1. Introduction

Optical metasurfaces have attracted considerable interest because of their powerful abilities to efficiently control the amplitude, phase, and polarization of electromagnetic waves [1–5]. By carefully designing subwavelength meta-atoms, the explorations of metasurfaces have produced many applications including beam splitters [6,7], holograms [8,9], metalenses [10,11], parity-dependent diffraction [12] and angularly asymmetric diffraction [13]. Recently, this concept has been extended to the nonlinear regime, providing a paradigm for studying nonlinear optics [14,15]. Thanks to its ultra-thin thickness, the nonlinear metasurface can achieve strong nonlinear optical response at more compact scales, and at the same time they do not require complex quasi-phase matching techniques in traditional methods based on nonlinear crystals [14,16]. Typical nonlinear effects have been demonstrated by nonlinear metasurfaces, such as second harmonic generation (SHG) [17], third-harmonic generation (THG) [18], and four-wave mixing (FWM) [19]. The nonlinear efficiency can be greatly enhanced through various resonance mechanisms, such as surface plasmon [20,21], the Mie resonance in high-index dielectric [22,23] and bound states in the continuums (BICs) [24,25]. However, the majority of these geometries have focused on the efficiency enhancement of nonlinear response; the wavefront control of the generated nonlinear light is seldom studied.

Inspired by the concept of phase gradient metasurfaces (PGMs) in linear optics, several nonlinear PGMs were theoretically proposed and experimentally fabricated [26–29], by which the wavefront control of THG can be effectively engineered to desired direction. In addition to producing the THG, the the designed nonlinear PGMs can also introduce an abrupt phase shift ${\varphi _{3\omega }}(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} )$ of fully covering $2\pi$ for THG that distributes spatially along an interface of two media. As an analogy to the linear case, the nonlinear abrupt phase shift enables an additional momentum, i.e., $\xi = \nabla {\varphi _{3\omega }}(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} )$, leading to nonlinear generalized Snell’s law (NGSL), i.e.,

In this work, we will examine this problem by designing and studying a reflection-typed nonlinear PGM with pronounced THG effect in the terahertz (THz) region. We will show that the NGSL is not correct al all for the FF incident angle beyond the critical angle. Due to the periodic nature of nonlinear PGMs, complex grating diffraction effect occurs in generated nonlinear waves. Inspired by recent advances in linear PGMs and revealed higher-order diffractions [30,31], we present a modified diffraction law involving the reciprocal lattice vector, which can completely describe the diffraction behavior of generated THG. This diffraction law offers more opportunities to freely manipulate the propagation of nonlinear light and allows one to achieve many interesting effects. For instance, we designed a nonlinear PGM-based retroreflector with high performance, which can direct the generated nonlinear light back to the original direction of the FF incidence. Alternatively, with the advantage of the electrically tunable properties of graphene, we designed and studied a tunable nonlinear PGM that can steer the reflection of a generated THG in several directions by simply changing the periodic arrangement of applied external voltage.

