Graphene-assisted dual-frequency third harmonic generation in nonlinear metamaterials for high-efficiency on-chip terahertz integration

Abstract

Nonlinear effects in the terahertz regime play a pivotal role in advancing terahertz wave generators with higher frequencies, particularly through harmonic generation processes. However, the development of efficient nonlinear terahertz materials that exhibit high conversion efficiency, compatibility with large-scale on-chip integration, and stable operation at room temperature remains a significant challenge. Graphene-assisted nonlinear metamaterials provide a promising platform for investigating nonlinear effects within the terahertz frequency regime. In this study, we present a transmission-mode nonlinear metamaterial-integrated device that synergistically combines the resonant characteristics of metamaterials with the nonlinear enhancement properties of graphene. This integrated structure enables efficient dual-frequency third harmonic generation (THG) at 9.535 THz and 10.959 THz, achieving a conversion efficiency of 0.127% under a pump intensity of 1 MW/cm². A comprehensive theoretical analysis is conducted to investigate both the linear and nonlinear operational characteristics of the integrated device. The enhancement mechanism of THG is systematically investigated by examining the electric field distributions and plasmonic resonance characteristics. Additionally, the influences of device structural parameters and terahertz wave incidence angle on the operational characteristics are thoroughly evaluated. The proposed graphene-based nonlinear metamaterial shows exceptional potential for broad applications in terahertz integrated systems and related photonic technologies.

Graphical abstract

Keywords

graphene metamaterial / third harmonic generation / nonlinear

1 Introduction

Metamaterials represent a unique class of artificially engineered composites composed of precisely arranged subwavelength structures. Their unique advantage lies in the design flexibility of their constituent elements, enabling the realization of extraordinary optical phenomena that are unattainable in natural materials [1]. Notable manifestations of these engineered properties include negative refractive index and perfect absorption [2−5]. In particular, metamaterials utilizing surface plasmon resonances (SPRs) at metallic interfaces have emerged as a highly promising research field, attracting considerable attention from the scientific community [6].

Terahertz (THz) waves exhibit distinctive characteristics that enable a broad range of applications across various fields, including spectroscopy [7], non-invasive imaging [8], and wireless communication [9]. Concurrently, nonlinear optical phenomena in the terahertz regime have progressively attracted significant research interest, offering promising prospects for ultrafast manipulation of light and frequency conversion [10]. Plasmonics can enhance the electric field by confining photons within nanoscale regions, thereby facilitating the mitigation of intrinsic weak nonlinear effects in practical systems [11]. Nevertheless, conventional plasmonic structures often face considerable limitations in nonlinear optical applications, primarily due to low conversion efficiency, poor scalability, and complex fabrication processes [12−17]. To address these challenges, the integration of graphene-based materials has emerged as a promising strategy for significantly enhancing nonlinear optical effects in the terahertz spectral regime.

