1. Introduction

Within the validity of Fermi’s Golden Rule, the decay rate of a quantum emitter is proportional to the photonic local density of states (LDOS) at the emitter’s location. When a quantum emitter is coupled to a plasmonic nanoantenna [1–3], the plasmonic modes of the antenna can offer additional LDOS that is much larger than the vacuum LDOS. This leads to two important consequences: the emitter decays predominantly to plasmons and the decay rate is strongly enhanced via the so-called Purcell effect [4–6]. With the emitter predominantly decaying to plasmons, one can further tailor the emission by engineering the plasmonic modes. For instance, the far-field radiation can be directed [7–9], the polarization can be tailored [6], and the radiation can be made multipolar [10]. With the decay rate strongly enhanced, one can cycle the emitter faster to make an ultrafast single photon source [11] or to enhance the gain for nanolasers [12–15]. When the emitter-plasmon coupling is in the strong-coupling regime, energy coherently oscillates between the emitter and the plasmon, which leads to Rabi energy splitting [16–25] and holds great potential for single-photon nonlinear optics [26].

When the emitter is close to the surface of the antenna, besides coupling to the dipolar plasmonic mode, the emitter also considerably couples to high-order plasmonic modes. In principle, the coupling rate to any mode can be rigorously computed using quasi-normal mode expansion [27,28]. When the emitter is close to the surface of the antenna, hundreds of modes are needed to describe the coupling. For high-order modes, one usually only cares about the superposition of them, which is referred to as a “pseudomode” [29]. The pseudomode channel competes with the dipolar mode channel and is usually unwanted in emitter-plasmon coupling. For example, in the weak coupling regime, the pseudomode channel causes fluorescence quenching [28,30,31]; and in the strong coupling regime, the competition from the pseudomode channel makes it harder to observe strong coupling between emitters and the dipolar mode [29,32,33]. In order to avoid the adverse effects of the pseudomode channel when an emitter strongly couples with a plasmonic nanoantenna, one should make the branching ratio of the pseudomode channel as low as possible, or in other words, make the branching ratio of the dipolar mode channel as high as possible.

It was found that antennas with elongated geometry with sharp ends are favorable for achieving a high coupling rate to the dipolar mode with weak fluorescence quenching [34–36]. More complicated structures were also proposed. Dimer antennas with the emitter placed in the gap were found to be favorable for coupling to the dipolar mode [34,37]. It was also found that embedding the emitter below the surface of a high index dielectric film can make the coupling to the dipolar mode more efficient [38]. For controlling the competition between the dipolar mode channel and the pseudomode channel, it is of great importance to understand the differences between them in their mechanisms, characteristics, and dependences on system parameters. However, comprehensive understandings of the differences are scarcely presented. The dependence of the coupling channels on emitter-antenna distance was investigated for spherical antennas [1,31,39–41]. For elongated antennas, the dependence on antenna shape was investigated [34–36,42], but the dependences on emitter-antenna distance and antenna end size have not been systematically studied.

In this work, we comprehensively investigate the differences between the dipolar mode channel and the pseudomode channel in a representative system where a dipole emitter couples to a silver nanorod. The two channels are shown to be distinct in their mechanisms, characteristics (including chromatic dispersion and field distribution), and dependences on system parameters (including emitter-antenna distance, antenna geometry, and material loss). The study reveals important design rules for controlling the competition between the two channels.

2. The emitter-antenna system

The emitter-plasmon coupling system contains a dipole emitter placed close to the end of a rod-shaped plasmonic nanoantenna as schematically shown in Fig. 1(a). The nanoantenna is a plasmonic nanorod with a square cross section $a \times a$ and a length of l. The relative permittivity $\varepsilon $ of the plasmonic material is given by a Drude model $\varepsilon (\omega )= {\varepsilon _{\inf }} - {{\omega _\textrm{p}^2} \mathord{\left/ {\vphantom {{\omega_\textrm{p}^2} {({{\omega^2} + i\omega {\gamma_\textrm{p}}} )}}} \right.} {({{\omega^2} + i\omega {\gamma_\textrm{p}}} )}}$, where ${\varepsilon _{\inf }}\textrm{ = }4$, ${\omega _\textrm{p}} = 1.4 \times {10^{16}}{\textrm{s}^{{ - }1}}$ and ${\gamma _\textrm{p}} = 1.6 \times {10^{14}}{\textrm{s}^{{ - }1}}$, which can approximately describe the permittivity of silver in the wavelength range 400-2000 nm, with the loss level about 4 times that in [43] considering nanofabrication-induced losses [44]. The quantum emitter here is assumed to be a linearly polarized dipole emitter with a unity quantum efficiency, oriented in the x-direction and located at a distance d from the end of the nanorod along the longitudinal axis of the nanorod.

 

Fig. 1. The emitter-antenna system. (a) The structure of the emitter-plasmon coupling system. (b) The electric displacement field distribution (in the xy plane of z = 0) at the resonance wavelength of the dipolar mode (i.e., at 808 nm) for the system with a = 40 nm, l = 190 nm and d = 6 nm.