2. Model and theory

Figure 1(a) shows the geometric schematic of considered nonlinear PGM, a periodic metal grating filled with PMMA and covered with periodically arranged graphene ribbons that service a nonlinear medium. The metal is gold with its optical properties modeled by the Drude model [32], i.e., ${\varepsilon _m} = 1.53 - f_p^2/({f^2} - i\gamma f)$, where ${f_p} = 2069\textrm{ }\textrm{THz}$ and $\gamma = 17.65\textrm{ }\textrm{THz}$. The permittivity of PMMA is 2.25 [21]. To introduce a nonlinear phase shift along the interface of the PGM, different voltages are applied to m graphene ribbons (m is integer), which is shown in Fig. 1(b). Because the graphene ribbons do not contact the metal grating, the chemical potential of graphene ribbons can be independently controlled by applying different voltages. The width and depth of the groove are w and h, respectively, and the width of the graphene ribbons is ${w_g} < w$. Therefore, the studied PGM is composed of periodically repeating supercells in the x-direction, with each supercell containing m identically corrugated grooves. The distance between two adjacent grooves is a, so the period of the supercell is $p = ma$. In operating terahertz (THz) frequency region, only graphene is considered as a nonlinear material, as the third-order nonlinear coefficient of gold and PMMA is much smaller than that of graphene and therefore can be ignored. The linear conductivity of graphene (i.e., ${\sigma _g}$) can be expressed by a Drude-like model [33], where the relaxation time $\tau $ is set as ${10^{ - 13}}\textrm{ }{s^{ - 1}}$. The designed graphene-based nonlinear metasurface could be fabricated by transferring prepared single-layer graphene ribbons [34] onto a metallic grating filled with PMMA. Here we take THG as an example for discussions, and this work mainly focuses on the wavefront control of THG, without considering how to improve efficiency, since the diffraction properties of THG in nonlinear PGMs are usually complex. Considering wavefront control and efficiency at the same time will make diffraction problems impossible to study. This is because it requires simultaneously controlling the amplitude and phase of both linear and nonlinear fields, which is very challenging.

 

Fig. 1. (a) A schematic diagram of the proposed nonlinear metasurface (one supercell). The metasurface is composed of a simple metallic grating filled with PMMA and periodically engineered graphene ribbons covering the grating. (b) The side view of the supercell, where different voltages are applied to the m groups of graphene ribbons denoted by V₁, V₂, …, V_m from left to right to achieve a nonlinear phase gradient. The period is $p = ma$.

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Note that due to its periodic nature, high-order diffraction effect exists in any nonlinear PGM. However, previously designed PGMs did not have a good angular response of phase gradient, then the high-order diffraction effect is too complex to reveal the behind mechanism. In proposed design, the metallic grooves employed can avoid wave coupling between unit cells and ensure that the generating THG can only excite the fundamental mode, which is reflected multiple times between the top and the bottom of the grooves, resulting in a wide-angle response of phase gradient. In this way, higher-order diffraction in nonlinear PGMs can be clearly distinguished, providing a platform to study to diffraction mechanism. In addition, similar to linear PGM [13], the grooves can effectively enhance the interaction between light and graphene through waveguide resonances [35].

Considering transverse-magnetic (TM) polarized fundamental frequency (FF) light (i.e., the magnetic field only along the y direction) normally incident to the PGM from air, a THG signal is generated immediately from the graphene ribbons due to photon interaction, and most of the generated nonlinear light will enter the dielectric grooves except for a small amount of radiation entering the air. Then, the THG experiences multiple reflections between the top and the bottom of the grooves and is finally reflected back into the air. The phase of THG radiated from each groove contains three parts: (i) the accumulated phase of propagating THG inside the groove, (ii) the abrupt phase induced by the graphene layer, and (iii) the additional phase from the multiple reflections between the top and bottom of the grooves. Because the depths and widths of the grooves are identical, the accumulated phase and the additional phase are the same for all grooves. Therefore, the phase difference (i.e., $\Delta \varphi = {\varphi _{i + 1}} - {\varphi _i}$) between two adjacent grooves is mainly determined by graphene ribbons with different chemical potentials. When $\Delta \varphi = 2\pi /m$, an abrupt phase shift fully covering $2\pi $ is satisfied, and a continuous THG wave can be generated. The required abrupt phase shift for THG can be obtained by engineering the appropriate chemical potential for each graphene ribbon by applying different voltages on the graphene ribbons, which are denoted by V₁, V₂, …, V_m from left to right in a supercell, as shown in Fig. 1(b). Once the abrupt phase shift of the THG wave is introduced along the interface of the designed metasurface, for incident FF light with an angle of ${\theta _i}$, the reflected angle ${\theta _r}$ of THG is not determined by NGSL of Eq. (1) but by the following equation instead,

