Abstract

The confined surface plasmon of fundamental wave and second harmonic wave (SHW) are investigated in graphene grating structure. The linear-optical absorption spectra with various fermi energy and carrier mobility are investigated with the finite difference time domain (FDTD) simulations and coupled mode theory (CMT). Based on the CMT, a theoretical model for the graphene grating is established to study the spectrum features of fundamental wave. The lifetimes of linear-optical resonant modes in theoretical model are investigated through the theoretical fitting of exact values in simulation, which are tunable with both the fermi energy and carrier mobility. We also have investigated the second-order nonlinearity of graphene grating by introducing the second-order nonlinear source. The proposed configuration and method are useful for research of the absorption, local field enhancement factor, lifetime of light, and nonlinear optical processes in highly integrated graphene photoelectric devices.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Graphene-based photonics and optoelectronics have the unique electronic properties and the excellent optical properties, such as high electron mobility, long mean free path, tunable band gap, broadband optical absorption, tunable optical conductivity, and ultrafast linear/nonlinear optical devices [1–3]. Compared with metal, the graphene can be physically considered as a 2D material and possesses unique features such as relatively low loss, broad tunability, and enhanced optical conductivity [4]. The graphene surface plasmon (GSP) trapped in graphene nanoribbons can break the classical diffraction limit and manipulate light in the nanoscale domain [5]. A single sheet of graphene can absorb approximately 2.3% energy of incident light. Periodic patterns in graphene not only increase the coupling between the incident field and the surface plasmon but also enhance the absorption [6]. The GSP has been studied with analytical circuit model in periodic patterns [7], graphene disk arrays [8], the interface between graphene and kerr-media [9,10], rainbow trapping [11], and optical force [12]. The graphene nanoribbon can act as a plasmonic slow light source due to its strong field confinement [13]. The coupling of field and GSP in graphene nanoribbon arrays is stronger than that in a single graphene nanoribbon [14]. The carrier density in graphene can be electrically adjusted by voltage with a field-effect transistor (FET) [15].

For a free-standing graphene, the second-order nonlinearity is forbidden because of the centrosymmetry of its structure [16]. Despite its center symmetry, second-order nonlinearity in graphene was also studied due to symmetry breaking induced by the presence of substrate [17–19], the interfacial effects [20], and the effective grating coupler [21]. The second-order nonlinearity is a powerful optical tool for probing multilayer graphene. The second-order nonlinear processes can be actualized for interacting graphene plasmonic modes with an effective grating coupler while the conversion efficiency is about 8.7 × 10−7 [21]. The impacts of the dielectric constant of core medium on the conversion efficiency (about 10−12) of metal were reported [22]. And the surface second harmonic generation (SHG) from the graphene/vicinal-SiC sample was studied with large second-order susceptibility (−1.99 × 10−10m/V) [20,23].The optical SHG on suspended single-layer and bi-layer graphene sheets was experimentally investigated [24,25]. The coupled plasmon resonator system in metal has been theoretically studied based on the temporal CMT [26–29]. The coupled graphene-cavity system was described in the framework of CMT [30]. The relations between the lifetimes and fermi energy or carrier mobility in graphene are seldom considered before.

In this paper, we investigate confined surface plasmon of fundamental wave and SHW in a graphene grating structure. And the lifetimes for linear-optical resonant modes changing with fermi energy or carrier mobility are investigated with both FDTD method and CMT analysis. The symmetry of graphene can be broken by the presence of periodic graphene nanoribbon grating. The CMT is also applied in the theoretical analysis of the linear-optical absorption under the change of fermi energy and carrier mobility. The enhanced linear fundament field in graphene grating can cause the local second-order nonlinearity. We investigate the second-order nonlinearity such as SHG, sum frequency generation (SFG) and difference frequency generation (DFG). And the local SHG, SFG and DFG in graphene nanoribbons can act as highly confined photon sources. The main results are shown as follows: (1) The GSP has strong field confinement. The GSP mode is seen to be mainly localized inside the graphene nanoribbon regions due to the short-range interaction effect and corner effect; (2) graphene has the broad tunability by changing the fermi energy or carrier mobility. The linear-optical absorption spectra for various fermi energy or carrier mobility with FDTD method agrees well with the theoretical CMT description. The expressions of the lifetimes changing with fermi energy or carrier mobility are obtained from theoretical fitting of exact values used in FDTD simulation; (3) The strongly localized field induces an expected increase of second-order nonlinearity in graphene nanoribbons. The distributions of charge density and electric field for fundamental light and the nonlinear SHG signals are also shown.

2. The coupled GSP system and theoretical analysis

We consider a periodic grating, formed by the graphene nanoribbon arrays, that has a lattice constant L = 200nm in xy-plane. The graphene grating is placed on the dielectric substrate as shown in Fig. 1(a). The graphene nanoribbons have the cuboid shape (length × width × thickness) with L1 × L2 × Δ. The parameters about the graphene nanoribbons are set as: length L1 = 160nm, width L2 = 40nm, and Δ = 1nm. To simplify the model, the dielectric substrate is considered as air. In the mid-infrared spectral region, the optical feature of graphene can be expressed by the surface conductivity σgra [3]:

σgra(ω)=ie2Ef/[πћ2(ω+iτ1)]
Here, e is electric charge, ħ is the reduced Planck's constant. The carrier relaxation time is τ = (μEf)/(eνf 2), μ is the carrier mobility, Ef is the fermi energy or chemical potential, νf is the fermi velocity. The parameters of the graphene are set as νf = 106m/s, μ = 10000cm2/(V·s), and Ef = 0.64eV [3,15].

 figure: Fig. 1

Fig. 1 (a) The cuboid-shaped graphene nanoribbon grating on dielectric substrate. The input light wave is polarized along the x direction and propagates along the z direction. (b) The CMT model with three resonant modes of the graphene grating.

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The equivalent dielectric tensor component of graphene is given by ε11 = ε22 = ε0 [1 + iσgra/(ε0ωΔ)] and ε33 = ε0 [3,15], where ε0 is the vacuum permittivity. After inserting Eq. (1) into dielectric tensor, the dielectric tensor component can be described by the drude model:

ε11(ω)=ε22(ω)=ε0[1ω2p,gra/(ω2+iωγgra)]
Here ωp,gra and γgra stand for the bulk plasma frequency and electron collision frequency of graphene grating, respectively. And these parameters for graphene grating can be set as ω2p,gra = e2Ef /(πћ2ε0Δ) and γgra = τ −1.

The characteristics of the GSP in graphene grating structure can be analyzed with CMT method as shown in Fig. 1(b). When the light wave S+,in passes through the graphene grating, the energy can be coupled into the graphene grating due to the GSP effect. The resonance modes of GSP can be inspired with the energy amplitude Am (m = 1,2,3) in the whole region of graphene grating, where dAm/dt = -iωAm. And Am is the m-th mode of resonant modes. The incoming and outgoing waves are depicted by S ± ,in(out). The energy amplitude A1, A2, and A3 of the GSP resonator can be expressed as

dA1dt=(iω11τi11τw1)A1+S+,in1τw1+S,in1τw1iμ12A2iμ13A3
dA2dt=(iω21τi21τw2)A2+S+,in1τw2+S,in1τw2iμ21A1iμ23A3
dA3dt=(iω31τi31τw3)A3+S+,in1τw3+S,in1τw3iμ32A2iμ31A1
where the 1/τim (m = 1,2,3) is the decay rate due to the intrinsic loss of the m-th mode in graphene grating and τim is the lifetime in decay process due to the intrinsic loss for the m-th mode. The 1/τwm is the decay rate due to the energy coupling from each mode into the light field and τwm is the lifetime in the process of energy coupling. μij (i,j = 1,2,3) are the coupling coefficients between the three resonant modes. The transmission function t(ω,Ef) = S+,out /S+,in and reflection function r(ω,Ef) = S-,out / S+,in of this graphene plasmonic grating system can be calculated by the following theoretical formula:
t(ω,Ef)=1(1τw1+1τi1)P1P0(1τw2+1τi2)P2P0(1τw3+1τi3)P3P0
r(ω,Ef)=1τw1P1P01τw2P2P01τw3P3P0P0=χ1χ4χ5+χ1χ3γ3+χ2χ5γ2+χ2χ3χ6+γ1χ4χ6γ1γ2γ3P1=(γ2γ3χ4χ6)1τw1+(χ1γ3+χ2χ6)1τw2+(χ2γ2+χ1χ4)1τw3P2=(χ4χ5+χ3γ3)1τw1+(γ1γ3χ2χ5)1τw2+(γ1χ4+χ2χ3)1τw3P3=(γ2χ5+χ3χ6)1τw1+(γ1χ6+χ1χ5)1τw2+(γ1γ2χ1χ3)1τw3
Where ωm2 = 2e2Ef /[ћ2ε0b,m + εt,m)Leff,m] (m = 1,2,3), the εb,m = εt,m = 1 are the effective dielectric constants of the bottom and top surfaces of graphene nanoribbon, respectively [15]. The parameter Leff,m is the effective width of the graphene nanoribbon grating for the m-th mode, which reflects the influence of the periodic graphene grating. δm = ω-ωm, γm = iδm −1/τim-1/τwm, χ1 = iμ12, χ2 = iμ13, χ3 = iμ21, χ4 = iμ23, χ5 = iμ31, χ6 = iμ32. The transmission, reflection, and absorption efficiency of the system can be expressed as T(ω,Ef) = \t\2, R(ω,Ef) = \r\2, and A(ω,Ef) = 1-T(ω,Ef)-R(ω,Ef).

