Third-harmonic generation from gold nanowires of rough surface considering classical nonlocal effect

Tingyin Ning; Shoubao Gao; Yanyan Huo; Shouzhen Jiang; Cheng Yang; Jian Li; Yuefeng Zhao; Baoyuan Man

doi:10.1364/OE.25.006372

1. Introduction

Nonlinear optical effects from metallic nanostructures have been widely studied for frequency conversion [1–5], biosensor [6], nanoruler [7,8], switching [9,10] and bistability [11,12]. Comparing with the dielectric materials, metallic nanostructures possess unique optical properties, such as the support of surface plasmon mode to confine the light so as to enhance the nonlinear response, and large nonlinear susceptibilities [1]. Moreover, the linear and nonlinear optical properties from metallic nanostructures can be easily modulated by changing the geometry and dielectric surroundings of metallic nanostructure [13–16]. These advantages make the metallic nanostructures promising for nonlinear photonic devices.

Metallic nanostructures are commonly prepared by chemical synthesis method or chemical/physical deposition combining with state-of-art micro-/nano-fabrication technique. Naturally, the structural surface defect of metallic nanostructures is unavoidable during the preparation [17–21]. Many investigations have shown that the structural defects can sensibly affect the optical properties of metallic nanostructures, especially the nonlinear optical response, such as the enhanced second harmonic generation (SHG) by metallic surface roughness [22,23], the suppressed higher-multipole effects of SHG in high-quality L-shape gold (Au) nanoparticle arrays [24], inhomogeneous broadening of SHG from the magnetic resonance [25], the different microscopy SHG and THG images from silver (Ag) nanocones with/without defects [26,27], and the completely different scattering SH intensity as a function of the detection angle for the idealized and the realistic nanoantennas [28], and so on [29,30]. Traditionally, the free-electron model, or the local model in the metal is used to describe the polarization of metal to light. Recent studies show that the strong electron-electron interaction in Thomas-Fermi length scale inside the metal, i.e. the nonlocal effect of metal dielectric response, may play very important roles in the linear and nonlinear optical properties from the metallic nanostructures [31–40]. Especially, when the feature size of metallic nanostructures is down to subnanometer scale, the classical nonlocal permittivity, and ultimately a quantum mechanical model of metal are exactly required to characterize the plasmonic behavior [31,32,41–43]. Although, the nonlocal effect, or so-called hydrodynamic model has been previously used to study the nonlinear optical properties from metallic nanostructures [33–35], the ideal surface, not the realistic surface of metallic nanostructures are considered.

In this paper, we utilize Au nanowires, which can be synthesized by chemical method or state-of-art nanofabrication technique, as a model to investigate how the realistic surface roughness affects the THG when considering classical nonlocal effect. We theoretically demonstrate that, THG from metal nanowires can be significantly enhanced by several orders of magnitude when the nonlocal effect is involved. The mechanism of the enhancement originate from the surface charges permeating into the metal, thus the corresponding electric field, to enlarge the volume which has contributions to the bulk third-order nonlinearity of metal. Further, we show that the THG intensities have obviously anisotropic distributions which depend on the angle of incidence due to the inhomogeneous surface morphology. Our results have a significance for the theoretical analysis of nonlinear optical properties from metal nanostructures to fit the experimental data. Moreover, in the article, we suggest two possible ways to verify the nonlocal model of free-electron which plays a crucial role in THG from the surface roughness of metallic nanostructures. We expect some possible experiments to be conducted to examine our findings.

2. Numerical model

The schematic configuration of cross section of Au nanowires is depicted in Fig. 1. We consider a typical size of Au nanowire with a radius R = 10 nm (Fig. 1(a)). The artificial roughness within ± 0.5 nm fluctuation with the maximum number of knots 100, 200 and 300 is generated using random functions embedded in the commercial software Comsol Multiphysics, as shown in Figs. 1(b)-1(d), respectively. For simplicity, we assume the nanowires have a translation invariance in the longitude direction. In the following, we will label the ideal nanowire without roughness as Perfect, and samples with rough surface of different numbers of knots as Rough100, Rough200, and Rough300, respectively. For THG simulation, we consider a fundamental wavelength λ₀ of 1064 nm (such as Nd:YAG laser), thus the THG wavelength is 355 nm. The polarization of fundamental light is set to be transverse-magnetic (TM) that is perpendicular to the longitude of nanowire, so as the same polarization of THG light is collected around the nanowires. The configuration of the light incidence is shown in the middle of Fig. 1.

