## 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^{−10}m/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 L_{1} × L_{2} × Δ. The parameters about the graphene nanoribbons are set as: length L_{1} = 160nm, width L_{2} = 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 σ_{gra} [3]:

*e*is electric charge,

*ħ*is the reduced Planck's constant. The carrier relaxation time is

*τ*= (μ

*E*)/(

_{f}*e*ν

_{f}^{2}), μ is the carrier mobility,

*E*is the fermi energy or chemical potential, ν

_{f}*is the fermi velocity. The parameters of the graphene are set as ν*

_{f}*= 10*

_{f}^{6}m/s, μ = 10000cm

^{2}/(V·s), and

*E*= 0.64eV [3,15].

_{f}The equivalent dielectric tensor component of graphene is given by ε_{11} = ε_{22} = ε_{0} [1 + iσ_{gra}/(ε_{0}*ω*Δ)] and ε_{33} = ε_{0} [3,15], where ε_{0} is the vacuum permittivity. After inserting Eq. (1) into dielectric tensor, the dielectric tensor component can be described by the drude model:

*ω*

_{p,gra}and γ

_{gra}stand for the bulk plasma frequency and electron collision frequency of graphene grating, respectively. And these parameters for graphene grating can be set as

*ω*

^{2}

_{p,gra}=

*e*

^{2}

*E*/(π

_{f}*ћ*

^{2}ε

_{0}Δ) and γ

_{gra}=

*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 *A*_{m} (m = 1,2,3) in the whole region of graphene grating, where d*A*_{m}/dt = -i*ωA*_{m}. And *A*_{m} is the m-th mode of resonant modes. The incoming and outgoing waves are depicted by S _{± ,in(out)}. The energy amplitude *A*_{1}, *A*_{2}, and *A*_{3} of the GSP resonator can be expressed as

*τ*

_{im}(m = 1,2,3) is the decay rate due to the intrinsic loss of the m-th mode in graphene grating and

*τ*

_{im}is the lifetime in decay process due to the intrinsic loss for the m-th mode. The 1/

*τ*

_{wm}is the decay rate due to the energy coupling from each mode into the light field and

*τ*

_{wm}is the lifetime in the process of energy coupling. μ

_{ij}(i,j = 1,2,3) are the coupling coefficients between the three resonant modes. The transmission function

*t*(

*ω*,

*E*) =

_{f}*S*

_{+,out}/

*S*

_{+,in}and reflection function

*r*(

*ω*,

*E*) =

_{f}*S*

_{-,out}/

*S*

_{+,in}of this graphene plasmonic grating system can be calculated by the following theoretical formula:

*ω*

_{m}^{2}= 2

*e*

^{2}

*E*/[ћ

_{f}^{2}ε

_{0}(ε

_{b,m}+ ε

_{t,m})L

_{eff,m}] (m = 1,2,3), the

*ε*=

_{b,m}*ε*= 1 are the effective dielectric constants of the bottom and top surfaces of graphene nanoribbon, respectively [15]. The parameter

_{t,m}*L*

_{eff}_{,}

*is the effective width of the graphene nanoribbon grating for the m-th mode, which reflects the influence of the periodic graphene grating. δ*

_{m}*=*

_{m}*ω*-

*ω*, γ

_{m}_{m}= iδ

*−1/*

_{m}*τ*

_{im}-1/

*τ*

_{wm}, χ

_{1}= iμ

_{12}, χ

_{2}= iμ

_{13}, χ

_{3}= iμ

_{21}, χ

_{4}= iμ

_{23}, χ

_{5}= iμ

_{31}, χ

_{6}= iμ

_{32}. The transmission, reflection, and absorption efficiency of the system can be expressed as

*T*(

*ω*,

*E*) = \t\

_{f}^{2},

*R*(

*ω*,

*E*) = \r\

_{f}^{2}, and

*A*(

*ω*,

*E*) = 1-

_{f}*T*(

*ω*,

*E*)-R(

_{f}*ω*,

*E*).