3. Results and discussion

To verify this proposed theory, we consider a simple example of a nonlinear PGM with two unit cells (i.e., $m = 2$). The working wavelength is ${\lambda _{\textrm{FF}}} = 60\,\mathrm{\mu}\textrm{m}$. The geometric parameters of the unit cell are optimized as $a = 10\,\mathrm{\mu}\textrm{m}$, $h = 7\,\mathrm{\mu}\textrm{m}$, $w = 6\,\mathrm{\mu}\textrm{m}$ and ${w_g} = 2\,\mathrm{\mu}\textrm{m}$. In this case, $G = 3{k_0}$ and $\xi = 3{k_0}$, which means that the critical angle is ${\theta _c} = {0^ \circ }$ for the $n = 0$ order. To present the THG in a designed PGM, numerical calculations are performed by COMSOL Multiphysics, in which nonlinear graphene is modeled by a nonlinear surface current with ${\textbf J} = {\sigma _g}{{\textbf E}_{\textrm{TH}}} + {\sigma ^{(3)}}{\textbf E}_{\textrm{FF}}^3$ [36–39], where ${{\textbf E}_{\textrm{FF}}}$ and ${{\textbf E}_{\textrm{TH}}}$ are the electric fields of the local linear FF light and the generated THG light, respectively, and ${\sigma ^{(3)}}$ is the third-order nonlinear surface conductivity of graphene. According to the simulation method used in Ref. [21], THG radiation can be obtained. In all calculations, the intensity of incident FF light is set to $10\textrm{ }\textrm{kW}/\textrm{c}{\textrm{m}^2}$ if there is no declaration.

We first consider the THG from the designed PGM with identical graphene ribbons to reveal the relationship between the optical properties (including the amplitude and phase) of the generated THG wave and the chemical potential applied to the graphene ribbons, with the obtained results shown in Fig. 2(a). When the apploied chemical potentials are ${E_{f1}} = 0.268\textrm{ }\textrm{eV}$ (point B₁) and ${E_{f2}} = 0.160\textrm{ }\textrm{eV}$ (point A₁), a nonlinear phase difference $\pi $ is obtained for two unit cells, which requires an abrupt phase ($2\pi /m$) in the nonlinear PGM with m = 2. Figure 2(b) shows the corresponding magnetic field distributions, clearly revealing the $\pi $ phase difference of the reflected THG waves in the two unit cells. With these two unit cells, a nonlinear PGM with $m = 2$ can be designed. To reveal the diffraction process, Fig. 2(c) shows the iso-frequency contour based on Eq. (2). When the incident angle is below the critical angle, i.e., ${\theta _i} < {0^ \circ }$, the reflected THG wave is governed by the NGSL [30], i.e., $n = 0$. This relationship is indicated by the magenta circle in Fig. 2(c). When the incident angle is beyond the critical angle, i.e., ${\theta _i} > {0^ \circ }$, the reflected THG wave will follow the $n = 2$ diffraction order (see the green circle). Specular reflection (see the black circle) always exists. In theory, the reflection angles of THG for each diffraction order can be calculated exactly using Eq. (2), which are shown by the solid curves in Fig. 2(d).

 

Fig. 2. (a) The amplitude and phase of the magnetic field of the reflected THG wave (${\lambda _{\textrm{TH}}} = 20\;\mathrm{\mu}\textrm{m}$) versus the chemical potential of graphene, where the incident wave is a TM polarized plane wave with ${\lambda _{\textrm{FF}}} = 60\;\mathrm{\mu}\textrm{m}$. (b) The magnetic field (H_y) distribution of the reflected THG wave at point A₁ (left) and point B₁ (right). (c) The isofrequency contour for $\xi = 3{k_0}$, where the magenta, black, and green circles represent the different diffraction orders of the reflected THG wave. (d) The reflection angle of the THG wave versus the incident angle, where the simulated results (solid dot) agree with the calculated results (solid line) based on Eq. (2).