The linear optical response of the graphene grating in Maxwell equations is given by:

B(1)t=×E(1)
E(1)t=c2×B(1)1ε0j(1)
j(1)=ε0ωp,gra2γgraiωE(1)=e2Efπ2Δ(γgraiω)E(1)
Here the E(1), B(1) and J(1) represent the electric field, magnetic flux intensity and current density vectors of fundamental frequency wave (FFW) which are marked with superior (1). The electric field and magnetic flux intensity of FFW can be obtained with discretizing Eq. (8) and Eq. (9) in space and time with using FDTD method. Rearranging Eq. (10), we can obtain the equation –iωj(1) = –γgra j(1) + βE(1). And then, taking inverse fourier transformation, we can obtain
j(1)/t=γgraj(1)+βE(1)
The term ˗γgraj(1) describes the current decay due to coulomb scattering with γgra = (eνf2)/(μEf). The parameter β = ε0ω2p,gra.

Now we rewrite Eq. (11) in terms of finite difference

j(1),n+1j(1),nΔt=γgraj(1),n+1j(1),n2+βE(1),n+1+E(1),n2
This equation can be simplified as
j(1),n+1=k1j(1),n+k2(E(1),n+1+E(1),n)
using k1 = (1-γgraΔt/2)/(1 + γgraΔt/2) and k2 = 0.5βΔt/(1 + γgraΔt/2).

The nonlinear response between light and the graphene grating is described by time-dependent Maxwell equations here. The nonlinear theory of second-order nonlinearity in graphene nanoribbons is based on the nonlinear Lorentz force acting on the electrons. The theory has been formulated with an approach based on the Vlasov-Maxwell equations [31]. The dynamics of second-order nonlinearity in graphene grating can be described as follows [5,32]:

B(2)t=×E(2)
E(2)t=c2×B(2)1ε0j(2)
j(2)t=γgraj(2)+βE(2)+S(2)
S(2)=krk(j(1)jk(1)en0)eme[ε0(E(1))E(1)+j(1)×B(1)]
Here, k represents the coordinates x, y, and z. And n0 and me stand for the graphene-ion density and effective electron mass in graphene grating, respectively. The surface charge density ρ(1) = ε0▽·E(1) is the net charge including the electrons and the graphene-ion background. J(2), E(2), and B(2) represent the current density vector, electric field vector, and magnetic flux intensity vector of harmonic wave which are marked with superscript (2). The S(2) is the nonlinear source for second-order nonlinearity. There are two second-order contributions to nonlinear optics. One is the electric part of the Lorentz force ρ(1)E(1), another is the magnetic part of the Lorentz force J(1) × B(1).

The FDTD approach is applied for the numerical calculation of the above first-order Eqs. (8)-(10), and the second-order Eqs. (13)-(16). There are two computational loops for the calculations of the fundamental and second harmonic fields in the program. Yee’s discretization scheme is utilized so that all electric and magnetic components can be defined in a cubic grid. The fields are temporally separated by a half time step and spatially interlaced by a half grid cell. The perfectly matched absorbing boundary conditions are employed at the below and top of the computational space along the z direction, and the periodic boundary conditions are used on the boundaries of x and y directions. Only one unit cell of the periodic graphene grating is considered in the computational space. The incident plane wave, polarized along the x direction by exciting a plane of identical dipoles in phase, propagates along the z-axis.

3. Linear-optical response of the graphene grating

The normalized linear-optical absorption (black circles) of FFW through the graphene grating is investigated with FDTD simulation as shown in Fig. 2(a). Here, the parameters are set as L = 200nm, L1 = 160nm, L2 = 40nm, and Δ = 1nm, μ = 10000cm2/(V·s), and Ef = 0.64eV. There are three different GSP resonance modes with the resonant wavelengths 3.88μm, 4.26μm and 4.82μm, respectively. The red line represents the theoretical CMT result as shown in Fig. 2(a). All the other parameters about CMT for the result in Fig. 2(a) are set as follows: Leff,1 = 5.7 × 40nm, Leff,2 = 6.8 × 40nm, Leff,3 = 8.7 × 40nm, τw1 = 7.2 × 108 rad/s, τw2 = 5.8 × 1010rad/s, τw3 = 5.3 × 1010rad/s, τi1 = 9.8 × 1011rad/s, τi2 = 1.1 × 1012rad/s, τi3 = 1.08 × 1012rad/s, μ12 = μ21 = 1011rad/s, μ13 = μ31 = 1011rad/s, μ23 = μ32 = 1012rad/s. We suppose that the coupling coefficients μ12, μ21, μ13, μ31, μ23 and μ32 are unchanged in the model later. The theoretical CMT results (red line) are in good agreement with the FDTD simulation (black circles).

 figure: Fig. 2

Fig. 2 (a) The simulated linear-optical absorption with FDTD simulation (black circles) and the theoretical CMT analysis results (red line). The distribution of (b) charge density σ(1) [labeled with ± ] and (c) electric field Ex(1) for the FFW at resonant wavelength 4.26 μm, (d) charge density σ(1) [labeled with ± ] and (e) electric field Ex(1) at wavelength 4.82 μm. (f) The plot of electric field Ex(1) along x direction in C1 region at wavelength 4.26 μm (black line) and in C2 region at wavelength 4.82 μm (blue line).

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Moreover, the electric and magnetic fields are coupled by Maxwell’s equations. For varying ε(r) (i.e.,▽ε ≠ 0), there is ▽·D(1,2) = ρ(1,2) ≠ 0 in region of graphene, where ρ(1,2) is the bulk charge density for fundamental wave and harmonic wave, respectively. The spectra stems from the fact that waves satisfy ▽·D(1,2) ≠ 0 and may induce electric charge oscillation ρ(1)(r)eiωt or ρ(2)(r)ei2ωt. Here the electric displacement component Dz(1,2) has abrupt discontinuity across the surface of graphene nanoribbons. According to the boundary condition n·Dz(1,2) = σ(1,2), where σ(1,2) is the surface charge density for FFW and SHW, respectively. And the normal direction is n = nz. Therefore, we are able to determine the magnitude of surface charges σ(1,2).

The distributions of charge density σ(1) and electric field Ex(1) for the FFW at the resonant wavelength with 4.26μm are shown in Figs. 2(b)-2(c), respectively. Inside a graphene nanoribbon, one dipole is arranged at the long-side of graphene nanoribbon due to the short-range interaction effect in Fig. 2(b). The patterns of charge density σ(1) at the long-side of graphene nanoribbon are nearly the same, but they have opposite signs which are labeled as ± . The distribution of surface charges density σ(1) in graphene grating is consistent with the local charge dipole oscillations. The electric field Ex(1) at wavelength 4.26μm is seen to be entirely localized inside the graphene nanoribbon region as shown in Fig. 2(c). Seemly, the “dipole” in Fig. 2(b) is flanged by opposite surface charge and the charges establish a standing GSP resonance inside the graphene nanoribbon region in Fig. 2(c). Strong localization in C1 region indicates that the C1 region is the principal channel of the energy transportation or absorption. After the absorption of incident wave, local field locates at the surface of graphene nanoribbon.

The charge density σ(1) for the FFW at resonant wavelength 4.82μm is seen to be strongly located at the corners of the graphene nanoribbon as shown in Fig. 2(d), which is a typical corner effect. Inside one graphene nanoribbon, two dipoles are arranged at the corners due to the corner effect. Seemly, the two “dipoles” are flanged by opposite surface charges which are labeled as ± . And these charges establish a standing wave resonance inside the short-side region of the graphene nanoribbon in Fig. 2(e). The plot of electric field Ex(1) along x direction in C1 region at wavelength 4.26μm (black line) and in C2 region at wavelength 4.82μm (blue line) are shown in Fig. 2(f). The electric field Ex(1) has the exponential decay out of the graphene nanoribbon. Figure 2(f) shows that the electric field Ex(1) is confined inside the graphene nanoribbon region with a decay length about 11nm for resonant wavelengths 4.26μm and 9nm for resonant wavelength 4.82μm, respectively.

There is a good broad tunability of the GSP in graphene nanoribbons by changing the fermi energy Ef. To get more insight into the GSP structure, the absorption spectra with different fermi energy (a) Ef = 0.74eV, (b) Ef = 0.64eV, (c) Ef = 0.54eV, and (d) Ef = 0.44eV are shown in Fig. 3 by using the CMT theory (red line) and FDTD method (black circles), respectively. The simulated absorption spectra with FDTD method are in good agreement with the theoretical CMT results. And the absorption for each resonant wavelength will get higher with the increment of fermi energy Ef as shown in Figs. 3(a)-3(d).

 figure: Fig. 3

Fig. 3 The absorption with different fermi energy (a) Ef = 0.74eV, (b) Ef = 0.64eV, (c) Ef = 0.54eV, and (d) Ef = 0.44eV by using the CMT theory (red line) and FDTD method (black circles), respectively. (e) The lifetimes for τw1, τw2, and τw3 with various fermi energy Ef. (f) The evolution of linear-optical absorption for different fermi energy with using CMT theory. The show in (g) is the top view of (f).

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All the parameters about CMT for the result in Figs. 3(a)-3(d) are set as follows: Leff,1 = 5.7 × 40nm, Leff,2 = 6.8 × 40nm, Leff,3 = 8.7 × 40nm, τw1 = 9 × 108 −3 × 1028Ef + 2.7 × 1047 Ef2, τw2 = −6.7 × 1010 + 1.9 × 1030Ef −6.4 × 1048 Ef2, τw3 = −5.2 × 1010 + 1.4 × 1030Ef −3.4 × 1048 Ef2, τi1 = 9.8 × 1011 rad/s, τi2 = 1.1 × 1012 rad/s, τi3 = 1.1 × 1012 rad/s. In the process of energy coupling, the lifetimes for τw1, τw2, and τw3 with various fermi energy Ef are shown in Fig. 3(e). The lifetimes τwm (m = 1,2,3) can be changed with the fermi energy Ef since 1/τwm represents the energy coupling between each mode and the light field. The parameters of lifetimes τi1, τi2 and τi3 are unchanged with various fermi energy Ef, which are the same as that in Fig. 2(a).

