1. Introduction

Recent advances that allow metals to be structured on nanoscale dimension have aroused great interest for designing different nanostructures and synthesizing various kinds of nanocomplexes for nanodevices [1]. Noble metals which support surface plasmons (SPs) have gained increasing attention due to their unusual optical properties and abilities to manipulate light in unique ways [2,3]. Metallic nanostructures possess geometry-dependent localized surface plasmon (LSP) resonances, which is one of the major reasons for the growing interest in developing nanoscale geometries including nanorice, nanosphere, nanorod, nanoring, nanocube, and nanoshell [4–9]. In addition to these basic structures, more hybrid nanostructures such as multimer clusters and array structures, which exhibit rich plasmonic properties, have been developed and applied in many fields [10–15].

Among the hybrid nanostructures, plasmonic dimers have gained an increasing aware of importance. The dimers can exhibit different plasmon modes resulting from surface plasmon hybridization [16]. Furthermore, the SPs coupling in a dimer structure can concentrate light in subwavelength volume and lead to strong electric field enhancement. For example, the localized electric field enhancement in the gap region of a coupled dimer such as bow-tie structure [17,18], is usually much larger than that associated with isolated nanoparticle. It is well known that the enhancement factor of SERS is proportional to ${\left|E/E_0\right|}^4$, where $E$ is the local electric field near the probe molecule and $E_{\text{0}}$ is the incident electric field. Therefore, an important application of dimer structure is for SERS. Although the optical properties of nanorod dimers with different configurations have been studied widely [19–23], the present work aims at providing new insight into some physical phenomena associated with plasmon resonance in three-dimensional chiral plasmonic dimers. In current paper, we design and study a tangent nanospheroid homodimer (TNSHD) structure that allows the generation of strong near-field interaction. The effects of the rotation angle and polarization direction on the surface plasmon resonance and electric field enhancement of the TNSHD are investigated in detail.

2. Model and Computational Methods

The schematic illustrations of the TNSHD and the referenced coordinates are shown in Figs. 1(a) and 1(b). The TNSHD consists of two tangent Au spheroids with the same size, which are denoted as nanospheroid 1 and nanospheroid 2. Figure 1(a) presents the case where the nanospheroid 1 is parallel to the nanospheroid 2 in a side-by-side manner, and Fig. 1(b) presents the case where the angle between the long axes of the two nanospheroids is θ. The side view and top view of the individual nanospheroid are shown in Figs. 1(c) and 1(d), respectively, where constant l represents the length of the long axis and w represents the lengths of two short axes. In the calculation, we take the constant l as 50 nm and w as 20 nm. All of the systems are assumed to be in vacuum. Figure 1(e) shows the process of changing the rotation angle θ, which assumes that the long axis of the nanospheroid 1 is along the y-axis all the time and the nanospheroid 2 rotates in an anti-clockwise direction. Here the z-axis is taken as its rotation axis. The incident wave vector is along z-axis in all of the cases.

 

Fig. 1 Schematic of the model: (a) Front view of the TNSHD consisting of two parallel nanospheroids in the y-z plane. The green circle with dotted line indicates that there is a touching point in the center of the circle. (b) Side view of the TNSHD and θ presents the rotation angle. (c) Side view and (d) top view of the individual nanospheroid. The constant l represents the length of long axis and w represents the lengths of two short axes of the individual nanospheroid.

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The effects of rotation angle θ and two kinds of polarization modes (the direction of incident polarization is parallel to y-axis and x-axis, respectively) on the surface plasmon resonance and electrical field enhancement of the TNSHD are considered in the work. The extinction and absorption spectra of the individual nanospheroid and TNSHD are calculated by employing the DDSCAT code package developed by Draine and Flatauthe [24]. DDSCAT is a freely available open-source software package based on the discrete dipole approximation (DDA) method [25,26]. In this method, the target nanoparticle is represented as a cubic array of virtual point dipoles, and its response to an applied light field is described by self-consistently determining the induced dipole moment in each element. The DDA method has been demonstrated to be a very powerful electrodynamic method for calculating optical spectra of particles with arbitrary geometries. The electric field distributions of the individual nanospheroid and TNSHD structure are simulated by using the finite difference time domain (FDTD) method [27]. This method is based on Yee lattices, in which the vector components of the electric field and magnetic field are spatially staggered proposing a leapfrog scheme.