Fig. 1 The schematic configuration of cross section of Au nanowires of radius 10 nm with an ideal surface (a) and rough surface within ± 0.5 nm fluctuation with the maximum number of knots 100 (b), 200 (c) and 300 (d). The configuration of light incidence is shown in the middle.

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To obtain the linear and nonlinear optical properties from metallic nanostructures, we will solve the classical Maxwell equations. We assume the light of time-harmonic propagation exp(iωt) and in the undepleted pump approximation for THG process, where the ω is angular frequency of light. The fundamental electric field at ω, i.e. E_ω, can be expressed as [44],

\nabla \times \nabla \times E_{ω} - k_{ω}^{2} E_{ω} = μ_{0} ω^{2} P_{ω}

where

k_{ω} = ω / c

is the wave vector of fundamental light with c the light velocity in vacuum, and P_ω is the polarization at fundamental frequency ω, μ₀ is the vacuum permeability. We divide P_ω into two contributions, P_ω = P_ωf + P_ωb, where P_ωf and P_ωb are related to free electron response and bound electron response, respectively [34]. In the local approximation, the P_ωf(r, ω) = D(r, ω)-ε₀E(r, ω) with the electric displacement field D(r, ω) = ε₀ε(r, ω)E(r, ω), where the ε(r, ω) is the permittivity of metal. Typically, we use the Drude model to describe the permittivity of metal from the free electron contribution as [37],

ε (ω) = ε_{\infty} - \frac{ω_{p}^{2}}{ω^{2} + i ω γ}

where

ε_{\infty} = 1

,

γ

the damping frequency, and the plasma frequency

ω_{p} = \sqrt{n e^{2} / ε_{0} m_{e}^{*}}

with n the electron density, e the electron charge, and

m_{e}^{*}

the effective electron mass. So basically, under the local approximation of the dielectric response, the P_ωf(r, ω) can be expressed as,

P_{ω f} = - \frac{ε_{0} ω_{p}^{2}}{ω^{2} + i ω γ} E_{ω}

While when consider the nonlocal response, the hydrodynamic model of metal should be employ to describe the permittivity [31,37],

ε (k_{ω}, ω) = ε_{\infty} - \frac{ω_{p}^{2}}{ω^{2} + i ω γ - β^{2} k_{ω}^{2}}

where

β = \sqrt{5 / 3} v_{f}

with the Fermi velocity v_f. Under such nonlocal approximation, the P_ωf(r, ω) is referring to an expression [31,33],

β^{2} \nabla (\nabla \cdot P_{ω f}) + (ω^{2} + i ω γ) P_{ω f} = - \frac{n_{0} e^{2}}{m_{e}^{*}} E_{ω}

Since the nonlocal response originates from the strong electron-electron interaction, so we only consider the nonlocal effect from free-electron polarization. The polarization of bound electrons P_ωb can be expressed using a multipole Lorentz oscillator model [34,39],

P_{ω b} = - \sum_{j} \frac{ε_{0} ω_{p}^{2}}{ω^{2} - ω_{0, j}^{2} + i ω γ_{j}} E_{ω}

where j is an index representing the electron transitions from d-band to sp-band at frequency

ω_{0, j}

[45].

Solving the coupled Eqs. (1), (3) and (6), we can obtain the linear optical properties, including the extinction and electric field distribution from metallic nanostructures under the traditional framework of local response. The nonlocal effect related linear optical properties can be obtained by solving the coupled Eqs. (1), (5) and (6). Note that we neglect the spill-out effect of electron from Au surface [46] and just consider a hard-wall boundary, i.e. $n \cdot P_{ω f} = 0$ with n the unit vector normal to the surface.

