1. Introduction

Plasmonic antennas can concentrate incident light into deeply subwavelength regions, known as hot spots, due to strong surface plasmon resonances. The near-field enhancements (NFEs) in these hot spot regions can reach up to ∼10² or even higher [1,2]. Subwavelength dielectric cavities, made of high-refractive index materials, can also exhibit strong far-field optical responses with considerable near-field enhancements due to their Mie-like resonances [3,4]. The near field enhancements of common resonances in dielectric cavities are usually lower compared to their plasmonic counterparts. Nonetheless, the NFEs of certain special modes in subwavelength dielectric cavities, such as whispering gallery modes (WGMs), can still be quite high [5]. The significant NFEs in those photonic structures are usually accompanied by highly enhanced Purcell factors (F_p), which refer to the enhancements in the spontaneous decay rates of an emitter. [5–14]. These near-field properties have enabled a wide range of applications such as enhanced fluorescence, Raman scattering, nanolasers, enhanced light-matter interactions and so on [15–24].

The enhanced near field properties of plasmonic antennas are associated with their highly concentrated field and small sizes, namely, small mode volumes (V) [6,10,25–28]. In contrast, the enhanced near field properties of dielectric cavities are achieved by resonances of high enough quality (Q) factors [5,6]. The F_p of a plasmonic or dielectric nanostructure can be expressed in terms of these factors as ${F_p} \propto Q/V$ [6]. However, the Q factors of common plasmonic antennas are usually low (∼10¹), and the field concentrations of common dielectric cavities are low (large mode volumes), meaning the fields are widely spread inside the whole cavities [4]. Recently, hybrid photonic cavities of plasmonic antennas and dielectric cavities have been drawing increasing interests [29–33]. These hybrids may offer the potential to combine the advantages of each component. A dielectric cavity can readily provide enhanced field environment for a small plasmonic antenna due to their differing geometric sizes. This arrangement has enabled increased NFEs around plasmonic antennas [29,34–39]. The relative increase in NFE for a plasmonic antenna can reach more than ∼10¹. F_p responses have also been studied in terms of large values, spectral modifications, and directional emissions [36,40–45]. Referring to the reports on NFE and F_P in individual dielectric or plasmonic nanostructures, it seems intuitive that a larger NFE would indicate a larger Fp. As for hybrid systems, the relation between NFE and F_P has been scarcely investigated.

Here, we theoretically show that enlarged NFEs in nanophotonic cavities do not necessarily indicate enlarged F_P. This is particularly evident in plasmonic-dielectric hybrid cavities. We consider a general plasmonic-dielectric hybrid cavity through a semi-analytical model. A small plasmonic antenna exhibits a dipolar surface plasmon resonance, while a large dielectric cavity has a Mie-like resonance. The antenna is assumed to be exposed to an effective field with a certain value provided by the dielectric cavity. The results based on analytical model and direct numerical simulations agree well. The NFE of a hybrid cavity (${|{E/{E_0}} |_{\textrm{tot}}}$) is closely related to the product of NFEs provided by the antenna and dielectric cavity. However, the enlarged NFE of the antenna does not lead to a simultaneous increase in the F_P of a hybrid cavity. This is due to the fact that extinction cross section ($\sigma$) also plays a crucial role in determining the F_P. Interestingly, the peak F_P of coupled system can be estimated by the NFE and $\sigma$ responses. Both the ${|{E/{E_0}} |_{\textrm{tot}}}$ and F_P values also depend on the antenna-cavity coupling strength. Nonetheless, the upper limit for F_P of a hybrid cavity is set by the sum of F_P provided by individual cavity and redshifted antenna (or modified antenna, if applicable). These findings are important for nanophotonic applications such as enhanced fluoresces, Raman scattering and strong light-matter couplings. These relevant impacts will also be discussed.

2. Theoretical model

We consider a general hybrid cavity of a plasmonic antenna and a dielectric cavity (Fig. 1). The antenna exhibits a common dipolar surface plasmon resonance, while the dielectric cavity has a Mie-like resonance. The whole hybrid structure is excited by a plane wave or a dipole emitter. The NFE of the hybrid system can be obtained by considering a plane wave as the excitation [Fig. 1(a)]. The total field felt by the antenna can be expressed as ${E_a} = {E_0} + X{P_c}$ [46], where E₀ is the electric field of the excitation plane wave, $X{P_c}$ is the contribution from the dielectric cavity, ${P_c}$ is the moment of an electromagnetic mode, and X represents the proportional coefficient between the electric field and the moment ${P_c}$ at the location of the antenna. The total field felt by the cavity can be written as ${E_c} = {E_0} + X{P_a}$, where ${P_a}$ is the dipole moment of the antenna, namely, ${P_a} = {\varepsilon _0}{\alpha _a}{E_a}$. ${\alpha _a}$ is the electric dipolar polarizability of the antenna, and ε₀ is the free space permittivity. Similarly, ${P_c}$ can be written as ${P_c} = {\varepsilon _0}{\alpha _c}{E_c}$, where ${\alpha _c}$ is the polarizability of the cavity. Then, one obtains

Now, if one considers the field of a point L in the hybrid [Fig. 1(a)]. It can be expressed as $E = {E_0} + Y{P_a} + Z{P_c}$, where Y (Z) represents the proportional coefficient between the electric field at point L and the moment of antenna (cavity). Thus, the total NFE ${\left|{\frac{{E\; }}{{{E_0}}}} \right|_{\textrm{tot}}}$ at a point L can be expressed as

 

Fig. 1. (a) Schematic of a general antenna-dielectric hybrid structure excited by a plane wave. The point L denotes the location where the NFE is investigated. (b) The same hybrid structure excited by a dipole emitter, which is located at point L.

