1. Introduction

Metallic nanoparticles exhibit abundant fascinating optical phenomena owing to the localized surface plasmons they support [1]. Localized surface plasmons are a result of confined collective oscillations of electrons caused by the coupling of light and conduction electrons in the vicinity of the metal surface. For individual nanoparticles, the optical properties of plasmon mode strongly depend on the shape and size of particles, ambience permittivity and polarization state of light, which have enabled applications ranging from biosensing [2], photocatalysis [3], data storage [4] to solar energy harvesting [5]. Owing to the electron density spill-out, the electromagnetic field of an individual plasmon mode can extend some distance away from the nanoparticle. When assembled closely, these individual plasmon modes of nanoparticles strongly interact with each other to form new collective modes, resembling the atomic orbitals hybridization in molecules [6]. Therefore, assembled metallic nanoparticles are also referred to as plasmonic molecules [7,8]. Recently, many concepts in chemistry have been readily applied to plasmonic molecules such as stereoisomers [9], vibrational coupling [8] and group theory [10], providing new insights into light-surface plasmon interactions. More interestingly, very much analogous to phenomena in molecular systems including electromagnetically induced transparency [11,12] and Fano resonance [13–15] have been observed in plasmonic molecules, rendering many promising applications in slow lights [11], chemical/bio-sensing [16,17], and nanolasers [18].

Recently, among collective modes of plasmonic molecules, dark plasmon modes are receiving a growing interest [14,19,20]. Interestingly, owing to the vanishing dipole moment, dark plasmon modes have a significant low radiation loss but a considerably strong field enhancement. As a result, dark plasmon modes are widely used to enhance the inherently weak optical processes such as Raman scattering [21] and second harmonic generations [22]. Lately, the plasmonic radial breathing mode (RBM), as a new type of dark modes, has been extensively studied in single nanoparticles [19,23,24] and plasmonic molecules [25–27] either with electron energy loss spectroscopy (EELS) or optical means. It has been also reported that RBM can be attainable by illuminating the metal tip with a tilted linear polarized plane wave, providing a facile way for tip enhanced Raman spectroscopy applications [28]. The RBM shows a character of dipole moment with radial symmetry, in analogy with the radial expansion-contraction vibration mode in benzene rings [29] and carbon nanotubes [30]. In particular, this RBM behaves as its counterpart in molecular systems, having interesting characteristics such as wavenumber confinement [31] and group symmetry [10]. As has been revealed in molecule systems like double walled carbon nanotubes, the coupling between vibrational RBMs governs the physical properties of nanotubes, reflecting Van der Waals couplings of nanotube sheets and the interference between electronic transitions pathways of inner and outer nanotubes [32]. Similarly, it can be envisioned that the coupling of plasmonic RBM would be equivalently important for fully understanding the properties of plasmonic molecules. In this work, we use a tightly focused radially polarized beam as the input field to excite RBMs of dual-hexamer. The RBM couplings are symmetrically investigated for two constituted hexamers with the different gap sizes, in-plane twisting and out-of-plane tilting angles. Our results provide an understanding of the RBM coupling in plasmonic molecules and a tool for tailoring the optical responses of plasmonic molecules. Also, it could serve as a reference for researches of RBM coupling in molecular systems.

2. Results and discussions

The metallic hexamer is a plasmonic analogue of the benzene molecule, in which the nanoparticles sit on the vertexes of hexagon. The interaction of light with metallic hexamer leads to a strong plasmon hybridization effect, exhibiting rich plasmonic modes including radial breathing mode and azimuthal mode [27]. Moreover, the metallic hexamer can readily inter-couple with the monomer to form the superradiant and subradiant modes, further enabling novel physics phenomena such as Fano resonance and electromagnetically induced transparency [27,33]. Figure 1(a) depicts our proposed plasmonic molecule which consists of two coaxially packed hexamers. These dual-hexamers are made of gold nanodisks with identical diameter of d = 200 nm and thickness of 20 nm. The center to center distance between nearby nanodisks is fixed to a constant of L₁ = 400 nm for inner hexamer and that of outer hexamer is defined by L₂ varying from 620 nm to 800 nm. To excite the RBM of hexamers, a radially polarized beam is employed and coaxially focused onto hexamers through an objective with NA = 0.7 as schematically shown in Fig. 1(b). The pupil function of the objective has the form [34]:

 

Fig. 1 (a) Schematic view of a plasmonic dual-hexamers structure. (b) Excitations of plasmonic radial breathing mode (RBM) with a tightly focused radially polarized beam. (c) Intensity distribution of the radially polarized beam at the top surface of hexamers, where the white arrows indicate the polarization states of beam.