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To verify the proposed diffraction law, Fig. 3 displays the numerically simulated magnetic field (H_y) patterns of the THG wave for an FF Gaussian beam with different incident angles. As mentioned above, for ${\theta _i} < {0^ \circ }$, below the critical angle, the THG wave will mainly take the diffraction order of $n = 0$ and $n = 1$. Figure 3(a)–3(c) shows the results when ${\theta _i} ={-} {15^ \circ }$, $- {30^ \circ }$ and $- {60^ \circ }$, respectively. In addition to the specular reflection, the anomalous reflection or radiation of THG mainly takes the n = 0 order in all cases, with corresponding reflection angles of ${\theta _r} = {47.8^ \circ }$, ${30^ \circ }$ and ${7.7^ \circ }$, respectively. The simulated reflection angles for ${\theta _i} < {0^ \circ }$ are indicated by magenta circles in Fig. 2(d), which are consistent with the calculated values from Eq. (2). For ${\theta _i} > {0^ \circ }$, i.e., ${\theta _i} = {15^ \circ }$, ${\theta _i} = {30^ \circ }$ and ${\theta _i} = {60^ \circ }$, as shown by Figs. 3(d), 3(e), and 3(f), respectively, the reflection angles of the THG waves take mainly the diffraction order of $n = 2$. The corresponding reflection angles are ${\theta _r} ={-} {47.8^ \circ }$(${\theta _i} = {15^ \circ }$), ${\theta _r} ={-} {30^ \circ }$(${\theta _i} = {30^ \circ }$) and ${\theta _r} ={-} {7.7^ \circ }$(${\theta _i} = {60^ \circ }$), which are indicated by the green circles in Fig. 2(d) and consistent with the calculated value from Eq. (2). In addition, for specular reflection, the simulated results (see the black circle in Fig. 2(d)) also agree well with the theoretical results (see the black curve in Fig. 2(d)).

 

Fig. 3. The magnetic field (H_y) pattern of the reflected THG wave for the nonlinear metasurface with two unit cells ($m = 2$), with incident angles of (a) ${\theta _i} ={-} {15^ \circ }$, (b) ${\theta _i} ={-} {30^ \circ }$, (c) ${\theta _i} ={-} {60^ \circ }$, (d) ${\theta _i} = {15^ \circ }$, (e) ${\theta _i} = {30^ \circ }$, and (f) ${\theta _i} = {60^ \circ }$ at FF.

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In principle, the modified diffraction law of Eq. (2) can work for arbitrary $m$. In this study, we designed and demonstrated different m values ranging from 2 to 5. The obtained results are almost the same in all these cases, with only a difference in the diffraction efficiency of THG in each channel (this will be discussed later). Here we take the case of $m = 3$ for illustrations. To be consistent, the phase gradient of the THG wave is kept unchanged, i.e., $\xi = 3{k_0}$. In this way, the period length of the supercell is still $a = 20\,\mathrm{\mu}\textrm{m}$, and the geometrical parameters of the unit cell are $a = 6.67\,\mathrm{\mu}\textrm{m}$, $h = 7\,\mathrm{\mu}\textrm{m}$, $w = 5\,\mathrm{\mu}\textrm{m}$ and ${w_g} = 2\,\mathrm{\mu}\textrm{m}$. The reflected phase difference of the two adjacent unit cells in each supercell is $2\pi /3$, which can also be achieved by designing a suitable chemical potential. Figure 4(a) shows the amplitude and phase of the reflected THG wave as a function of the chemical potential of graphene for normal incident FF light. The required phase difference of $2\pi /3$ is achieved when ${E_{f1}} = 0.290\textrm{ }\textrm{eV}$ (point C₂), ${E_{f2}} = 0.218\textrm{ }\textrm{eV}$ (point B₂), and ${E_{f3}} = 0.120\textrm{ }\textrm{eV}$ (point A₂). Such phase differences in three unit cells are clearly demonstrated by the corresponding magnetic field patterns (H_y) of the reflected THG waves (see Fig. 4(b)). As $\xi = 3{k_0}$, the critical angle of the designed metasurface with $m = 3$ is also ${\theta _c} = {0^ \circ }$. For ${\theta _i} < {0^ \circ }$, the reflected THG wave would be diffracted into two diffraction orders, i.e., $n = 0$ and $n = 1$, corresponding to the magenta and black circles in Fig. 4(c), respectively. For ${\theta _i} > {0^ \circ }$, the reflected THG wave is diffracted into two diffraction orders, i.e., $n = 1$ and $n = 2$, corresponding to the black and green circles in Fig. 4(c), respectively. The diffraction rule is identical in the case of $m = 2$ (see Fig. 2(c)), and accordingly, the reflection angles for all incidences are the same for both cases of $m = 2$ and $m = 3$ (see Fig. 2(d) and Fig. 4(d)).

