1. Introduction

Plasmonic nanostructures [1] stand in the heart of nano-optics owing to its extraordinary both linear [2, 3] and nonlinear optical properties [4, 5]. Among them, second harmonic generation (SHG) [6] is of particular interest in nonlinear plasmonics due to its sensitivity to symmetry [7–9] and relatively high efficiency. However, the intensity of SHG is still negligible compared with that of linear response due to the small value of nonlinear coefficient ${\chi }^{\left(2\right)}$ [10], even though the free charges in plasmonic nanostructure resonate with incident light. Taking the advantage of near-field resonance of plasmonic nanostructure, a large variety of mechanisms have been investigated to boost the SHG efficiency, including gap resonance [11, 12], overlapping of plasmonic resonance and SHG [13], low-radiated magnetic resonance [14–17], Fano resonances [18–20], and so on [21–23].

Theoretical analysis unveils that the dominating contributions to SHG would be a nonlocal electric-dipole moment and a local electric-quadrupole moment, which are bright and dark, respectively [24, 25]. It has been widely shown that interference between bright and dark modes produce a Fano dip in the extinction spectrum, thereby leading to highly concentration of near-field intensity, which could be utilized for fundamental study of electromagnetic and application of optical devices [26–38]. Recently, plasmonic heterodimers with both i- and T-shaped morphologies have been investigated [39–46], where a short nanorod serves as a bright dipole source to give rise to a dark quadrupole resonance supported by a long nanorod [42, 44]. In general, the smaller the separation between nanorods is, the stronger the nanorods capacitively couples, and the larger the local field is [47]. Moreover, in analogy to the directional scattering in linear regime [48–51], the contributions of plasmonic SH sources and their distinct radiation direction [52] provide us an opportunity to control the far-field pattern of SH light.

Using coupled dipole resonances, T-shaped heterodimer consisting of short nanorods has been used to enhance SHG [22]. In this paper, we show that the SHG efficiency, in T-shaped heterodimer consisting of a pair of a long and a short gold nanorods, can be efficiently boosted the Fano resonance which associates with electric dipole-quadrupole hybridization (EDQH). We first briefly describe the numerical method and design principle of our structure. By changing the gap between two nanorods, we numerically analyze the dependence of both linear and nonlinear responses on the capacitive coupling between electric dipole and quadrupole. It is found that for strongly coupled modes, hybridization of bright and dark modes gives rise to a near-field concentration within the gap, thus resulting in enhancement of SHG. Furthermore, far-field patterns of SHG confirm that EDQH plays a key role in enhancement of SHG emission.

2. Numerical methods and design principle

Let us start by introducing numerical method and design principle. All numerical simulations were carried out by using the commercial software COMSOL Multiphysics. A two-steps simulation has been adopted [8, 10]. First, a total-field-scattering background source was employed to remove the incident light with power density of 1 MW/m² from scattering signals. For simplicity, we do not consider the fundamental power for detectable SHG, the damage threshold of the plasmonic structures and so on [19, 68]. Second, an effective nonlinear current sheet at the surface of nanostructure has been applied as the source of second harmonic scattering process with the use of weak form. Then, both the near-field and far-field distributions can be computed with scattering model at SH frequency. In our case, only the dominant component ${\chi }_{\perp \perp \perp }$ of the second order surface susceptibility tensor was considered, where $\perp$ represents the orientation perpendicular to the surface of the structure. Note that, there are some other theoretically allowed but negligible contributions to SH signal [10]. The optical constant of gold was taken from Johnson and Christy’s work [53]. For simplicity, the surrounding dielectric material is set to be air.

In order to get an insightful picture of the underlying hybridization between modes, we start with developing the basic analyze of individual gold nanorods, as shown in Fig. 1. In our simulation, the gold nanorods were modeled as cylindrical rods with radius $R=20$nm and hemispherical end caps to approach the real experimental conditions. The total lengths of short and long gold nanorods, including the end caps, were $L_1\text{=100}$nm and $L_2=220$nm, respectively. Figure 1(a) shows the scattering cross section (SCS) of electric quadrupole moment of the long nanorod under both TE and TM plane wave, indicating an electric quadrupole resonance around wavelength of 620nm. However, due to the dark nature of electric quadrupole mode, the interaction between this mode and a plane wave is weak, and it could be excited by using a near-field electric dipole source [42]. Figure 1(b) shows the SCS of electric dipole moment of the short nanorod under a TM plane wave, therefore demonstrating it may function as a near-field source to excite the electric quadrupole in the long nanorod. The electric dipole and quadrupole moments are obtained from $p={\displaystyle \int {\epsilon }_0({\epsilon }_r-1)E(r^{\text{'}})\mathrm{d}{\mathrm{V}}^{\text{'}}}$and $Q={\displaystyle \int {\epsilon }_0({\epsilon }_{\text{r}}-1)[E(r^{\text{'}})r^{\text{'}}+r^{\text{'}}E(r^{\text{'}})]}\mathrm{d}{\text{V}}^{\text{'}}$, respectively.