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The total decay rate of the emitter can be expressed as ${\gamma _{\textrm{tot}}} = {\gamma _0} + \sum\limits_{i = 1}^N {{\gamma _i}} + \sum\limits_{j = N + 1}^\infty {{\gamma _j}}$, where ${\gamma _0}$ denotes the coupling rate to free-space, ${\gamma _i}$ denotes the modal coupling rates to the several low order plasmonic modes that are in the spectral range of interest, and ${\gamma _j}$ denotes the modal coupling rates to the high order plasmonic modes that are far-detuned from the spectral range of interest. The superposition of the high order modes can be defined as a pseudomode, whose decay rate is ${\gamma _{\textrm{pm}}} = \sum\limits_{j = N + 1}^\infty {{\gamma _j}}$. In this work, only the dipolar mode (the 1^st order mode) has its resonant frequency in the spectral range of interest. Figure 1(b) shows the electric displacement field distribution at the resonance wavelength of the dipolar mode (i.e., at 808 nm) for the emitter-antenna system with a = 40 nm, l = 190 nm and d = 6 nm. The field globally distributed in the antenna is attributed to the dipolar mode, while the field spatially localized at the coupling end of the nanorod is attributed to the pseudomode.

3. Theoretical analysis of the coupling channels

The normalized total coupling rate ${{{\gamma _{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{tot}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ can be classically determined using the relation ${{{\gamma _{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{tot}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}} = {{{P_{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{P_{\textrm{tot}}}} {{P_0}}}} \right.} {{P_0}}}$, where the emitter is modelled as a classical dipole ${\boldsymbol{\mu}}$ located at ${{\textbf r}_0}$ and ${P_{\textrm{tot}}}$ $({P_0})$ is the power coming out from the classical dipole in presence (absence) of the nanoantenna [45]. ${P_{\textrm{tot}}}$ can be written in terms of the Green’s function as

Although the coupling rate to any mode can be rigorously computed using quasi-normal mode expansion [27,28], the result is still a numerical result, and therefore the underlying mechanism or the key parameters in play cannot be highlighted. In this section we use an “optical nanocircuit” model to describe the dipolar mode channel and use quasi-electrostatic approximation to describe the pseudomode channel. The models can not only reveal the underlying mechanisms of the coupling channels, but can also infer the characteristics (including the chromatic dispersions and field distributions) of the coupling channels and the dependences of the coupling rates on the system parameters (including the emitter-antenna distance, antenna geometry, and material loss).

3.1 Theoretical analysis of the dipolar mode channel

Here we use the concept of “optical nanocircuit” [46–48] to qualitatively investigate the property of the imaginary part of the Green’s function associated with the dipolar mode. In the nanocircuit model, the displacement current in the nanoantenna is modelled as the nanocircuit current I, the external driving for the antenna is modelled as the nanocircuit voltage source U, and the plasmonic nanorod antenna is modelled as an effective impedance composed of lumped nanocircuit elements

 

Fig. 2. Influences of the geometric parameters on the interaction factor. (a) The interaction factor ${{{\mathop{\rm Re}\nolimits} \{{{G_{{\mathop{\rm int}} }}} \}} \mathord{\left/ {\vphantom {{{\mathop{\rm Re}\nolimits} \{{{G_{{\mathop{\rm int}} }}} \}} S}} \right.} S}$ as a function of a and d. (b) The factor $a{[{{{{\mathop{\rm Re}\nolimits} \{{{G_{{\mathop{\rm int}} }}} \}} \mathord{\left/ {\vphantom {{{\mathop{\rm Re}\nolimits} \{{{G_{{\mathop{\rm int}} }}} \}} S}} \right.} S}} ]^2}$ as a function of a and d. The black dots highlight the maximal values of the factor for different d.

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The factor $\textrm{Re}\{{{1 \mathord{\left/ {\vphantom {1 Z}} \right.} Z}} \}$ in Eq. (12) accounts for the resonant nature of the dipolar mode channel. At the characteristic frequency ${{{\omega _0} = 1} \mathord{\left/ {\vphantom {{{\omega_0} = 1} {\sqrt {LC} }}} \right. } {\sqrt {LC} }}$, this factor equals ${1 \mathord{\left/ {\vphantom {1 {({R + \omega_0^2{R^{\textrm{rad}}}} )}}} \right.} {({R + \omega_0^2{R^{\textrm{rad}}}} )}}$. So decreasing the material loss can increase the coupling rate of the dipolar mode channel by decreasing the effective dissipative resistance R. As we will see later, decreasing the material loss decreases the coupling rate of the pseudomode channel. Therefore, decreasing the material loss should favor the dipolar mode channel in channel competition.

The interaction factor ${{{\mathop{\rm Re}\nolimits} \{{{G_{{\mathop{\rm int}} }}} \}} \mathord{\left/ {\vphantom {{{\mathop{\rm Re}\nolimits} \{{{G_{{\mathop{\rm int}} }}} \}} S}} \right.} S}$ increases with decreasing a. At the same time, the factor $\textrm{Re}\{{{1 \mathord{\left/ {\vphantom {1 Z}} \right.} Z}} \}$ also depends on a. Here we qualitatively look into the dependence of the normalized coupling rate [Eq. (12)] on a. In the quasi-electrostatic limit, the resonance frequency is determined by the aspect ratio ${l \mathord{\left/ {\vphantom {l a}} \right.} a}$ of the plasmonic nanorod. So the effective dissipative resistance R, which is proportional to ${l \mathord{\left/ {\vphantom {l {{a^2}}}} \right.} {{a^2}}}$, should be proportional to ${1 \mathord{\left/ {\vphantom {1 a}} \right.} a}$. Therefore, at resonance the normalized coupling rate should be proportional to $a{[{{{{\mathop{\rm Re}\nolimits} \{{{G_{{\mathop{\rm int}} }}} \}} \mathord{\left/ {\vphantom {{{\mathop{\rm Re}\nolimits} \{{{G_{{\mathop{\rm int}} }}} \}} S}} \right.} S}} ]^2}$. Figure 2(b) plots the dependence of this factor on a for different d. For any finite d, there is a finite optimal a that maximizes this factor. This feature will be confirmed by rigorous full-wave numerical simulation in Section 4.2.