 

Fig. 4. (a) The amplitude and phase of the magnetic field (H_y) of the reflected THG wave versus the chemical potential of graphene. (b) The magnetic field pattern of the reflected THG wave corresponding to point A₂ (left), point B₂ (middle), and point C₂ (right). (c) The isofrequency contour for $\xi = 3{k_0}$, where the magenta, black and green circles represent the different diffraction orders of the reflected THG wave. (d) The calculated (solid line) and simulated (five-pointed star) reflection angle of the THG wave for the incident angle at FF.

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Figure 5 shows the numerically simulated results of the THG diffraction patterns when a Gaussian beam of FF light with different incident angles is incident on the designed nonlinear metasurface with $m = 3$. Similarly, when ${\theta _i} < {0^ \circ }$, incident angles of ${\theta _i} ={-} {15^ \circ }$, $- {30^ \circ }$ and $- {60^ \circ }$ are considered for demonstrations. As shown in Fig. 5(a)-(c), the reflected THG waves indeed are diffracted into the order of $n = 0$ and $n = 1$. In particular, for the $n = 0$ order, the corresponding reflection angles of the THG wave are ${\theta _r} = {47.8^ \circ }$(${\theta _i} ={-} {15^ \circ }$), ${\theta _r} = {30^ \circ }$(${\theta _i} ={-} {30^ \circ }$), and ${\theta _r} = {7.7^ \circ }$(${\theta _i} ={-} {60^ \circ }$), which are also consistent with the theoretical results given by Eq. (2) (see the magenta pentacle in FIG. 4(d)). In the case of ${\theta _i} > {0^ \circ }$, Fig. 5(d)–5(f) displays the corresponding field patterns of the generated THG when ${\theta _i} = {15^ \circ }$, ${30^ \circ }$ and ${60^ \circ }$. Likewise, the reflected THG waves are mainly diffracted into the diffraction order of $n = 1$ and $n = 2$. For the $n = 2$ order, the corresponding reflection angles of the THG wave are ${\theta _r} ={-} {47.8^ \circ }$(${\theta _i} = {15^ \circ }$), ${\theta _r} ={-} {30^ \circ }$(${\theta _i} = {30^ \circ }$), and ${\theta _r} ={-} {7.7^ \circ }$(${\theta _i} = {60^ \circ }$), which are also shown with green five-pointed stars in Fig. 4(d). The numerical results agree well with the theoretical results.

 

Fig. 5. The magnetic field (H_y) patterns of the reflected THG wave for the nonlinear metasurface with three unit cells ($m = 3$), with incident angles of (a) ${\theta _i} ={-} {15^ \circ }$, (b) ${\theta _i} ={-} {30^ \circ }$, (c) ${\theta _i} ={-} {60^ \circ }$, (d) ${\theta _i} = {15^ \circ }$, (e) ${\theta _i} = {30^ \circ }$, and (f) ${\theta _i} = {60^ \circ }$ at FF.