Initial studies on graphene primarily focused on its mechanical characteristics [18]. As research progresses, the scientific community gradually recognizes the significant potential of graphene’s distinctive nonlinear optical properties. Particularly noteworthy manifestations encompass the remarkable electromagnetic field enhancement enabled by surface plasmon polariton (SPP) excitations, as well as the realization of dynamically tunable graphene-based nonlinear metamaterials through precise chemical doping and gate voltage modulation [19]. In the terahertz frequency range, graphene exhibits a notably strong third-order susceptibility (χ⁽³⁾), which facilitates efficient third-harmonic generation (THG) processes in graphene-based materials. Consequently, this characteristic has stimulated extensive investigations for enhanced THG efficiency [20−23]. Furthermore, graphene possesses excellent tunability characteristics, as its THG efficiency can be precisely modulated through various parameters including layer thickness, temperature, and applied gate voltage. Additionally, fabrication methodologies for graphene-based nanostructures have witnessed remarkable advancements in recent years [24−31]. Thus, graphene emerges as a highly promising and versatile nonlinear material with significant potential. In 2017, Jin et al. [32] proposed an innovative ultrathin nonlinear metamaterial design based on patterned graphene microstrips, which significantly enhanced THG processes in the far-infrared and terahertz frequencies. The excitation of highly localized plasmonic modes leads to a stringent field confinement and substantial enhancement of the incident wave within the metamaterial. Incorporating a metallic substrate beneath the graphene microstrips reduces the bandwidth of the resonant response, leading to zero transmission and the formation of standing waves in the intermediate dielectric layer. At an incident intensity of approximately 0.1 MW/cm², the conversion efficiency can reach around −26 dB. In 2023, Wang et al. [33] incorporated graphene into grating-based metamaterials and numerically investigated the enhancement of terahertz THG by leveraging bound states in the continuum (BIC) within continuous media. The system exhibited a significant THG conversion efficiency of 3.1% under an incoming light intensity of 100 kW/cm². They also introduced a graphene-metal hybrid metamaterials supporting BICs, specifically designed to enhance terahertz THG [23]. In 2024, Yan et al. [34] proposed an innovative approach utilizing graphene strip-assisted ultrathin nonlinear grating metamaterials to significantly enhance THG at terahertz frequencies. This configuration facilitates the excitation of strongly confined surface plasmon resonance, resulting in pronounced field confinement and substantial enhancement of the incident pump light along the graphene surface. At a low incident intensity of 10 kW/cm², the THG conversion efficiency may attain about 2.8%. By tuning the Fermi level of graphene, the parameter-tuned BIC supported by the graphene-metal metamaterial transitions into different quasi-BIC states, allowing for the formation of strongly confined localized fields within a single graphene layer. Currently reported research on terahertz THG in graphene-based metamaterials primarily focuses on reflective-mode configurations, which pose certain challenges for device fabrication and experimental implementation. Moreover, most existing studies are limited to achieving THG at single frequency, thereby constraining their applicability across broader spectral ranges in terahertz nonlinear devices.

This paper introduces a transmission-mode graphene-based nonlinear metamaterial designed for THG in the terahertz frequency range. The enhancement mechanism of THG in graphene-stacked strip metamaterials is theoretically investigated. Compared to previous studies, the designed structure exhibits two notable innovations: (i) The integration of graphene and silicon dioxide within the metamaterial enables dual-band THG in the terahertz regime. (ii) The metamaterial operates in transmission mode and achieves dual-frequency third-harmonic signals at 9.535 THz and 10.959 THz, with a conversion efficiency of 0.127% under a pump light intensity of 1 MW/cm². Additionally, this study examines the effect of the geometric properties of the device on the transmission and absorption spectra, as well as on the conversion efficiency of THG. Furthermore, the impact of the incident angle of incoming terahertz waves on the overall harmonic conversion performance of the metamaterial is explored. The results suggest that the proposed graphene-based nonlinear metamaterial holds significant promise for applications involving terahertz frequency converters and other nonlinear components in integration systems.

2 Theoretical approaches and methodologies

The design of the structure presented in this study draws initial inspiration from the work of Xing et al. [35]. The specific structural details are illustrated in Fig.1, which shows a layered structure consisting of three graphene strips, a silicon dioxide dielectric layer, a continuous graphene film, and a silicon dioxide substrate, arranged sequentially from top to bottom. Fig.1(a) presents a three-dimensional diagram of the full periodic metamaterial, where the top graphene strips are periodically arranged along the x-axis and extend infinitely along the y-axis. Fig.1(b) presents the side-view schematic of the unit cell, facilitating a clearer view of the entire configuration. In the illustration, the black horizontal lines represent the graphene strips, while the gray layer denotes silicon dioxide with a relative permittivity of 2.09 [35]. The entire nonlinear metamaterial structure operates in transmission mode. The research findings indicate that, within the terahertz frequency range, the graphene strips in the structure exhibit pronounced third-order nonlinearity, primarily due to intraband electronic transitions and localized electric field enhancement [32]. As shown in Fig.1(b), the three graphene strips at the top are symmetrically arranged with respect to the central axis, forming a balanced configuration conducive to field confinement and harmonic generation. The outer graphene strips have equal lengths of w₁ = 0.6 μm, while the central graphene strip has a length of w₂ = 1.6 μm. The thickness of the silicon dioxide dielectric layer located under the three graphene strips is d = 0.5 μm. A continuous graphene layer is uniformly positioned between the dielectric layer and the underlying silicon dioxide substrate. The thickness of the silicon dioxide substrate is l = 2.3 μm. The periodicity of each graphene-based nonlinear metamaterial unit cell along the x-axis is P = 8.1 μm.