With the using the parameters about CMT above, the evolution of simulated linear-optical absorption spectra for different fermi energy is investigated with CMT theory as shown in Fig. 3(f). Here the carrier mobility is set as μ = 10000cm2/(V·s). Figure 3(g) is the top view of Fig. 3(f). The three resonance modes have the blue-shift with increasing fermi energy Ef. This quasi-linear response characteristic in Fig. 3(g) between the fermi energy Ef and resonant modes is especially valuable for the application in the optoelectronic devices of graphene.

The lower mobility in the graphene nanoribbons corresponds to higher loss. The fermi energy Ef = 0.64eV is also used here. We can suppose the lifetimes τim(m = 1,2,3) are inversely proportional to carrier mobility μ because the decay rate 1/τim represents the intrinsic loss of the m-th mode in graphene grating. To get more insight into the loss in GSP structure, The absorption spectra with different carrier mobility (a) μ = 1000cm2/(V·s), (b) μ = 5000cm2/(V·s), and (c) μ = 10000cm2/(V·s) are depicted in Fig. 4 by using the CMT theory (red line) and FDTD method (black circles), respectively. The simulated absorption spectra with FDTD method show good agreement with the theoretical CMT results. The absorption spectra become broader and shallower when the carrier mobility μ decreases from μ = 10000cm2/(V·s) to μ = 1000cm2/(V·s). As the carrier mobility decreases, the absorption for the resonant wavelength 4.26μm drops from 46.7% to 14% while the absorption for the resonant wavelength 4.82μm drops from 43.2% to 11% as shown in Figs. 4(a)-4(c).

 figure: Fig. 4

Fig. 4 The absorption spectra with different carrier mobility (a) μ = 1000 cm2/(V·s), (b) μ = 5000 cm2/(V·s), and (c) μ = 10000 cm2/(V·s) by with using both the CMT theory (red line) and the FDTD simulation (black circles). (d) The evolution of linear-optical absorption spectra with different carrier mobility μ with using CMT theory. The data fitting for (e) τw1,τw2,τw3, and (f) τi1,τi2,τi3 with various carrier mobility μ.

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The evolution of linear-optical absorption spectra for different carrier mobility μ is investigated with CMT theory as shown in Fig. 4(d). With the decrease of carrier mobility μ, the absorption of the three resonance modes in GSP system possesses the exponential decay. This exponential decay response characteristic in Fig. 4(d) between the carrier mobility μ and resonance modes is especially valuable for the application of conversion efficiency in the optoelectronic devices of graphene.

All the parameters about the CMT for the result in Figs. 4(a)-4(d) are set as follows: Leff,1 = 5.7 × 40nm, Leff,2 = 6.8 × 40nm, Leff,3 = 8.7 × 40nm, τw1 = −6.1 × 107 + 8.0 × 108μ-2.3 × 107μ2rad/s, τw2 = 3.5 × 1010 + 1.5 × 1011μ-1.3 × 1011μ2rad/s, τw3 = 1.9 × 1010 + 1.3 × 1011μ-1.1 × 1011μ2rad/s, τi1 = 9.8/μ × 1011 rad/s, τi2 = 1.1/μ × 1012 rad/s, τi3 = 1.1/μ × 1012 rad/s. The lifetimes for τw1, τw2, τw3, τi1, τi2 and τi3 with various carrier mobility μ are shown in Figs. 4(e)-4(f). The lifetimes τi1, τi2 and τi3 are inversely proportional to the change of carrier mobility μ.

4. Second-order nonlinearity of the graphene grating

Within the electric dipole approximation, the electric field of SHW can be related to the electric field of the FFW such that [33,34]:

E(2)(2ω)χ(2)E(1)(ω)E(1)(ω)
where χ(2) stands for a generalized second-order nonlinear susceptibility.

For an incident monochromatic wave with the frequency ω0, the linear electric field can be given by

E(1)(r,t)=E0(1)(r)eiω0t
with the amplitude E0(1)(r). Then, the magnetic field and current for linear optical response of graphene nanoribbons can be given by B(1) = -i▽ × E(1)/ω0 and -iωj(1) = -γgra j(1) + βE(1)j(1) = iβE(1)/(ω0 + iγgra). Since every contribution to S(2) in Eq. (16) is the form of a product A(1)B(1) between two first-order terms, these products can be expressed as: A(1)B(1) = A0(1)e-iω0t × B0(1)e-iω0t = A0(1)B0(1)e-i2ω0t. Thus, the second-order source S(2) in Eq. (16) can be expressed as
S(2)=S0(2)ei2ω0t
And the similar expresses for nonlinear optical response of graphene nanoribbons can be obtained as follows: j(2)(r,t) = j0(2)(r)e-i2ω0t, E(2)(r,t) = E0(2)(r)e-i2ω0t, and B(2)(r,t) = -i▽ × E(2) (r,t)/(2ω0). Here j0(2)(r) and E0(2)(r) are the amplitude. So j(2)(r,t), E(2)(r,t), and B(2)(r,t) have the second-order nonlinearity with second harmonic mode 2ω0 such as SHG.

For an incident wave with wavelength λ0 = 4.26μm, one can see the amplitude of fourier spectrum for FFW Ex(1) with resonant wavelength 4.26μm in Fig. 5(a). The amplitude of fourier spectrum for SHW Ex(2) and Ey(2) with the wavelength 2.13μm are shown in Figs. 5(b)-5(c). The SHG conversion efficiency η is defined in the nonlinear optical process as the expression η = \ E(2)(2ω0)/ E(1)(ω0)\. The y-polarized SHG conversion efficiency is about 10−11 while the x-polarized SHG conversion efficiency is about 10−10 for the FFW at the wavelength 4.26μm. The numerical results show that conversion efficiency is about 10−10 for the rectangle-shaped graphene nanoribbons corresponding to second-order nonlinear susceptibility χ(2)∽10−10 m/V.

 figure: Fig. 5

Fig. 5 (a) The amplitude of fourier spectrum for FFW Ex(1) with wavelength 4.26μm. The amplitude of fourier spectrum for SHW (b) Ex(2) and (c) Ey(2) with wavelength 2.13μm. The distributions of (d) charge density σ(2)[labeled with ± ] and (e) electric field Ex(2) at wavelength 2.13μm. (f) The plot of electric field Ex(2) along x direction in C1 region at resonant wavelength 2.13 μm.

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The strongly localized field and noncentrosymmetry can induce an increase of second harmonic nonlinearity signals. The distributions of charge density σ(2) and electric field Ex(2) for SHG at wavelength 2.13μm are also shown in Figs. 5(d)-5(e). The charge density σ(2) is labeled as ± in Fig. 5(d). Seemly, these charges in Fig. 5(d) establish a standing wave resonance inside the graphene nanoribbon region in Fig. 5(e). The radiation of second-order nonlinearity at the wavelength 2.13μm points into the same direction as the FFW Ex(1). The plot of second-harmonic field Ex(2) along x direction in C1 region with wavelength 2.13μm is shown in Fig. 5(f).

Under the incident monochromatic wave with wavelength λ0 = 4.82μm, the amplitude of fourier spectrum for FFW Ex(1) at wavelength 4.82μm is shown in Fig. 6(a). The amplitude of fourier spectrum for SHG Ex(2) and Ey(2) at wavelength 2.41μm are also shown in Figs. 6(b) and 6(c), respectively. The SHG conversion efficiency is about 10−10 for x polarization or y polarization at wavelength 2.41μm. The distributions of charge density σ(2) and electric field Ex(2) for SHG at wavelength 2.41μm are shown in Figs. 6(d) and 6(e). Seemly, these charges along the edge of graphene nanoribbon in Fig. 6(d) establish a standing wave resonance along the surface of graphene nanoribbon in Fig. 6(e). The plot of the second-harmonic field Ex(2) along the short side in graphene nanoribbon with wavelength 2.41μm is shown in Fig. 6(f).

 figure: Fig. 6

Fig. 6 (a) The amplitude of fourier spectrum for FFW Ex(1) with wavelength 4.82μm. The amplitude of fourier spectrum for SHG (b) Ex(2) and (c) Ey(2) with wavelength 2.41μm. The distributions of (d) charge density σ(2)[labeled with ± ] and (e) electric field Ex(2) at wavelength 2.41μm. (f) The plot of electric field Ex(2) along the short side of graphene at resonant wavelength 2.41μm.