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Furthermore, we will show that the diffraction efficiency (i.e., conversion efficiency) in each diffraction channel, particularly on the order of n = 0 and n = 2 corresponding to anomalous radiation, is m-dependent. The efficiency is defined by ${C_{\textrm{eff}}} = {P_{\textrm{TH}}}/{P_{\textrm{FF}}}$, where ${P_{\textrm{FF}}}$ is the input power of the incident wave and ${P_{\textrm{TH}}}$ is the output power of the reflected THG wave of each diffraction channel. Figure 6 shows the conversion efficiency of THG light in two diffraction orders of n = 0 and n = 2 versus the incident angle for m = 2, 3, 4 and 5. Here, $m = 2$ is a special case, as the designed nonlinear PGM has mirror symmetry for the $yoz$ plane. Therefore, the diffraction response of the THG light has angular symmetry for full incidences (see Fig. 6(a)). For $m \ge 3$, such mirror symmetry is broken, leading to an angularly asymmetric response (see Fig. 6(b)–6(d)). In simulations and calculations, for $m = 4$, the parameters of the unit cell are $a = 5\,\mathrm{\mu}\textrm{m}$, $h = 7\,\mathrm{\mu}\textrm{m}$, $w = 4\,\mathrm{\mu}\textrm{m}$ and ${w_g} = 2\,\mathrm{\mu}\textrm{m}$. To obtain the required nonlinear phase difference of $\pi /2$ between adjacent unit cells, the chemical potentials of graphene ribbons are ${E_{f1}} = 0.338\textrm{ }\textrm{eV}$, ${E_{f2}} = 0.266\textrm{ }\textrm{eV}$, ${E_{f3}} = 0.206\textrm{ }\textrm{eV}$, and ${E_{f4}} = 0.120\textrm{ }\textrm{eV}$. For $m = 5$, the parameters are $a = 4\,\mathrm{\mu}\textrm{m}$, $h = 7\,\mathrm{\mu}\textrm{m}$, $w = 3\,\mathrm{\mu}\textrm{m}$ and ${w_g} = 2\,\mathrm{\mu}\textrm{m}$; the chemical potentials of graphene ribbons are ${E_{f1}} = 0.444\textrm{ }\textrm{eV}$, ${E_{f2}} = 0.340\textrm{ }\textrm{eV}$, ${E_{f3}} = 0.278\textrm{ }\textrm{eV}$, ${E_{f4}} = 0.212\textrm{ }\textrm{eV}$ and ${E_{f5}} = 0.120\textrm{ }\textrm{eV}$, which introduce a nonlinear phase difference of $2\pi /5$ between the adjacent unit cells.

 

Fig. 6. Conversion efficiency of the THG light in diffraction order of n = 0 and 2 versus the incident angle in the nonlinear PGM with (a) $m = 2$, (b) $m = 3$, (c)$m = 4$, and (d) $m = 5$. In all cases, the input intensity of the incident FF light is $10\;{{\textrm{KW}} / {\textrm{c}{\textrm{m}^2}}}$.

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With increasing m, the conversion efficiency in the n = 2 order gradually decreases and eventually becomes small. The physical mechanism of this asymmetric THG conversion is mainly due to the multiple reflection effect in higher-order diffraction [12,13]. Note that the n = 0 order is the lowest diffraction order, while n = 2 is a higher-order order. Similar to results in the linear PGM [12,13], the THG in the $n = 0$ order undergoes one round trip inside the grooves, and for higher-order diffraction (i.e., $n = 2$ order), the THG experiences multiple reflections inside the grooves with round trips given by $L = m - n + 1$. This means that the round trip is m-dependent, in particular $L = 1$ for $m = 2$ and $L = 2$ for $m = 3$. Owing to the ohmic loss in Au and graphene, more round trips in the higher-order diffraction lead to more energy dissipation. Thus, for $m = 2$, the diffraction response is symmetric because the round trips are the same for both $n = 0$ and $n = 2$ orders. For $m = 3$, the round trips are $L = 1$ for $n = 0$ and $L = 2$ for $n = 2$ order; then, the THG efficiency for the incident wave above the critical angle is smaller than that below the critical angle. Furthermore, as m increases, the degree of asymmetry response becomes more severe (see Fig. 6(c) and 6(d)). A large tolerance of incident angle of the metasurface is seen in the diffraction and conversion efficiency, which is mainly due to the subwavelength metallic grooves used in in the designed PGM. The subwavelength metallic groove can be regarded as a waveguide, in which only fundamental mode, i.e., TEM mode, survives. For the light of any incidences reaching the groove, the generating THG can only excite the fundamental mode that experiences multiple reflections between the top and the bottom of the grooves, leading to wide angle response.