In graphene-based nonlinear metamaterials, the graphene layer plays a critical role in the entire process of nonlinear harmonic generation. The incident pump light is normally directed perpendicularly along the −z direction toward the surface of the nonlinear metamaterial. By enhancing the nonlinear responses of the graphene, efficient THG process can be achieved. The conversion efficiency of THG can be defined as

(1)CE=P_{\mathrm{T}\mathrm{H}\mathrm{G}}/P_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}},

where P_THG represents the output power of the generated third harmonic, and P_pump denotes the input power of the incident pump light [36]. Additionally, in this study, the normalized conversion efficiency η_THG of THG is additionally defined as

(2){\eta }_{\mathrm{T}\mathrm{H}\mathrm{G}}={\displaystyle \frac{P_{\mathrm{T}\mathrm{H}\mathrm{G}}}{{\left(P_{\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{p}}\right)}^3}}.

This definition facilitates a more comprehensive evaluation of the relationship between pump power and the efficiency of THG conversion [37]. In the context of graphene-based metamaterials, some studies treat graphene as a material with finite thickness for theoretical analyses [38−41], while others consider graphene as an equivalent surface current density, effectively modeling it as a one-dimensional conductive layer and neglecting its physical thickness [42−45]. In this work, graphene material is modeled as a surface current density, effectively treated as a one-dimensional conductive material. The research reveals that the nonlinear surface current in graphene is jointly determined by the linear conductivity σ_L and the third-order nonlinear conductivity. Within the investigated terahertz frequency range, the conditions for interband transitions in graphene are not satisfied. As a result, the Drude model is employed to describe its electromagnetic behavior [46]:

(3){\sigma }_L\approx -{\displaystyle \frac{\mathrm{i}D}{\pi \left(\omega -\mathrm{i}/\tau \right)}}.

In this context, the expression D = q²E_f/{\hslash }^2 is utilized, and other key parameters are defined as follows: τ = 1 ps [22] represents the electron relaxation time in graphene, q is the electron charge, ω denotes the angular frequency, and \hslash is the reduced Planck constant [47]. The expression for the third-order nonlinear surface conductivity of graphene in the terahertz frequency range is provided as follows [21]:

(4){\sigma }^{\left(3\right)}={\displaystyle \frac{\mathrm{i}{\sigma }_0{\left(\hslash v_{F}q\right)}^2}{48\pi {\left(\hslash \omega \right)}^4}}T\left({\displaystyle \frac{\hslash \omega }{2E_{\mathrm{f}}}}\right),

where σ₀= q²/(4\hslash), v_F= 1 × 10⁶ m/s is the Fermi velocity, T(x) = 17G(x) − 64G(2x) + 45G(3x), and G(x) = ln|(1 + x)/(1 − x)| + iπθ(|x| − 1), with θ(x) being the Heaviside step function [48]. In Fig.2, the dependence of the third-order nonlinear surface conductivity of graphene on both Fermi energy and frequency is demonstrated. It can be observed that, in the frequency interval between 9 and 11.5 THz, the nonlinear surface conductivity decreases rapidly as the Fermi energy increases. This behavior supports the proposal that tuning the Fermi energy of graphene can enhance device performance. Moreover, under a fixed Fermi energy, the nonlinear surface conductivity of graphene increases with higher frequency.