_{f}The linear optical response of the graphene grating in Maxwell equations is given by:

**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 –i

*ω*

**j**

^{(1)}= –γ

_{gra}

**j**

^{(1)}+

*β*

**E**

^{(1)}. And then, taking inverse fourier transformation, we can obtainThe term ˗

*γ*

_{gra}

**j**

^{(1)}describes the current decay due to coulomb scattering with γ

_{gra}= (

*e*ν

_{f}

^{2})/(μ

*E*). The parameter

_{f}*β*= ε

_{0}

*ω*

^{2}

_{p,gra}.

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

*k*

_{1}= (1-γ

_{gra}Δt/2)/(1 + γ

_{gra}Δt/2) and

*k*

_{2}= 0.5

*β*Δt/(1 + γ

_{gra}Δt/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]:

_{0}and m

_{e}stand for the graphene-ion density and effective electron mass in graphene grating, respectively. The surface charge density ρ

^{(1)}= ε

_{0}▽·

**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, L_{1} = 160nm, L_{2} = 40nm, and Δ = 1nm, μ = 10000cm^{2}/(V·s), and *E _{f}* = 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:

*L*

_{eff}_{,1}= 5.7 × 40nm,

*L*

_{eff}_{,2}= 6.8 × 40nm,

*L*

_{eff}_{,3}= 8.7 × 40nm,

*τ*

_{w1}= 7.2 × 10

^{8}rad/s,

*τ*

_{w2}= 5.8 × 10

^{10}rad/s,

*τ*

_{w3}= 5.3 × 10

^{10}rad/s,

*τ*

_{i1}= 9.8 × 10

^{11}rad/s,

*τ*

_{i2}= 1.1 × 10

^{12}rad/s,

*τ*

_{i3}= 1.08 × 10

^{12}rad/s, μ

_{12}= μ

_{21}= 10

^{11}rad/s, μ

_{13}= μ

_{31}= 10

^{11}rad/s, μ

_{23}= μ

_{32}= 10

^{12}rad/s. We suppose that the coupling coefficients μ

_{12}, μ

_{21}, μ

_{13}, μ

_{31}, μ

_{23}and μ

_{32}are unchanged in the model later. The theoretical CMT results (red line) are in good agreement with the FDTD simulation (black circles).

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**)*e*^{i}^{ω}^{t} or *ρ*^{(2)}(**r**)*e*^{i2}^{ω}^{t}. Here the electric displacement component D_{z}^{(1,2)} has abrupt discontinuity across the surface of graphene nanoribbons. According to the boundary condition n·D_{z}^{(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** = n**z**. Therefore, we are able to determine the magnitude of surface charges σ^{(1,2)}.

The distributions of charge density σ^{(1)} and electric field E_{x}^{(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 E_{x}^{(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 C_{1} region indicates that the C_{1} 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 E_{x}^{(1)} along x direction in C_{1} region at wavelength 4.26μm (black line) and in C_{2} region at wavelength 4.82μm (blue line) are shown in Fig. 2(f). The electric field E_{x}^{(1)} has the exponential decay out of the graphene nanoribbon. Figure 2(f) shows that the electric field E_{x}^{(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 *E _{f}*. To get more insight into the GSP structure, the absorption spectra with different fermi energy (a)

*E*= 0.74eV, (b)

_{f}*E*= 0.64eV, (c)

_{f}*E*= 0.54eV, and (d)

_{f}*E*= 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

_{f}*E*as shown in Figs. 3(a)-3(d).

_{f}All the parameters about CMT for the result in Figs. 3(a)-3(d) are set as follows: *L _{eff}*

_{,1}= 5.7 × 40nm,

*L*

_{eff}_{,2}= 6.8 × 40nm,

*L*

_{eff}_{,3}= 8.7 × 40nm,

*τ*

_{w1}= 9 × 10

^{8}−3 × 10

^{28}

*E*+ 2.7 × 10

_{f}^{47}

*E*

_{f}^{2},

*τ*

_{w2}= −6.7 × 10

^{10}+ 1.9 × 10

^{30}

*E*−6.4 × 10

_{f}^{48}

*E*

_{f}^{2},

*τ*

_{w3}= −5.2 × 10

^{10}+ 1.4 × 10

^{30}

*E*−3.4 × 10

_{f}^{48}

*E*

_{f}^{2},

*τ*

_{i1}= 9.8 × 10

^{11}rad/s,

*τ*

_{i2}= 1.1 × 10

^{12}rad/s,

*τ*

_{i3}= 1.1 × 10

^{12}rad/s. In the process of energy coupling, the lifetimes for

*τ*

_{w1},

*τ*

_{w2}, and

*τ*

_{w3}with various fermi energy

*E*are shown in Fig. 3(e). The lifetimes

_{f}*τ*

_{wm}(m = 1,2,3) can be changed with the fermi energy

*E*since 1/

_{f}*τ*

_{wm}represents the energy coupling between each mode and the light field. The parameters of lifetimes

*τ*

_{i1},

*τ*

_{i2}and

*τ*

_{i3}are unchanged with various fermi energy

*E*, which are the same as that in Fig. 2(a).