This proposed diffraction law and the efficiency response of THG in each diffraction order provide us with a route to design nonlinear devices with fascinating functionalities. In particular, in our PGM design, the graphene used is electrically tunable, which offers much convenience for the design of tunable devices by only changing the chemical potential of graphene. For illustrations, we design a nonlinear retroreflector, which means that the reflected THG wave can be redirected back to its original direction [40–42], i.e., ${\theta _r} ={-} {\theta _i}$. Here, we take the case of $m = 2$ to demonstrate the function of nonlinear retroflection. The geometric parameters of the designed nonlinear retroreflector are set as $a = 10\,\mathrm{\mu}\textrm{m}$, $h = 7\,\mathrm{\mu}\textrm{m}$, $w = 6\,\mathrm{\mu}\textrm{m}$ and ${w_g} = 2\,\mathrm{\mu}\textrm{m}$. As $p = {\lambda _{\textrm{TH}}} = 20\,\mathrm{\mu}\textrm{m}$, the nonlinear phase gradient is $\xi = 3{k_0}$. According to Eq. (2), when the angle of the incident wave at ${\lambda _{\textrm{FF}}} = 60\,\mathrm{\mu}\textrm{m}$ is ${\pm} {30^ \circ }$, nonlinear retroreflection can be achieved. To attain nonlinear retroreflection, the chemical potentials of the graphene ribbons are selected as ${E_{f1}} = 0.268\textrm{ }\textrm{eV}$ and ${E_{f2}} = 0.160\textrm{ }\textrm{eV}$ (see Fig. 2(a)) to create a nonlinear phase difference of $\pi $. To reveal the performance of this designed nonlinear retroreflector, we show the simulated reflected fields of the THG wave in Fig. 7, where the incident waves with incident angles of ${\pm} {30^ \circ }$ illuminate the metasurface. It is clearly shown that when ${\theta _i} ={-} {30^ \circ }$, the reflected THG wave can be redirected back to its original direction with ${\theta _r} = {30^ \circ }$ (see Fig. 7(a)). Similarly, for ${\theta _i} = {30^ \circ }$, the reflection angle of the THG wave is ${\theta _r} ={-} {30^ \circ }$ (see Fig. 7(b)).

 

Fig. 7. Nonlinear retroreflector. Magnetic field (H_y) patterns of the reflected THG wave for incident angles ${\theta _i} ={-} {30^ \circ }$ (a) and ${\theta _i} = {30^ \circ }$ (b) for the designed nonlinear retroreflector.