Download Full Size | PDF

In Fig. 2(a), we show the normalized scattering spectrum of the individual inner hexamer. A strong resonance peak is found at the wavelength of 690 nm. We derive the charge plot of the hexamer at the resonant wavelength [Fig. 2(b)] by calculating the difference of the normal component of the electric field above and below the Au surface according to Gauss's law. This charge pattern indicates a radial oscillation mode assigned to be a RBM. Due to the charge distribution, the electric field of hexamer is strongly confined along the radial direction of the hexamer with the maximum amplitude at the periphery as shown the electric field enhancement map in Fig. 2(c). Figure 2(d) illustrates the far field radiation pattern of the RBM. Owing to the charge symmetry, it shows a typical cone-like pattern with a nearly zero radiation at the vertical direction [36], which indicates that this mode cannot be excited by the normal incident plane wave and has an optically dark nature.

 

Fig. 2 (a) Scattering spectrum of inner hexamer. (b) Charge plot and (c) Electric field enhancement map of inner hexamer at wavelength of 690 nm. (d) Radiation pattern of the plasmonic RBM.

Download Full Size | PDF

We further obtain the scattering spectrum of a dual-hexamer with geometry parameters L₁ = 400 nm and L₂ = 620 nm (inter-hexamers gap size of 20 nm) as shown by the red dots in Fig. 3(a). Two obvious resonant peaks are found in the scattering spectrum of the dual-hexamer. To reveal the origin of these peaks, a coupled oscillator model is introduced to describe the interactions between the two RBMs of the inner and outer hexamers. We assume that the two interacting modes have harmonic terms of x₁₍₂₎ = C₁₍₂₎(ω)e^iωt, where C₁₍₂₎ are the oscillation amplitudes. The coupling between these two RBMs is governed by [37]:

 

Fig. 3 (a) Scattering spectrum of dual-hexamers with L₁ = 400 nm and L₂ = 620 nm, where red dots and the black solid line are simulation and fitting results, respectively. (b,c) Charge plots and (d,e) Electric field enhancement maps of coupled RBMs1 (b,d) and RBMs2 (c,e) in the dual-hexamers. (f) Charge distributions along the horizontal lines as labeled in (b) and (c). (g) Electric field enhancement along the horizontal lines as labeled in (d) and (e). The black and red solid lines in (f,g) are for coupled RBMs1 and RBMs2, respectively.

Download Full Size | PDF

Here, ω₁₍₂₎, γ₁₍₂₎, and η₁₍₂₎ are the resonance frequency, nonradiative damping and radiation coupling efficiency of inner and outer RBMs, respectively. The parameter g denotes the coupling constant between two modes, and E represents the electrical field of the excitation beam with the form of E = E₀e^iωt. After substituting x₁₍₂₎ to Eq. (3), expressions for C₁₍₂₎ can be analytically derived [38]. Then, α (ω) = I₀ + |C₁(ω) + C₂(ω)|² with I₀ to account for the background is used to fit the scattering spectrum of the plasmonic dual-hexamers as shown the black solid line in Fig. 3(a). It can be seen that a good agreement is achieved between the simulations and the coupled oscillator model. Resonant wavelengths of the two coupled RBMs are determined to be 650 nm and 810 nm from the theoretical model. We further examine the charge and near field distributions of the dual-hexamer at these two resonant wavelengths. As seen from Figs. 3(b) and 3(c), the charge plots reveal two similar in-phase bonded RBMs. The difference lies in the coupled RBMs with shorter resonant wavelength [coupled RBMs 1 shown in Fig. 3(b)] holds more charge in the inner hexamer, while, that with longer resonant wavelength [coupled RBMs 2 shown in Fig. 3(c)] contains more charge in the outer hexamer. This is further examined in Fig. 3(f), where the black and red solid lines correspond to the charge distributions along the dash lines indicated in Figs. 3(b) and 3(c). The differences between coupled RBMs1 and RBMs2 are also found in the electric field enhancement maps shown in Figs. 3(d) and 3(e), where the near field map of coupled RBMs 1 shows most of electric field is trapped in-between the gap of inner and outer hexamers and that of coupled RBMs 2 indicates considerable of electric field is confined both in the gap and periphery of outer hexamer, which can be closely examined in Fig. 3(g).