3. Results and discussion

3.1 The longitudinal and transverse plasmon modes of the individual nanospheroid

When the polarization direction of the incident light is along the long axis of the individual nanospheroid, only one dipole peak corresponding to the longitudinal mode is observed and the resonant peak appears at ~586 nm, which is shown in Fig. 2(a). For the transverse excitation, the resonance peak corresponding to the transverse mode appears at ~500 nm, which is shown in Fig. 2(b). The DDA simulations of the far field extinction spectra of the individual nanospheroid reveal that the longitudinal plasmon mode has a much stronger extinction strength than the transverse mode, while the transverse mode has a higher energy resonance than the longitudinal mode. The electric field distributions of the individual nanospheroid at the longitudinal and transverse resonance wavelength are shown in the inset of Figs. 2(a) and 2(b), respectively.

 

Fig. 2 Extinction cross section of (a) the longitudinal polarization mode and (b) the transverse polarization mode. The insets in the top-left corners show the directions of the incident wave vector and polarization. The insets in the top-right corners show the corresponding electric field distributions at the resonance wavelength. The color bars represent the amplitude of $\left|E/E_0\right|$, where $E$ is the local electric field near the TNSHD, and $E_{\text{0}}$ is the incident electric field.

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3.2 Plasmonic response of the TNSHD when the incident polarization is along the y-axis

Figures 3(a) and 3(b) show the extinction and absorption cross sections of the TNSHD as a function of the rotation angle when the incident polarization is parallel to y-axis, namely along the long axis of nanospheroid 1 as shown in the inset of Fig. 3(a). From Fig. 3(a), we can observe that two distinct modes appear in the extinction spectra (one is at a higher energy level, ~570 nm, and the other is at a lower energy level, ~650 nm) when $\theta ={\text{0}}^{°}$. With the increase of the rotation angle θ, the higher-energy peak decreases in intensity and exhibits a gradual red shift, while the lower-energy peak decreases gradually and blue-shifts slightly. The double-peak spectral feature vanishes when θ increases to 90° and only one peak at about 590 nm is observed. It can also be found that with the increase of the rotation angle θ, the spectra shoulder becomes more and more narrower, resulting from the diminishing dipole-dipole interaction. Fig. 3(b) indicates that the absorption dominates the extinction for the TNSHD. The extinction and absorption cross sections of the individual nanospheroid for the longitudinal mode are shown as a reference in Figs. 3(a) and 3(b), respectively.

 

Fig. 3 (a) Extinction and (b) absorption cross sections of the TNSHD as a function of the rotation angle θ. The extinction and absorption cross sections of the individual nanospheroid (green dashed lines) are shown in Figs. 3(a) and 3(b), respectively. The incident wave vector and polarization are illustrated in the inset of Fig. 3(a).

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In order to explain the spectral responses in the TNSHD, we employ the exciton coupling theory [28–32] to provide some physical insight into the dipole-dipole interaction between the two nanospheroids. Because the dipole moment contribution of the transverse mode is minor, this TNSHD can be idealized to a dimer of two dipoles ${\stackrel{\rightharpoonup }{p}}_1$ and ${\stackrel{\rightharpoonup }{p}}_2$ oriented along the long axis of each nanospheroid. Incident light interacting with the TNSHD causes each dipole to radiate an electromagnetic field, which in turn affects its adjacent dipole. This dipole-dipole interaction is associated with a splitting energy:

where ${\stackrel{\rightharpoonup }{p}}_1$ and ${\stackrel{\rightharpoonup }{p}}_2$ stand for the dipole moments of nanospheroids 1 and 2, respectively. ${\stackrel{\rightharpoonup }{R}}_{12}$ is the vector connecting the point dipoles and $\left|{\stackrel{\rightharpoonup }{R}}_{12}\right|=r$. $n$ is the refractive index of the medium ($n=1$ for vacuum). For this side-by-side configuration investigated in this paper, ${\stackrel{\rightharpoonup }{p}}_1\cdot {\stackrel{\rightharpoonup }{R}}_{12}=0$ and ${\stackrel{\rightharpoonup }{p}}_2\cdot {\stackrel{\rightharpoonup }{R}}_{12}=0$. When $\theta ={\text{0}}^{°}$ (the nanospheroids are parallel to each other, which results in ${\stackrel{\rightharpoonup }{p}}_1//{\stackrel{\rightharpoonup }{p}}_2$), the splitting energy reaches its maximum. In this case, two different modes can be observed clearly in the optical spectra as shown in Fig. 3, corresponding to the hybridized antibonding mode $|{\psi }_{+}〉$ (570 nm) and the hybridized bonding mode $|{\psi }_{\text{-}}〉$ (650 nm). When $\theta =9{\text{0}}^{°}$ (the nanospheroids are perpendicular to each other which results in ${\stackrel{\rightharpoonup }{p}}_1\perp {\stackrel{\rightharpoonup }{p}}_2$), the splitting energy reduces to zero and the spectral response degenerates to the longitudinal mode of an individual nanospheroid.

To understand the plasmon hybridization in detail, the hybridization picture for the case when $\theta ={\text{0}}^{°}$ is illustrated in Fig. 4. It can be observed that the antibonding mode $|{\psi }_{+}〉$ locates at a higher energy level and exhibits an in-phase charge distribution. This antibonding mode possess finite dipole moments and can couple to the incident light directly, and thus is a bright mode. Its resonance is spectrally broadened due to radiative damping. The bonding mode $|{\psi }_{\text{-}}〉$ locates at a lower energy level and exhibits an out-of-phase charge distribution. It has a zero net dipole moment (for the homodimer, $\left|{\stackrel{\rightharpoonup }{p}}_1\right|=\left|{\stackrel{\rightharpoonup }{p}}_2\right|=p$) and cannot be coupled to incident electromagnetic field efficiently, and thus is termed as a dark mode. Similar phenomenon can be observed in metal-insulator-metal (MIM) nanodisks aligned vertically [33].

 

Fig. 4 Scheme of the plasmon hybridization of the TNSHD structure when $\theta ={\text{0}}^{°}$. The dipole resonance wavelength of the individual nanospheroid is 580 nm. The resonance wavelengths of the antibonding mode $|{\psi }_{+}〉$ and the bonding mode $|{\psi }_{\text{-}}〉$ are 570 nm and 650 nm, respectively. The inset in the top-right corner shows the directions of the incident wave vector and polarization.

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Figure 5(a) shows the electric field distributions of the TNSHD in the y-z cross section as the rotation angle θ increases from 15° to 90°. Strong electric field enhancements can be observed at the ends of nanospheroid 1 and in the gap regions between nanospheroids 1 and 2. Figure 5(b) shows the electric field distributions in the plane determined by the z-axis and the long axis of nanospheroid 2. The electric fields at the ends of nanospheroid 2 and in the gap regions between nanospheroids 1 and 2 are enhanced strongly except when $\theta ={\text{90}}^{°}$. The reason is that the transverse plasmon mode of nanospheroid 1 and the longitudinal plasmon mode of nanospheroid 2 are not excited at $\theta ={\text{90}}^{°}$. The Electric field distributions of the TNSHD in the x-y plane at the center of nanospheroid 1 and that of nanospheroid 2 as a function of rotation angle are provided in the Appendix A.