The THG process can be described using the nonlinear wave equation. For the electric field oscillating at 3ω, E_3ω, can be expressed as [44],

\nabla \times \nabla \times E_{3 ω} - k_{3 ω}^{2} E_{3 ω} = μ_{0} {(3 ω)}^{2} P_{3 ω} + μ_{0} {(3 ω)}^{2} P^{N L}

where

k_{3 ω} = 3 ω / c

is the wave vector of THG light, and P_3ω is the linear polarization at third-harmonic frequency 3ω, the expressions of P_3ω have the same form as P_ω, just substituting 3ω for ω, in local and nonlocal models, respectively. Au as an isotropic material has only one independent component of the susceptibility tensor,

χ_{i j k l}^{(3)} (3 ω; ω, ω, ω) = \frac{1}{3} χ^{(3)} (δ_{i j} δ_{k l} + δ_{i k} δ_{j l} + δ_{i l} δ_{k j})

with the value

χ^{(3)} = 1 \times 10^{- 18}

m²/V² [44]. So the element of third-order nonlinear polarization P^NL can be written as,

P_{i}^{N L} = ε_{0} \sum_{j k l} χ_{i j k l}^{(3)} (3 ω; ω, ω, ω) E_{ω j} E_{ω k} E_{ω l}

All the above coupled equations are numerically solved by implementing the finite element method (Comsol Multiphysics). Our model is first tested by the simple case of THG from bare silver layers, which were already calculated using nonlinear Lorentz-Duffing oscillator model [47] and compared with the experimental measurements [48]. Our results for the reflected and transmitted THG conversion efficiency versus the silver layer thickness at normal incidence agree well with the results of Fig. 4 in Ref [47]. Moreover, the THG from bare metal layer using local and nonlocal models of free-electron almost overlaps with each other, as reported in refs [34,47].

In the following sections about the simulation of nanowires, we will assume the incident electric field E₀ = 1 × 10⁷ V/m, and the corresponding intensity I₀ is around 13.3 MW/cm², which is feasible output intensity from most of lasers. The parameters for dielectric constant of Au, including the effective electron mass, the equilibrium charge intensity, the electron collision rate, the Fermi energy, and the bound electrons polarization of are employed as in the refs [33,45].

3. Results and discussion

We first study the THG from the nanowire of perfect surface considering the local- and nonlocal-response of free-electron. We vary the value of β, ranging from 0 (Local model) to β = 5 × 10⁶ m/s, to obtain the corresponding THG powers as shown in Fig. 2(a). Although the precise value of β is unknown, within the reported orders of 10⁶ m/s, the THG power calculated using the nonlocal model of free-electron is two to three orders of magnitude larger than that is calculated by traditional local Drude model. In order to explain such large deviation of the two models, we calculate the induced charge and local electric-field distributions under different values of β, as shown in the inset of Fig. 2(a) and Figs. 2(c)-2(f). For the traditional local Drude model, the induced charge density has a profile of a Dirac delta function which is centered at the metal–air interface (as shown in Fig. 2(c)). The singular charge profile at metal-air interface leads to the discontinued and rapidly decayed electric field inside the metal (as shown in Figs. 2(d) and 2(e)). While in the framework of hydrodynamic model of free-electron, the induced charge can permeate into the metal since the longitudinal wave can exist when considering the electron pressure, so as the electric field will go decay smoothly inside the metal, as shown in Figs. 2(d) and 2(f). The penetrated electric field inside the metal will thus boost the bulk third order nonlinearity of metals with the increased effective volume.

Fig. 2 (a) The THG power from the perfect nanowire versus the value of β at the angle of incidence 0°. The inset shows the induced charge density beneath the metal surface under different values of β. (b) Schematic configuration of metal-air interface with a cut-line. (c) The induced charge density and (d) electric field along the cut line in (b). The local field distributions near the top-part of nanowire in local (e) and nonlocal (f) models of free-electron are calculated with the typical value of β = 1.26 × 10⁶ m/s of Au. All calculations are conducted under the fundamental wavelength of 1064 nm and the incident electric field E₀ = 1 × 10⁷ V/m.

