normalized SHG are recorded at distances z = 5 μm and 40 μm away from the surface of the metalens, and shown in Figs. 4(d) and 4(e), respectively. It is clearly seen that the diameter of the SHG is ~14 μm at the z = 5 μm plane [Fig. 4(d)], and narrows to ~8 μm at the z = 40 μm plane [Fig. 4(e)].

 

Fig. 4 (a) SEM image of a CSRR-based metasurface acting as a nonlinear metalens with a desired focal length f = 40 μm. The scale bar is 5 μm. Dashed line represents a concentric ring with a radius of r on which all the CSRRs have the same phase shift. (b) and (c) Calculated and measured SHG focusing effect of the metalens upon illumination by normally incident FW (λ_FW = 1400 nm) with its polarization along the CSRR arm, respectively. (d) and (e) Images of the normalized SHG (λ_SHG = 700 nm) recorded at z = 5 μm and z = 40 μm cross-section planes, respectively.

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3. Conclusions

In summary, we demonstrate that the adjustment of the geometrical parameters, such as the arm length and width, coupled with the mirror inversion of the CSRRs enable a continuous control over the phase change from zero to 2π for the locally generated SHG upon illumination by linearly polarized FW. By designing a nonlinear metasurface consisting of eight-CSRR supercells with a phase gradient of Δφ = π/4, SHG beam bending behavior is experimentally demonstrated and agrees well with the theoretical predictions. We show that the bending angle of the SHG beam can be tailored by varying either the wavelength of FW or the phase gradient. Furthermore, we show that concentrically arranged CSRRs that possess a particular phase shift on each concentric ring can act as a flat nonlinear metalens to focus SHG beam. We hope the complete phase control over the nonlinear emission presented here can be extended to construct three-dimensional nonlinear phase-gradient metamaterials due to the relatively high transmittance through the CSRRs at both fundamental and SHG wavelengths, and can be exploited in the applications, such as nonlinear holograms, data storages, and the coherent generation and manipulation of nonlinear signals in integrated nonlinear photonic circuits.