3.2 Theoretical analysis of the pseudomode channel

As long as d is much smaller than the wavelength and the end size a, the quasi-electrostatic approximation can be applied for the analysis of the pseudomode channel and the surface of the end of the nanoantenna seen by the emitter can be viewed as an infinite plane interface [31,41,45]. With the quasi-electrostatic approximation, the electric field inside the antenna is simply the attenuated electric field of the emitter dipole ${\boldsymbol{\mu}}$ (in the x direction)

Equation (17) indicates that the pseudomode channel relies on the imaginary part of the permittivity. Decreasing the material loss can decrease the coupling rate of the pseudomode channel. Recalling that decreasing the material loss can increase the coupling rate of the dipolar mode channel, we see that decreasing the material loss favors the dipolar mode channel in channel competition. Equation (17) also indicates that ${\gamma _{\textrm{pm}}}$ scales as ${d^{ - 3}}$. So ${\gamma _{\textrm{pm}}}$ rises (drops) very quickly as d decreases (increases).

3.3 Understanding the difference in the mechanisms

The differences between the mechanisms of the two coupling channels can be understood from the origin of the phase shift during the near field scattering of the dipole field. From Eq. (4) one can see that for a coupling channel to have a large coupling rate, the field at the dipole emitter must have a large imaginary part. The imaginary part can have two contributions. One is the imaginary part of the primary dipole field and the scattering (by the environment) of it. The imaginary part of the primary dipole field is very weak, so the imaginary part of the primary dipole field itself and the scattering of it can only provide very limited coupling rate. Another contribution is the scattering of the real part of the primary dipole field. Since the real part of the primary dipole field can have extremely large amplitude in the near field region, this contribution can provide large coupling rates as long as the near field scattering is strong and introduces significant phase shift. The phase shift is crucial. Without phase shift, no matter how strong the scattering is, the scattering of the real part of the primary dipole field would not contribute to the coupling.

For small emitter-antenna distance, both the dipolar mode channel and the pseudomode channel predominantly rely on the near field scattering of the real part of the primary dipole field. However, they have distinct origins of the required phase shifts during the scattering. For the dipolar mode channel, the phase shift comes from the resonance of the antenna [cf. Equation (10)], while for the pseudomode channel, the phase shift comes from the imaginary part of the permittivity of the metal [cf. Equation (15)].

4. Numerical study of the coupling channels

In this section we perform full-wave numerical simulations using a commercial finite-difference time-domain solver (FDTD solutions, Lumerical) to rigorously investigate how the emitter-antenna distance, the antenna geometry, and the material loss influence the two coupling channels.

The normalized total coupling rate ${{{\gamma _{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{tot}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ is obtained by using the relation ${{{\gamma _{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{tot}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}} = {{{P_{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{P_{\textrm{tot}}}} {{P_0}}}} \right.} {{P_0}}}$, where ${P_{\textrm{tot}}}$ (${P_0}$) is the calculated power emitted from the dipole source in the presence (absence) of the nanoantenna. One simulation is need for each position of emitter (i.e., for each emitter-antenna distance d).

The normalized coupling rate of the dipolar mode channel ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ at the resonance wavelength is obtained by using the concept of generalized mode volume [50], which allows one to obtain ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ as a function of the emitter’s spatial position within a small number of simulations (see Appendix B for the method).

The normalized coupling rate of the pseudomode channel ${{{\gamma _{\textrm{pm}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{pm}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ is calculated using Eq. (17) for small d. For large d, the coupling rate of the pseudomode is much smaller than that of the dipolar mode channel or the free-space channel, and therefore can be neglected.

4.1 Influence of the emitter-antenna distance

Figure 3(a) shows the normalized coupling rates ${{{\gamma _{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{tot}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ (solid data points), ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ (solid curves) and ${{{\gamma _{\textrm{pm}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{pm}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ (dashed curve) at the resonance wavelength as a function of d for different nanoantennas with different end sizes but the same resonance wavelength. As d decreases, the coupling rate increases for both the dipolar mode channel and the pseudomode channel. However the two channels have different rising rates. For ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$, the rising slope gradually decreases as d decreases and eventually the trend flattens, and the larger the end size a, the earlier the trend flattens. This indicates that placing the emitter closer does not help too much when the distance is already much smaller than the end size of the nanoantenna. On the contrary, ${{{\gamma _{\textrm{pm}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{pm}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ scales as ${d^{ - 3}}$ and therefore the rising slope of ${{{\gamma _{\textrm{pm}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{pm}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ becomes larger and larger as d decreases. Therefore the ratio between the two channels changes with d. For large d, the dipolar mode channel is dominant, and therefore ${{{\gamma _{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{tot}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ well coincides with ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ (note that for very large d that far exceeds the plot range, ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ can decrease to well below one so that the free-space channel dominates). Whereas for small d, the pseudomode channel starts to show its effect so that the total coupling rate ${{{\gamma _{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{tot}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ deviates from ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ more and more as d decreases. For small enough d, ${{{\gamma _{\textrm{pm}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{pm}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ may become larger than ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ and even dominate ${{{\gamma _{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{tot}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$. This characteristic is more prominent for nanoantennas with larger end size a [see Fig. 3(a), as will be discussed in Section 4.2] or with larger material loss [see Fig. 5(a), as will be discussed in Section 4.3], where ${{{\gamma _{\textrm{pm}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{pm}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ starts to exceed ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ from a relatively larger value of d. This indicates that one should never place the emitter too close to the surface of the nanoantenna, but how close is too close significantly depends on the end size and the material loss of the nanoantenna.