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For realistic systems, If $Z$ satisfies $Z \ll Y$ and the direct contributions from the dielectric cavities can be ignored, Eq. (1) reduces to

For the F_P, the hybrid system is excited by an electric dipole emitter [Fig. 1(b)]. The emitter is located at the same point L. The total field felt by the antenna can be expressed as ${E_a} = YD + X{P_c}$, where D is the electric dipole moment of the emitter. The other parameters are the same as before. The total field felt by the cavity can be expressed as ${E_c} = X{P_a} + ZD$. Then, one obtains

The total field felt by the quantum emitter E′ can be expressed as $E\mathrm{^{\prime}} = Y{P_a} + Z{P_c}$. So, the power radiated from the emitter can be written as

Here, we have assumed that the direction of the dipole moment D is the same as that of the E′ for simplicity. Thus, the F_P of such a system (denoted by F_P−tot) is

For the sake of later discussion, Eq. (5) can be divided into the coupling part $\textrm{Im}\left( {\frac{{{Y^2}{\varepsilon_0}{\alpha_a}}}{{1 - {X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}}}} \right)$ and frequency part $\frac{{6\pi {\varepsilon _0}{c^3}}}{{{\omega _0}^3}}$. We define the term ${X^2}{\varepsilon _0}^2{\alpha _a}{\alpha _c}$ as the coupling parameter, and its value will largely affect the behaviors of the coupled system. If the coupling between antenna and cavity disappears, namely, X = 0. F_P-tot reduces to the F_P of an individual antenna (denoted by F_P-A)

The above parameters X, Y, Z, ${\alpha _a}$,and ${\alpha _c}$ are highly dependent on the geometries of antennas and cavities. Generally, they cannot be obtained by analytical solutions. Thus, we use numerical simulations to calculate these parameters. The numerical simulations are carried out by a finite-difference in time-domain (FDTD solutions) method (see Supplement 1). We employ the multipole decomposition method (see Supplement 1) [48] to numerically obtain ${\alpha _a}$ and ${\alpha _c}$. The near fields $X{P_c}$, $Y{P_a}$, and $Z{P_c}$ can be directly obtained by the FDTD simulations. Then, one can obtain X, Y, and Z. The extinction cross sections can also be calculated straightforwardly via the FDTD simulations.

3. Results and discussion

3.1 Relations between near-field enhancements and Purcell factors

Let us now consider a hybrid system consisting of a gold nanorod (antenna) and a silicon (Si) disk (Fig. 2). The nanorod has a broad dipolar plasmon response around the considered spectral region. The disk has a narrow WGM of m = 5 at λ = 2403 nm (see Fig. S1 in Supplement 1) [5]. The surface distance between antenna and disk is d = 32 nm. The sizes of the nanorod and disk are chosen to make their resonances spectrally around each other. Such a distance between the rod and disk is chosen based on two facts. First, there is an effective coupling between the plasmon and the WGM. Second, a relatively large distance will facilitate the direct comparison with a latter case where an emitter is placed in between the rod and disk (see section 3.3 below). We consider the total NFE (${|{E/{E_0}} |_{\textrm{tot}}}$) around the end of antenna at the location L [Fig. 2(a)]. Here, the direct contribution from WGM to the ${|{E/{E_0}} |_{\textrm{tot}}}$ at the position L is only ∼5. Although such a value is comparable to that of an antenna (∼6), the role of Z in Eq. (1) can still be ignored based on the following two facts. First, Z / Y is smaller than ∼ 10² due to the huge difference between ${\alpha _a}$ and ${\alpha _c}$ (Fig. S1). Second, the direct contribution from cavity (Z${\alpha _c}$∼5) is much smaller than the total NFE. Thus, Eq. (1) reduces to Eq. (2). The directly simulated NFE agrees well with the analytical result based on Eq. (2) [Figs. 2(b) and 2(c)]. Numerical results also verify that the direct contributions from cavity can be ignored (see Fig. S2 in Supplement 1), namely, Z can be taken as Z ≈ 0.The spectrum shows a Fano lineshap due to the Fano resonance of the system [41,49,50]. The peak value of ${|{E/{E_0}} |_{\textrm{tot}}}$ is about 8 times of the antenna at this position (${|{E/{E_0}} |_\textrm{A}}$ ≈ 6). The relative enhancement is smaller than that provided by the disk cavity (∼20 times), which is attributed to the broadened response brought by a relatively large coupling effect. Similar coupling effect has been reported in other nanophotonic structures [51]. The NFEs, which are away from the WGM position, obtained based on direct simulations and analytical model are slightly different. This is due to the fact that there are contributions from other modes in simulations (see Fig. S1) that are not included in the analytical mode.