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For the incident continue wave with two frequencies ω1 and ω2, the linear electricmagnetic field can be given by

E(1)(r,t)=E0(1)(r)(eiω1t+eiω2t)
Then, the similar magnetic field and current for linear optical response of graphene can be given by B(1)(r,t) = B0(1)(r)(e-iω1t + e-iω2t) and j(1)(r,t) = j0(1)(r)(e-iω1t + e-iω2t) with the amplitude B0(1)(r) and j0(1)(r). Since every contribution to S(2) in Eq. (16) is the form of a product A(1)B(1), these products can be expressed as: A(1)B(1) = A0(1) (e-iω1t + e-iω2t) × B0(1) (e-iω1t + e-iω2t) e-iω0t = A0(1)B0(1)[-2 + e-i2ω1t + e-i2ω2t + 2e-i(ω1+ω1)t-cos(ω2-ω1)t-sin(ω2-ω1)t]. Thus, the second order source S(2) in Eq. (16) can be described as
S(2)=2S0(2)+S1(2)ei2ω1t+S2(2)ei2ω2t+2S3(2)ei(ω1+ω1)tS4(2)cos(ω2ω1)tS5(2)sin(ω2ω1)t
The similar expresses for nonlinear optical response of graphene nanoribbons can be obtained by using Eq. (21) into Eqs. (13)-(15):
E(2)=2E0(2)+E1(2)ei2ω1t+E2(2)ei2ω2t+2E3(2)ei(ω1+ω1)tE4(2)cos(ω2ω1)tE5(2)sin(ω2ω1)t
B(2)=2B0(2)+B1(2)ei2ω1t+B2(2)ei2ω2t+2B3(2)ei(ω1+ω1)tB4(2)cos(ω2ω1)tB5(2)sin(ω2ω1)t
j(2)=2j0(2)+j1(2)ei2ω1t+j2(2)ei2ω2t+2j3(2)ei(ω1+ω1)tj4(2)cos(ω2ω1)tj5(2)sin(ω2ω1)t
Where E(2)0,1,2,3,4,5, B(2)0,1,2,3,4,5 and j(2)0,1,2,3,4,5 and are the amplitude. There are frequency components 2ω1, 2ω2, ω2 + ω1 and ω2-ω1 in the expressions of E(2), B(2), and j(2); that is to say, it has the second-order nonlinearity such as SHG, SFG and DFG.

In the case of the incident wavelengths λ1 = 4.26μm and λ2 = 4.82μm, we can obtain the second-order nonlinearity including SHG as well as the SFG and DFG. The amplitude of fourier spectrum for FFW Ex(1) with two different wavelengths λ1 = 4.26μm and λ2 = 4.82μm is shown in Fig. 7(a). The amplitude of fourier spectrum for SHW Ex(2) and Ey(2) are shown in Figs. 7(b) and 7(c). It is noted that there are four peaks for second-order nonlinear spectrum with the four resonant wavelengths λ3 = 2.13μm, λ4 = 2.27μm, λ5 = 2.41μm, and λ6 = 36.9μm in Figs. 7(b) and 7(c), respectively.

 figure: Fig. 7

Fig. 7 (a) The amplitude of fourier spectrum for FFW Ex(1) with two wavelengths λ1 = 4.26 μm and λ2 = 4.82 μm, amplitude of fourier spectrum for SHW (b) Ex(2) and (c) Ey(2) with the four wavelengths λ3, λ4, λ5, and λ6.

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We find that the second-order nonlinearity modes at wavelengths λ3 = 2.13μm and λ5 = 2.41μm are obtained from the incident monochromatic wave with wavelengths λ1 = 4.26μm and λ2 = 4.82μm due to the SHG effect, respectively. The second-order nonlinearity mode at wavelength λ4 = 2.27μm is the sum-frequency field signals for the incident continuous wave with wavelengths λ1 and λ2. The second-order nonlinearity mode at wavelength λ6 = 36.9μm is the difference-frequency field for the FFW with two resonant wavelengths λ1 and λ2. And the second-order nonlinear conversion efficiency for the FFW with two resonant wavelengths is about 10−10.

5. Conclusion

We have investigated the confined surface plasmon of fundamental wave and SHW in graphene grating. The expressions of the lifetimes for linear-optical resonant modes in CMT are obtained from theoretical fitting of exact values used in the FDTD simulation. The theoretical descriptions and data fitting will make it useful in applying the methods for future graphene applications. The linear-optical absorption spectra with various fermi energy and carrier mobility simulated FDTD method agree well with the CMT analysis. The strongly confined graphene plasmonic waves in graphene nanoribbons result from an enhancement of the local field. The strongly localized field induces a desired increase of the second-order nonlinearity including SHG as well as the SFG and DFG signals. The proposed configuration and results could provide the guidance for designing highly integrated graphene plasmonic devices.

Funding

National Natural Science Foundation of China (NSFC) (51675174), Natural Science Foundation of Hunan Province (2017JJ2097), and Education Department of Hunan Province (16A067).

References and links

1. P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014). [PubMed]  

2. T. Xiao, L. Gan, and Z. Y. Li, “Graphene surface plasmon polaritons transport on curved substrates,” Photon. Res. 3(6), 300–307 (2015).

3. X. Chen and W. Fan, “Study of the interaction between graphene and planar terahertz metamaterial with toroidal dipolar resonance,” Opt. Lett. 42(10), 2034–2037 (2017). [PubMed]  

4. Z. Shi, L. Gan, T. Xiao, H. Guo, and Z. Y. Li, “All-optical modulation of a graphene-cladded silicon photonic crystal cavity,” ACS Photonics 2, 1513–1518 (2015).

5. Y. Zeng and J. V. Moloney, “Volume electric dipole origin of second-harmonic generation from metallic membrane with noncentrosymmetric patterns,” Opt. Lett. 34(18), 2844–2846 (2009). [PubMed]  

6. M. Amin, M. Farhat, and H. Bağcı, “An ultra-broadband multilayered graphene absorber,” Opt. Express 21(24), 29938–29948 (2013). [PubMed]  

7. S. B. Parizi, B. Rejaei, and A. Khavasi, “Analytical circuit model for periodic arrays of graphene disks,” IEEE J. Quantum Electron. 51(9), 1–7 (2015).

8. S. Thongrattanasiri, F. H. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012). [PubMed]  

9. L. Wang, W. Cai, X. Zhang, and J. Xu, “Surface plasmons at the interface between graphene and Kerr-type nonlinear media,” Opt. Lett. 37(13), 2730–2732 (2012). [PubMed]  

10. T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15(11), 990–996 (2013).

11. L. Chen, T. Zhang, X. Li, and G. Wang, “Plasmonic rainbow trapping by a graphene monolayer on a dielectric layer with a silicon grating substrate,” Opt. Express 21(23), 28628–28637 (2013). [PubMed]  

12. Y. Yang, Z. Shi, J. Li, and Z. Y. Li, “Optical forces exerted on a graphene-coated dielectric particle by a focused Gaussian beam,” Photon. Res. 4(2), 65–69 (2016).

13. H. Lu, C. Zeng, Q. Zhang, X. Liu, M. M. Hossain, P. Reineck, and M. Gu, “graphene-based active slow surface plasmon polaritons,” Sci. Rep. 5, 8443 (2015). [PubMed]  

14. B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012). [PubMed]  

15. W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012). [PubMed]  

16. M. Glazov, “Second harmonic generation in graphene,” JETP Lett. 93(7), 366–371 (2011).

17. S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).

18. M. L. Ren, S. Y. Liu, B. L. Wang, B. Q. Chen, J. Li, and Z. Y. Li, “Giant enhancement of second harmonic generation by engineering double plasmonic resonances at nanoscale,” Opt. Express 22(23), 28653–28661 (2014). [PubMed]  

19. D. Smirnova, I. Shadrivov, A. Miroshnichenko, A. Smirnov, and Y. Kivshar, “Second-harmonic generation by a graphene nanoparticle,” Phys. Rev. B 90(3), 93–97 (2014).

20. M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing graphene χ(2) using a gold photon sieve,” Nano Lett. 16(1), 48–54 (2016). [PubMed]  

21. D. Smirnova and Y. Kivshar, “Second-harmonic generation in subwavelength graphene waveguides,” Phys. Rev. B 90(16), 165433 (2014).

22. A. Rahmati and N. Granpayeh, “Enhancement of the second and third harmonic generations in nanorods and nanotubes based on metal electron nonlinear responses and dielectric nonlinearity,” Plasmonics 10, 1201 (2015).

23. Y. An, J. Rowe, D. Dougherty, U. Ji, and A. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 106 (2014).

24. K. Lin, S. Weng, P. L. Yu, T. Tsai, and W. Su, “Observation of optical second harmonic generation from suspended single-layer and bi-layer graphene,” Appl. Phys. Lett. 105, 151605 (2014).

25. A. Hadarig, C. Vázquez, M. Fernández, S. V. Hoeye, G. R. Hotopan, R. Camblor, and F. Heras, “Experimental analysis of the high-order harmonic components generation in few-layer graphene,” Appl. Phys., A Mater. Sci. Process. 118(1), 83–89 (2015).

26. Q. Li, T. Wang, Y. Su, M. Yan, and M. Qiu, “Coupled mode theory analysis of mode-splitting in coupled cavity system,” Opt. Express 18(8), 8367–8382 (2010). [PubMed]  

27. G. Cao, H. Li, Y. Deng, S. Zhan, Z. He, and B. Li, “Plasmon-induced transparency in a single multimode stub resonator,” Opt. Express 22(21), 25215–25223 (2014). [PubMed]  

28. S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002).

29. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light 2nd ed. (Princeton University, 2008).

30. X. Gan, K. F. Mak, Y. Gao, Y. You, F. Hatami, J. Hone, T. F. Heinz, and D. Englund, “Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity,” Nano Lett. 12(11), 5626–5631 (2012). [PubMed]  

31. Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).