 

Fig. 3. Influence of the emitter-antenna distance on the coupling rates. (a) The normalized coupling rates at the resonance wavelength as a function of d for three different nanoantennas with three different end sizes but the same resonance wavelength (black: a = 40 nm, l = 190 nm; blue: a = 20 nm, l = 130 nm; red: a = 10 nm, l = 72 nm). (b) The dispersion curves of the total coupling rates for different d (the end size a = 40 nm).

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We note that for small d the rising slope of ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ that is here rigorously obtained using full-wave numerical simulation is significantly smaller than predicted by the theoretical analysis in Section 3.1 where a simple nanocircuit model is applied. This is because the dipolar mode actually has a sinusoidal current density distribution [51] as shown in Fig. 1(b), with larger values in the middle and smaller values at the ends, whereas in the nanocircuit model a uniform current density distribution is assumed since as a lumped parameter the current I cannot give the information of the field distributions within the nanoantenna.

Figure 3(b) plots the dispersion curves of the total coupling rates for different d. The more and more prominent baseline in the dispersion curve indicates that the branching ratio of the pseudomode channel becomes larger and larger as d decreases.

4.2 Influence of the antenna geometry

The structure parameters of the nanorod antenna have little influence on the pseudomode channel since for the pseudomode the nanoantenna seen by the emitter can be viewed as an infinite plane. Whereas, they strongly influence the dipolar mode channel by affecting the resonance wavelength and linewidth of the dipolar mode, and through the “lightning rod effect” as well. Here we focus on the “lightning rod effect”, that is, the influence of the end size a on the coupling rate of the dipolar mode channel.

From Fig. 3(a) we see that by reducing the end size a, ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ can be remarkably enhanced as long as d is significantly smaller than a (as we will point out later, if d is significantly larger than a, reducing a may decrease ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ on the contrary). This is also clearly shown by the field distributions plotted in Figs. 4(a)–4(c) for d = 2 nm. For the antenna with a = 40 nm, the globally distributed field of the dipolar mode is much weaker than the spatially localized field of the pseudomode, as shown in Fig. 4(a). As a reduces, the field of the dipolar mode becomes stronger while the field of the pseudomode keeps almost unchanged. As a reduces to 10 nm, the field of the dipolar mode is already at the same level as the field of the pseudomode, as shown in Fig. 4(c). We further plot the dispersion curves of the total coupling rates in Fig. 4(d) for different end sizes. As a reduces, the resonance amplitude increases. Noting that the black curve in Fig. 4(d) is just the red curve in Fig. 3(b), we see that as a reduces the baseline becomes insignificant (due to the fact that the amplitude of the baseline keeps almost unchanged while the dipolar mode becomes stronger), which indicates that the dipolar mode channel dominates.

 

Fig. 4. Influence of the end size on the coupling rates. (a-c) The field distributions at resonance for the three different nanoantennas with different end sizes but the same resonance wavelength. The emitter-antenna distance d = 2 nm. They share the size scale and color scale. (d) The dispersion curves of the total coupling rates for different end sizes. The emitter-antenna distance d = 4 nm.

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As have been pointed out in Section 4.1, the rising slope of ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ gradually decreases as d decreases and eventually the trend becomes rather flat. The larger the end size a, the earlier the trend flattens. In other words, reducing the end size can extend the rising trend till smaller d.

Due to the benefits from reducing the end size, we consider controlling the end size as the most important strategy for controlling channel competition. However, one should keep in mind that in the discussion above, a has been kept larger than d. For any finite d, there should be a finite optimal a that maximizes ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$, as we have already pointed out in the theoretical analysis referring to Fig. 2(b) (Section 3.1). For instance in Fig. 3(a) the red solid curve and the blue solid curve cross at around 24 nm. This means that when d is larger than 24 nm, ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ decreases (rather than increases) as a is reduced from 20 nm to 10 nm. In practice, if emitters of very small physical size are used, d can always be made significantly smaller than practical end size, then one should reduce the end size as much as possible. If emitters of large physical size are used, then one should carefully find the optimal end size.