The composite nonlinear surface current in graphene is introduced through the following equation [23]:

(5)J={\sigma }_L{E}_{\mathrm{T}\mathrm{H}}+{\sigma }^{(3)}{E}_{\mathrm{F}\mathrm{F}}^3.

Here, E_FF is the fundamental electric field of the incident pump light, and E_TH is the third-harmonic electric field generated by the structure. The linear and nonlinear characteristics of the metamaterial are theoretically explored utilizing the finite element method (FEM) in COMSOL Multiphysics. To represent the periodic arrangement of unit cells along both the x and y directions, periodic boundary conditions are applied. In addition, a perfectly matched layer (PML) is introduced along the z-axis to absorb outgoing waves and prevent artificial reflections.

To better elucidate the mechanism behind the emergence of third-harmonic peaks, we derive the equivalent permittivity and permeability from the S-parameters using the Smith retrieval method [49]. This approach reveals the electromagnetic characteristics of the metamaterial structure discussed as follows:

(6)n={\displaystyle \frac{1}{kD}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left[{\displaystyle \frac{1}{2S_{21}}}(1-S_{11}^2+S_{21}^2)\right],

(7)Z=\pm \sqrt{{\displaystyle \frac{{(1+S_{11})}^2-S_{21}^2}{{(1-S_{11})}^2-S_{21}^2}}},

(8)\epsilon ={\displaystyle \frac{n}{Z}},

(9)\mu =nZ,

where n represents the refractive index, and Z denotes the wave impedance. D is the thickness of the metamaterial, and k = 2π/λ (λ is the wavelength). S₁₁ and S₂₁ denote the reflection and transmission coefficients at the metamaterial interface, respectively.

3 Results and discussion

In this study, the incident pump electromagnetic field is directed normally onto the xy-plane, with the electric field polarized along the x-axis, perpendicular to the orientation of the graphene strips. To facilitate a more accurate evaluation of the linear optical response across the entire metamaterial structure, the graphene-centered nonlinear metamaterial is first divided into a top layer consisting of graphene gratings and a bottom layer composed of continuous graphene. The absorption and transmission characteristics of each component are calculated separately and then compared to the overall structure. In this study, the Fermi level is set to 0.6 eV. In Fig.3, when only the bottom graphene layer is present, absorption is nearly negligible, and the incident light primarily experiences transmission loss. When only the top grating is present, a noticeable difference is observed compared to the case with only the bottom graphene layer. A distinct absorption peak appears near 9.1 THz, indicating single-frequency absorption behavior. In contrast, the full graphene-based nonlinear metamaterial structure exhibits significantly enhanced linear responses, demonstrating superior broadband absorption performance. Within the frequency range of 9 to 11.5 THz, the complete graphene-based metamaterial structure exhibits dual-frequency absorption peaks, confirming effective energy absorption at two distinct frequencies. At these points, the electric field is strongly enhanced and localized around the nonlinear graphene elements of the metamaterial.

