Abstract

Strong coupling (SC) between two resonant plasmon modes can result in the formation of new hybrid modes exhibiting Rabi splitting with strong energy exchange at the nanoscale. However, normal Rabi splitting is often limited to 50320meV due to the short lifetime of the plasmon mode. Here, we theoretically demonstrate a record Rabi splitting energy as large as 805 meV arising from the SC between the high-Q plasmonic whispering gallery mode and high-Q cavity plasmon resonance supported by a spherical hyperbolic metamaterial cavity, which consists of a dielectric nanosphere core wrapped in 7 alternating layers of silver/dielectric materials. In addition, the new hybrid modes formed by the SC are shown to exhibit an extralong lifetime of up to 71.9–81.6 fs, with the large electric field intensity enhancement at both the dielectric core and the dielectric layers. More importantly, the spectral ranges of SC can be tuned across an ultrabroad range from the visible to the near-IR by simply changing the dielectric core size. These findings may have potential applications in bright single-photon sources.

© 2021 Chinese Laser Press

1. INTRODUCTION

Localized surface plasmon resonances (LSPRs) arising from the collective electron oscillation at the surfaces of metallic nanoparticles have the ability to concentrate light into the nanometric scale, produce highly localized fields, and promote light–matter interactions [1], which enables their use in a wide range of nanophotonic devices, such as nanolasers [2,3] and nanosensors [4,5], as well as in Purcell enhancement [6,7]. In recent years, LSPR-based strong coupling (SC) characterized by the realization of Rabi oscillation and the formation of anti-crossing behavior (new hybrid states) has become the focus of research, divided into three main categories. In the first category, the realization of SC between gap plasmons supported by metal nanoparticle-on-mirror structures (LSPR–mirror coupled modes) and different kinds of quantum emitters, including semiconductor quantum dots (QDs) [8], molecules [914], J-aggregates [1517], and two-dimensional materials [1822], has been intensively explored. In particular, more recently it has been observed experimentally that the SC between gap plasmons and a single molecule [23,24] or single QD [25,26] successfully forms part-light/part-matter states (mixed states) at the quantum limit, which paves the road toward quantum entangled single-photon sources. In the second category, the SC between LSPR or magnetic plasmons and optical modes (waveguide modes or Fabry–Perot nanocavity modes) enables high electric and magnetic intensity enhancement [27,28] and assisted hot-electron transfer [28], which has important applications in photovoltaic devices [29,30]. In addition, the SC between LSPR or magnetic plasmons and propagating surface plasmon polaritons (SPPs) (the third category) can also achieve high electric and magnetic intensity enhancement [31,32], assisted hot-electron transfer [33], enhanced dephasing time [34], and improved sensor performance [35,36].

The common shortcoming of LSPR is the low quality factor (Q-factor) caused by intrinsic metal absorption (nonradiative) and radiative (scattering) losses, which severely shortens the lifetime of only 7–8 fs and subsequently limits the interaction time with the matter [3739]. In other words, for LSPR-based SC systems, the LSPR corresponds to a short-lifetime oscillator (78fs), while the quantum emitter (QD, molecule, J-aggregate, two-dimensional material), the optical mode (waveguide mode, Fabry–Perot nanocavity mode), and the SPP correspond to a long-lifetime oscillator. As a result, the Rabi splitting energies of LSPR-based SC systems are limited to less than 320 meV [836]. To the best of our knowledge, the largest Rabi splitting, with a record value of up to 700 meV, has only been demonstrated between plasmonic Fabry–Perot cavity modes and dye molecules [40], which reveals that using high-Q plasmonic modes could be an effective way for improving Rabi splitting.

In this paper, we aim to explore the SC effect between two high-Q plasmonic modes in a spherical hyperbolic metamaterial (HMM) cavity formed by seven alternating silver/dielectric layers wrapping around a dielectric nanosphere core. First, we theoretically demonstrate a very large Rabi splitting energy of up to 805 meV arising from the SC between a high-Q plasmonic whispering gallery mode (WGM) and high-Q cavity plasmon resonance in the spherical HMM cavity. In addition, the new hybrid modes formed by the SC are demonstrated to exhibit a longer lifetime of up to 71.9–81.6 fs and achieve electric field intensity enhancement at both the dielectric core and the dielectric layers. More importantly, by simply adjusting the dielectric core size, the SC is demonstrated to occur at the same dielectric core index and to be tuned across a broad spectral range (from visible to near-IR).