32. Y. Shen, The principles of nonlinear optical (Lohn Wiley & Sons, 1984).

33. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011). [PubMed]  

34. T. Boyd and J. Sanderson, The Physics of Plasmas (Cambridge University, 2003).

References

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  1. P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
    [PubMed]
  2. T. Xiao, L. Gan, and Z. Y. Li, “Graphene surface plasmon polaritons transport on curved substrates,” Photon. Res. 3(6), 300–307 (2015).
  3. X. Chen and W. Fan, “Study of the interaction between graphene and planar terahertz metamaterial with toroidal dipolar resonance,” Opt. Lett. 42(10), 2034–2037 (2017).
    [PubMed]
  4. Z. Shi, L. Gan, T. Xiao, H. Guo, and Z. Y. Li, “All-optical modulation of a graphene-cladded silicon photonic crystal cavity,” ACS Photonics 2, 1513–1518 (2015).
  5. Y. Zeng and J. V. Moloney, “Volume electric dipole origin of second-harmonic generation from metallic membrane with noncentrosymmetric patterns,” Opt. Lett. 34(18), 2844–2846 (2009).
    [PubMed]
  6. M. Amin, M. Farhat, and H. Bağcı, “An ultra-broadband multilayered graphene absorber,” Opt. Express 21(24), 29938–29948 (2013).
    [PubMed]
  7. S. B. Parizi, B. Rejaei, and A. Khavasi, “Analytical circuit model for periodic arrays of graphene disks,” IEEE J. Quantum Electron. 51(9), 1–7 (2015).
  8. S. Thongrattanasiri, F. H. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
    [PubMed]
  9. L. Wang, W. Cai, X. Zhang, and J. Xu, “Surface plasmons at the interface between graphene and Kerr-type nonlinear media,” Opt. Lett. 37(13), 2730–2732 (2012).
    [PubMed]
  10. T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15(11), 990–996 (2013).
  11. L. Chen, T. Zhang, X. Li, and G. Wang, “Plasmonic rainbow trapping by a graphene monolayer on a dielectric layer with a silicon grating substrate,” Opt. Express 21(23), 28628–28637 (2013).
    [PubMed]
  12. Y. Yang, Z. Shi, J. Li, and Z. Y. Li, “Optical forces exerted on a graphene-coated dielectric particle by a focused Gaussian beam,” Photon. Res. 4(2), 65–69 (2016).
  13. H. Lu, C. Zeng, Q. Zhang, X. Liu, M. M. Hossain, P. Reineck, and M. Gu, “graphene-based active slow surface plasmon polaritons,” Sci. Rep. 5, 8443 (2015).
    [PubMed]
  14. B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
    [PubMed]
  15. W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
    [PubMed]
  16. M. Glazov, “Second harmonic generation in graphene,” JETP Lett. 93(7), 366–371 (2011).
  17. S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).
  18. M. L. Ren, S. Y. Liu, B. L. Wang, B. Q. Chen, J. Li, and Z. Y. Li, “Giant enhancement of second harmonic generation by engineering double plasmonic resonances at nanoscale,” Opt. Express 22(23), 28653–28661 (2014).
    [PubMed]
  19. D. Smirnova, I. Shadrivov, A. Miroshnichenko, A. Smirnov, and Y. Kivshar, “Second-harmonic generation by a graphene nanoparticle,” Phys. Rev. B 90(3), 93–97 (2014).
  20. M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing graphene χ(2) using a gold photon sieve,” Nano Lett. 16(1), 48–54 (2016).
    [PubMed]
  21. D. Smirnova and Y. Kivshar, “Second-harmonic generation in subwavelength graphene waveguides,” Phys. Rev. B 90(16), 165433 (2014).
  22. A. Rahmati and N. Granpayeh, “Enhancement of the second and third harmonic generations in nanorods and nanotubes based on metal electron nonlinear responses and dielectric nonlinearity,” Plasmonics 10, 1201 (2015).
  23. Y. An, J. Rowe, D. Dougherty, U. Ji, and A. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 106 (2014).
  24. K. Lin, S. Weng, P. L. Yu, T. Tsai, and W. Su, “Observation of optical second harmonic generation from suspended single-layer and bi-layer graphene,” Appl. Phys. Lett. 105, 151605 (2014).
  25. A. Hadarig, C. Vázquez, M. Fernández, S. V. Hoeye, G. R. Hotopan, R. Camblor, and F. Heras, “Experimental analysis of the high-order harmonic components generation in few-layer graphene,” Appl. Phys., A Mater. Sci. Process. 118(1), 83–89 (2015).
  26. Q. Li, T. Wang, Y. Su, M. Yan, and M. Qiu, “Coupled mode theory analysis of mode-splitting in coupled cavity system,” Opt. Express 18(8), 8367–8382 (2010).
    [PubMed]
  27. G. Cao, H. Li, Y. Deng, S. Zhan, Z. He, and B. Li, “Plasmon-induced transparency in a single multimode stub resonator,” Opt. Express 22(21), 25215–25223 (2014).
    [PubMed]
  28. S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002).
  29. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light 2nd ed. (Princeton University, 2008).
  30. X. Gan, K. F. Mak, Y. Gao, Y. You, F. Hatami, J. Hone, T. F. Heinz, and D. Englund, “Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity,” Nano Lett. 12(11), 5626–5631 (2012).
    [PubMed]
  31. Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
  32. Y. Shen, The principles of nonlinear optical (Lohn Wiley & Sons, 1984).
  33. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
    [PubMed]
  34. T. Boyd and J. Sanderson, The Physics of Plasmas (Cambridge University, 2003).

2017 (1)

2016 (2)

Y. Yang, Z. Shi, J. Li, and Z. Y. Li, “Optical forces exerted on a graphene-coated dielectric particle by a focused Gaussian beam,” Photon. Res. 4(2), 65–69 (2016).

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing graphene χ(2) using a gold photon sieve,” Nano Lett. 16(1), 48–54 (2016).
[PubMed]

2015 (6)

A. Rahmati and N. Granpayeh, “Enhancement of the second and third harmonic generations in nanorods and nanotubes based on metal electron nonlinear responses and dielectric nonlinearity,” Plasmonics 10, 1201 (2015).

A. Hadarig, C. Vázquez, M. Fernández, S. V. Hoeye, G. R. Hotopan, R. Camblor, and F. Heras, “Experimental analysis of the high-order harmonic components generation in few-layer graphene,” Appl. Phys., A Mater. Sci. Process. 118(1), 83–89 (2015).

H. Lu, C. Zeng, Q. Zhang, X. Liu, M. M. Hossain, P. Reineck, and M. Gu, “graphene-based active slow surface plasmon polaritons,” Sci. Rep. 5, 8443 (2015).
[PubMed]

Z. Shi, L. Gan, T. Xiao, H. Guo, and Z. Y. Li, “All-optical modulation of a graphene-cladded silicon photonic crystal cavity,” ACS Photonics 2, 1513–1518 (2015).

T. Xiao, L. Gan, and Z. Y. Li, “Graphene surface plasmon polaritons transport on curved substrates,” Photon. Res. 3(6), 300–307 (2015).

S. B. Parizi, B. Rejaei, and A. Khavasi, “Analytical circuit model for periodic arrays of graphene disks,” IEEE J. Quantum Electron. 51(9), 1–7 (2015).

2014 (7)

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

M. L. Ren, S. Y. Liu, B. L. Wang, B. Q. Chen, J. Li, and Z. Y. Li, “Giant enhancement of second harmonic generation by engineering double plasmonic resonances at nanoscale,” Opt. Express 22(23), 28653–28661 (2014).
[PubMed]

D. Smirnova, I. Shadrivov, A. Miroshnichenko, A. Smirnov, and Y. Kivshar, “Second-harmonic generation by a graphene nanoparticle,” Phys. Rev. B 90(3), 93–97 (2014).

G. Cao, H. Li, Y. Deng, S. Zhan, Z. He, and B. Li, “Plasmon-induced transparency in a single multimode stub resonator,” Opt. Express 22(21), 25215–25223 (2014).
[PubMed]

Y. An, J. Rowe, D. Dougherty, U. Ji, and A. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 106 (2014).

K. Lin, S. Weng, P. L. Yu, T. Tsai, and W. Su, “Observation of optical second harmonic generation from suspended single-layer and bi-layer graphene,” Appl. Phys. Lett. 105, 151605 (2014).

D. Smirnova and Y. Kivshar, “Second-harmonic generation in subwavelength graphene waveguides,” Phys. Rev. B 90(16), 165433 (2014).

2013 (4)

M. Amin, M. Farhat, and H. Bağcı, “An ultra-broadband multilayered graphene absorber,” Opt. Express 21(24), 29938–29948 (2013).
[PubMed]

S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15(11), 990–996 (2013).

L. Chen, T. Zhang, X. Li, and G. Wang, “Plasmonic rainbow trapping by a graphene monolayer on a dielectric layer with a silicon grating substrate,” Opt. Express 21(23), 28628–28637 (2013).
[PubMed]

2012 (5)

S. Thongrattanasiri, F. H. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[PubMed]

L. Wang, W. Cai, X. Zhang, and J. Xu, “Surface plasmons at the interface between graphene and Kerr-type nonlinear media,” Opt. Lett. 37(13), 2730–2732 (2012).
[PubMed]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[PubMed]

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[PubMed]

X. Gan, K. F. Mak, Y. Gao, Y. You, F. Hatami, J. Hone, T. F. Heinz, and D. Englund, “Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity,” Nano Lett. 12(11), 5626–5631 (2012).
[PubMed]

2011 (2)

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[PubMed]

M. Glazov, “Second harmonic generation in graphene,” JETP Lett. 93(7), 366–371 (2011).

2010 (1)

2009 (2)

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).

Y. Zeng and J. V. Moloney, “Volume electric dipole origin of second-harmonic generation from metallic membrane with noncentrosymmetric patterns,” Opt. Lett. 34(18), 2844–2846 (2009).
[PubMed]

2002 (1)

S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002).

Alonso-González, P.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

Amin, M.

An, Y.

