Enhanced terahertz frequency mixing in all-dielectric metamaterial with multiple surface plasmon polaritons resonances

Nonlinear metamaterials hold a promising platform for generating terahertz (THz) waves. In this paper, we present an all-dielectric metamaterial with multiple surface plasmon polariton (SPP) resonances for enhanced THz frequency mixing. The metamaterial is composed of graphene ribbons, a dielectric layer, and a one-dimensional photonic crystal, displaying the multiple absorptions with simultaneous excitation of three SPP resonances. Taking advantage of SPP resonances with high Q factor and strong localized field at the input frequency, the third-order nonlinear processes are remarkably enhanced, including third-harmonic generation and four-wave mixing, producing a variety of frequencies in the THz range. The proposed efficient nonlinear metamaterials offer promising applications for THz frequency synthesis.


Introduction
Terahertz (THz) waves provide unique opportunities for a broad spectrum of important applications, such as wireless communications [1], label-free biosensing [2], and security imaging [3].Metamaterials, artificially designed structural materials, are utilized to manipulate THz waves by exploiting their exceptional electromagnetic properties.Metamaterials have facilitated the realization of essential functionalities in the THz regime including holograms [4,5], vortex beam generators [6,7], metalenses [8][9][10], and perfect anomalous diffractors [11,12].Moreover, metamaterials have emerged as a crucial platform for investigating and exploring THz nonlinear phenomena, propelling the field of nonlinear photonics [13].Metamaterials present many advantages over nonlinear bulk crystals, for example, enhanced localized density of states * Authors to whom any correspondence should be addressed.and relaxation of phase-matching techniques, causing stronger nonlinear phenomena [14].Such properties make metamaterials greatly promising for THz nonlinear applications.
Nonlinear optical frequency mixing for generating new frequencies has been widely used in various emerging research areas [15,16].Graphene has been experimentally demonstrated to exhibit an exceptionally strong third-order nonlinear susceptibility in the THz region [17,18], originating from the intraband electron transitions and the resonant nature of the interactions between light and graphene [19].Graphene can support propagating or localized surface plasmon polaritons (SPPs) like metals and 3D Dirac semimetal [20][21][22].The excitation of these highly localized SPPs in is accompanied by much stronger increase in the surface localized field, and the nonlinear optical processes are further strongly enhanced due to the dependence of the nonlinearity on the intensity.Therefore, the introduction of nonlinear graphene into metamaterials has been successfully used to enhance third harmonic generation (THG) [23][24][25][26][27][28], four-wave mixing (FWM) [29,30] and high harmonic generation (HHG) [31,32].The graphene diffraction gratings were proposed to enhance THG by strong plasmonic resonances at fundamental frequency and third-harmonic [24].The metallic grating-graphene hybrid metamaterial is presented, THG and FWM was enhanced benefiting from the robust localization and enhancement of electric field [29].Extremely high THz HHG was realized by exploiting the thermodynamic response of free electrons in the monolayer graphene [32].FWM is a common third-order nonlinear effect among various optical frequency mixing processes [33,34].In the case of FWM processes, three incident photons with different frequencies ω 1 , ω 2 , and ω 3 interacts with the system, generating a photon with frequency ω 4 governed by the equation Different from the harmonic generation processes, the multiple incident frequencies are involved in FWM process, thus the simultaneous resonances of the incident frequencies are required to enhance the efficiency [35].Although resonancebased enhanced THz FWM processes have been investigated in metamaterial platforms [29], the enhanced THz FWM processes by metamaterials with multiple simultaneous resonances have rarely been explored.
In this paper, an all-dielectric metamaterial is proposed for high-efficiency THG and FWM processes by simultaneously exciting multiple SPP resonances at the input frequencies.Due to the SPP resonances excited by the graphene ribbons, the incident waves are tightly confined and greatly boosted along the graphene surface.Additionally, the introduction of onedimensional photonic crystals improves the resonance absorption, leading to strong multiple SPP resonances with high Q factor and strong localized field.Combined with the strong third-order nonlinear susceptibility of graphene, the frequency mixing process can generate various THz frequencies.These nonlinear processes reach peaks when the pump beams correspond to the frequency at the SPP resonances.