Next, we analyze and discuss the nonlinear THG process in the designed graphene-based metamaterials. In this study, the power density of the incoming pump light is set to 1 MW/cm². According to Eq. (5), the incident wave is constrained by the graphene strips positioned on the metamaterial surface, resulting in a nonlinear surface conductivity and the generation of a strongly localized field, accompanied by dual-band SPRs. The numerical results of the nonlinear process are presented in Fig.4(a), where the entire graphene-based nonlinear metamaterial exhibits dual-frequency THG with a conversion efficiency of 0.127% at the two frequency points of 9.535 THz (Peak 1) and 10.959 THz (Peak 2). Furthermore, using Eq. (2), the normalized conversion efficiency of the THG is calculated to be 1.94 × 10⁻¹³ (W⁻²). It can be observed that when only the bottom graphene layer is present, absorption is negligible, resulting in an almost nonexistent THG response. In contrast, when only the top graphene grating is present, the third-harmonic conversion efficiency is higher than that of the entire metamaterial structure, but the dual-frequency characteristic is absent. Fig.4(b) and (c) illustrate the distributions of the fundamental electric field intensity at the two frequency peaks, Peak 1 and Peak 2, respectively. These figures reveal that the generation mechanisms of the two peaks differ slightly. As shown in Fig.4(b), at 9.535 THz, a strong electric field is localized at the central graphene strip of the top grating, in conjunction with the bottom graphene layer. The side graphene strips also interact with the bottom graphene layer, further enhancing the electric field and contributing to the formation of a localized plasmonic resonance. The surface arrows representing the electric field vectors indicate a ring-like distribution around the side strips, with a relatively uniform electric field surrounding both the central and side graphene strips. In contrast, Fig.4(c) shows that at 10.959 THz, the side graphene strips in the top grating contribute minimally to the plasmonic resonance response. Only the central graphene strip interacts significantly with the bottom graphene layer, leading to strong electric field confinement. Moreover, the surface arrows − representing electric field vectors − are highly concentrated around the central strip, indicating a substantially enhanced localized field. These observations suggest that both the upper graphene grating and the bottom graphene layer play essential roles in shaping the overall resonance behavior of the metamaterial.

Fig.5 shows the refractive index, equivalent impedance, equivalent permittivity, and equivalent permeability (including both real and imaginary parts) calculated for the THG around Peak 1 using S-parameter retrieval. The parameters Z, n, ε_eff and μ_eff of the metamaterial structure are computed based on Eqs. (6) to (9). At the Peak 1 frequency of 9.535 THz, the structure can be classified as a metamaterial when the real part of the equivalent permeability or the real part of the equivalent permittivity is less than 1, or when both values are simultaneously less than 1 [50]. The nonlinear metamaterials discussed in this study exhibit diamagnetic behavior at the resonant frequency, where the near-zero permeability enhances the ability of the graphene layer to modulate the magnetization within the underlying silica dielectric layer. Furthermore, in double-positive metamaterials − where both the real parts of the equivalent permittivity and equivalent permeability are positive − both transverse electric (TE) and transverse magnetic (TM) SPP modes can be supported.

Fig.6 presents the refractive index, equivalent impedance, equivalent permittivity, and equivalent permeability (including both real and imaginary components) of the metamaterial around the Peak 2 frequency range. Within this range, the metamaterial exhibits a negative refractive index, as clearly given in Fig.6(a). Fig.6(b) illustrates the impedance characteristics of the metamaterial, where the imaginary part exhibits a negative value at the resonant frequency, reflecting the passive properties of the metamaterial [51]. Furthermore, Fig.6(c) and (d) reveal that the real part of ε_eff is negative, while the real part of μ_eff is positive, indicating that the metamaterial behaves as an epsilon-negative (ENG) medium [52]. From Fig.5, it can be seen that within the frequency range of Peak1, both the equivalent permittivity and permeability are greater than 0, satisfying the excitation conditions for SPP. Therefore, under illumination by the incident pump light, the structure supports localized plasmonic resonances.

An investigation is carried out to examine the effects of the structural parameters of graphene-based nonlinear metamaterials on the operational performances, aiming to obtain the optimal structural parameters and acceptable fabrication tolerances. The structure primarily consists of four key parameters: the thickness of dielectric separation layer (d), the widths of the upper graphene grating strips (w₁ and w₂), the thickness of dielectric substrate (l), and the period of the fundamental unit cell structure (P). From Fig.7(a), it can be observed that the dual-frequency characteristics are optimal when d = 0.58 μm. Although the conversion efficiency at Peak 1 is higher for d = 0.6 μm and d = 0.62 μm, the conversion efficiency at Peak2 does not match that achieved at d = 0.58 μm. For d = 0.54 μm and d = 0.56 μm, the dual-frequency characteristics are still present but exhibit reduced conversion efficiency compared to larger dielectric spacer thicknesses. Due to constraints imposed by the periodicity of the full metamaterial unit cell, the width of the central top graphene strip (w₂) is fixed at 1.6 μm. To systematically adjust the widths of the three top graphene strips, the differential width is defined as Δw = (w₂ − w₁)/2. As illustrated in Fig.7(b), variations in widths of the graphene strips have a substantial impact on the THG conversion efficiency. Effective dual-frequency THG occurs only when Δw = 0.5 μm. For other values of Δw, the dual-frequency characteristic disappears, and the conversion efficiency drops to less than half of that achieved at Δw = 0.5 μm.