4.3 Influence of the material loss

In Fig. 5(a) we compare the coupling rates (as a function of d) for nanoantennas with the same geometry but with different material losses (i.e., different damping rates ${\gamma _\textrm{p}}$). The coupling rates plotted with black color and blue color are for nanoantennas with geometry parameters a = 40 nm, l = 190 nm. The damping rate of the metal for one antenna is ${\gamma _\textrm{p}} = 1.6 \times {10^{14}}{\textrm{s}^{{ - }1}}$ (black color), which is the default damping rate used in this work. The damping rate of the metal for the other antenna is reduced to 1/4 of the default value, i.e., ${\gamma _\textrm{p}} = 0.4 \times {10^{14}}{\textrm{s}^{{ - }1}}$ (blue color). We can see that by reducing the material loss, the solid curve $({{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}})$ lifts by a constant factor, while the dashed curve ${{({\gamma _{\textrm{pm}}}} \mathord{\left/ {\vphantom {{({\gamma_{\textrm{pm}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}})$ drops by a constant factor, just as predicted by the theoretical analysis in Section 3. This indicates that with smaller material loss, ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ can dominate ${{{\gamma _{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{tot}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ until a smaller d. For instance at d = 4 nm, with small material loss, ${{{\gamma _{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{tot}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ (blue data point) is still very close to ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ (blue solid curve), whereas with large material loss, ${{{\gamma _{\textrm{tot}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{tot}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ (black data point) is already very far from ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ (black solid curve). The effect of reducing the material loss is also apparent in the dispersion curve of the total coupling rate as shown in Fig. 5(b). The baseline in the dispersion curve of the total coupling rate decreases significantly, which indicates that the coupling rate of the pseudomode channel decreases. With significantly weaker baseline, the peak coupling rate at the resonance wavelength increases slightly, which indicates that the coupling rate of the dipolar mode channel increases significantly. Moreover, the linewidth of the resonance becomes narrower, indicating reduced damping of the dipolar mode.

 

Fig. 5. Influence of the material loss on the coupling rates. (a) The normalized coupling rates (at the resonance wavelength) as a function of d for nanoantennas with different material losses (i.e., different damping rates ${\gamma _\textrm{p}}$). The coupling rates plotted with black color and blue color are for nanoantennas with geometry parameters a = 40 nm, l = 190 nm (black: ${\gamma _\textrm{p}} = 1.6 \times {10^{14}}{\textrm{s}^{{ - }1}}$; blue: ${\gamma _\textrm{p}} = 0.4 \times {10^{14}}{\textrm{s}^{{ - }1}}$). The coupling rates plotted with red color and green color are for nanoantennas with geometry parameters a = 10 nm, l = 72 nm (red: ${\gamma _\textrm{p}} = 1.6 \times {10^{14}}{\textrm{s}^{{ - }1}}$; green: ${\gamma _\textrm{p}} = 0.4 \times {10^{14}}{\textrm{s}^{{ - }1}}$). (b) The dispersion curves of the total coupling rates for nanoantennas with different material losses but with the same geometric parameters a = 40 nm, l = 190 nm (the emitter-antenna distance d = 4 nm).

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With the material loss reduced to 1/4 of the default value, ${{{\gamma _{\textrm{pm}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{pm}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ drops to about 1/4 (actually, as long as the imaginary part of the permittivity is significantly smaller than its real part, ${{{\gamma _{\textrm{pm}}}} \mathord{\left/ {\vphantom {{{\gamma_{\textrm{pm}}}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ scales nearly linearly with the damping rate of the material). However ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ increases to only 1.2 times (rather than 4 times). This is because the damping of the dipolar mode include both dissipative damping (where energy is converted to heat) and radiation damping (where energy is converted to free-space photon radiations) [cf. Equation (5)], and here for the nanoantenna of relatively large size (a = 40 nm, l = 190 nm) the radiation damping is significantly larger than the dissipative damping (cf. Table 1 in Appendix A). If a nanoantenna of a smaller size is used, the radiation damping can be smaller and ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ can be increased more significantly by reducing material loss. For instance, in Fig. 5(a) we also plot (with red and green colors) for nanoantennas with geometry parameters a = 10 nm, l = 72 nm. We see that by reducing the material loss to 1/4, ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ increases to about 3 times (comparing between the green curve and the red curve).

Table 1. Fitting results for the lumped nanocircuit elements.

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5. Discussions and conclusions

In this work, we have studied the differences between the dipolar mode channel and the pseudomode channel in a representative system where a dipole emitter couples to a silver nanorod. The mechanisms of the coupling channels were studied in the framework of Green’s function. The Green’s function for the dipolar mode channel was analyzed using a resonant nanocircuit model, while the Green’s function for the pseudomode channel was analyzed using the quasi-electrostatic image dipole approximation. Then we came up with an understanding of the mechanism difference from the distinct origin of the phase shift during the near field scattering of the dipole field: for the dipolar mode channel, the phase shift comes from the resonance of the antenna; while for the pseudomode channel, the phase shift comes from the imaginary part of the permittivity of the metal.

