Second-Order Nonlinearity Assisted by Dual Surface Plasmon Resonance Modes in Perforated Gold Film

We have studied analytically the reflection assisted with surface plasmon through the square lattice perforated gold film. Under the excitation of the external electromagnetic field with one or two different frequencies, the second-order nonlinearity exists in this noncentrosymmetric metal-based metamaterial. We employed the two surface plasmon resonances modes with different lattice periods. With the excitation of two different plasmon resonances modes, the strong local field induces an expected increase of the second-order nonlinearity including second harmonic generation as well as the sum (difference) frequency generation. The field distributions results also indicate that the enhancement of sum frequency signals and difference frequency signals strongly depends on surface plasmon resonance effect.


Introduction
The optical second harmonic generation (SHG) was first observed for the one-dimensional metal surfaces in 1965 [1].The optical transmission of light through metal film has been shown to be orders of magnitude higher than that expected by the classical aperture theory due to the surface plasmon (SP) resonances [2][3][4][5].The SHG in such a perforated metal film with different topographical nanoscale apertures has also been the focus of many researchers [6,7].Zeng et al. has found that the SHG effect mainly results from the convective derivative of the continuous electron current [7].The enhanced linear fundamental field gives rise to the local SHG at the tip surface, thereby creating a highly confined photon source of SHG.The SHG efficiency has been enhanced in a circularly symmetric structure.It has been shown that the SHG is a symmetry-sensitive process.The arrangement of the nanoholes in a random way can break out the symmetry and has been found to favor SHG [8].In addition, the less-symmetric double-hole array has been studied experimentally and found to cause the enhancement of SHG when the sharp tips are formed by folding the double holes [6].The SHG has been also investigated in geometric configurations such as split-ring resonators, T-shaped, and Lshaped nanoparticles [9][10][11][12][13].
For the ideally infinite metal surfaces, it is well known that the dominant second harmonic electric dipole source occurs only at the interface between centrosymmetric media, in which the inversion symmetry is broken.The higher order multipole sources merely provide a relatively small bulk SHG polarization density.On the other hand, for low-symmetric or even asymmetric nanoparticles, such as gold split-ring resonators, SHG dipolar polarizability may be presented in the whole volume and not limited at the interface.Consequently, the overall shape of the nanoparticles plays a significant role in determining the second harmonic response [14,15].In this paper, we have investigated the SHG, sum frequency generation (SFG), and difference frequency generation (DFG) from gold film with a periodic subwavelength air nanohole patterns by means of the three-dimensional (3D) finite-difference time-domain (FDTD) algorithm.The enhanced linear fundament field causes the local secondorder nonlinearity.Thereby the local SFG, DFG, and SHG can act as highly confined photon sources.The main results are shown as follows: (1) the reflection of the fundamental light results from the enhancement of the local field due to the SP

Structure and Calculation Method
In the second-order nonlinearity of gold nanostructures, the classical theoretical method of the second-order nonlinearity calculation has been proposed in [7].The interaction between light and Drude-type metal is generally described by the time-dependent Maxwell equations which are coupled to an equation that describes the light induced oscillation of quasifree electrons in the metal.The linear response of the gold nanostructures is given by [15]:  (1)   = −∇ ×  (1) , =  2 ∇ ×  (1) − 1  0  (1) ,  (1) = − (1) = − 0 (  − 1)  (1) , = − (1) +  2  0    (1) . ( The second-order nonlinearity of the gold film can be considered as follows: (2)   = −∇ ×  (2) , 2) ,  (2) = − (2) = − 0 (  − 1)  (2) +  (2) , 2) +  (2) , (1) )  (1) +  (1) ×  (1) ] . ( Here,  represents the , , and  coordinates. (1) and  (2) represent the current density vectors of fundamental and harmonic waves, respectively. (1) and  (2) ,  (1) and  (2)  are the electric field and magnetic flux intensity vectors of fundamental and harmonic waves, respectively. 0 is the ion density,   is the electron mass,  the light speed in vacuum air,  is the elementary charge, and  (2) is the nonlinear source of the plasma for second-order nonlinearity, respectively.It includes three terms of different physical mechanisms.The first term represents a generalized divergence originating from the convective time derivative of the electron velocity field.The second and third terms are the electric and magnetic components of the Lorentz force, respectively [7].There has the unit cell shape with  = 219 nm,  = 131 nm, and  = 97 nm.The input light wave is polarized along the  direction and propagates along the  direction.
The permittivity   () of the gold nanostructures has the form In the expression, the  is the phenomenological collision frequency, and   = √ 2  0 /   0 is the bulk plasma frequency according to the well-known Drude model of gold.The bulk plasma frequency and phenomenological collision frequency of gold are taken as   = 1.367 × 10 16 s −1 and  = 6.478 × 10 13 s −1 , respectively.The permittivity   () can become negative at frequencies below   .The  0 is the charge carrier concentration;  is the effective mass of the charge carriers.
The FDTD approach is applied for the numerical calculation of the above first-order and second-order equations.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  direction, and the periodic boundary conditions are used on the boundaries of  and  directions.Only one unit cell of the periodic holes array is considered in our computational space.
The structure of a perforated gold film is shown in Figure 1.The incident wave, polarized along the  direction, propagates along the -axis which is generated by a total field/scattering field technique.Then, we calculate the temporal reflection of the fundamental light and second-order nonlinearity by considering the normal incident light on the perforated semiconductor film.
In our calculation, the size of the spatial cell is set as 3.05 nm.We consider the structure of a perforated gold film with the thickness ℎ = 30.5 nm.The structure of a perforated gold film has a lattice constant  = 305 nm.The lattice constant is  =  = 305 nm along the -axis and -axis directions, respectively.There has the unit cell shape with  = 219 nm,  = 131 nm, and  = 97 nm.Each unit of the perforated gold film is noncentrosymmetric as shown The spectrum for  (1)   component of FFW with wavelength 630 nm.The spectrum for (b)  (2)   and (c)  (2)   component of SHG with wavelength 315 nm.
in Figure 1.The input plane wave is polarized along the  direction by exciting a plane of identical dipoles in phase and propagates along the  direction.