 

Fig. 2. NFE and F_P of a hybrid structure. The diameter and height of the disk are 2020 and 730 nm, respectively. The length, width and height of the gold nanorod are 246, 10 and 10 nm, respectively. The surface distance between disk and nanorod is 32 nm. (a) Schematic of the hybrid structure excited by a plane wave. The point L denotes a location where the NFE is investigated, and it is 21 nm away from the rod end. (b,c) The NFE at L calculated based on direct simulations (b) and the semi-analytical model (c). (d) Schematic of the same hybrid structure excited by a dipole emitter. The emitter is located at the same point of L. (e,f) The F_P calculated based on direct simulations (e) and the semi-analytical model (f).

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Let us turn to the F_P of the hybrid system F_P-tot, where an emitter is taken as the excitation source [Fig. 2(d)]. The emitter is located at the same point L with its polarization aligned along the antenna. The simulated F_P-tot also agrees well with the analytical result based on Eq. (5) [Figs. 2(e) and 2(f)]. Here, the direct contribution from disk cavity to the F_P-tot is ignorable (Z ≈ 0, see Fig. S1). The F_P-tot spectral lineshape is similar to that of the above NFE response. On the other hand, the peak F_P-tot (denoted by $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$) is only less than 3 times of the antenna (F_P-A ≈ 750). This enhancement value is far away from the relative field-intensity enhancement at L, namely $|{E/{E_0}{|_{\textrm{tot}}}^2\; /} |E/{E_0}{|_\textrm{A}}^2$ ≈ 64. Note that some small peaks appear in the simulations [Figs. 2(e)]. This is caused by the finite mesh size in such simulations. These small peaks will disappear with higher accuracy. On the other hand, they have ignorable effects on the results of the main peak. Generally, one can balance between the simulation accuracy and time consumption

Figure 3 shows more results about the peak values of NFE and F_P-tot as a function of the coupling parameter ${|{{X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}} |^{1/2}}$. Here, the variation of coupling parameter is carried out by changing the NFE provided by the WGM ${|{E/{E_0}} |_\textrm{c}}$ while the other parameters remain that same as that of Fig. 2. Thus, the X varies correspondingly. This can be done by narrowing the gap between antenna and cavity in a realistic system. We still ignore Z for simplicity. The behavior of F_P clearly does not follow that of the field intensity in hybrid structures [Figs. 3(a) and 3(c)]. This comes from the fact that the extinction cross section $\sigma$ should also be considered for F_P in addition to NFE. Note that even within an individual component, the relation between the NFE and F_P (F_P$\propto $ NFE²) holds under the default condition that the extinction $\sigma$ of the antenna is fixed. This could correspond to a common situation that an emitter moves around a given antenna (see Supplement 1, Fig. S1). The full relation in an antenna is Eq. (7), namely, F_P $\propto $ NFE²/ $\sigma$. Thus, the relation F_P $\propto $ NFE² without considering the $\sigma$ is only partially correct. This can be verified by comparing the cases of individual antenna and cavity (see Fig. S1). As for a hybrid structure, the NFE can be enlarged significantly while the $\sigma$ of the coupled system is also much larger than that of the individual antenna [Figs. 3(b) and S1).

 

Fig. 3. Relations between NFE and F_P of hybrid structures. (a-c) The peak NFE (a), $\sigma$ (b) and F_P-tot (c) of the hybrid system as a function of the coupling parameter ${|{{X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}} |^{1/2}}$ and resonant position of WGM. Here, the variation of coupling parameter ${|{{X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}} |^{1/2}}$ is carried out by changing the NFE provided by the WGM (${|{E/{E_0}} |_\textrm{c}}$) as demonstrated by the right axis in (a). (d) The estimated $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ as a function of ${|{{X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}} |^{1/2}}$ and resonant position of WGM. The results above and below the dotted white line are based on Eq. (8) and (7), respectively. (e) The estimated $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ with three different ${|{{X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}} |^{1/2}}$ $({{{|{E/{E_0}} |}_\textrm{c}}} )$. The exact $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ are also shown for comparison. The data are also marked by the corresponding dotted blue lines in (d).

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The peak F_P-tot (= $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$) in a hybrid system can be estimated by the peak NFE and the $\sigma$ responses. Let us first consider the case of a weak enough coupling where the term $|{{X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}} |$<< 1. Here, we still ignore the direct contribution from the cavity for simplicity (Z ≈ 0). Under such a condition, Eq. (5) reduces to Eq. (6), namely, $F_{\textrm{P} - \textrm{tot}}^\textrm{p} \approx F_{\textrm{P} - \textrm{A}}^\textrm{p}$. This indicates that $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ is largely determined by that of the uncoupled antenna under weak enough coupling. Interestingly, for the case with strong enough coupling where the coupling parameter satisfies $|{{X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}} |$≥ 3, the $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ can be well estimated by

Figures 3(c) and 3(d) shows the exact and estimated results of $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ for a case with varying the coupling strength, respectively. For weak and strong couplings, the estimated results based on Eqs. (7) and (8) show good agreements with the exact values, respectively. It is also found that for moderate couplings (0.3 <$|{{X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}} |$< 3), the $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ can also be roughly estimated by Eq. (8), while the value is underestimated. The estimated values based on Eq. (8) are about 30% to 80% of the exact values, corresponding to the region between the white and black lines in Fig. 3(d). Further calculations show that the above estimations and the corresponding accuracies also apply for cases with different ${\alpha _a}$ in the coupling parameter (see Fig. S3 in Supplement 1). The results indicate that if one takes the hybrid system as a whole, the enlarged NFE will always be accompanied by a larger effective $\sigma$ involved for the $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$. This effective $\sigma$ is larger than that of the uncoupled antenna.