 

Fig. 5 Electric field distributions of the TNSHD in (a) the y-z cross section and (b) the plane determined by the z-axis and the long axis of nanospheroid 2 as a function of the rotation angle θ which increases from 15° to 90°. The corresponding excitation wavelength is 574 nm, 575 nm, 577 nm, 583 nm, 593 nm, and 595 nm, respectively. The incident wave vector and polarization are illustrated inside nanospheroid 1 (for (a)), where the wave vector and polarization direction are the same for all panels and inside nanospheroid 2 (for (b)), where the wave vector direction is constant but polarization direction is rotated out of this plane gradually. The color bars represent the amplitude of $\left|E/E_0\right|$, where $E$ is the local electric field near the TNSHD, and $E_{\text{0}}$ is the incident electric field.

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3.3 Plasmonic response of the TNSHD when the incident polarization is along the x-axis

Figure 6(a) shows the extinction cross section of the TNSHD with different rotation angles when the incident polarization is parallel to x-axis, namely along the short axis of nanospheroid 1. It can be observed clearly that the extinction cross section of the TNSHD is very similar to the transverse extinction cross section of the individual nanospheroid at $\theta ={\text{0}}^{°}$, while it is similar to the longitudinal extinction cross section of the individual nanospheroid at $\theta ={\text{90}}^{°}$. When $\theta ={\text{0}}^{°}$, only the transverse modes of nanospheroids 1 and 2 are excited and their weak coupling results in the spectral response similar to the transverse mode of individual nanospheroid. When $\theta =9{\text{0}}^{°}$, the amplitude of the longitudinal mode of nanospheroid 2 increases to its maximum. The coupling between the weak transverse mode of nanospheroid 1 and the strong longitudinal mode of nanospheroid 2 is responsible for the spectral response similar to the longitudinal mode of individual nanospheroid. Surprisingly, we notice that a distinct Fano resonance appears when θ increases to 30° and it becomes unobvious when θ increases to 60°. For further investigation of the effect of θ on Fano resonance, we calculate the absorption spectra of the TNSHD with a varied θ from 25° to 50° as shown in Fig. 6(b). The appearance of the distinct Fano resonance is caused by the interaction between the bright mode and the dark mode of the nanospheroid monomers [34]. The spectral overlap and destructive interference of these two modes lead to the formation of the Fano resonance. When the polarization of the incident light is along the long axis of the individual nanospheroid, the dipole bright mode can be excited. In the TNSHD, when θ is not equal to 0° or 90°, the y-axis component of the bright mode of nanospheroid 2 (the rotated one) can excite a dark mode in nanospheroid 1. This dark mode is along the long axis of nanospheroid 1, namely the y-axis. Then, the dark mode of nanospheroid 1 couples with the bright mode of nanospheroid 2 by means of electromagnetic near-field interaction. As a result, the Fano resonance is formed. From Fig. 6(b), we can see that the higher-energy peak red-shifts from 560 nm at $\theta ={\text{25}}^{°}$ to 580 nm at $\theta ={\text{50}}^{°}$, whereas the lower-energy peak blue-shifts from 641 nm at $\theta ={\text{25}}^{°}$ to 619 nm at $\theta ={\text{50}}^{°}$. Furthermore, we can find that the Fano profile becomes narrower gradually and the Fano dip enhances first then reduces as θ increases. At $\theta ={\text{15}}^{°}$ and $\theta ={\text{75}}^{°}$, no distinct Fano resonance lineshape is observed because that the y-axis component of the bright mode of nanospheroid 2 is very small and only a weak dark mode is excited in nanospheroid 1.

 

Fig. 6 (a) Extinction cross section of the TNSHD as a function of the rotation angle θ which increases from 0° to 90°. (b) Absorption cross section as a function of θ which increases from 25° to 50°. The green dotted lines are for eye guiding. The extinction cross section of the individual nanospheroid (salmon pink dashed lines) is shown as a reference in Fig. 6(a). The incident wave vector and polarization are illustrated in the inset of Fig. 6(a).