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Moreover, using the electrically tunable property of graphene, we are able to design tunable devices [43] that can steer the generated THG in several directions by only changing the chemical potential of graphene. Based on Eq. (2), for normal incident FF light, the outgoing direction of the generated THG wave is ${\theta _r} = ar\sin ({\lambda _{\textrm{FF}}}/3p)$. By adjusting the supercell size p at a target FF, the outgoing direction of the generated THG could be controlled. As the supercell covering the $2\pi $ phase range is discretized by m groups of unit cells, the supercell size of $p = ma$ is determined by the number of unit cells m for a given size of a unit cell. In conventional design, it is very difficult to change the supercell size p once the configuration of PGMs is fixed. However, in this design with graphene, it becomes quite convenient to simply arrange and combine different voltages applied to the graphene ribbons. For example, the geometrical parameters of the metasurface (unit cell) are chosen as $a = 10\,\mathrm{\mu}\textrm{m}$, $h = 7\,\mathrm{\mu}\textrm{m}$, $w = 6\,\mathrm{\mu}\textrm{m}$ and ${w_g} = 2\,\mathrm{\mu}\textrm{m}$. When identical voltage is applied on all of the graphene ribbons (the chemical potential of graphene ribbons is ${E_f} = 0.100\textrm{ }\textrm{eV}$), there is no phase gradient along the metasurface. Then, the THG wave will be reflected by obeying Snell’s law, i.e., ${\theta _r} = {\theta _i} = {0^ \circ }$, as shown in Fig. 8(a). The applied voltage features with the arrangement of “AAAAAA”. When two different voltages are periodically applied on the graphene ribbons (${E_{f1}} = 0.268\textrm{ }\textrm{eV}$ and ${E_{f2}} = 0.160\textrm{ }\textrm{eV}$) to achieve a phase gradient $\xi = 3{k_0}$, the reflection angle of the THG wave is ${\theta _r} = {90^ \circ }$ owing to $p = 2a = 20\;\mu m$, as shown in Fig. 8(b). In this case, the arrangement of the applied voltage becomes “ABABAB”. When three different voltages are periodically applied on the graphene ribbons, i.e., ${E_{f1}} = 0.270\textrm{ }\textrm{eV}$, ${E_{f2}} = 0.202\textrm{ }\textrm{eV}$ and ${E_{f3}} = 0.100\textrm{ }\textrm{eV}$, the nonlinear phase difference of the adjancent unit cells is $2\pi /3$, and a phase gradient $\xi = 2{k_0}$ is obtained by $p = 3a = 30\,\mathrm{\mu}\textrm{m}$. As a result, the reflection angle of the THG wave is ${\theta _r} = {41.8^ \circ }$ (Fig. 8(c)). In this case, the arrangement of the applied voltage is “ABCABC”. In this way, a tunable device for wavefront control is obtained. Furthermore, as m increases and when the unit cell goes to the deep subwavelength scale, the pixel of PGM becomes very small. By applying voltage permutation and combination in the nonlinear metasurface, the nonlinear wavefront could be steered in a flexible way.

 

Fig. 8. Nonlinear tunable metasurface. The magnetic field (H_y) patterns of the reflected THG wave for the designed metasurface containing different unit cell numbers: (a) $m = 1$, (b) $m = 2$, and (c) $m = 3$. For normally incident waves at FF, the reflection angle of the THG wave is ${\theta _r} = {0^ \circ }$ ($m = 1$), ${\theta _r} = \textrm{9}{0^ \circ }$ ($m = 2$), and ${\theta _r} = \textrm{41}\textrm{.}{\textrm{8}^ \circ }$ ($m = 3$).

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4. Conclusion

We have demonstrated a modified diffraction law for controlling the nonlinear wavefront by designing and studying a graphene-based nonlinear phase-gradient metasurface in the THz region. This law provides a more complete way to manipulate nonlinear waves beyond the NGSL limit. The existence of higher-order diffraction results in an angularly asymmetric conversion efficiency of generated THG light, and the asymmetry is highly dependent on the number m of unit cells. Furthermore, by reasonably tuning the chemical potential of graphene, we realized multifunctional control of nonlinear waves, including retroreflection and beam steering. Based on recent advances in linear PGMs [12], similar discussions can be extended transmission-type nonlinear PGMs. Accordingly, more interesting nonlinear effects could be revealed in a transmission-type nonlinear metasurface, such as asymmetric transport of nonlinear light. Our work and the proposed diffraction mechanism provide a way to control nonlinear waves and design nonlinear metadevices.