3.2 Limits for Purcell factors of hybrid systems with different coupling strengths

Now let us consider the limit for F_P of a hybrid system. Here, we assume a system with a given antenna (F_P-A) and study the effects of mode couplings on the total F_P of the system. We are not intended to optimize the F_P value of a general nanophotonic system [52,53]. To get more insight into the limit for ${F_P}$ of a system with Z ≈ 0, we shall focus on the antenna-cavity coupling part of Eq. (5), namely, $\textrm{Im}\left( {\frac{{{\alpha_a}}}{{1 - {X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}}}} \right)$. As the Q factor of a dielectric resonance (WGM) is much higher than the plasmon resonance, the ${\alpha _c}$ at a wavelength far away from its resonance can be taken as ${\alpha _c} \approx 0$. One obtains $\textrm{Im}\left( {\frac{{{\alpha_a}}}{{1 - {X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}}}} \right) \approx \textrm{Im}({{\alpha_a}} )$. This means that ${F_P}$ is determined by the antenna for wavelengths far away from the cavity mode.

For the wavelengths around cavity resonance, ${\alpha _a}$ can be taken as a fixed value as its variation with wavelength is relatively much slower than that of ${\alpha _c}$. The response of the cavity mode can be approximately taken as a Lorentz resonance. Then, ${\alpha _c}$ can be written as $\left\{ {\begin{array}{l} {\textrm{Re}({{\alpha_c}} )= {D_c}\; sin{\theta_c}\cdot cos{\theta_c}}\\ {\textrm{Im}({{\alpha_c}} )= {D_c}\; co{s^2}{\theta_c}\; \; \; \; \; \; \; \; \; \; \; } \end{array}} \right.$, where ${\theta _c}$ is a parameter that is related to wavelength and ${D_c}$ is the maximal value of $\textrm{Im}({{\alpha_c}} )$, namely, ${D_c}$=$\textrm{Max}[{\textrm{Im}({{\alpha_c}} )} ]$ (see Fig. S4 in Supplement 1). The peak value of $\textrm{Im}\left( {\frac{{{\alpha_a}}}{{1 - {X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}}}} \right)$ can be approximately written as (see Supplement 1 for the relevant derivation)

To obtain the maximal value of the peak $\textrm{Im}\left( {\frac{{{\alpha_a}}}{{1 - {X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}}}} \right)$, the response of the dipolar plasmon mode can also be taken as a Lorentz lineshape. ${\alpha _a}$ can be written as $\left\{ {\begin{array}{l} {\textrm{Re}({{\alpha_a}} )= {D_a}\; sin{\theta_a}\cdot cos{\theta_a}}\\ {\textrm{Im}({{\alpha_a}} )= {D_a}\; co{s^2}{\theta_a}\; \; \; \; \; \; \; \; \; \; \; } \end{array}} \right.$, where ${\theta _a}$ is a parameter that is also related to wavelength and ${D_a}$ is the maximal value of $\textrm{Im}({{\alpha_a}} )$ (${D_a}$=$\textrm{Max}[{\textrm{Im}({{\alpha_a}} )} ]$) which occurs at ${\theta _a} = 0$. Then, one can obtains (see Supplement 1 for the relevant derivation)

Based on Eq. (10), the peak value of coupling part reaches its maximum at the resonance of the antenna (${\theta _a} = 0$). More importantly, Eq. (10) indicates that both the coupling between antenna and cavity (X) and the properties of a dielectric cavity (D_c) do not affect the maximal value of ${F_P}$.

Figure 4(a) shows the results of coupled systems with WGMs of different resonant positions varying around the broad response of antenna. Here, the direct contributions from the WGMs are ignorable (Z ${\approx} $ 0). We first fix the coupling parameter X for simplicity. For each case, $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ can be obtained based on Eq. (5), and the $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ as a function of the WGM position is shown [Fig. 4(a)]. The redline in Fig. 4(a) shows the results based on the analytical expression of Eq. (9). They agree well with the results obtained from Eq. (5), which verifies the validity of the analytical formula of Eq. (9). Figure 4(b) shows the frequency $\frac{{6\pi {\varepsilon _0}{c^3}}}{{{\omega _0}^3}}$ and coupling $\textrm{Im}\left( {\frac{{{\alpha_a}}}{{1 - {X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}}}} \right)$ contributions to $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ as the resonant position of WGM varies. The coupling part indeed cannot exceed the ${D_a}$ antenna, as expected (Eq. (10)). The contribution of the frequency part increases with wavelength. Thus, for $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ that are larger than that of an individual antenna ($F_{\textrm{P} - \textrm{A}}^\textrm{p}$), the contribution only comes from the frequency shift.