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In the next, we will consider the THG from the Au nanowires of rough surface in reality with the typical value of β = 1.26 × 10⁶ m/s. The THG power dependence on the angle of incidence from the Au nanowires considering the local and nonlocal effects is shown in Figs. 3(a) and 3(b), respectively. We can conclude that, firstly, under the framework of local response approximate of free-electron (Fig. 3(a)), the THG responses from nanowires of rough surfaces are obviously enhanced when compared with THG from perfect nanowire, but by only several times. The enhancement can be understood due to the enhanced local field around the tiny structures of rough surface. While when the nonlocal response of free-electron is considered (Fig. 3(b)), the THG efficiency from rough nanowires has dramatic enhancement compared with that from the perfect nanowire. The enhancement factor is up to two orders of magnitude for the samples Rough200 and Rough300. This could be explained by the enlarged effective volume for THG due to the penetrated electric field into the fine nanostructure on metal surface. The more details will be shown below. The so significant difference of enhancement for THG from nanowires of rough and smooth surface probably offers us a way to experimentally examine the nonlocal model of free-electron. Secondly, the THG efficiencies from nanowires of surface roughness have obviously anisotropic distribution under different angles of incidence whatever local or nonlocal effect is considered. This could be ascribed to the anisotropic local field distribution due to the inhomogeneous surface, and the details will be discussed below.

Fig. 3 Calculated THG power from Au nanowires dependence on the angle of incidence under (a) local and (b) nonlocal model, respectively. The parameters: the fundamental wavelength of 1064 nm, the incident electric field E₀ = 1 × 10⁷ V/m and β = 1.26 × 10⁶ m/s.

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In order to explain the above THG behaviors more clearly and firmly, we will study the linear optical extinction and the local field distributions for nanowires of different rough surfaces. The dimensionless scaled optical extinction is defined by σ_ext = P_ext/(I₀D), where the P_ext is extinction power integrated around the nanowire nanostructure, and D = 2R the diameter of the nanowire. The extinctions of different samples dependence on the wavelength at the angle of incidence 0° considering the local and nonlocal effects of Au are shown in Figs. 4(a) −4(c), respectively. We can see that there is only slight difference for the extinctions from the sample with local and nonlocal models since the radius of R = 10 nm on the operational edge of nonlocal mode [49]. So the linear extinction will not or only slightly affect the THG efficiency at the same angle of incidence. The local field distributions near the metal should be responsible for the difference of THG from different samples under different models. We choose the sample Rough300, which has the maximum enhancement of THG compared with perfect nanowire, to study the local field affection. The same conclusions should be arrived for the other samples. The local fields at fundamental and THG wavelengths for Rough300 are shown in Fig. 5. It is clear that the rough surface indeed enhances the local field significantly more, for example, the maximum local fields at the angle of incidence 0° are of around 12 and 23 times enhancement (compared to the incident electric field), which are much larger than the enhancement factor of 2 (Figs. 2(e) and 2(f)) for the perfect nanowire under local and nonlocal models, respectively. This is responsible for the enhanced nonlinear process from rough surface, and agrees well with the previous reports for the enhanced nonlinear process [22,23,29]. Further, we can see from the Fig. 5(b), the local electric field distribution is only near the surface of metal under the local-response approximation, the same as the field distribution around perfect nanowire (Fig. 2(e)), while the electric field can penetrate into the protrusions on the surface with the subnanometer scale to dramatically enlarge the effective volume for the bulk THG. This is why the nonlocal model can produce much stronger THG in surface rough nanowires, so as the prefect nanowire. Moreover, since the effective area in nonlocal model is in a volume form, and the more penetrated electric field confined inside the protrusions on the rough surface when compared with the penetrated electric field inside the prefect nanowire, so the enhancement factor, which is the THG from surface rough nanowires compared with that from perfect nanowire, could be much larger than that obtained by local model. Lastly, in order to understand the THG dependence on the angle of incidence clearly, we calculated the extinction of sample Rough300 at angle of incidence 0° to 180°. We just plotted the extinction spectra calculated under the nonlocal model in Fig. 4(d), and the almost overlapped extinctions in local mode at the same angle of incidence are not shown. We can see that the extinction spectra at different angle of incidenceare almost overlapped, indicating the linear optical property are not sensitive to the surface roughness we studied. So the linear extinction has minor affection on the THG efficiency at the different angle of incidence. We further calculated the local field distribution at different angle of incidence. Here we focus on the local field at the angle of incidence 0 ^o and 90°, as shown in Fig. 5. The 90° is just the angle of minimum THG from Rough300. It is clear that the maximum local field near the metal at 90° is indeed much smaller than that at 0 ^o at the fundamental wavelength. Since the THG is related to the intensity of pumping local field to the power three, so the difference of local field will be amplified to get much more different THG efficiency. This could also be read from the local field near the metal at the generated third-harmonic wavelength. The different pumping local field distribution induced by the inhomogeneous rough surface makes the significantly anisotropic distribution under different angles of incidence, as shown in Fig. 3.