Next, we investigate the influence of the thickness of dielectric substrate layer (l) and the periodicity (P) of the unit cell on the THG conversion efficiency. Fig.8(a) illustrates the impact of the dielectric substrate thickness (l) on the THG conversion efficiency. It can be observed that the conversion efficiency reaches its peak value when l = 2.3 μm. Any deviation from this optimal thickness − either an increase or a decrease − leads to a reduction in conversion efficiency, with larger deviations causing more pronounced declines. Additionally, the dual-frequency characteristic becomes slightly degraded under non-optimal substrate thicknesses. Fig.8(b) explores the influence of the periodicity (P) of the unit cell structure on THG conversion efficiency. It can be observed that the optimal conversion efficiency is achieved when P = 8.1 μm. Similar to the effect observed with variations in dielectric substrate thickness (l), deviations from this optimal periodicity − whether increased or decreased − lead to a noticeable decline in THG efficiency, highlighting the sensitivity of the structure to periodicity tuning.

Apart from the intrinsic properties of the graphene-based nonlinear metamaterial structure, it is crucial to examine the impact of changes in the Fermi level of the graphene material on both the absorption spectrum and THG conversion efficiency. Fig.9(a) depicts the absorption spectra of the metamaterial at different Fermi levels. It is evident that the Fermi level impacts the resonant features of the whole structure, altering the resonance frequency points. Changes in the Fermi level have a pronounced but non-monotonic effect on the absorption response of the graphene-based nonlinear metamaterial. With changes in the Fermi level, the peaks of the absorption curves are weakened to varying degrees. For instance, when the Fermi level is at 1 eV, only one absorption peak appears. According to Fig.2, it can be observed that across a certain Fermi level range, the third-order nonlinear surface conductivity of decreases rapidly with an increase in the Fermi level. This result is also well illustrated in Fig.9(b). To ensure clarity and facilitate better comparison of the different curves, the conversion efficiency corresponding to the Fermi level of 0.3 eV has been scaled down by a factor of 100. It is evident that as the Fermi level rises, the THG conversion efficiency decreases significantly. When the Fermi level falls below 0.6 eV, the conversion efficiency does not exhibit dual-band characteristics. Additionally, when the Fermi level exceeds 0.6 eV, the dual-band effect also decreases to a certain degree. The observed dependence of THG conversion efficiency on the Fermi level can be attributed to the behavior of the third-order nonlinear surface conductivity σ⁽³⁾ of graphene. As shown in Fig.2, the real part of σ⁽³⁾ decreases significantly with increasing E_f, which is consistent with the analytical model described in Eq. (4). This trend arises from the reduction in the argument \hslashω/(2E_f) in the function T(x), leading to a weakened nonlinear response. Physically, increasing E_f raises the carrier density in graphene and suppresses interband transitions, thereby reducing the strength of nonlinear polarization. As described by the nonlinear current equation Eq. (5), this reduction in σ⁽³⁾ directly limits the THG output, emphasizing the critical role of Fermi level tuning in optimizing nonlinear terahertz performance. This strong dependence is also supported by previous studies [33, 53].

