Resonant Properties of Modified Triangular Plasmonic Nanoparticles with Higher Field Concentration

In this paper, we present an analysis of the resonant response of modified triangular metallic nanoparticles with polynomial sides. The particles are illuminated by an incident plane wave and the method of moments is used to solve numerically the electromagnetic scattering problem. We investigate spectral response and near field distribution in function of the length and polynomial order of the nanoparticles. Our results show that in the analyzed wavelength range (0.5-1.8) μm these particles possess smaller number of resonances and their resonant wavelengths, near field enhancement and field confinement are higher than those of the conventional triangular particle with linear sides.

In the conventional triangular nanoparticles, the sides are straight lines.In this paper, we suggest and analyze triangular nanoparticles which have two of their sides in the form of polynomial curves.
We shall call them modified triangular nanoparticles with polynomial sides.We investigate the optical response of the proposed gold nanoparticles and the dependence of this response on the length and polynomial order of the side curves.Near field distributions of electric field are also calculated.
Numerical analysis is fulfilled by our computational code based on the method of moments (MoM) [23].The Lorentz-Drude formula with one interband term is used to describe the complex permittivity of the gold.
We show below that in the analyzed wavelength range (0.The particles are centered in the origin of the rectangular coordinate system with one tip oriented along the x-axis.The thickness of the particles is w (it is not shown in Fig. 1).With these parameters, the right tip on the x-axis is positioned at the point (0.5h; 0; 0).In what follows, we shall investigate the particles made of gold (Au) with four values α = 1,2,3,4 of the polynomial order of the side variation.The particles shown in Fig. 1 are illuminated by an E x -polarized +z-directed plane wave.
We used E x polarization because this field excites more efficiently the resonant modes with larger charge oscillations along the x direction.These oscillations produce strong charge accumulation in the right tip of the nanoparticles, and consequently higher field concentration near this tip.This behavior is similar to that observed in RF-microwave dipole antennas in the receiving mode, where the polarization of the incident wave along the axis of the dipole leads to a larger resonant response.
The numerical analysis of the scattering problem is fulfilled by our MoM code based on the model proposed in [23], where the equivalent polarization current inside the volume of the particles is determined by solving the tensor integral equation for the electric field.In this model, the volume of a particle is divided in the N small cubic subvolumes, where the total electric field is approximately constant.With this approximation, the integral equation is transformed into a linear system with N t =3N equations because there are three electric field components in each subvolume, i.e.N t is the total matrix size of the linear system.This MoM code was validated by comparison with the results obtained by the exact Mie solution for sphere [16] and with the simulations performed by commercial softwares [17].The Lorentz-Drude model with one interband term was used to describe the complex permittivity = 0  r of the gold particles, where  0 is permittivity of the vacuum and  r is defined by . ( The parameters of this equation are  ∞ =8,  p1 =13.8×10 15 s −1 , =1.075×10 14 s −1 ,  0 =2c/ 0 ,  0 =450nm,  p2 =45×10 14 s −1 , and =9×10 14 s −1 [1].

III. NUMERICAL RESULTS
Using the developed computational code we simulated the electromagnetic response of the nanoparticles with different sizes and parameter α.The discretization of the nanoparticles with α = 1, 2, 3, 4 are N t = 4380, 5310, 4410 and 5328, respectively.With these discretizations we obtained a good accuracy of the results.
For each nanoparticle with a given α, eight simulations with different values of length L, namely, L=50, 100, 150, 200, 250, 300, 350 and 400 (all in nanometers) were performed.The thickness w of the particles with L=50nm are w=N z ×L=4×2nm=8.0nm(N t =4380), w=N z ×L=5×1.65nm=8.3nm(N t =5310), w=N z ×L=5×1.55nm=7.8nm(N t =4410) and w=N z ×L=6×1.41nm=8.5nm(N t =5328) for the particles with α = 1, 2, 3, 4, respectively, where L=x=y=z is the size of the cubic subvolumes and N z the number of cubic subvolumes in the z direction.In this discretization, for the particles with higher values of α, we used finer mesh (smaller L) and higher N z in order to discretize better the geometries with more acute tips.The other particles with higher L possess higher thicknesses with values proportional to the thickness of the case L=50nm.For example, the particle with α = 1 and L=2×50=100nm has the thickness w=2×8=16nm.
It is important to note here that the particles with different α and the same dimension L have the thickness slightly different because we used in our MoM code cubic discretization (L=x=y=z) and different values of L and N z for each case of α.To make the particles with the same thickness, one can use, for example, a discretization like L=x=y≠z and choose the number of subvolumes in the z direction N z and the size z in order to maintain the thickness w=N z ×z constant.