Y. An, J. Rowe, D. Dougherty, U. Ji, and A. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 106 (2014).

Bagci, H.

Cai, W.

Camblor, R.

A. Hadarig, C. Vázquez, M. Fernández, S. V. Hoeye, G. R. Hotopan, R. Camblor, and F. Heras, “Experimental analysis of the high-order harmonic components generation in few-layer graphene,” Appl. Phys., A Mater. Sci. Process. 118(1), 83–89 (2015).

Cao, G.

Casanova, F.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

Cecchet, F.

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing graphene χ(2) using a gold photon sieve,” Nano Lett. 16(1), 48–54 (2016).
[PubMed]

Centeno, A.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

Chen, B. Q.

Chen, J.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

Chen, L.

Chen, X.

Colomer, J. F.

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing graphene χ(2) using a gold photon sieve,” Nano Lett. 16(1), 48–54 (2016).
[PubMed]

Deng, Y.

Diebold, A.

Y. An, J. Rowe, D. Dougherty, U. Ji, and A. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 106 (2014).

Dougherty, D.

Y. An, J. Rowe, D. Dougherty, U. Ji, and A. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 106 (2014).

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[PubMed]

Englund, D.

X. Gan, K. F. Mak, Y. Gao, Y. You, F. Hatami, J. Hone, T. F. Heinz, and D. Englund, “Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity,” Nano Lett. 12(11), 5626–5631 (2012).
[PubMed]

Fan, S. H.

S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002).

Fan, W.

Farhat, M.

Fernández, M.

A. Hadarig, C. Vázquez, M. Fernández, S. V. Hoeye, G. R. Hotopan, R. Camblor, and F. Heras, “Experimental analysis of the high-order harmonic components generation in few-layer graphene,” Appl. Phys., A Mater. Sci. Process. 118(1), 83–89 (2015).

Gan, L.

Z. Shi, L. Gan, T. Xiao, H. Guo, and Z. Y. Li, “All-optical modulation of a graphene-cladded silicon photonic crystal cavity,” ACS Photonics 2, 1513–1518 (2015).

T. Xiao, L. Gan, and Z. Y. Li, “Graphene surface plasmon polaritons transport on curved substrates,” Photon. Res. 3(6), 300–307 (2015).

Gan, X.

X. Gan, K. F. Mak, Y. Gao, Y. You, F. Hatami, J. Hone, T. F. Heinz, and D. Englund, “Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity,” Nano Lett. 12(11), 5626–5631 (2012).
[PubMed]

Gao, W.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[PubMed]

Gao, Y.

X. Gan, K. F. Mak, Y. Gao, Y. You, F. Hatami, J. Hone, T. F. Heinz, and D. Englund, “Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity,” Nano Lett. 12(11), 5626–5631 (2012).
[PubMed]

García de Abajo, F. J.

S. Thongrattanasiri, F. H. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[PubMed]

García-Vidal, F. J.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[PubMed]

Glazov, M.

M. Glazov, “Second harmonic generation in graphene,” JETP Lett. 93(7), 366–371 (2011).

Golmar, F.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

Granpayeh, N.

A. Rahmati and N. Granpayeh, “Enhancement of the second and third harmonic generations in nanorods and nanotubes based on metal electron nonlinear responses and dielectric nonlinearity,” Plasmonics 10, 1201 (2015).

Gu, M.

H. Lu, C. Zeng, Q. Zhang, X. Liu, M. M. Hossain, P. Reineck, and M. Gu, “graphene-based active slow surface plasmon polaritons,” Sci. Rep. 5, 8443 (2015).
[PubMed]

Guo, H.

Z. Shi, L. Gan, T. Xiao, H. Guo, and Z. Y. Li, “All-optical modulation of a graphene-cladded silicon photonic crystal cavity,” ACS Photonics 2, 1513–1518 (2015).

S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).

Hadarig, A.

A. Hadarig, C. Vázquez, M. Fernández, S. V. Hoeye, G. R. Hotopan, R. Camblor, and F. Heras, “Experimental analysis of the high-order harmonic components generation in few-layer graphene,” Appl. Phys., A Mater. Sci. Process. 118(1), 83–89 (2015).

Hatami, F.

X. Gan, K. F. Mak, Y. Gao, Y. You, F. Hatami, J. Hone, T. F. Heinz, and D. Englund, “Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity,” Nano Lett. 12(11), 5626–5631 (2012).
[PubMed]

He, Z.

Heinz, T. F.

X. Gan, K. F. Mak, Y. Gao, Y. You, F. Hatami, J. Hone, T. F. Heinz, and D. Englund, “Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity,” Nano Lett. 12(11), 5626–5631 (2012).
[PubMed]

Heras, F.

A. Hadarig, C. Vázquez, M. Fernández, S. V. Hoeye, G. R. Hotopan, R. Camblor, and F. Heras, “Experimental analysis of the high-order harmonic components generation in few-layer graphene,” Appl. Phys., A Mater. Sci. Process. 118(1), 83–89 (2015).

Hillenbrand, R.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

Hoeye, S. V.

A. Hadarig, C. Vázquez, M. Fernández, S. V. Hoeye, G. R. Hotopan, R. Camblor, and F. Heras, “Experimental analysis of the high-order harmonic components generation in few-layer graphene,” Appl. Phys., A Mater. Sci. Process. 118(1), 83–89 (2015).

Hone, J.

X. Gan, K. F. Mak, Y. Gao, Y. You, F. Hatami, J. Hone, T. F. Heinz, and D. Englund, “Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity,” Nano Lett. 12(11), 5626–5631 (2012).
[PubMed]

Hossain, M. M.

H. Lu, C. Zeng, Q. Zhang, X. Liu, M. M. Hossain, P. Reineck, and M. Gu, “graphene-based active slow surface plasmon polaritons,” Sci. Rep. 5, 8443 (2015).
[PubMed]

Hotopan, G. R.

A. Hadarig, C. Vázquez, M. Fernández, S. V. Hoeye, G. R. Hotopan, R. Camblor, and F. Heras, “Experimental analysis of the high-order harmonic components generation in few-layer graphene,” Appl. Phys., A Mater. Sci. Process. 118(1), 83–89 (2015).

Hoyer, W.

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).

Huang, L.

S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).

Hueso, L. E.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

Ji, U.

Y. An, J. Rowe, D. Dougherty, U. Ji, and A. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 106 (2014).

Khavasi, A.

S. B. Parizi, B. Rejaei, and A. Khavasi, “Analytical circuit model for periodic arrays of graphene disks,” IEEE J. Quantum Electron. 51(9), 1–7 (2015).

Kivshar, Y.

D. Smirnova and Y. Kivshar, “Second-harmonic generation in subwavelength graphene waveguides,” Phys. Rev. B 90(16), 165433 (2014).

D. Smirnova, I. Shadrivov, A. Miroshnichenko, A. Smirnov, and Y. Kivshar, “Second-harmonic generation by a graphene nanoparticle,” Phys. Rev. B 90(3), 93–97 (2014).

Koch, S.

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).

Koppens, F.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

Koppens, F. H.

S. Thongrattanasiri, F. H. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[PubMed]

Li, B.

Li, H.

Li, J.

Li, Q.

S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).

Q. Li, T. Wang, Y. Su, M. Yan, and M. Qiu, “Coupled mode theory analysis of mode-splitting in coupled cavity system,” Opt. Express 18(8), 8367–8382 (2010).
[PubMed]

Li, X.

Li, Z. Y.

Y. Yang, Z. Shi, J. Li, and Z. Y. Li, “Optical forces exerted on a graphene-coated dielectric particle by a focused Gaussian beam,” Photon. Res. 4(2), 65–69 (2016).

T. Xiao, L. Gan, and Z. Y. Li, “Graphene surface plasmon polaritons transport on curved substrates,” Photon. Res. 3(6), 300–307 (2015).

Z. Shi, L. Gan, T. Xiao, H. Guo, and Z. Y. Li, “All-optical modulation of a graphene-cladded silicon photonic crystal cavity,” ACS Photonics 2, 1513–1518 (2015).

M. L. Ren, S. Y. Liu, B. L. Wang, B. Q. Chen, J. Li, and Z. Y. Li, “Giant enhancement of second harmonic generation by engineering double plasmonic resonances at nanoscale,” Opt. Express 22(23), 28653–28661 (2014).
[PubMed]

S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).

Lin, K.

K. Lin, S. Weng, P. L. Yu, T. Tsai, and W. Su, “Observation of optical second harmonic generation from suspended single-layer and bi-layer graphene,” Appl. Phys. Lett. 105, 151605 (2014).

Lis, D.

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing graphene χ(2) using a gold photon sieve,” Nano Lett. 16(1), 48–54 (2016).
[PubMed]

Liu, J.

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).

Liu, S.

S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).

Liu, S. Y.

Liu, X.

H. Lu, C. Zeng, Q. Zhang, X. Liu, M. M. Hossain, P. Reineck, and M. Gu, “graphene-based active slow surface plasmon polaritons,” Sci. Rep. 5, 8443 (2015).
[PubMed]

Lobet, M.

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing graphene χ(2) using a gold photon sieve,” Nano Lett. 16(1), 48–54 (2016).
[PubMed]

Lu, H.

H. Lu, C. Zeng, Q. Zhang, X. Liu, M. M. Hossain, P. Reineck, and M. Gu, “graphene-based active slow surface plasmon polaritons,” Sci. Rep. 5, 8443 (2015).
[PubMed]

Mak, K. F.

X. Gan, K. F. Mak, Y. Gao, Y. You, F. Hatami, J. Hone, T. F. Heinz, and D. Englund, “Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity,” Nano Lett. 12(11), 5626–5631 (2012).
[PubMed]

Martin-Moreno, L.