Finally, we explore the effect of the incident pump light, focusing on its intensity and angle of incidence, on the overall performance of graphene-based nonlinear metamaterials. To maintain clarity and focus, our analysis of the incident light is limited to the behavior of Peak 1. Fig.10 demonstrates the relationship between THG conversion efficiency and the intensity of the incident light, with the Peak 1 centered at 9.535 THz and the Fermi level set to 0.6 eV. It is clear that higher incident intensity causes an enhancement in THG conversion efficiency. The inset presents a logarithmic graph depicting THG conversion efficiency in relation to incident light intensity, where the fitted slope is 2. This result suggests that the THG conversion efficiency is proportional to the square of the incident intensity, highlighting the nonlinear nature of the THG process [53]. An increase in incident light intensity results in a stronger electric field within the graphene material, resulting in a higher localized electric field, thereby enhancing the localized field and significantly improving the THG conversion efficiency. Based on prior studies [20, 23], it is reasonable to infer that the THG conversion efficiency tends to saturate as incident intensity continues to rise. At high excitation levels, the significant depletion of charge carriers in graphene leads to the suppression of nonlinear THG response within the saturation regime. From a practical standpoint, the proposed design − leveraging a tailored metamaterial structure − achieves high conversion efficiency even at relatively low pump intensities, offering strong potential for scalable and energy-efficient integration into future terahertz photonic systems.

In addition to examining the strength of the incident pump light, it is also essential to explore the impact of the incidence angle on the linear and nonlinear behaviors of the graphene-based nonlinear metamaterials. Fig.11 presents the variation in absorption, reflection, and THG conversion efficiency with respect to different incidence angles, focusing specifically on the behavior near the Peak 1 frequency. Since the entire graphene-based nonlinear metamaterial is viewed from a side perspective, the three top graphene strips are symmetrically arranged around the center. Therefore, the four curves exhibit symmetry with respect to an incident angle of 0°. Notably, the transmission curve reveals that the transmission gradually increases as the incidence angle becomes larger. The absorption and reflection exhibit similar behaviors, remaining relatively stable within a certain range when the incidence angle varies between approximately −25° and 25°. As the angle of incidence continues to rise, the absorption value actually increases while the reflection value decreases. The angle of incidence has a significant influence on THG conversion efficiency, which shows a marked decline with increasing incidence angles.

To more effectively evaluate the influence of the incident angle on the dual-frequency THG performance of the complete graphene-based nonlinear metamaterial structure, Fig. 12 illustrates the relationship between conversion efficiency, frequency, and angle of incidence. It is apparent that maximum THG conversion efficiencies occur at normal incidence and at the frequency points of 9.535 THz and 10.959 THz. Consistent with the trends presented in Fig.11, the THG conversion efficiency gradually diminishes as the incident angle increases under the same frequency conditions. In Fig.12, the optimal dual-frequency THG peak points are specifically highlighted using dashed lines and pentagram symbols for clarity.

Existing studies have confirmed that graphene-assisted metamaterials can achieve harmonic conversion [54, 55], and the fabrication processes for their metamaterial structures have also been systematically described. The graphene-based metamaterial proposed in this study can be fabricated using standard techniques. First, a continuous graphene sheet is synthesized via chemical vapor deposition (CVD) and transferred onto a silica dielectric substrate. Next, a silica dielectric spacer layer is deposited on top of the graphene using plasma-enhanced chemical vapor deposition (PECVD). Finally, graphene strips of varying widths − also prepared via CVD − are transferred onto the spacer layer to form the top graphene pattern. This structure holds great promise for applications in the emerging field of nonlinear metamaterials. Future work may focus on experimentally validating its THG performance.

4 Conclusions

In conclusion, we present a graphene-based nonlinear metamaterial operating in transmission mode that enables dual-frequency THG within the terahertz range. Under a pump light intensity of 1 MW/cm², the designed metamaterial achieves efficient dual-frequency THG, displaying a conversion efficiency of 0.127% at 9.535 THz and 10.959 THz frequencies. By optimizing the structural parameters of the device, the Fermi level of graphene, along with the incident pump light intensity, the characteristics of dual-frequency THG are enhanced, leading to improved nonlinear conversion efficiency. The presented graphene-based nonlinear metamaterial exhibits considerable promise for integration into frequency converters and advanced nonlinear devices operating within the terahertz range.

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