The difference in the mechanism is the fundamental difference between the two coupling channels, which leads to the differences in their characteristics (including the chromatic dispersions and field distributions) and their dependences on system parameters (including the emitter-antenna distance, antenna geometry, and material loss). Here we briefly summarize these differences. (1) The two coupling channels have very different chromatic dispersion. The dipolar mode channel is resonant, with strong chromatic dispersion around its resonance wavelength. Whereas the pseudomode channel is nonresonant in the wavelength range of interest. (2) The two coupling channels have very different field distributions. The dipolar mode is globally distributed in the antenna. Whereas the pseudomode is spatially localized at the coupling end of the nanoantenna. In scenarios where multiple separated emitters couple with the nanoantenna, the dipolar mode channels can mediate the interaction among the emitters, while the pseudomodes couple separately with the emitters [29]. (3) The two coupling channels have very different dependences on emitter-antenna distance. For the dipolar mode channel, the rising slope of the coupling rate gradually decreases as d decreases and eventually the trend flattens. Whereas, for the pseudomode channel, the coupling rate scales as ${d^{ - 3}}$ and therefore the rising slope of the coupling rate becomes larger and larger as d decreases. (4) The two coupling channels have very different dependences on the geometry of the nanorod antenna. The geometry of the nanorod antenna strongly influence the dipolar mode channel by affecting the resonance wavelength and linewidth, and through the “lightning rod effect” as well. Whereas, they have little influence on the pseudomode channel since for small emitter-antenna distance the end of the nanoantenna seen by the emitter can be viewed as an infinite plane interface. (5) The two coupling channels have very different dependences on materials loss. Lower material loss provide stronger resonance and therefore leads to larger coupling rates for the dipolar mode channel. Whereas for the pseudomode channel, lower material loss means smaller imaginary part of the permittivity and therefore smaller coupling rates.

With these understandings on the differences between the dipolar mode channel and the pseudomode channel, we propose some important design rules for controlling the channel competition. (1) To optimize the emitter-antenna distance, one may reduce the emitter-antenna distance until the rising trend (with decreasing emitter-antenna distance) of the dipolar mode flattens or the branching ratio of the pseudomode channel becomes non-negligible. The optimal emitter-antenna distance shall depend on both the end size and the material loss of the nanoantenna. (2) The end size of the antenna should be optimized according to the emitter-antenna distance. For any finite emitter-antenna distance, there is a finite optimal end size that maximizes the coupling rate of the dipolar mode channel. In practice, if the emitter-antenna distance can be made significantly smaller than practical end size, one should reduce the end size as much as possible and then optimize the emitter-antenna distance according to the smallest end size available. If the emitter-antenna distance is limited by the physical size of large emitters, then one should carefully find the optimal end size. (3) Material loss should be reduced as much as possible, since reducing material loss enhances the dipolar mode channel while suppresses the pseudomode channel.

Note that the dipole orientation considered in this work is the most favorable orientation to achieve a strong coupling rate and a high branching ratio to the dipolar mode channel. If the dipole emitter is transverse (oriented in the yz plane), it would no longer couple to the longitudinal dipolar mode but only to the transverse dipolar mode, with a much smaller coupling rate. And then the emitter would be easily quenched by the pseudomode, noting that the coupling rate to the pseudomode reduces only by half [31]. For a dipole emitter in any orientation, one may treat it as a superposition of longitudinal and transverse dipoles. For a practical emitter, such as a colloidal quantum dot, the electronic transition system may be more complicated than a two-level system. In this case, one may treat the emitter as a combination of several dipoles that may have different dipole moments and different orientations [6], rather than as a single dipole. No matter how complicated the electronic transition system is and what the orientation is, as long as the emitter has a significant longitudinal dipole component, the understandings and design rules obtained in this work are relevant.

Although the study in this work is based on the representative rod-type antenna, it can be a reference for more complicated structures. Note that for dimer structures or particle-on-mirror structures with emitters placed in the nanogap, besides considering the dipolar mode channel and the pseudomode channel, one should also consider the coupling to the gap-plasmon mode, whose coupling rate can be comparable to that of the pseudomode channel for very small gap sizes [52,53]. The findings in this work shall give physical insight into channel competition and guide the design of emitter-plasmon hybrid nanosystems.

Appendices

A. Determine the lumped nanocircuit elements

The values of the lumped nanocircuit elements in Eq. (5) can be determined by fitting the spectra of the absorption and scattering cross sections. Consider a plane wave with electric field ${E_0}$ polarized along the nanorod as the excitation. The driving voltage for the nanocircuit is $U = {E_0}l$, where l is the length of the nanorod antenna. The current is $I = {U \mathord{\left/ {\vphantom {U Z}} \right.} Z}$. Then we can get the absorbed power ${P^{\textrm{abs}}} = {\textstyle{1 \over 2}}{\mathop{\rm Re}\nolimits} \{{RI \cdot {I^\ast }} \}$ and the scattered power ${P^{\textrm{scat}}} = {\textstyle{1 \over 2}}{\mathop{\rm Re}\nolimits} \{{{\omega^2}{R^{\textrm{rad}}}I \cdot {I^\ast }} \}$. Finally, the absorption and scattering cross section can be expressed in the form of the nanocircuit elements as

(18)$${\sigma _{\textrm{abs}}}(\omega )= \frac{{{P^{\textrm{abs}}}(\omega )}}{{{\textstyle{1 \over 2}}{\varepsilon _0}E_0^2c}} = \frac{{{l^2}}}{{{\varepsilon _0}c}}\frac{R}{{{{\left( {\frac{1}{{\omega C}} - \omega L} \right)}^2} + {{({R + {\omega^2}{R^{\textrm{rad}}}} )}^2}}}$$

and

(19)$${\sigma _{\textrm{rad}}}(\omega )= \frac{{{P^{\textrm{rad}}}(\omega )}}{{{\textstyle{1 \over 2}}{\varepsilon _0}E_0^2c}} = \frac{{{l^2}}}{{{\varepsilon _0}c}}\frac{{{\omega ^2}{R^{\textrm{rad}}}}}{{{{\left( {\frac{1}{{\omega C}} - \omega L} \right)}^2} + {{({R + {\omega^2}{R^{\textrm{rad}}}} )}^2}}}.$$

Using the target functions Eq. (18) and Eq. (19) one can fit the spectra ${\sigma _{\textrm{abs}}}(\omega )$ and ${\sigma _{\textrm{rad}}}(\omega )$ obtained from numerical simulation, with the lumped nanocircuit elements L, C, R and ${R^{\textrm{rad}}}$ as shared fitting parameters. Table 1 gives the fitting results for several nanorod antennas studied in this paper.