 

Fig. 4. Limit for F_P of a hybrid system with different couplings. The system is based on the structure in Fig. 2 with Z ≈ 0. (a) The F_P spectra of hybrid structures (F_P-tot) with different WGM resonances (blue). The corresponding extracted peak value ($F_{\textrm{P} - \textrm{tot}}^\textrm{p}$) as a function of WGM resonance is shown by the dotted black line. The $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ obtained by the analytical expression Eq. (9) are plotted by the red line. The F_P spectrum of an individual antenna (F_P-A) is also shown for comparison. (b) Contributions from the coupling and frequency parts to the above $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$. The values are normalized to the peak value of Im(${\alpha _a}$), namely ${D_a}$. The normalized Im(${\alpha _a}$) spectrum is also shown. (c) The coupling part with varying the coupling coefficient X. The values are also normalized by ${D_a}$. The 1${\times} $X₀ case corresponds to the coupling strength in (a,b). (d) The F_P-tot spectrum of a hybrid structure (blue) compared to that of a redshifted antenna (dotted pink). For the redshifted antenna, its resonance is moved to that of the WGM while the other parameters are the same as the individual one. The F_P spectrum of an individual antenna is also shown.

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The coupling-part contributions with different coupling strengths are also considered [Fig. 4(c)]. With increasing the coefficient X, the coupling-part contributions that are spectrally distant from the antenna's resonance also increase. However, there is an upper limit which equals to ${D_a}$ of the antenna. This means that if the coupling between an antenna and a WGM cavity is not large enough, the coupling part will be smaller than ${D_a}$. Considering the frequency part, one finds that the $F_{\textrm{P} - \textrm{tot}}^\textrm{p}$ of hybrid system will be smaller than that of the redshifted antenna F_P−rA [Fig. 4(d)]. Here, the F_P-rA corresponds to an individual antenna whose resonance position is redshifted to align with that of the WGM. The other parameters of the antenna are kept the same as the original one for simplicity. Note that it is easy to obtain a redshifted antenna in a practical plasmonic system.

For the cases with direct contributions from the WGMs, Z cannot be ignored anymore. Figure 5(a) shows the results with a certain Z (F_P-C ≈ 700), and the other parameters are the same as that in Fig. 2. The F_P of hybrid system is relatively larger than that with Z = 0 due to the direct contribution from the WGM [Figs. 2(c) and 5(a)]. Figure 5(b) shows the peak value $F_{P - tot}^p$ as a function of the WGM-resonance position. The $F_{P - tot}^p$ may also exceed the sum of an individual antenna and cavity (F_P-A+ F_P-C). As discussed above, the frequency shift plays an important role in $F_{\textrm{P} - \textrm{tot}}^p$. Thus, if we compare $F_{P - tot}^p$ with the sum of an individual cavity and a redshifted antenna (F_P-rA+ F_P-C), the $F_{\textrm{P} - \textrm{tot}}^p$ will still not exceed the later one as shown in Fig. 5(b). Similarly, $F_{\textrm{P} - \textrm{tot}}^p$ cannot be relatively enlarged by increasing the antenna-cavity coupling strength (X) [Fig. 5(c)]. The reason is similar to that in the previously discussed case of Z = 0 [Fig. 4(c)]. Similarly, By increasing the F_P-C (Z), the $F_{P - tot}^p$ increases while it can still not exceed (F_P-rA+ F_P-C) as expected [Fig. 5(c)]. Note that in Fig. 5(c), both X and Z increase simultaneously with ${|{E/{E_0}} |_\textrm{c}}$. This is in consistent with a realistic situation by narrowing the gap between cavity and antenna. The antenna-cavity coupling strength can also be turned by varying the polarizability of antenna ${\alpha _a}$ [Fig. 5(d)]. The coupling strength ($|{{X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}} |$) increases with ${\alpha _a}$, while the $F_{P - tot}^p$ still keeps no larger than (F_P-rA+ F_P-C).

 

Fig. 5. The cases with direct contributions from the WGMs (Z ≠ 0). The other parameters are the same as that in Fig. 2. (a) The F_P-tot spectrum (blue) with a certain F_P-C directly provided by the WGM (cyan). The other parameters are the same as that in Fig. 2. The cases for the individual (yellow) and redshifted (pink) antenna are also shown. (b) Extracted peak value $F_{P - tot}^p$ normalized by (F_P-rA+ F_P-C) with varying the resonance of a WGM. The $F_{P - tot}^p$ normalized by (F_P-A+ F_P-C) are also show for comparison. The dots denote the case in (a). (c) The ratio $F_{P - tot}^p$ / (F_P-rA+ F_P-C) as a function of the WGM resonance and ${|{E/{E_0}} |_\textrm{c}}$. The ${|{E/{E_0}} |_\textrm{c}}$ value is normalized by that in (a,b). The dotted line corresponds to the results in (b). (d) The ratio $F_{P - tot}^p$ / (F_P-rA+ F_P-C) as a function of the WGM resonance and the ${\alpha _a}$. The ${\alpha _a}$ value is normalized by that in (a,b). The dotted line also corresponds to the results in (b).

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3.3 Cases with relatively enlarged Purcell factors

We have shown that a relatively enlarged NFE of an antenna in hybrids does not bring an enlarged ${F_P}$, and the limit of ${F_P}$ is determined by the uncoupled antenna and cavity. However, it should be noted that ${F_P}$ can be relatively enhanced if the near field properties of an antenna (e.g. Y) is affect by a dielectric cavity (Fig. 6). The only difference between the configuration here and that of Fig. 2 is that the emitter is located at the left side of the antenna, while the other parameters are the same. In this case, the NFE at L is additionally increased due to the appearance of a high-index dielectric material [Figs. 6(a)-6(c)]. This can be explained by the fact that the dielectric material performs a mirror-like function and induces imaginary charges [54]. Thus, the NFE at the emitter location is additionally enlarged. Note that the contributions from other modes to the baseline of NFE increase here [Fig. 6(a)] compared to that of Fig. 2, while this will not be included in the analytical model.