Fig. 4 Extinction from Au nanowires calculated using the (a) local and (b) nonlocal model under the angle of incidence 0°. In order to compare the differences of the two models, the nanowire of ideal surface (Perfect) and rough surface with maximum knots 300 (Rough300) are simultaneously plotted in (c). Extinction spectra from the sample Rough300 versus angle of incidence 0° to 180° are plotted in (d) considering the nonlocal model. The insets in Figs. 4(a)-4(c) show the extinction around the surface plasmon resonance wavelength.

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Fig. 5 (a) The distributions of fundamental and THG electric field from Au nanowires of rough surface (Rough300) considering local and nonlocal model at angle of incidence 0° and 90°, respectively. In order to observe the local field distribution clearly, we enlarge the top-part of sample at the angle of incidence 0°. The parameters: the fundamental wavelength of 1064 nm, the incident electric field E₀ = 1 × 10⁷ V/m and β = 1.26 × 10⁶ m/s.

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We further investigate the THG from nanowires with the rougher surface. We set the artificial roughness within ± 1 nm, ± 1.5 nm and ± 2 nm fluctuations with a fixed maximum number of knots 300. The local and nonlocal effects on the THG from nanowires, including the previous results of roughness within ± 0.5 nm, are shown in Figs. 6(a) and 6(b), respectively. Basically, the THG efficiency is increasing with the increment of surface roughness, which is in agreement with the previous statements [22,29]. The anisotropy of THG efficiency dependence on the angle of incidence is still obviously observed from different surface roughness, and the profiles of THG power versus the angle of incidence are exactly microstructure dependent. Comparing with the THG calculated using local model, the nonlocal model further and dramatically modulates the THG with an enhancement factor up to five orders of magnitude as the roughness of ± 2 nm fluctuations. This can be ascribe to the more penetration of light into the metal volume because of the more protrusions and depressions of subnanometer size on the nanowire surface, as shown in Figs. 6(c) and 6(d) for the electric field distribution calculated under local and nonlocal model, respectively. Moreover, it is clear that for the same nanowire of rough surface, the THG dependence on the angle of incidence has obviously different profile under local and nonlocal description of free-electron. This probably offers us another way to verify the crucial roles of nonlocal model for the THG from the rough nanowires.

Fig. 6 The dependence of THG power on the angle of incidence from Au nanowires of roughness within ± 0.5 nm, ± 1 nm, ± 1.5 nm and ± 2 nm fluctuations with the fixed maximum number of knots 300 using (a) local and (b) nonlocal model, respectively. The electric field distribution on the top-part of Au nanowire of roughness within ± 1 nm at angle of incidence 0° using (c) local and (d) nonlocal model, respectively. The parameters: the fundamental wavelength of 1064 nm, the incident electric field E₀ = 1 × 10⁷ V/m and β = 1.26 × 10⁶ m/s.

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4. Conclusions

We numerically investigate the third-order harmonic generation from perfect and rough surface nanowires in the presence of classical nonlocal model. Comparing with the traditional local model, the THG efficiency in nonlocal model is significantly enhanced by a factor of several orders of magnitude dependent on the surface roughness. The nonlocal model, which considers the electrical field penetrating into the metal surface to enhance the effective volume which now contributes to bulk third-order nonlinearities of metal, is probably more exact to describe the THG, especially for the rough surface with the sub-nanometer microstructures. The anisotropy THG efficiency dependent on the angle of incidence is also demonstrated. Additionally, the results indicate the surface morphology of metallic nanostructures indeed significantly influences the nonlinear process, and is an essential factor that could not be ignored when the numerical results are used to fit the experimental data. Moreover, we suggest two possible ways to verify the roles of nonlocal model to THG. Finally, the artificially introduced fine microstructure on the surface of metallic nanostructure could enhance the nonlinear response for nonlinear photonics devices.

Funding

National Natural Science Foundation of China (Grant No. 11404195, 11474187, 11574185, 11504209); China Postdoctoral Science Foundation (2015M582127).

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Third-harmonic generation from gold nanowires of rough surface considering classical nonlocal effect

Abstract

1. Introduction

2. Numerical model

3. Results and discussion

4. Conclusions

Funding

References and links

Cited By

Figures (6)

Equations (8)

Optics Express