Structure design and methods
Figure 1 illustrates the all-dielectric metamaterial illuminated by three pump beams for nonlinear optical frequency mixing processes.The designed structure is a hybrid system consisting of periodic graphene ribbons, calcium fluoride (CaF 2 ) layer and one-dimensional photonic crystal, which can be called metamaterial due to the periodic ribbons of graphene.The all-dielectric metamaterial structure is inspired by the [36] with the appropriate scaling of its dimensions to obtain resonances in the desired THz bands.The alternating Si and SiO 2 layers with N = 5 pairs form the employed onedimensional photonic crystal, enabling nearly-perfect reflection (see section 1 in the supplemental information).The refractive indices of CaF 2 , Si, and SiO 2 are 1.3, 3.47, and 1.44, respectively.The period of the designed metamaterial is P = 9.6 µm, the width of graphene ribbons is w = 8.4 µm, and the thickness of CaF 2 layer is t 0 = 3.8 µm.For the onedimensional photonic crystal with alternate dielectric layers of thicknesses a and b, refractive indices n A and n B , the central frequency ω 0 of nearly-perfect photonic reflection bandgap satisfies n A a = n B b = πc/2ω 0 [37].At the central frequency of 15 THz, the thicknesses of the Si and SiO 2 layers are set to be t 1 = 1.44 µm and t 2 = 3.46 µm.Experimentally, the alldielectric metamaterial can be realized by current technology.Electron beam induced deposition can be used to fabricate the one-dimensional photonic crystal.The graphene ribbons can be grown by chemical vapor deposition and transferred into the structure.The facile preparation of the designed metamaterials represents an advantageous feature for practical applications.The finite element method is carried out numerical simulations for the linear and nonlinear interaction between light and all-dielectric metamaterial.The periodic boundary conditions are placed in x direction, the y direction of the all-dielectric metamaterial is considered infinite.Additionally, the perfectly matched layers are set at the top and bottom of the unit cell to absorb all incident waves.
During the numerical simulation, graphene is modelled as the surface current in both linear and nonlinear responses.The linear conductivity of graphene can be estimated by the Kubo formula, which includes the intraband transition and the interband transition, as follows [38,39] where e is the electron charge, k B is the Boltzmann constant, T = 300 K is the temperature, E F = 1 eV is the graphene Fermi level, ℏ is the reduced Planck's constant, and τ is the carrier relaxation time, following equation τ = µE F /(ev F 2 ), where v F = 10 6 m s −1 is the Fermi velocity, and µ is the carrier mobility.In the previous report, the carrier mobility of graphene on silica substrate can be as high as 40 000 cm 2 (V s) −1 at room temperature [40].Considering both performance and practical feasibility of the proposed metamaterials, the carrier mobility µ is chosen to be 30 000 cm 2 (V s) −1 .Due to the 0.34 nm ultra-thin thickness of monolayer graphene, it can be modelled with surface current boundary condition to describe its optical properties with the expression J = σE, where σ is the linear conductivity of graphene and E is the electric field along the graphene surface.At THz frequencies and room temperature E F ≫ hω, k B T, the linear conductivity of graphene can be expressed as [41,42] ( The real part and imaginary part of the permittivity of graphene can be achieved.In our simulation, graphene exhibits the properties of the metallic material (see section 2 in the supplemental information).In the graphene nanoribbon structure, the excited resonance frequency of the graphene SPP mode can be expressed as [43] where ε 1 and ε 2 are the permittivity of the air and CaF 2 .

Multiple SPPs
For the proposed metamaterial without graphene ribbons, the one-dimensional photonic crystal can produce a nearly-perfect photonic reflection bandgap ranging from 10.0 to 20.2 THz shown in figure 2(a).