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15(11), 990–996 (2013).

Meng, Z.

S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).

Miroshnichenko, A.

D. Smirnova, I. Shadrivov, A. Miroshnichenko, A. Smirnov, and Y. Kivshar, “Second-harmonic generation by a graphene nanoparticle,” Phys. Rev. B 90(3), 93–97 (2014).

Moloney, J.

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).

Moloney, J. V.

Navickaite, G.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

Nesterov, M. L.

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15(11), 990–996 (2013).

Nikitin, A. Y.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15(11), 990–996 (2013).

Parizi, S. B.

S. B. Parizi, B. Rejaei, and A. Khavasi, “Analytical circuit model for periodic arrays of graphene disks,” IEEE J. Quantum Electron. 51(9), 1–7 (2015).

Pesquera, A.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

Qiu, C.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[PubMed]

Qiu, M.

Rahmati, A.

A. Rahmati and N. Granpayeh, “Enhancement of the second and third harmonic generations in nanorods and nanotubes based on metal electron nonlinear responses and dielectric nonlinearity,” Plasmonics 10, 1201 (2015).

Reckinger, N.

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing graphene χ(2) using a gold photon sieve,” Nano Lett. 16(1), 48–54 (2016).
[PubMed]

Reineck, P.

H. Lu, C. Zeng, Q. Zhang, X. Liu, M. M. Hossain, P. Reineck, and M. Gu, “graphene-based active slow surface plasmon polaritons,” Sci. Rep. 5, 8443 (2015).
[PubMed]

Rejaei, B.

S. B. Parizi, B. Rejaei, and A. Khavasi, “Analytical circuit model for periodic arrays of graphene disks,” IEEE J. Quantum Electron. 51(9), 1–7 (2015).

Ren, M. L.

Rowe, J.

Y. An, J. Rowe, D. Dougherty, U. Ji, and A. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 106 (2014).

Sarrazin, M.

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing graphene χ(2) using a gold photon sieve,” Nano Lett. 16(1), 48–54 (2016).
[PubMed]

Shadrivov, I.

D. Smirnova, I. Shadrivov, A. Miroshnichenko, A. Smirnov, and Y. Kivshar, “Second-harmonic generation by a graphene nanoparticle,” Phys. Rev. B 90(3), 93–97 (2014).

Shi, Z.

Y. Yang, Z. Shi, J. Li, and Z. Y. Li, “Optical forces exerted on a graphene-coated dielectric particle by a focused Gaussian beam,” Photon. Res. 4(2), 65–69 (2016).

Z. Shi, L. Gan, T. Xiao, H. Guo, and Z. Y. Li, “All-optical modulation of a graphene-cladded silicon photonic crystal cavity,” ACS Photonics 2, 1513–1518 (2015).

S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).

Shu, J.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[PubMed]

Slipchenko, T. M.

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15(11), 990–996 (2013).

Smirnov, A.

D. Smirnova, I. Shadrivov, A. Miroshnichenko, A. Smirnov, and Y. Kivshar, “Second-harmonic generation by a graphene nanoparticle,” Phys. Rev. B 90(3), 93–97 (2014).

Smirnova, D.

D. Smirnova, I. Shadrivov, A. Miroshnichenko, A. Smirnov, and Y. Kivshar, “Second-harmonic generation by a graphene nanoparticle,” Phys. Rev. B 90(3), 93–97 (2014).

D. Smirnova and Y. Kivshar, “Second-harmonic generation in subwavelength graphene waveguides,” Phys. Rev. B 90(16), 165433 (2014).

Su, W.

K. Lin, S. Weng, P. L. Yu, T. Tsai, and W. Su, “Observation of optical second harmonic generation from suspended single-layer and bi-layer graphene,” Appl. Phys. Lett. 105, 151605 (2014).

Su, Y.

Teng, J.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[PubMed]

Thongrattanasiri, S.

S. Thongrattanasiri, F. H. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[PubMed]

Tsai, T.

K. Lin, S. Weng, P. L. Yu, T. Tsai, and W. Su, “Observation of optical second harmonic generation from suspended single-layer and bi-layer graphene,” Appl. Phys. Lett. 105, 151605 (2014).

Vakil, A.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[PubMed]

Vázquez, C.

A. Hadarig, C. Vázquez, M. Fernández, S. V. Hoeye, G. R. Hotopan, R. Camblor, and F. Heras, “Experimental analysis of the high-order harmonic components generation in few-layer graphene,” Appl. Phys., A Mater. Sci. Process. 118(1), 83–89 (2015).

Vélez, S.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

Vlad, A.

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing graphene χ(2) using a gold photon sieve,” Nano Lett. 16(1), 48–54 (2016).
[PubMed]

Wang, B.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[PubMed]

Wang, B. L.

Wang, C.

S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).

Wang, G.

Wang, L.

Wang, T.

Weng, S.

K. Lin, S. Weng, P. L. Yu, T. Tsai, and W. Su, “Observation of optical second harmonic generation from suspended single-layer and bi-layer graphene,” Appl. Phys. Lett. 105, 151605 (2014).

Xiao, T.

Z. Shi, L. Gan, T. Xiao, H. Guo, and Z. Y. Li, “All-optical modulation of a graphene-cladded silicon photonic crystal cavity,” ACS Photonics 2, 1513–1518 (2015).

T. Xiao, L. Gan, and Z. Y. Li, “Graphene surface plasmon polaritons transport on curved substrates,” Photon. Res. 3(6), 300–307 (2015).

Xu, H.

S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).

Xu, J.

Xu, Q.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[PubMed]

Yan, M.

Yang, Y.

You, Y.

X. Gan, K. F. Mak, Y. Gao, Y. You, F. Hatami, J. Hone, T. F. Heinz, and D. Englund, “Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity,” Nano Lett. 12(11), 5626–5631 (2012).
[PubMed]

Yu, P. L.

K. Lin, S. Weng, P. L. Yu, T. Tsai, and W. Su, “Observation of optical second harmonic generation from suspended single-layer and bi-layer graphene,” Appl. Phys. Lett. 105, 151605 (2014).

Yuan, X.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[PubMed]

Zeng, C.

H. Lu, C. Zeng, Q. Zhang, X. Liu, M. M. Hossain, P. Reineck, and M. Gu, “graphene-based active slow surface plasmon polaritons,” Sci. Rep. 5, 8443 (2015).
[PubMed]

Zeng, Y.

Y. Zeng and J. V. Moloney, “Volume electric dipole origin of second-harmonic generation from metallic membrane with noncentrosymmetric patterns,” Opt. Lett. 34(18), 2844–2846 (2009).
[PubMed]

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).

Zhan, S.

Zhang, Q.

H. Lu, C. Zeng, Q. Zhang, X. Liu, M. M. Hossain, P. Reineck, and M. Gu, “graphene-based active slow surface plasmon polaritons,” Sci. Rep. 5, 8443 (2015).
[PubMed]

Zhang, T.

Zhang, X.

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[PubMed]

L. Wang, W. Cai, X. Zhang, and J. Xu, “Surface plasmons at the interface between graphene and Kerr-type nonlinear media,” Opt. Lett. 37(13), 2730–2732 (2012).
[PubMed]

Zurutuza, A.

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

ACS Nano (1)

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[PubMed]

ACS Photonics (1)

Z. Shi, L. Gan, T. Xiao, H. Guo, and Z. Y. Li, “All-optical modulation of a graphene-cladded silicon photonic crystal cavity,” ACS Photonics 2, 1513–1518 (2015).

Appl. Phys. Lett. (2)

K. Lin, S. Weng, P. L. Yu, T. Tsai, and W. Su, “Observation of optical second harmonic generation from suspended single-layer and bi-layer graphene,” Appl. Phys. Lett. 105, 151605 (2014).

S. H. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002).

Appl. Phys., A Mater. Sci. Process. (1)

A. Hadarig, C. Vázquez, M. Fernández, S. V. Hoeye, G. R. Hotopan, R. Camblor, and F. Heras, “Experimental analysis of the high-order harmonic components generation in few-layer graphene,” Appl. Phys., A Mater. Sci. Process. 118(1), 83–89 (2015).

IEEE J. Quantum Electron. (1)

S. B. Parizi, B. Rejaei, and A. Khavasi, “Analytical circuit model for periodic arrays of graphene disks,” IEEE J. Quantum Electron. 51(9), 1–7 (2015).

J. Opt. (1)

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Y. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15(11), 990–996 (2013).

J. Phys. Chem. C (1)

S. Liu, L. Huang, J. Li, C. Wang, Q. Li, H. Xu, H. Guo, Z. Meng, Z. Shi, and Z. Y. Li, “Simultaneous excitation and emission enhancement of fluorescence assisted by double plasmon modes of gold nanorods,” J. Phys. Chem. C 117, 10636–10642 (2013).

JETP Lett. (1)

M. Glazov, “Second harmonic generation in graphene,” JETP Lett. 93(7), 366–371 (2011).

Nano Lett. (2)

M. Lobet, M. Sarrazin, F. Cecchet, N. Reckinger, A. Vlad, J. F. Colomer, and D. Lis, “Probing graphene χ(2) using a gold photon sieve,” Nano Lett. 16(1), 48–54 (2016).
[PubMed]

X. Gan, K. F. Mak, Y. Gao, Y. You, F. Hatami, J. Hone, T. F. Heinz, and D. Englund, “Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity,” Nano Lett. 12(11), 5626–5631 (2012).
[PubMed]

Opt. Express (5)

Opt. Lett. (3)

Photon. Res. (2)

Phys. Rev. B (4)

D. Smirnova, I. Shadrivov, A. Miroshnichenko, A. Smirnov, and Y. Kivshar, “Second-harmonic generation by a graphene nanoparticle,” Phys. Rev. B 90(3), 93–97 (2014).