 

Fig. 1 Schematics of (a) a long and (b) a short gold nanorod functioning as electric quadrupole and dipole, respectively. The gold nanorods were modeled as cylindrical rods with radius $R=20$nm and hemispherical end caps. The total lengths of short and long gold nanorods, including the end caps, were $L_1=100$nm and $L_2=220$nm, respectively. The SCS of electric quadrupole and dipole from the long and short nanorod, respectively.

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Previous studies show that by arranging the short and long nanorod in i- and T-shaped heterodimers, the capacitive coupling between them hybridizes the bright dipole mode of short nanorod and the dark quadrupole mode of long nanorod [39–46]. This hybridization forms a Fano-like resonance with a high local density of state, with charges at both ends of long nanorod are of the same sign whereas charges in the middle are opposite. Due to the law of conservation of electric charge, the charges accumulated in the middle are twice of that at each end. Therefore, we choose the T-shaped heterodimer as the investigated scheme instead of the i-shaped configuration because the capacitive coupling between nanorods is stronger in it than the latter [45]. The detailed schematic is shown in Fig. 2(a).

 

Fig. 2 (a) The schematic of proposed T-shaped heterodimer composed of a short and a long gold nanorod with lengths of $L_1\text{=100}$nm and $L_2=220$nm, respectively, and radius of $R=20$nm. A plane wave is normally incident along z-axis with electric field polarizing along y-axis. (b) Simulated linear extinction cross section of the T-shaped heterodimer with gap size of $g=2,4,6,8,10,20,50$ and 100 nm. (c) Near field distributions of normalized electric field intensity at wavelengths of 575, 625 and 695 nm. The top and bottom column represents $g=2$ and $g=100$nm, respectively. (d) The schematic of surface charge distribution in T-shaped heterodimer with gap size of $g=2$ and $g=100$nm. (e) Far field distribution of electric field for T-shaped heterodimer with gap size of $g=2$ and $g=100$nm at wavelength of 695nm and 585nm, respectively.

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3. Results and discussion

In order to illustrate the effect of EDQH in the proposed T-shaped heterodimer on SHG emission, we firstly perform a comprehensive analysis on the dependence of optical responses in linear regime on gap size $g$. Generally, to enhance the capacitive coupling strength so as to increase the near-field intensity, it is desirable to adopt extremely narrow interparticle gaps [54]. However, the quantum tunneling may come into play below sub-nanometer gap size and the nearby nanorods become conductive, whereby the classical description of optical properties of metal and air could be invalid [55–57]. Thus, we suppose the smallest gap as $g\text{=}2$nm to ensure no quantum effect happening between nanorods. Here, we choose several different separations between two nanorods $g=2,4,6,8,10,20,50$ and 100 nm. To provide a clear insight into the nature of mode hybridization in this heterodimer, in the following we firstly calculate the linear optical responses of the T-shaped heterodimer, including extinction cross section (ECS), near-field distribution and far-field pattern, as shown in Figs. 2(b-e).

Figure 2(b) plots the simulated ECSs of the heterodimers as a function of different gap size $g$under y-polarized plane wave. When the two nanorods are separated with a wavelength-sized distance ($g=100$nm), there is only a broad resonance around 590 nm, indicating the dominant of dipolar response of short nanorod and the absence of EDQH. By reducing the separation to $g=\text{5}0$nm, interparticle coupling is still negligible, thereby only leveling the dipolar resonance peak. Further decreasing the gap size to $g=\text{2}0$nm, that is the radius of nanorod, the capacitive coupling between nanorods appears, thus producing a Fano-like resonance with two peaks at 580 nm, 625 nm and a shallow dip at 600 nm. With strong near-field capacitive coupling ($g=2,4,6,8,10$), the EDQH plays a dominant role in optical extinction spectrum of the T-shaped heterodimer in the linear regime, exhibiting a Fano-like resonance, and the eigen-frequency gradually shifts to lower energy due to the increased charge storage capacity in the capacitive gap. The simulated dependence of eigen-frequency is in a good agreement with the optical circuit model prediction, i.e. red-shift of eigen-resonances in case the capacitive gap increases [58, 59].