_{f}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 = 10000cm^{2}/(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 *E _{f}*. This quasi-linear response characteristic in Fig. 3(g) between the fermi energy

*E*and resonant modes is especially valuable for the application in the optoelectronic devices of graphene.

_{f}The lower mobility in the graphene nanoribbons corresponds to higher loss. The fermi energy *E _{f}* = 0.64eV is also used here. We can suppose the lifetimes

*τ*

_{im}(m = 1,2,3) are inversely proportional to carrier mobility μ because the decay rate 1/

*τ*

_{im}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 = 1000cm

^{2}/(V·s), (b) μ = 5000cm

^{2}/(V·s), and (c) μ = 10000cm

^{2}/(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 = 10000cm

^{2}/(V·s) to μ = 1000cm

^{2}/(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).

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: *L _{eff}*

_{,1}= 5.7 × 40nm,

*L*

_{eff}_{,2}= 6.8 × 40nm,

*L*

_{eff}_{,3}= 8.7 × 40nm,

*τ*

_{w1}= −6.1 × 10

^{7}+ 8.0 × 10

^{8}μ-2.3 × 10

^{7}μ

^{2}rad/s,

*τ*

_{w2}= 3.5 × 10

^{10}+ 1.5 × 10

^{11}μ-1.3 × 10

^{11}μ

^{2}rad/s,

*τ*

_{w3}= 1.9 × 10

^{10}+ 1.3 × 10

^{11}μ-1.1 × 10

^{11}μ

^{2}rad/s,

*τ*

_{i1}= 9.8/μ × 10

^{11}rad/s,

*τ*

_{i2}= 1.1/μ × 10

^{12}rad/s,

*τ*

_{i3}= 1.1/μ × 10

^{12}rad/s. The lifetimes for

*τ*

_{w1},

*τ*

_{w2},

*τ*

_{w3},

*τ*

_{i1},

*τ*

_{i2}and

*τ*

_{i3}with various carrier mobility μ are shown in Figs. 4(e)-4(f). The lifetimes

*τ*

_{i1},

*τ*

_{i2}and

*τ*

_{i3}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]:

^{(2)}stands for a generalized second-order nonlinear susceptibility.

For an incident monochromatic wave with the frequency *ω*_{0}, the linear electric field can be given by

**E**

_{0}

^{(1)}(

**r**). Then, the magnetic field and current for linear optical response of graphene nanoribbons can be given by

**B**

^{(1)}= -i▽ ×

**E**

^{(1)}/

*ω*

_{0}and -i

*ω*

**j**

^{(1)}= -γ

_{gra}

**j**

^{(1)}+

*β*

**E**

^{(1)}→

**j**

^{(1)}= i

*β*

**E**

^{(1)}/(

*ω*

_{0}+ iγ

_{gra}). 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)}=

*A*

_{0}

^{(1)}

*e*

^{-i}

^{ω}^{0t}×

*B*

_{0}

^{(1)}

*e*

^{-i}

^{ω}^{0t}=

*A*

_{0}

^{(1)}

*B*

_{0}

^{(1)}

*e*

^{-i2}

^{ω}^{0t}. Thus, the second-order source

**S**

^{(2)}in Eq. (16) can be expressed asAnd the similar expresses for nonlinear optical response of graphene nanoribbons can be obtained as follows:

**j**

^{(2)}(

**r**,t) =

**j**

_{0}

^{(2)}(

**r**)

*e*

^{-i2}

^{ω}^{0t},

**E**

^{(2)}(

**r**,t) =

**E**

_{0}

^{(2)}(

**r**)

*e*

^{-i2}

^{ω}^{0t}, and

**B**

^{(2)}(

**r**,t) = -i▽ ×

**E**

^{(2)}(

**r**,t)/(2

*ω*

_{0}). Here

**j**

_{0}

^{(2)}(

**r**) and

**E**

_{0}

^{(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 2

*ω*

_{0}such as SHG.