B. Simulating the spatial distribution of the normalized coupling rate

One may obtain the spatial distribution of the normalized coupling rate to a specific mode following the established method based on quasi-normal mode computation [27,28]. In this work, we propose a method that allows one to use FDTD technique to obtain ${{{\gamma _1}} \mathord{\left/ {\vphantom {{{\gamma_1}} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ as a function of the emitter’s spatial position by simply performing a small number of FDTD simulations. The method uses the concept of generalized mode volume [50]. At the resonance wavelength of the dipolar mode, the normalized decay rate of the emitter located at ${{\textbf r}_i}$ can be expressed as [50]

(20)$${{{\gamma _1}({{{\textbf r}_i}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_i}} )} {{\gamma_0}}}} \right.} {{\gamma _0}}} = \frac{3}{{4{\pi ^2}}}{\lambda _0}^3{\mathop{\rm Re}\nolimits} \left[ {\frac{{{Q_1}}}{{{V_1}({{{\textbf r}_i}} )}}} \right],$$

where ${\lambda _0}$ is the resonance wavelength, ${Q_1}$ is the quality factor of the resonance and ${V_1}({{{\textbf r}_i}} )$ is the generalized mode volume of the dipolar mode at ${{\textbf r}_i}$. The mode volume ${V_1}({{{\textbf r}_i}} )$ is complex and can be expressed as [50]

(21)$${V_1}({{{\textbf r}_i}} )= \frac{{\int {\left[ {{{\textbf E}_1}({\textbf r} )\cdot {\varepsilon_0}\frac{{\partial [{\omega \cdot \varepsilon ({\textbf r} )} ]}}{{\partial \omega }}{{\textbf E}_1}({\textbf r} )- {{\textbf H}_1}({\textbf r} )\cdot {{{{\mu}}}_0}\frac{{\partial ({\omega {{{\mu}}} ({\textbf r} )} )}}{{\partial \omega }}{{\textbf H}_1}({\textbf r} )} \right]{\textrm{d}^3}{\textbf r}} }}{{2{\varepsilon _0}{{[{{{\textbf E}_1}({{{\textbf r}_i}} )\cdot {\textbf u}} ]}^2}}},$$

where the fields are that of the dipolar mode excited by a dipole source ${{\boldsymbol{\mu}}} $ at some position ${{\textbf r}_0}$, ${\textbf u} = {{{\boldsymbol{\mu}} } \mathord{\left/ {\vphantom {{{\boldsymbol{\mu}} } {|{{\boldsymbol{\mu}} } |}}} \right.} {|{{\boldsymbol{\mu}} } |}}$ is the unit vector of the dipole. From Eq. (21), we note that the only factor that depends on ${{\textbf r}_i}$ is ${[{{{\textbf E}_1}({{{\textbf r}_i}} )\cdot {\textbf u}} ]^2}$. So the mode volume can simply be expressed with Green’s function as ${V_1}({{{\textbf r}_i}} )= {{A{e^{i\phi }}} \mathord{\left/ {\vphantom {{A{e^{i\phi }}} {{{[{G_1^{xx}({{{\textbf r}_i}{\textbf ,}{{\textbf r}_0}} )} ]}^2}}}} \right.} {{{[{G_1^{xx}({{{\textbf r}_i}{\textbf ,}{{\textbf r}_0}} )} ]}^2}}}$, where the factor A and the phase $\phi$ are real and have no dependence on ${{\textbf r}_i}$. Here we write only the xx component of the Green’s function since in this work the dipole source is oriented in the x direction. Then using Eq. (20), one can simply express ${{{\gamma _1}({{{\textbf r}_i}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_i}} )} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ as

(22)$${{{\gamma _1}({{{\textbf r}_i}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_i}} )} {{\gamma_0}}}} \right.} {{\gamma _0}}} = B{|{G_1^{xx}({{{\textbf r}_i}{\textbf ,}{{\textbf r}_0}} )} |^2}\cos[{2\varphi ({{{\textbf r}_i},{{\textbf r}_0}} )- \phi } ],$$