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The electric field distributions in the y-z cross section and in the plane determined by z-axis and long axis of nanospheroid 2 as θ varies from 15° to 90° are shown in Figs. 7(a) and 7(b), respectively. We can find that strong electric field enhancements can be achieved at the ends and in the gap regions of the TNSHD due to the coupling between the longitudinal and transverse plasmon modes. It is worth noting that the electric field enhancement in y-z plane is weak, whereas the electric enhancement in the plane determined by z-axis and long axis of nanospheroid 2 is strong when $\theta ={\text{90}}^{°}$. The reason is that the longitudinal plasmon mode of nanospheroid 1 and the transverse plasmon mode of nanospheroid 2 are not excited while the transverse plasmon mode of nanospheroid 1 and the longitudinal plasmon mode of nanospheroid 2 are excited. The Electric field distributions of the TNSHD in the x-y plane at the center of nanospheroid 1 and that of nanospheroid 2 as a function of rotation angle are provided in the Appendix A.

 

Fig. 7 Electric field distributions of the TNSHD in (a) the y-z cross section and (b) the plane determined by the z-axis and the long axis of nanospheroid 2 as a function of the rotation angle θ which increases from 15° to 90°. The corresponding excitation wavelength is 650 nm, 640 nm, 580 nm, 588 nm, 590 nm, and 590 nm, respectively. The incident wave vector and polarization are illustrated inside nanospheroid 1 (for (a)), where the wave vector and polarization direction are the same for all panels and inside nanospheroid 2 (for (b)), where the wave vector direction is constant but polarization direction is rotated into this plane gradually. The color bars represent the amplitude of $\left|E/E_0\right|$, where $E$ is the local electric field near the TNSHD, and $E_{\text{0}}$ is the incident electric field.

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By comparing the electric field spatial distributions in two different polarization directions, we find an interesting phenomenon that the optimal electric field enhancement effect of the TNSHD emerges when the rotation angle θ equals 60°, where stronger electric field enhancements and larger hot spot (where the electric field is enhanced strongly) areas can be gained simultaneously.

4. Conclusion

The plasmon resonances of the TNSHD excited with two different polarization modes have been investigated by using the DDA method and the surrounding electric field distributions have been simulated by employing the FDTD method. Fano resonances which arise from the coherent coupling between the bright mode and dark mode have been observed in the optical spectra, and the strong electric field enhancements which result from the surface plasmon coupling have been realized. Interestingly, the optimal electric field enhancement effect emerges when $\theta =6{\text{0}}^{°}$, where stronger electric field enhancements and larger hot spot areas can be gained simultaneously. The TNSHD provides a promising strategy to produce prominent Fano resonance signals and to achieve strong electric field enhancements. The tunability of this Fano resonance could be utilized in the biosensing. The strong electric field enhancement can be used for enhancing Raman scattering and light trapping in solar cells. It is also expected that the TNSHD might have a high potential to serve as platforms for subwavelength optics and other relevant frontier research fields.

Appendix A

Electric field distributions of the TNSHD in the x-y plane at the center of nanospheroid 1 and that of nanospheroid 2 as a function of rotation angle in two different polarization directions are provided in Fig. 8 to help readers to understand the spatial electric field distributions in detail. Incident wave vector is perpendicular to the x-y plane (along the z-axis) all the time. Panel (a) corresponds to the case where the polarization direction is along the y-axis and panel (b) corresponds to the case where the polarization direction is along the x-axis.

 

Fig. 8 Electric field distributions of the TNSHD in the x-y plane at the center of nanospheroid 1 and that of nanospheroid 2 as a function of rotation angle. Incident wave vector is along the z-axis all the time. The polarization direction is (a) along the y-axis (the excitation wavelength is 570 nm, 574 nm, 575 nm, 577 nm, 583 nm, 593 nm, and 595 nm, respectively) and (b) along the x-axis (the excitation wavelength is 500 nm, 650 nm, 640 nm, 580 nm, 588 nm, 590 nm, and 590 nm, respectively). The color bars represent the amplitude of $\left|E/E_0\right|$, where $E$ is the local electric field near the TNSHD, and $E_{\text{0}}$ is the incident electric field.

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Acknowledgments

This work was supported by the National Science Foundation of China (Grant No. 11174190) and the Fundamental Research Funds for the Central Universities (Grant No. GK201101006).

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