THz frequency mixing
We now investigate the THz frequency mixing of the proposed all-dielectric metamaterial when the nonlinear properties of graphene are introduced into the nonlinear response.
The nonlinear response is simulated by two-step sequential computations.Pulsed excitation is generally impossible to do it.Initially, the linear response is calculated in the first step.And the nonlinear current density is induced by employing the linear electric field as the source to obtain the nonlinear response, which can be given by the following formula [26,46] where E FF is the electric fields at the fundamental frequency and E TO is the electric fields at the frequency of the third order nonlinear processes.σ (3) is third order conductivity of graphene at THz frequencies following by [47,48] where σ 0 = e 2 /4ℏ, and is the Heaviside step function.The all-dielectric metamaterial is illuminated by pump beams with the relatively low optical intensity of 100 kW cm −2 under normal incidence.The physical mechanism of enhanced THz frequency mixing for the all-dielectric metamaterial is understood.Due to the SPP resonances excited by the graphene ribbons, the incident waves are tightly confined and greatly boosted along the graphene surface.The introduction of one-dimensional photonic crystals improves the resonance absorption, resulting in multiple SPP resonances with high Q factor and strong localized field.The strong localized field that interacts with the nonlinear graphene leads to the enhanced nonlinear process.To quantify the power conversion efficiency of the third-order nonlinear processes, the conversion efficiency is calculated as η = P out /P in [49,50], where P out represents the output power of frequency of the third-order nonlinear processes obtained by integrating the power outflow from the metamaterial, and P in represents the incident power of the THz pump beam.
To start with, we investigate the THG by the all-dielectric metamaterial illuminated by a single THz pump beam.In the simulations, the one physical process at the fundamental frequencies and the second physical process at the THG frequency are used.The simulation results are shown in figure 3, where the THG conversion efficiency exhibits three distinct peaks at 10.85, 14.61, and 17.55 THz, respectively.The insets in figure 3 plot the electric-field-enhancement distributions at the SPP resonances by calculating the ratio |E/E 0 |, where E 0 represents the amplitude of the incident electric field.The electric field can be enhanced by approximately 50 times along graphene surface.The strong localized field at the SPP resonances can significantly enhances the nonlinear process, as explained by equation ( 4), leading to the excellent agreement observed between the three peaks of THG conversion efficiency at the three SPP resonances.Lower or higher frequencies lead to lower THG conversion efficiencies.Although the SPP resonance M1 displays a weaker localized field enhancement at the fundamental frequency 10.85 THz compared to M3 at 17.55 THz, the THG conversion efficiency of M1 is approximately twice as high as that of M3.This arises from the higher third-order nonlinear conductivity associated with the lower frequency given by equation (5).
FWM is an another attractive third-order nonlinear optical effect, commonly requiring extremely high intensity of pump beams.To improve the FWM conversion efficiency, one approach is to enhance the localized field intensity at the multiple simultaneous resonances by exploiting an artificially designed structure.The excitation of multiple SPP resonances in the all-dielectric metamaterial allows for the investigation of FWM processes at THz frequencies.Figure 4 shows the conversion efficiency of sum frequency generation of FWM responses for mixing different excitation frequencies at the three SPP resonances of ω M1 , ω M2 and ω M3 , respectively, to generate a THz frequency ω 4 = ω 1 + ω 2 + ω 3 .In figures 4(a)-(c), the degenerate sum frequency generation is considered, where the two pump beams are fixed at a frequency ω M1 , ω M2 or ω M3 , respectively.The simulation employs two physical processes at the fundamental frequencies and a third physical process at the FWM frequency.In figure 4(d), we explore the nondegenerate sum frequency generation, realized through the pump beams with two fixed frequencies at ω M1 and ω M2 .In the simulation, three physical processes at the fundamental frequencies and the fourth physical processes at the FWM frequency are utilized.Notably, the peak of FWM conversion efficiency displays a significant enhancement by fully exploiting the three SPP resonances for the all-dielectric metamaterial at the input frequency.
The above sum frequency generation of FWM signals generated in the all-dielectric metamaterial can be classified into three distinct groups.The first group is the THG processes resulting from pump beams with identical frequency.Clearly, the THG conversion efficiencies in this case are higher than those shown in figure 3, owing to the three identical pump beams indirectly increasing the input pump intensity.The second group comprises the degenerate FWM processes, including 2ω respectively.The third group corresponds to the nondegenerate FWM process with three different pump bumps to generate a new frequency of ω M1 + ω M2 + ω M3 .Due to the simultaneous excitation of three SPP resonances at the input beam frequency, the peak of FWM conversion efficiency can be up to 0.006.As shown in figure 4, the enhancement of THG process in this case is better than that of the FWM processes.This can be understood that for the THG process, three fundamental frequencies ω 1 = ω 2 = ω 3 , the localized fields are the same and completely overlap, while for the FWM processes, the overlap among the three localized fields depends on the excited resonances, and they do not overlap completely [51].This also indicates that the conversion efficiency of the FWM processes is not simply determined by the product of Q factors of the relevant SPP resonances, but also by the overlap of the localized fields of the SPP resonances.As an example, figure 4(a) demonstrates that the FWM conversion efficiency at 2ω M1 + ω M2 is higher than that at 2ω M1 + ω M3 , with an inverse relationship to the corresponding Q factors (Q factor of 126 for M2 and Q factor of 180 for M3 in section 2).
Besides the above cascaded sum frequency generation, FWM also includes the difference frequency generation.Figure 5 shows the conversion efficiency of difference frequency generation excited by the mixing frequencies at the three SPP resonances of ω M1 , ω M2 and ω M3 , producing a frequency ω 4 = ω 1 + ω 2 − ω 3 .The differential frequency generation with 2ω M1 − ω M1 = ω M1 , 2ω M2 − ω M2 = ω M2 and 2ω M3 − ω M3 = ω M3 is not considered, which is not considered, which are e special degenerate FWM accompanied by the phase conjugate.It can be regarded as a nonlinear correction to the linear optical process, without any new frequency generation.Thus, the difference frequency generation can be divided into two groups, the first group with two identical pump beams and the second group with three different pump beams.The first group of difference frequency generation includes 2ω and 2ω M3 − ω M2 , for a kind of degenerate FWM.The second group is ω M1 + ω M2 − ω M3 .Overall, the conversion efficiency of differential frequency generation is higher than that of cascaded sum frequency generation.Taking 2ω M1 + ω M2 and 2ω M1 − ω M2 as examples, which have the same localized field at the fundamental frequency, and the frequency of FWM 2ω M1 − ω M2 is relatively low, resulting in a strong third-order nonlinear conductivity according to equation (5).However, the conversion efficiency of 2ω M3 − ω M1 is lower than that of 2ω M3 + ω M1 , giving the opposite phenomenon.After a systematic analysis, it is found that there is a strong absorption resonance at the frequency of 2ω M3 + ω M1 , leading to an improvement of the sum frequency generation with 2ω M3 + ω M1 .The conversion efficiency of the differential frequency generation 2ω M3 − ω M2 is the maximum, reaching 0.03.By employing SPP resonances with high Q factor and strong localized field at the input frequencies, the third-order nonlinear processes are remarkably enhanced, including THG and two kinds of FWM.Compared to the previously reported nonlinear graphene metamaterials and gratings [23,[25][26][27][28]50], the all-dielectric metamaterial not only achieves high nonlinear conversion efficiencies, but also generates a range of new THz frequencies.It thus offers a powerful platform for significantly enhancing and manipulating nonlinear optical processes in the THz range.
Finally, the third-order nonlinear processes as a function of the optical intensity of the pump beams are studied.Figure 6 depicts the log-log plot of the nonlinear conversion efficiency versus the pump intensity illuminated on the all-dielectric metamaterial.The THG, and two FWM processes with frequencies of 3ω M1 , 2ω M1 + ω M2 , and 2ω M1 − ω M2 are exemplified as representative cases among the three cases.The nonlinear conversion efficiency exhibits a remarkable enhancement as the incident intensity increases.As anticipated for third-order nonlinear processes, a slope value of 2 is observed between the conversion efficiency and the input pump intensity.This confirms the expected behavior and characteristics of third-order nonlinear processes, and reinforces the understanding that third-order nonlinear processes are highly sensitive to changes in optical intensity.