A.
Spectral Responses Figs.2-5 present the spectral responses of the particles.The field enhancement in these figures is defined by (E/E 0 ) 2 , where E is the total electric field (incident and scattered) near the particle and E 0 is the amplitude of the incident plane wave.In all these results, the field enhancement is calculated at the distance 10nm from the tip's particles along the-x axis, i.e. at the point (x = 0.5h+10nm; y = 0; z = 0).
We observe in these figures some characteristic resonances where the respective resonant wavelengths are λ i , i = 1, 2, ..8.Some of these resonances for determined values of L are very weak, for example in Fig. 2 the resonance λ 1 is significant only for L = 50nm and L = 100nm and practically zero for other values of L. These weak resonances are not shown in the figures.
In all the cases shown in Figs.2-5, we observe an approximately linear increasing of the wavelengths λ i with L. This variation is presented in Figs.6-9, where we note also increasing of λ i with α.For example, in case of L = 50nm, λ 1 =660, 754, 903 and 1096nm for α = 1, 2, 3, 4, respectively.In Figs.2-5, we see also that the field enhancement (E/E 0 ) 2 of each resonance is gradually increased and decreased with the variation of L, so that when one resonance is decreased, the next right resonance is increased.This behavior can be observed for example in the resonances λ 2 and λ 3 in Fig. 3, where for lower L (L = 50, 100, 150 and 200nm) the field enhancement of λ 2 is higher than that of λ 3 but for L=250nm, 300 nm, 350 nm and 400 nm the field enhancement of λ 3 is higher than that of λ 2 .
There exists a value of L for which the maximum of the field intensity (E/E 0 ) 2 is achieved.Figs.9-12 shows this variation of the field enhancement at resonances λ i , i = 1, 2, ...8 versus L for the modified triangular nanoparticles with =4,1,2,3, respectively.In general we note from these figures that the field enhancements is increased for larger values of , i.e. the modified triangular nanoparticles with polynomial sides possess higher field concentration in comparison with the conventional triangular particle with linear sides.For example, the conventional triangular particle with L=150nm and α=1 (Fig. 2) possesses maximum near field enhancement of about (E/E 0 ) 2 ≈110, and the modified triangular particle with the same size L=150nm and α=3 (Fig. 4) has this parameter (E/E 0 ) 2 ≈300.

B. Near Field Distributions
Figs. 13-16 present the spatial distributions of the fields near the particles with =1,2,3,4, respectively.The results show the intensity of the normalized total electric field (E/E 0 ), and the components x (E x /E 0 ), y (E y /E 0 ), and z (E z /E 0 ) at the plane z=9nm (−50nm<x,y<50nm).This plane is approximately 5nm above the particle's surface, because the thickness of them are variable, i.e.
We see from these results that the modified triangular nanoparticles with higher values of  (Figs. 14-16) produce the higher and more confined total fields than the fields of the conventional triangular particles (Fig. 13).We believe that is due the electrostatic effect of more acute tips, where the electric field of the incident plane wave excites charges oscillations along the x direction with strong charge accumulation in the right tip, and consequently produces higher field enhancements near this tip.≠ 0. This occurs because the electric field lines are oriented in radial directions away from the charges.These results show again that there is a higher concentration of charges in the right tips of the particles, and this concentration is more pronounced for larger values of .

IV. CONCLUSIONS
In this paper, we presented a theoretical analysis of the resonances of modified triangular gold nanoparticles with polynomial sides.We showed that in the analyzed wavelength range, the number of resonances of the proposed particles with larger values of polynomial order is reduced.For example, the conventional triangular particle has eight resonances, and the modified one with polynomial order equals to four has two resonances.We also demonstrated that the modified particles have resonant wavelength, near field enhancement and confinement higher than those of the conventional triangular particle with linear sides.These novel particles can also be useful in design of nanoantennas.
5-1.8) μm, modified triangular nanoparticles with polynomial sides have smaller number of resonances, longer resonant wavelength, near field enhancement and confinement in comparison with those of the conventional triangular particles with straight sides.II.THEORY The geometries of the investigated nanoparticles are shown in Fig. 1.The polynomial function used to model the curvature of the lower side is x = (y/k)  , where k = L/2h 1/ , h = 0.5L(3) 0.5 , L is the length of the conventional equilateral triangle.The upper side is obtained by symmetry reflection of the lower side in the plane y=0 and the left side is a straight line with length L. The conventional particle with the linear sides corresponds to the case α = 1.For higher values of α, the tips of the particles are more acute.