 

Fig. 6. The case for a hybrid system with a modified antenna. The geometries of the disk and antenna are the same as that in Fig. 2. (a) Schematic of a hybrid system excited by a plane wave. The investigated point L is between the antenna and disk, and it is 21 nm away from the rod end. (b,c) The NFE at L calculated based on direct simulations (b) and the analytical model (c). (d) Schematic of the hybrid system excited by a dipole emitter at the same point L. (e,f) F_P calculated based on direct simulations (e) and the analytical model (f).

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The additional enhancement of the NFE can be considered as a result of the increase in Y. The increase ratio can be estimated by the peak NFE responses with subtracting the effects from baselines (Figs. 2(b), 6(b) and Supplement 1, S1). It can be obtained that the Y here in the gap [Fig. 6(b)] is increased ∼1.4 times compared to the case in Fig. 2. By taken this this factor in to account, one can obtain the NFE response based on the analytical expression Eq. (1) as shown in Fig. 6(c). The results agree well with the direct simulations after the effects from the baselines have been subtracted. Note that the only assumption is the increased Y, while the other parameters used in the model are totally based on that in Fig. 2.

The analytical results of ${F_{\textrm{P} - \textrm{tot}}}$ agree well with the simulations [Figs. 6(e) and 6(f)], which again verifies the validity of the assumption of increased Y. It should be noted that the behavior of increased ${F_{\textrm{P} - \textrm{mA}}}$ remain unchanged when the disk is replaced by a semi-infinite dielectric material (Fig. S5 in Supplement 1). Thus, the mirror-like effect is induced by the material not the cavity mode (WGM). It is verified that the relation between the modified ${F_{\textrm{P} - \textrm{A}}}$ (denoted by ${F_{\textrm{P} - \textrm{mA}}}$) and ${|{E/{E_0}} |_{\textrm{mA}}}$ still holds, namely, ${F_{\textrm{P} - \textrm{mA}}} \propto $ $|E/{E_0}|_{mA}^2$ (Fig. S5). Thus, the ${F_{\textrm{P} - \textrm{mA}}}$ is increased here compared to the unmodified one ${F_{\textrm{P} - \textrm{A}}}$ [Figs. 2(c) and 6(d)]. It can also be easily verified that the direct mode couplings between cavity and modified antenna will not relatively increase ${F_{\textrm{P} - \textrm{tot}}}$, namely, ${F_{\textrm{P} - \textrm{tot}}}$ of hybrid system can still not exceed the modified antenna ${F_{\textrm{P} - \textrm{mA}}}$. The reason is the same that discussed in Fig. 4. On the other hand, the ${F_{\textrm{P} - \textrm{tot}}}$ of a hybrid system can indeed be further enhanced compared to the original components (F_P-A+ F_P-C or F_P-rA+ F_P-C) as long as the antenna response (e.g. Y) is modified.

3.4 Hybrid systems with broad cavity modes

In the above discussion, the response of antenna in the coupled system is spectrally much broader than that of the individual WGM. Let us now consider a cavity whose response is spectrally broader than that of antenna [ Fig. 7(a)]. The dielectric cavity has a magnetic dipolar mode [34] and an antenna still has a dipolar plasmon resonance (see Fig. S6 in Supplement 1). The coupling between the cavity and the small plasmonic antenna now is much weaker compared to that in Fig. 2. ($|{{X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}} |$ << 1, see Fig. S7 in Supplement 1). The direct contribution from WGM can also be ignored Z ${\approx} $ 0 [Fig. 7(b)]. Furthermore, the plasmon and cavity modes are in the same resonant position. Thus, based on Eq. (2), the peak ${|{E/{E_0}} |_{\textrm{tot}}}$ of the hybrid system can be well reproduced by the product of the antenna ${|{E/{E_0}} |_\textrm{A}}$ and cavity ${|{E/{E_0}} |_\textrm{C}}$, namely,

This is numerically verified as shown in Fig. 7(a). As expected, the F_P-tot of such a system is not relatively enlarged compared to that of an individual antenna [Fig. 7(b)]. This is due to the fact that the coupling in this system is weak enough. Thus, F_P-tot of such a system reduces to ∼ F_P-A. Note that F_P-tot is slightly larger than F_P-A. This due to the small redshift of the antenna as discussed before. The additional peak around 500 nm in simulations [Fig. 7(b)] is due to the Au material response which is ignored in our analytical model.

 

Fig. 7. A hybrid system with a broad resonance of a dielectric cavity. The diameter and height of the disk are 160 and 160 nm, respectively. The refractive index of the disk is n = 3.3 (e.g. GaP). The length and diameter of the gold nanorod are 26 and 10 nm, respectively. The surface distance between disk and nanorod is 5 nm. The spacer between antenna and disk is a material of n = 1.5. (a) Left: schematic of a hybrid structure excited by a plane wave. The point L denotes a location where the NFE is investigated, and it is 2.5 nm away from the rod end. Right: The NFE at L calculated based on direct simulations and the analytical model. The results of individual antenna and disk are also shown for comparisons. (b) Left: schematic of the same hybrid system excited by a dipole emitter. The dipole is located at the same point L. Right: F_P calculated based on direct simulations and the analytical model. The results of individual antenna and disk are also shown for comparisons. (c,d) The ${|{E/{E_0}} |_{\textrm{tot}}}$ (c) and F_P-tot (d) spectra with varying ${|{E/{E_0}} |_C}$ while the other parameters remain the same as before. (e) Simulated quantum efficiencies for the configuration of (b) and an individual antenna.