Figure 4(a) depicts the scattering spectra of the dual-hexamer as a function of the gap between inner and outer hexamers. It is suggested that with increasing of gap size, the resonant wavelength of coupled RBMs 1 (λ₁) has a slight red shift and that of coupled RBMs 2 (λ₂) first shows a blue shift then is gradually merged to the peak of coupled RBMs 1. To provide deeper insight to the entire process, we have fitted the scattering spectra of the dual-hexamer in Fig. 4(a) by using the coupled oscillator model. In Fig. 4(b), we show the resonant wavelengths of RBMs against the gap size. Here, the lower and upper triangles are λ₁ and λ₂, respectively. The lower and upper black dashed lines correspond to the intrinsic wavelengths of the individual inner (λ₁₀) and outer RBMs (λ₂₀). Also, the coupling constant (g) between inner and outer RBMs are derived from the theory model displayed as the red dots in Fig. 4(c). It is apparent that with increasing gap size, g decreases dramatically. As a result, λ₁ and λ₂ have red shift and blue shift, respectively. Eventually, these two modes approach to intrinsic wavelengths of individual inner and outer RBMs, showing uncoupled states. It is also worth noting that due to the strong Ohmic losses of dark modes, the coupling constant has an obvious deviation from the exponential law [black solid in Fig. 4(c)] within the small gap range, indicating a faster decay rate of RBM than that of bright mode [39].

 

Fig. 4 (a) Scattering spectra of dual-hexamer as a dependence of the gap between inner and outer hexamers. (b) Resonant wavelengths of RBMs as a function of the gap size, where red triangles represent the resonant wavelengths of coupled RBMs 1 (λ₁) and coupled RBMs 2 (λ₂), and black dashed lines are the intrinsic resonant wavelengths of individual inner (λ₁₀) and outer hexamers (λ₂₀). (c) Coupling constant between inner and outer RBMs versus the gap size, where the red dots and black solid line are theoretical results and the guide of the exponential decay, respectively.

Download Full Size | PDF

Subsequently, we examine the RBM coupling of dual-hexamer when subjected to in-plane twist as illustrated in Fig. 5(a), where the dual-hexamer is defined by the geometry parameters: L₁ = 400 nm, L₂ = 620 nm, and the twist angle of θ. Figure 5(b) shows the scattering spectra of dual hexamer as θ varying from 0° to 60°. It is found an obvious peak blue shift and then recover with twist angle increasing. The twist angle dependent λ₁, λ₂, and g are then obtained from the coupled oscillator model as shown the red dots in Figs. 5(c)-5(e), respectively. It is clear that variations of resonant wavelengths and coupling constant can be well reproduced by cosine curves with a period of 60° [black solid lines in Figs. 5(c)-5(e)], reflecting the C6 point group of the hexamer. The peak change in Fig. 5(b) can be interpreted as: the misalignment between radial dipole moments of inner and outer RBMs occurs when twisting the outer hexamer. This results in the decrease of coupling constant and leads to the convergence of λ₁ and λ₂. Such wavelength convergence finally reaches its minimum for twist angle of 30°. Then, the coupling constant increases with further increasing the twist angle. As a result, λ₁ and λ₂ become divergent and come back to their initial positions when the twist angle is set to 60°. Notably from Figs. 5(c) and 5(d), one can also detect the twist angle change of a dual-hexamer from its scattering spectrum, offering an averaged sensitivity of 1.1°/nm.