To understand the mode evolution of heterodimer resonances during the different gaps, we further calculate the near-field distribution of electric field intensity of heterodimer with smallest gap $g=2$nm at wavelengths of 575, 625 and 695nm, in comparison with the case of largest gap $g=\text{100}$nm, as shown in Fig. 2(c). For the latter case, the EDQH is minimal due to weak capacity coupling, thus leading to a significant amplification of electric field at the short gold nanorod with electric-dipolar character (upper panel). Meanwhile the quadrupole mode of long gold nanorod is not excited by either the incident plane wave or the dipole source (i.e. the short gold nanorod). In contrast to the case of $g=\text{100}$nm, the intense capacitive coupling at the smallest gap $g=2$nm results in a giant EDQH and near-field enhancement of more than two-orders of magnitude at the surface of long gold nanorod (lower panel).

According to the hybridization theory of plasmonic mode [60–62], optical property of the heterodimer is determined by the mode hybridization between the short and long gold nanorods. As an example, Fig. 2(d) shows the evolution of mode hybridization scheme of heterodimer with large gap $g=100$nm (upper panel) and small gap $g=\text{2}$nm (lower panel). In the upper panel, the separation between two nanorods is in wavelength-level and relatively large, charge distributions in both short and long nanorod show electric dipole characteristic. In the lower panel, near-field capacitive coupling happens and the short nanorod functions as a near-field dipole source to excite the quadrupolar charge distribution in the long nanorod. Due to the mode hybridization nature, the optical response of heterodimer exhibits a Fano-like resonance. It should be noted while the electric quadrupole mode of an isolate nanorod is low-radiative, for two tightly coupled nanorods the EDQH may change the radiative nature of the quadrupole [63, 64].

Besides, the simulated far field distribution of electric field for T-shaped heterodimer with gap size of $g=2$ and $g=100$nm at wavelength of 695nm and 585nm, respectively, are shown in upper and lower panel of Fig. 2(e). For the large gap, the T-shaped heterodimer radiates into electric dipolar far-field pattern, indicating negligible EDQH. In contrast, the intense capacitive coupling at the smallest gap results in a strong EDQH which is in good agreement with the far-field pattern.

Overall, because of the strong plasmon coupling within the gap, EDQH and giant near-field enhancement are realized with the proposed heterodimer, as shown in previous studies [39–46]. Therefore, the T-shaped heterodimer can be a promising platform for boosting SHG signal which can be amplified by the enhancement of localized electric field at fundamental frequency (FF).

Hereafter, we are going to unveil the influence of EDQH on the SHG responses. To this end, we simulate the SHG emission of the heterodimer with varied gaps size under y-polarized incident plane wave with the help of weak form equation, as shown in Fig. 3(a). The SHG efficiency is calculated by ${\eta }^{\text{SHG}}\text{=}SC{S}^{\text{SHG}}/EC{S}^{\text{FH}}$, where $SC{S}^{\text{SHG}}$ and $EC{S}^{\text{FH}}$ are the scattering and extinction cross sections for the second and fundamental harmonic cases, respectively, following [19].The heterodimers with large gap size of $g=100$ and 50nm, which have negligible EDQH, show broadband resonance peak on SHG efficiency spectrum around wavelength of 585 and 595nm, respectively, originating from the localized electric dipole resonance of the short nanorod. As expected, when we reduce the gap size to 20nm and further 10, 8, 6, 4, 2nm, the emitted SHG signal gradually increases due to EDQH and resulting strong enhancement of localized electric field in linear regime. Besides, we found that the maximum enhancement of SHG emission slightly shifts to low energy in comparison with Fano-like resonance, due to energy shift between near and far-field properties caused by the intrinsic damping loss in plasmonic nanostructures [65–67].

 

Fig. 3 (a) Simulated SHG efficiency of the T-shaped heterodimer with gap size of $g=2,4,6,8,10,20,50,100$nm. (b) Near field distributions of normalized SH electric field intensity at the SHG efficiency peaks. (c) Far field distribution of SH electric field in x-y plane. I: $g=2$nm, $\lambda =695$ nm; II: $g=4$nm, $\lambda =675$nm; III: $g=6$nm, $\lambda =660$nm; IV: $g=8$nm, $\lambda =655$nm; V: $g=10$nm, $\lambda =650$nm; VI: $g=20$nm, $\lambda =625$nm; VII: $g=50$nm, $\lambda =590$nm; VIII: $g=100$nm, $\lambda =585$nm.