Conclusion
In conclusion, an all-dielectric metamaterial with multiple SPP resonances is proposed for enhanced THz frequency mixing.In the designed system, the three SPP resonances are simultaneously excited for improving nonlinear effects.By employing SPP resonances with high Q factor and strong localized field at the input frequency, the third-order frequency mixing processes are remarkably enhanced, resulting in a range of THz frequencies originating from THG and FWM processes.The proposed nonlinear metamaterials greatly expand the application in THz frequency conversion.

Figure 1 .
Figure 1.Schematic of the all-dielectric metamaterial, (a) side view of the and (b) front view.the all-dielectric metamaterial is illuminated normally by three pump beams in the THz regime to generate a variety of THz frequencies.

Figure 2 .
Figure 2. (a) Reflection spectra of the all-dielectric metamaterial without graphene, showing a nearly-perfect photonic reflection bandgap.(b) Reflection and absorption spectra of the all-dielectric metamaterial.M1, M2 and M3 represent three SPP resonances.(c) Distribution of the electric field Ey of three SPP resonances M1, M2 and M3, respectively.
Figure 2(b)  shows the reflection and absorption spectra of the all-dielectric metamaterial.With the introduction of graphene into the proposed metamaterial, the graphene localized SPP resonance is excited when the TMpolarized wave is incident into the all-dielectric metamaterial.By modifying the parameters of the graphene ribbons based on equation(3), three SPPs resonances with absorption better than 98% can be achieved in the nearly-perfect photonic reflection bandgap region.Compared to metallic layers, onedimensional photonic crystals can completely reflect the incident waves without absorption in the nearly-perfect photonic reflection bandgap region.Moreover, a stronger standing wave effect based on Fabry-Perot resonance is produced, leading to an enhancement of the electric field and high Q-factor for the resonance modes.The Q factor can be given by Q = 2πf 0 P S /P L = f 0 /∆f[44,45], where P S and P L are the stored energy and dissipated energy, f 0 is the resonant frequency, and ∆f is the full width at half maximum.The Q factor of the of SPP resonances M1, M2 and M3 are 118, 126 and 180, respectively.To get insight into the resonances absorption, figure 2(c) illustrates the spatial distribution of the electric field (Ey) at the SPP resonances.In one metamaterial period, the component of the electric field Ey shifts by 2π, 4π and 6π for the three SPP resonances, leading to strong optical absorptions.The electromagnetic energy of incident wave can be efficiently guided along the monolayer graphene due to the localized SPP resonances, achieving tightly confined and greatly enhanced localized field.The SPP resonances with high Q factor and strong localized field can significantly boost the interaction between light and nonlinear graphene, inducing a strong nonlinear response.

Figure 3 .
Figure 3.For all-dielectric metamaterial illuminated by a single THz pump beam with the optical intensity of 100 kW cm −2 .THG conversion efficiency of as a function of fundamental frequency.The inset represents the electric-field-enhancement distribution of fundamental frequency at the SPP resonances M1, M2 and M3.