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It should be noted that Eq. (11) does not mean the peak value of ${|{E/{E_0}} |_{\textrm{tot}}}$ reaches maximal under a weak coupling condition. This is due to the fact that the coupling strength ($|{{X^2}{\varepsilon_0}^2{\alpha_a}{\alpha_c}} |$) relatively increases with ${|{E/{E_0}} |_\textrm{C}}$. Now if one has two cases with two NFEs for the cavity, namely ${|{E/{E_0}} |_{\textrm{C} - 1}}$ and ${|{E/{E_0}} |_{\textrm{C} - 2}}$. It is also assumed that ${|{E/{E_0}} |_{\textrm{C} - 1}} < {|{E/{E_0}} |_{\textrm{C} - 2}}$ and ${|{E/{E_0}} |_{\textrm{tot} - 1}} \approx {|{E/{E_0}} |_\textrm{A}} \times {|{E/{E_0}} |_{\textrm{C} - 1}}$, where ${|{E/{E_0}} |_{\textrm{tot} - 1}}$ corresponds to the NFE of first case. Even if the NFE of second case is ${|{E/{E_0}} |_{\textrm{tot} - 2}} < {|{E/{E_0}} |_\textrm{A}} \times {|{E/{E_0}} |_{\textrm{C} - 2}}$, it can still be larger than ${|{E/{E_0}} |_{\textrm{tot} - 1}}$. This is confirmed as shown in Fig. 7(c). When the coupling strength is strong enough by increasing ${|{E/{E_0}} |_\textrm{C}}$, the ${|{E/{E_0}} |_{\textrm{tot}}}$ spectrum begins to split. We can define this as a critical coupling strength. ${|{E/{E_0}} |_{\textrm{tot}}}$ reaches a maximal value at the critical coupling. The peak ${|{E/{E_0}} |_{\textrm{tot}}}$ approximately keeps this value with stronger coupling. In fact, the frequency part also plays a role in the NFE responses. A larger wavelength allows relatively larger NFE [27], namely NFE∼$\sqrt \lambda $. This means that the relative near-field concentration ability ${|{E/{E_0}} |_{\textrm{tot}}}/\sqrt \lambda $ reaches maximal near the critical coupling. For stronger couplings, ${|{E/{E_0}} |_{\textrm{tot}}}/\sqrt \lambda $ response drops due to the splitting effect. Similar critical phenomena have been reported in other nanophtonics systems [51].

Figure 7(d) shows the calculated F_P spectra with varying the coupling strength. The peak value $F_{P - tot}^p$ does not follow that of NFE as expected. The explanations are similar to that discussed before. The $F_{P - tot}^p$ can also be estimated by the relations between it and the NFE (Eqs. (7) and (8), see Fig. S6). Based on Eq. (11), $F_{P - tot}^p$ under weak enough coupling (Eq. (7)) can also be expressed as

From the point view of a whole system, the enlarged NFE are accompanied by an enlarged effective ($\sigma$_eff) involved in F_P based on Eqs. (8) and (12). We have $\sigma$_eff ≈ ${\mathrm{\sigma }_A} \times |E/{E_0}{|_\textrm{C}}^2$ for weak couplings, and $\sigma$_eff ≈ ${\mathrm{\sigma }_{\textrm{tot}}}$ for strong couplings. From a formal perspective, the term $|E/{E_0}{|_{tot}}^2/{\mathrm{\sigma }_{\textrm{eff}}}$ is also closely related to the well-known local density of states (LDOS) in F_P [8,9].

The coupling strength is highly related to the parameter X. This feature indicates that if one intends to obtain a large enough ${|{E/{E_0}} |_{\textrm{tot}}}$ with a given antenna, a proper way is to find a cavity with large enough ${|{E/{E_0}} |_\textrm{C}}$ while its extinction cross section is huge (the larger the better). Then, the coupling is weak enough and ${|{E/{E_0}} |_{\textrm{tot}}}$ will satisfy Eq. (11). Note that the F_P-tot should still remain around that of an individual antenna F_P-tot ≈ F_P-A. In realistic systems, the large ${|{E/{E_0}} |_{\textrm{tot}}}$ can be obtained by putting a small antenna in a large enough microcavity with an enlarged field environment. From a general physical perspective, such a system works as a cascaded lens for NFE, namely, a multiplicative feature. While for the F_P, the far field response of the whole system ($\sigma$_eff) also must be considered. Thus, the F_P response brought by the antenna-cavity coupling will not be relatively increased at all compared to an uncoupled antenna (F_P-A or F_P-rA). If the direct contribution from a microcavity to the F_P is also considered, the F_P-tot then arises from the sum of antenna and cavity.