 

Fig. 5 (a) Schematics of in-plane twist of the dual hexamer with geometry parameters L₁ = 400 nm and L₂ = 620 nm. (b) Scattering spectra of dual-hexamer as a dependence of twist angle. Resonant wavelengths of (c) coupled RBMs 1, (d) coupled RBMs 2 and (e) coupling constant as a function of the twist angle, where the red dots are theoretical results and the black solid lines are guides to show cosine curves.

Download Full Size | PDF

Finally, we study the RBM coupling of dual-hexamers under the out-of-plane tilting as shown in Fig. 6(a), where the tilt angle of the outer hexamer is denoted as α. Figure 6(b) displays the evolution of scattering spectra of a dual-hexamer (L₁ = 400 nm, L₂ = 620 nm) with increasing of α from 0° to 15°. We further use the theoretical model to derive the resonant wavelengths of two coupled RBMs. As seen from Fig. 6(c), with increasing of α, the resonant wavelength of coupled RBMs 2 (λ₂) has an obvious blue shift, whilst that of coupled RBMs 1 (λ₁) gets gradual red shift to the intrinsic RBM wavelength of inner hexamer [shown as the lower dashed line in Fig. 6(c)]. These wavelength shifts can be attributed to the reduction of the coupling constant [see in Fig. 6(d)], which generally has a cosine-like variation with the tilt angle [solid line in Fig. 6(d)] due to the fact that the RBMs coupling is dominated by the in-plane interaction between the RBM of the inner hexamer and the horizontal projection of the RBM of the outer hexamer. Additionally, it is interesting to note that due to the phase retardation effect [40], the RBM wavelength of the tilted individual outer hexamer (λ₂₀) has a slightly red shift for large α, leading to a cross point with λ₂ at the tilt angle of 10° [see in Fig. 6(c)]. It is also worth noting that under the excitation of azimuthally polarized beam, the magnetic dipoles could be induced in the dual-hexamers structure [27]. Then, the magnetic dipole-dipole coupling would dominate optical properties of the system, which could add more degrees of freedom to tune the resonant behavior of hexamer structures for new plasmonic functionalities [41].

 

Fig. 6 (a) Schematics of out-of-plane tilt of the dual hexamer with geometry parameters L₁ = 400 nm and L₂ = 620 nm. (b) Scattering spectra of dual-hexamer as a dependence of tilt angle. (c) Resonant wavelengths of RBMs as a function of the tilt angle, where red triangles represent the resonance wavelengths of coupled RBMs 1 (λ₁) and coupled RBMs 2 (λ₂), and black dashed lines are the intrinsic resonance wavelengths of individual inner (λ₁₀) and tilted outer hexamers (λ₂₀). (d) Coupling constant as a function of tilt angle, where the red dots are theoretical results and the black solid line is a guide to show the cosine trend.

Download Full Size | PDF

3. Conclusions

To summarize, we have theoretically investigated the radial breathing mode (RBM) coupling in a plasmonic dual-hexamer. Due to the zero net dipole moment, RBM is hard to be excited by a linear polarized illumination, however, can be easily realized by a tightly focused radially polarized beam. The coupling of RBM is systematically studied for two constituted hexamers that have different gap sizes, in-plane twisting and out-of-plane tilting angles. A simple coupled oscillator model was introduced to corroborate the numerical results, providing valuable insights into the RBM coupling of the system. Specifically, with increasing inter-hexamers separation, the coupling strength of RBM deceases faster than its counterpart of bright mode, showing a nonexponential decay law. Additionally, under the in-plane twisting, the symmetry of the hexamer structure governs the RBM coupling, resulting in cosine-like variations of resonant wavelengths and coupling constant with the twist angle. Furthermore, when subjected to the out-of-plane tilting, the RBM coupling is dominated by the in-plane interaction, leading to the convergence of resonant wavelengths of two coupled RBMs with increasing of the tilt angle. Our results facilitate the understanding of dark mode couplings in plasmonic molecules and we envision it could pave the way for adding beneficial functionalities to nanophotonic structures.