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In addition, to reveal the physical nature of amplified nonlinear source, we calculate the SH near-field distribution for the eight SHG emission peaks, as plotted in Fig. 3(b), where I: $g=2$nm, $\lambda =695$nm; II: $g=\text{4}$nm, $\lambda =6\text{7}5$nm; III: $g=\text{6}$nm, $\lambda =6\text{60}$nm; IV: $g=\text{8}$nm, $\lambda =6\text{5}5$nm; V: $g=\text{10}$nm, $\lambda =6\text{50}$nm; VI: $g=\text{20}$nm, $\lambda =6\text{2}5$nm; VII: $g=\text{50}$nm, $\lambda =\text{590}$nm; VIII: $g=\text{100}$nm, $\lambda =\text{58}5$nm. For case I, the normalized SH near-field intensity is strongest, especially inside the gap, due to the largest capacitive coupling and EDQH. Subsequently, the SHG emission is increased by two orders of magnitude. By widening the gap to $g=\text{4,6,8,10}$nm (i.e. cases II, III, IV, V), the SH near-field distributions show similar characters but weaker intensities since the capacity of light confinement and field enhancement inside the gap weakens. Further enlarging the gap size to $g=20,50,100$nm (i.e. cases VI, VII, VIII), we can see the capacitive coupling between two nanorods gradually disappears.

Moreover, the corresponding SH far-field scattering patterns in the E-H plane are shown in Fig. 3(c). Let us remind that in SHG regime, both electric dipole and electric quadrupole moments contribute to the leading order SH responses, while magnetic-dipole emission is absent due to the symmetry of nanostructure [24]. Usually, the most efficient mechanism for SHG emission is the linear dipole$\to$SH quadrupole because electric dipole dominates the optical response in linear regime and higher mode, i.e. electric quadrupole is negligible. However, in our proposed heterodimer, the EDQH significantly arises from the strong capacitive coupling between two nanorods. The SHG emission far-field pattern is completely specified by two coefficients ${\chi }_{\text{1}}$ and ${\chi }_{\text{2}}$, corresponding to the contributions of electric dipole and quadrupole terms to SHG signal [24, 68]. Therefore, besides the quadrupolar SH far-field scattering pattern, we may also observe dipolar and dipole-quadrupole hybrid SH far-field pattern for different linear EDQH mechanisms, which is well presented in Fig. 3(c).

To further figure out how the EDQH could modify the SHG far-field pattern, we choose the case of heterodimer with 2nm gap as an example and calculate the ratio of electric quadrupole to electric dipole in linear regime, as shown in Fig. 4(a). It is clearly shown that the EDQH is significantly excited in the proposed heterodimer due to the strong coupling effect between two nanorods. The amplitude of electric quadrupole reaches maximum around wavelength of 624nm, revealing that the electric quadrupole originates from the long gold nanorod rather than the short one. The SHG emission far-field patterns are simulated at wavelength of 560, 600, 624 and 700nm. Both the 3D and 2D (E-H plane) patterns are plotted in Figs. 4(b) and 4(c), respectively. Because of the different EDQH intensities, the SH emission can be modified into patterns beyond the well-known quadrupolar pattern. These results confirm that EDQH is a key factor in the enhancement of SHG emission efficiency. Indeed, the plasmon mode hybridization is a complex physical process and will significantly affect its nonlinear response, including both near and far field properties.

 

Fig. 4 (a) Simulated ratio of linear EQ to ED. The black circles show the wavelength at which the SHG far field patterns are studied. (b) Three-dimensional and (c) two-dimensional (in x-y plane) far field pattern of SH electric field from T-shaped heterodimer with gap size of $g=2$ nm for different incident wavelengths: 560, 600, 624 and 700nm.

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4. Conclusion

In summary, we have shown that SHG efficiency in T-shaped gold nanorods heterodimer can be efficiently boosted with EDQH. A two-steps numerical simulation has been performed by using the finite element method. By changing the size of gap between two nanorods, we numerically analyze the dependence of both linear and nonlinear responses on the capacitive coupling between electric dipole and quadrupole. It is found that for strongly coupled modes, hybridization of bright and dark modes gives rise to a near-field concentration within the gap, thus resulting in enhancement of the SHG by a factor >100 in comparison with that achieved by isolate gold nanorod. Furthermore, we have investigated the far-field patterns of SHG and confirmed that EDQH is a key factor in the enhancement of SHG emission efficiency. Our study could not only pave the way towards the enhancement of working efficiency of nonlinear optical devices but also establish a platform to control the direction of nonlinear signals.