D. Smirnova and Y. Kivshar, “Second-harmonic generation in subwavelength graphene waveguides,” Phys. Rev. B 90(16), 165433 (2014).

Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).

Y. An, J. Rowe, D. Dougherty, U. Ji, and A. Diebold, “Optical second-harmonic generation induced by electric current in graphene on Si and SiC substrates,” Phys. Rev. B 89(11), 106 (2014).

Phys. Rev. Lett. (2)

S. Thongrattanasiri, F. H. Koppens, and F. J. García de Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[PubMed]

B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[PubMed]

Plasmonics (1)

A. Rahmati and N. Granpayeh, “Enhancement of the second and third harmonic generations in nanorods and nanotubes based on metal electron nonlinear responses and dielectric nonlinearity,” Plasmonics 10, 1201 (2015).

Sci. Rep. (1)

H. Lu, C. Zeng, Q. Zhang, X. Liu, M. M. Hossain, P. Reineck, and M. Gu, “graphene-based active slow surface plasmon polaritons,” Sci. Rep. 5, 8443 (2015).
[PubMed]

Science (2)

P. Alonso-González, A. Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L. E. Hueso, and R. Hillenbrand, “Controlling graphene plasmons with resonant metal antennas and spatial conductivity patterns,” Science 344(6190), 1369–1373 (2014).
[PubMed]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[PubMed]

Other (3)

T. Boyd and J. Sanderson, The Physics of Plasmas (Cambridge University, 2003).

Y. Shen, The principles of nonlinear optical (Lohn Wiley & Sons, 1984).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light 2nd ed. (Princeton University, 2008).

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Figures (7)

Fig. 1
Fig. 1 (a) The cuboid-shaped graphene nanoribbon grating on dielectric substrate. The input light wave is polarized along the x direction and propagates along the z direction. (b) The CMT model with three resonant modes of the graphene grating.
Fig. 2
Fig. 2 (a) The simulated linear-optical absorption with FDTD simulation (black circles) and the theoretical CMT analysis results (red line). The distribution of (b) charge density σ(1) [labeled with ± ] and (c) electric field Ex(1) for the FFW at resonant wavelength 4.26 μm, (d) charge density σ(1) [labeled with ± ] and (e) electric field Ex(1) at wavelength 4.82 μm. (f) The plot of electric field Ex(1) along x direction in C1 region at wavelength 4.26 μm (black line) and in C2 region at wavelength 4.82 μm (blue line).
Fig. 3
Fig. 3 The absorption with different fermi energy (a) Ef = 0.74eV, (b) Ef = 0.64eV, (c) Ef = 0.54eV, and (d) Ef = 0.44eV by using the CMT theory (red line) and FDTD method (black circles), respectively. (e) The lifetimes for τw1, τw2, and τw3 with various fermi energy Ef. (f) The evolution of linear-optical absorption for different fermi energy with using CMT theory. The show in (g) is the top view of (f).
Fig. 4
Fig. 4 The absorption spectra with different carrier mobility (a) μ = 1000 cm2/(V·s), (b) μ = 5000 cm2/(V·s), and (c) μ = 10000 cm2/(V·s) by with using both the CMT theory (red line) and the FDTD simulation (black circles). (d) The evolution of linear-optical absorption spectra with different carrier mobility μ with using CMT theory. The data fitting for (e) τw1,τw2,τw3, and (f) τi1,τi2,τi3 with various carrier mobility μ.
Fig. 5
Fig. 5 (a) The amplitude of fourier spectrum for FFW Ex(1) with wavelength 4.26μm. The amplitude of fourier spectrum for SHW (b) Ex(2) and (c) Ey(2) with wavelength 2.13μm. The distributions of (d) charge density σ(2)[labeled with ± ] and (e) electric field Ex(2) at wavelength 2.13μm. (f) The plot of electric field Ex(2) along x direction in C1 region at resonant wavelength 2.13 μm.
Fig. 6
Fig. 6 (a) The amplitude of fourier spectrum for FFW Ex(1) with wavelength 4.82μm. The amplitude of fourier spectrum for SHG (b) Ex(2) and (c) Ey(2) with wavelength 2.41μm. The distributions of (d) charge density σ(2)[labeled with ± ] and (e) electric field Ex(2) at wavelength 2.41μm. (f) The plot of electric field Ex(2) along the short side of graphene at resonant wavelength 2.41μm.
Fig. 7
Fig. 7 (a) The amplitude of fourier spectrum for FFW Ex(1) with two wavelengths λ1 = 4.26 μm and λ2 = 4.82 μm, amplitude of fourier spectrum for SHW (b) Ex(2) and (c) Ey(2) with the four wavelengths λ3, λ4, λ5, and λ6.

Equations (25)

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σ gra ( ω ) = i e 2 E f /[ π ћ 2 ( ω+i τ 1 ) ]
ε 11 ( ω ) = ε 22 ( ω ) = ε 0 [ 1 ω 2 p,gra /( ω 2 +iω γ gra ) ]
d A 1 dt =(i ω 1 1 τ i1 1 τ w1 ) A 1 + S +,in 1 τ w1 + S ,in 1 τ w1 i μ 12 A 2 i μ 13 A 3
d A 2 dt =(i ω 2 1 τ i2 1 τ w2 ) A 2 + S +,in 1 τ w2 + S ,in 1 τ w2 i μ 21 A 1 i μ 23 A 3
d A 3 dt =(i ω 3 1 τ i3 1 τ w3 ) A 3 + S +,in 1 τ w3 + S ,in 1 τ w3 i μ 32 A 2 i μ 31 A 1
t(ω, E f )=1( 1 τ w1 + 1 τ i1 ) P 1 P 0 ( 1 τ w2 + 1 τ i2 ) P 2 P 0 ( 1 τ w3 + 1 τ i3 ) P 3 P 0
r(ω, E f )= 1 τ w1 P 1 P 0 1 τ w2 P 2 P 0 1 τ w3 P 3 P 0 P 0 = χ 1 χ 4 χ 5 + χ 1 χ 3 γ 3 + χ 2 χ 5 γ 2 + χ 2 χ 3 χ 6 + γ 1 χ 4 χ 6 γ 1 γ 2 γ 3 P 1 =( γ 2 γ 3 χ 4 χ 6 ) 1 τ w1 +( χ 1 γ 3 + χ 2 χ 6 ) 1 τ w2 +( χ 2 γ 2 + χ 1 χ 4 ) 1 τ w3 P 2 =( χ 4 χ 5 + χ 3 γ 3 ) 1 τ w1 +( γ 1 γ 3 χ 2 χ 5 ) 1 τ w2 +( γ 1 χ 4 + χ 2 χ 3 ) 1 τ w3 P 3 =( γ 2 χ 5 + χ 3 χ 6 ) 1 τ w1 +( γ 1 χ 6 + χ 1 χ 5 ) 1 τ w2 +( γ 1 γ 2 χ 1 χ 3 ) 1 τ w3
B (1) t =× E (1)
E (1) t = c 2 × B (1) 1 ε 0 j (1)
j (1) = ε 0 ω p,gra 2 γ gra iω E (1) = e 2 E f π 2 Δ( γ gra iω) E (1)
j (1) /t= γ gra j (1) +β E (1)
j (1),n+1 j (1),n Δt = γ gra j (1),n+1 j (1),n 2 +β E (1),n+1 + E (1),n 2
j ( 1 ),n+1 = k 1 j ( 1 ),n + k 2 ( E ( 1 ),n+1 + E ( 1 ),n )
B (2) t =× E (2)
E (2) t = c 2 × B (2) 1 ε 0 j (2)
j (2) t = γ gra j (2) +β E (2) + S (2)
S (2) = k r k ( j (1) j k (1) e n 0 ) e m e [ ε 0 ( E (1) ) E (1) + j (1) × B (1) ]
E ( 2 ) ( 2ω ) χ (2) E ( 1 ) ( ω ) E (1) ( ω )
E ( 1 ) ( r,t )= E 0 (1) ( r ) e iω0t
S ( 2 ) = S 0 (2) e i2ω0t
E ( 1 ) ( r,t )= E 0 (1) ( r ) ( e iω1t + e iω2t )
S ( 2 ) =2 S 0 (2) + S 1 (2) e i2ω1t + S 2 (2) e i2ω2t +2 S 3 (2) e i( ω1+ω1 )t S 4 (2) cos( ω 2 ω 1 )t S 5 (2) sin( ω 2 ω 1 )t
E ( 2 ) =2 E 0 (2) + E 1 (2) e i2ω1t + E 2 (2) e i2ω2t +2 E 3 (2) e i(ω1+ω1)t E 4 ( 2 ) cos( ω 2 ω 1 )t E 5 ( 2 ) sin( ω 2 ω 1 )t
B ( 2 ) =2 B 0 (2) + B 1 (2) e i2ω1t + B 2 (2) e i2ω2t +2 B 3 (2) e i(ω1+ω1)t B 4 ( 2 ) cos( ω 2 ω 1 )t B 5 ( 2 ) sin( ω 2 ω 1 )t
j ( 2 ) =2 j 0 (2) + j 1 (2) e i2ω1t + j 2 (2) e i2ω2t +2 j 3 (2) e i(ω1+ω1)t j 4 ( 2 ) cos( ω 2 ω 1 )t j 5 ( 2 ) sin( ω 2 ω 1 )t

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