For an incident wave with wavelength λ_{0} = 4.26μm, one can see the amplitude of fourier spectrum for FFW E_{x}^{(1)} with resonant wavelength 4.26μm in Fig. 5(a). The amplitude of fourier spectrum for SHW E_{x}^{(2)} and E_{y}^{(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)}(2*ω*_{0})/ **E**^{(1)}(*ω*_{0})\. 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.

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 E_{x}^{(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 E_{x}^{(1)}. The plot of second-harmonic field E_{x}^{(2)} along x direction in C_{1} region with wavelength 2.13μm is shown in Fig. 5(f).

Under the incident monochromatic wave with wavelength λ_{0} = 4.82μm, the amplitude of fourier spectrum for FFW E_{x}^{(1)} at wavelength 4.82μm is shown in Fig. 6(a). The amplitude of fourier spectrum for SHG E_{x}^{(2)} and E_{y}^{(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 E_{x}^{(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 E_{x}^{(2)} along the short side in graphene nanoribbon with wavelength 2.41μm is shown in Fig. 6(f).

For the incident continue wave with two frequencies *ω*_{1} and *ω*_{2}, the linear electricmagnetic field can be given by

**B**

^{(1)}(

**r**,t) =

**B**

_{0}

^{(1)}(

**r**)(

*e*

^{-i}

^{ω}^{1t}+

*e*

^{-i}

^{ω}^{2t}) and

**j**

^{(1)}(

**r**,t) =

**j**

_{0}

^{(1)}(

**r**)(

*e*

^{-i}

^{ω}^{1t}+

*e*

^{-i}

^{ω}^{2t}) with the amplitude

**B**

_{0}

^{(1)}(

**r**) and

**j**

_{0}

^{(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)}=

*A*

_{0}

^{(1)}(

*e*

^{-i}

^{ω}^{1t}+

*e*

^{-i}

^{ω}^{2t}) ×

*B*

_{0}

^{(1)}(

*e*

^{-i}

^{ω}^{1t}+

*e*

^{-i}

^{ω}^{2t})

*e*

^{-i}

^{ω}^{0t}=

*A*

_{0}

^{(1)}

*B*

_{0}

^{(1)}[-2 +

*e*

^{-i2}

^{ω}^{1t}+

*e*

^{-i2}

^{ω}^{2t}+ 2

*e*

^{-i(}

^{ω}^{1+}

^{ω}^{1)t}-cos(

*ω*

_{2}-

*ω*

_{1})t-sin(

*ω*

_{2}-

*ω*

_{1})t]. Thus, the second order source

**S**

^{(2)}in Eq. (16) can be described as

**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 2

*ω*

_{1}, 2

*ω*

_{2},

*ω*

_{2}+

*ω*

_{1}and

*ω*

_{2}-

*ω*

_{1}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 λ_{1} = 4.26μm and λ_{2} = 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 E_{x}^{(1)} with two different wavelengths λ_{1} = 4.26μm and λ_{2} = 4.82μm is shown in Fig. 7(a). The amplitude of fourier spectrum for SHW E_{x}^{(2)} and E_{y}^{(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 λ_{3} = 2.13μm, λ_{4} = 2.27μm, λ_{5} = 2.41μm, and λ_{6} = 36.9μm in Figs. 7(b) and 7(c), respectively.

We find that the second-order nonlinearity modes at wavelengths λ_{3} = 2.13μm and λ_{5} = 2.41μm are obtained from the incident monochromatic wave with wavelengths λ_{1} = 4.26μm and λ_{2} = 4.82μm due to the SHG effect, respectively. The second-order nonlinearity mode at wavelength λ_{4} = 2.27μm is the sum-frequency field signals for the incident continuous wave with wavelengths λ_{1} and λ_{2}. The second-order nonlinearity mode at wavelength λ_{6} = 36.9μm is the difference-frequency field for the FFW with two resonant wavelengths λ_{1} and λ_{2}. 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).

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