3.5 Hybrid structures for some nanophotonic applications

It is well known that both NFE and emission property are important for applications of enhanced Raman scattering, fluoresces and light-matter interactions. A most common situation for a hybrid structure is that a dielectric provides an enlarged-field environment for antennas. For simplicity, we shall focus on Z ${\approx} $ 0. In terms of Raman scattering, the enhancement is usually approximated as $|E/{E_0}{|^4}$ for simplicity. This is due to the fact that both excitation and emission parts are enhanced by $|E/{E_0}{|^2}$. This works well in many plasmonic or dielectric cavities [17,18,23]. However, in an antenna-dielectric hybrid structure, the Raman scattering enhancement is not $|E/{E_0}|_{\textrm{tot}}^4$ anymore as the emission part is not enhanced by $|E/{E_0}|_{\textrm{tot}}^2$. As for fluorescence enhancement, the excitation part is still determined by $|E/{E_0}|_{\textrm{tot}}^2$, while the emission part is related to quantum efficiency not Purcell factor itself [15]. Pure plasmonic antennas and dielectric cavities usually surfer from low quantum efficiencies and NFEs, respectively. In this regard, an antenna-dielectric hybrid structure could work as an outstanding platform. This is because the excitation part ($|E/{E_0}{|^2}$) is easily enlarged. The quantum efficiency of a hybrid system can be largely enhanced compared to an individual plasmonic antenna [55,56]. For example, the quantum efficiency of the above hybrid structure reaches an order of magnitude higher than that of an individual antenna [Fig. 7(e)].

For light-matter interactions, the coupling strength between emitter and cavity can also be obtained by the F_P [47,57,58]. Compared to an individual antenna, the effective mode volume of a hybrid system generally increases if the F_P is not increased, since the effective Q factor usually does not decrease. This means that simply placing a plasmon-emitter system into a dielectric cavity with enlarged-field environment cannot boost the coupling strength between an emitter and cavity, unless the property of antenna (Y) is modified by an antenna-dielectric configuration. Note that the relationship between F_P and the emitter-cavity coupling strength is predicated on a common condition: the emitter has a spectral width that is much narrower than that of a cavity mode, and the excitation occurs in the linear region [47]. In the nonlinear region, a hybrid system would exhibit obvious differences because the nonlinear response is highly dependent on the excitation power [47].

4. Conclusion

In conclusion, we have theoretically investigated the NFE and F_P of antenna- dielectric hybrid systems. The antenna exhibits a dipolar plasmon resonance and the cavity has Mie-like mode. Our theoretical model and direct numerical simulations agree well. The NFE around an antenna can be efficiently enlarged due to an enlarged-field environment provided by a dielectric cavity. The enlargement value is highly dependent on the ${|{E/{E_0}} |_\textrm{C}}$ and the coupling strength. The peak NFE value of a resonant hybrid system under weak coupling (${X^2}{\varepsilon _0}^2{\alpha _a}{\alpha _c}$<< 1) can reach the product of the antenna ${|{E/{E_0}} |_\textrm{A}}$ and cavity ${|{E/{E_0}} |_\textrm{C}}$, namely, ${|{E/{E_0}} |_\textrm{A}}$ ${\times} $ ${|{E/{E_0}} |_\textrm{C}}$.

In contrast to the NFE behavior, the F_P-tot of a hybrid system cannot be additionally enlarged through mode couplings between antenna and cavity. This is due to the fact that the $\sigma$ response should also be considered in F_P-tot. Interestingly, the peak F_p of coupled system $F_{P - tot}^p$ can be estimated by the NFE and $\sigma$ responses. For weak coupling, the $F_{P - tot}^p$ approaches that of an individual antenna ( $\propto |E/{E_0}{|_A}^2/{\mathrm{\sigma }_A}$). For strong enough couplings, $F_{P - tot}^p$ can be estimated by F_P $\propto |E/{E_0}{|_{tot}}^2/{\mathrm{\sigma }_{tot}}$. From the view point of a whole system, the enlarged NFE are accompanied by an enlarged effective $\sigma$_eff involved in F_P. Especially, if one takes the enlarged NFE with the original $\sigma$ for a small antenna, the F_P of a hybrid cavity is largely overestimated. Furthermore, the upper limit for $F_{P - tot}^p$ of a system is also considered. The upper limit for $F_{P - tot}^p$ is set by the sum of the F_P of individual cavity and the redshifted (or modified, if any) antenna, namely, F_P-rA+ F_P-C (or F_P-mA+ F_P-C). Note that the F_P-mA of a modified antenna can be larger than an individual one F_P-rA due to the mirror-like effect brought by a dielectric structure.

The increase in the F_P is not as easy as that of the NFE in hybrid systems. This will bring different impacts on relevant nanophotonic applications. For Raman scattering, the enhancement value is smaller than $|E/{E_0}|_{\textrm{tot}}^4$. The relatively low F_P shall not directly affect the fluorescence enhancement. The readily enlarged NFE and quantum efficiency would make antenna-dielectric hybrid cavities outstanding platforms for fluoresce enhancements. As for light-matter interaction, the emitter-cavity coupling strength cannot be boosted by a dielectric cavity that only provides an enlarged-field environment. This situation may improve if the Y of the antenna is modified. Note that we have restricted our system with a dipolar plasmon antenna. If the mode of an antenna involved in the near-field couplings goes beyond the dipolar approximation. The ${|{E/{E_0}} |_\textrm{A}}$ and corresponding Y may be modified correspondingly. A more comprehensive understanding of this phenomenon may require more detailed investigations in the future.