where the factor B is real and have no dependence on ${{\textbf r}_i}$, $\varphi ({{{\textbf r}_i},{{\textbf r}_0}} )$ is the phase of $G_1^{xx}({{{\textbf r}_i}{\textbf ,}{{\textbf r}_0}} )$. To get ${{{\gamma _1}({{{\textbf r}_i}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_i}} )} {{\gamma_0}}}} \right.} {{\gamma _0}}}$, one just needs to get $G_1^{xx}({{{\textbf r}_i}{\textbf ,}{{\textbf r}_0}} )$, B and $\phi$. We first get $G_{\textrm{tot}}^{xx}({{{\textbf r}_i}{\textbf ,}{{\textbf r}_0}} )$ from a simulation with the antenna and get $G_0^{xx}({{{\textbf r}_i}{\textbf ,}{{\textbf r}_0}} )$ from a simulation without the antenna, where the dipole source ${\boldsymbol{\mu}} $ (x-oriented) is located at ${{\textbf r}_0}$ that is far enough from the antenna so that the pseudomode channel can be safely neglected. Then $G_1^{xx}({{{\textbf r}_i}{\textbf ,}{{\textbf r}_0}} )$ can be obtained using $G_1^{xx}({{{\textbf r}_i}{\textbf ,}{{\textbf r}_0}} )= G_{\textrm{tot}}^{xx}({{{\textbf r}_i}{\textbf ,}{{\textbf r}_0}} )- G_0^{xx}({{{\textbf r}_i}{\textbf ,}{{\textbf r}_0}} )$. Note that $G_1^{xx}({{{\textbf r}_i}{\textbf ,}{{\textbf r}_0}} )$ is a distribution as a function of the position ${{\textbf r}_i}$. Then the remaining task is to determine the factor B and the phase $\phi$. For this task, one just needs to get ${{{\gamma _1}({{{\textbf r}_i}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_i}} )} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ at two different positions to form two equations, where B and $\phi$ can be solved. We perform FDTD simulations with the dipole source ${\boldsymbol{\mu}} $ (x-oriented) located at two different positions ${{\textbf r}_1}$ and ${{\textbf r}_2}$ where the dipolar mode channel is dominant while the pseudomode channel and free-space channel can be safely neglected. From these two simulations we can get ${{{\gamma _1}({{{\textbf r}_1}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_1}} )} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ and ${{{\gamma _1}({{{\textbf r}_2}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_2}} )} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ to form the equations, where the factor B and the phase $\phi$ can be solved as

(23)$$\phi = \arctan \left[ {\frac{{{{{\gamma_1}({{{\textbf r}_2}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_2}} )} {{\gamma_0}}}} \right.} {{\gamma_0}}}{{|{G_\textrm{1}^{xx}({{{\textbf r}_1},{{\textbf r}_0}} )} |}^2}\cos [{2\varphi ({{{\textbf r}_1},{{\textbf r}_0}} )} ]- {{{\gamma_1}({{{\textbf r}_1}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_1}} )} {{\gamma_0}}}} \right.} {{\gamma_0}}}{{|{G_1^{xx}({{{\textbf r}_2},{{\textbf r}_0}} )} |}^2}\cos [{2\varphi ({{{\textbf r}_2},{{\textbf r}_0}} )} ]}}{{{{{\gamma_1}({{{\textbf r}_1}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_1}} )} {{\gamma_0}}}} \right.} {{\gamma_0}}}{{|{G_1^{xx}({{{\textbf r}_2},{{\textbf r}_0}} )} |}^2}\sin [{2\varphi ({{{\textbf r}_2},{{\textbf r}_0}} )} ]- {{{\gamma_1}({{{\textbf r}_2}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_2}} )} {{\gamma_0}}}} \right.} {{\gamma_0}}}{{|{G_1^{xx}({{{\textbf r}_1},{{\textbf r}_0}} )} |}^2}\sin [{2\varphi ({{{\textbf r}_1},{{\textbf r}_0}} )} ]}}} \right]$$

and

(24)$$B = \frac{{{{{\gamma _1}({{{\textbf r}_1}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_1}} )} {{\gamma_0}}}} \right.} {{\gamma _0}}}}}{{{{|{G_1^{xx}({{{\textbf r}_1},{{\textbf r}_0}} )} |}^2}\cos[{2\varphi ({{{\textbf r}_1},{{\textbf r}_0}} )- \phi } ]}}.$$

Knowing $\phi$, B and $G_1^{xx}({{\textbf r,}{{\textbf r}_0}} )$, we can finally get ${{{\gamma _1}({{{\textbf r}_i}} )} \mathord{\left/ {\vphantom {{{\gamma_1}({{{\textbf r}_i}} )} {{\gamma_0}}}} \right.} {{\gamma _0}}}$ at any position ${{\textbf r}_i}$ according to Eq. (22).

The proposed method is validated by comparing to the results calculated using the quasi-normal mode method given by [28], as shown in Fig. 6. However, one should keep in mind that the proposed method is based on the precondition that one could somehow predominantly excite the target mode (here the dipolar mode) without significantly exciting other modes. In this work, the spatial distribution of the normalized coupling rate of the dipolar mode is calculated at its resonant wavelength, so we can predominantly excite the dipolar mode without significantly excite high-order modes as long as the emitter is not too close to the nanoantenna. For situations where the target mode cannot be exclusively excited, one should then resort to the rigorous quasi-normal mode methods [27,28].

 

Fig. 6. Comparison between the results computed with the method described in Appendix B (solid curves, which are just the results shown in Fig. 3a) and the results computed with the quasi-normal mode (QNM) method given by [28] (dashed curves).

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Funding

National Key Research and Development Program of China (2017YFA0205700); National Natural Science Foundation of China (11621101, 61774131, 91833303); Fundamental Research Funds for the Central Universities (2017FZA5001, 2018FZA5001); China Postdoctoral Science Foundation (2017M621920, 2017M622722); Science and Technology Department of Zhejiang Province; Guangdong Innovative Research Team Program (201001D0104799318).

Acknowledgments

The authors would like to thank Pu Zhang and Qiang Zhou in Huazhong University of Science and Technology for the helpful discussion and technical support on quasi-normal mode analysis.

Disclosures

The authors declare no conflicts of interest.

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