Results and Discussions
First, the normalized reflection of fundamental frequency wave (FFW) through the square lattice perforated gold film for different lattice constants along  direction  =  (black line), 1.1 (blue line), and 1.2 (red line) is investigated here, and the calculation results are shown in Figure 2(a).There are two different SP resonance modes at the wavelengths 630 nm and 1295 nm for lattice constants  =  (black line).When the lattice constant along  direction increases while the holes shape is fixed, the short-wavelength SP resonance mode does not move significantly while the long-wavelength SP resonance mode is blue-shifted dramatically.The normalized fundamental frequency wave reflection for different lattice constants along y direction  = 1.0, 1.1, and 1.2 is also shown in Figure 2(b).When the lattice constant along the  direction increases, the short-wavelength SP resonance mode does not move significantly while the long-wavelength SP resonance mode red-shifts slightly.
To obtain the SHG of the gold film with a periodic subwavelength air nanohole patterns, the incident FFW  (1)    with wavelengths  has the form  (1)   =  0 sin(2/), where  0 is amplitude.
Under the continuous incident FFW  (1)   with wavelengths  = 630 nm, one can see spectrum of FFW with the resonance wavelength 630 nm in Figure 3(a).It is also shown that the SHG spectrum of the  (2)   and  (2)    with the wavelength 315 nm is shown in Figures 3(b)-3(c),  and (b)  (1)   , and electric field distribution of SHG (c)  (2)   and (d)  (2)   at wavelengths 630 nm and 315 nm, respectively.800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 E (1)   x 0.0 0.5 The spectrum for  (1)   component of fundamental frequency wave at the wavelength 1295 nm.Spectrum for (b)  (2)   and (c)  (2)    component of SHG at the wavelength 648 nm.respectively.The SHG conversion efficiencies  are defined in the second-order nonlinear optical process as the expression Where  0 is the frequency of the incident FFW.The -polarized SHG conversion efficiencies are about 10 −5 while the -polarized SHG conversion efficiencies is about 10 −7 for the FFW at the wavelength 630 nm as shown in Figures 3(b)-3(c).
The transmission of the FFW results from an enhancement of the local field.The strong local field and noncentrosymmetry induces an increase of second harmonic nonlinearity signals.The electric field distribution of  (1)    and  (1)   for FFW above the gold film region at wavelengths 630 nm is also shown in Figures 4(a)-4(b), respectively.And the electric field distribution of  (2)   and  (2)   for SHG above the gold film region at wavelengths 315 nm is also shown in Figures 4(c)-4(d).It is noted that the different distribution of fundamental frequency and second harmonic is shown in Figures 4(a)-4(d).
Under the continuous incident wave  (1)   with wavelengths  = 1295 nm, one can see spectrum of FFW at wavelength 648 nm is in Figure 5(a).It is also shown that the spectrum of the  (2)   and the  (2)   component of SHG at wavelength 648 nm in Figures 5(b)-5(c), respectively.The SHG conversion efficiencies for polarization are about 10 −5 while they are about 10 −6 for  polarization at wavelength 1295 nm in Figures 6(b)-6(c).The electric field distribution of  (1)   and  (1)   for FFW above the gold film region at wavelengths 1295 nm is also shown in Figures 6(a)-6(b), respectively.The electric field distribution of  (2)   and  (2)   for SHG above the gold film region at wavelengths 648 nm is also shown in Figures 6(c)-6(d).

Figure 1 :
Figure 1: The structure of a perforated gold film with the thickness ℎ = 30.5 nm and the lattice periods  =  =  = 305 nm.There has the unit cell shape with  = 219 nm,  = 131 nm, and  = 97 nm.The input light wave is polarized along the  direction and propagates along the  direction.

Figure 6 :
Figure 6: The electric field distribution of fundamental frequency field and second harmonic field above the gold film region at wavelengths 1295 nm and 648 nm, respectively.