2. RESULTS AND DISCUSSION

A. Mode Analysis of a Spherical HMM Cavity

The schematic of the spherical HMM cavity is illustrated in Fig. 1(a), and consists of a dielectric nanosphere core (radius rcore, refractive index ncore) wrapped with seven alternating silver/dielectric layers. The thicknesses of the silver layers are s1, s2, s3, and s4 from outer to inner layer. The thicknesses and refractive indices of the dielectric layers are d1, d2, d3, and nd1, nd2, nd3, respectively, from outer to inner layer. For investigation simplicity, we keep s1=s2=s3=s4=s=15nm, d1=d2=d3=d, and nd1=nd2=nd3=nd in the following discussion unless otherwise specified. Due to the perfect spherical symmetry of the proposed spherical concentric core–shell structure, the absorption (scattering) of a plane wave by the proposed HMM cavity can be solved analytically based on Mie theory [41]. In the calculation, the permittivity of the silver is expressed by a Drude model:

εAg=εωp2/[ω(ω+iω)],
where the parameters (ω=3.8355, ωp=1.3937×1016rad/s, and ωc=2.9588×1013rad/s) of the Drude model are extracted by fitting the experimental data of Johnson and Christy [42]. Previous studies have demonstrated that the spherical HHM cavity could support multipolar WGMs with the resonant energy highly localized into the different dielectric layers [43,44]. Unlike in these papers, we first calculate the absorption efficiency spectrum for the HMM cavity with a relatively large dielectric core (rcore=50nm); other structural/material parameters are ncore=2.0, s=15nm, d=20nm, and nd=1.4. One of the main advantages of the analytical Mie theory is the ability to obtain a decomposed spectrum, which is characterized by electrical (al) and magnetic (bl) absorption (scattering) coefficients, where the subscript l is the angular mode number [41]. Figure 1(b) displays the absorption efficiency spectra for the first two electric terms (a1, a2) and the first magnetic term (b1) of the Mie expansion, and it is clear that the electric terms of both a1 and a2 have four sharped absorption peaks, while the magnetic term of b1 has no sharp absorption peak, with ultralow intensity in the displayed energy range. As in Ref. [44], the three sharp and strong absorption peaks of a1 or a2 at lower resonant energies correspond to the excitations of three WGMs in the HMM cavity, and thus the total six absorption peaks marked as WGM1,1, WGM1,2, WGM1,3, WGM2,1, WGM2,2, and WGM2,3 are observed at resonant energies of E0.602eV, 0.873 eV, 1.305 eV, 1.021 eV, 1.446 eV, and 2.024 eV, respectively. In addition to the excitations of three WGMs for each electric term (a1 or a2) in the lower energy range, one additional sharp absorption peak marked as TM1,1 (a1) or TM2,1 (a2) is also observed at the higher resonant energy of E2.827eV or E3.217eV, which corresponds to the excitations of plasmonic cavity modes in the HMM cavity. Note that both the WGMs and the TM modes in the spectrum can be identified by two mode numbers (WGMl,m and TMl,m), where l denotes the angular mode number and m is the order of the mode.
 figure: Fig. 1.

Fig. 1. (a) Schematic of a spherical HMM cavity composed of a dielectric nanosphere core (radius rcore; refractive index ncore) and seven alternating layers of silver (thickness: s1, s2, s3, and s4) and dielectric layers (thickness d1, d2, d3, and d4; refractive index nd1, nd2, nd3, and nd4). (b) Calculated decomposed absorption efficiency spectrum for the first two electric terms (a1, a2) and the first magnetic term (b1) of the Mie expansion for an HMM cavity with the parameters rcore=50nm, ncore=2.0, s1=s2=s3=s4=15nm, d1=d2=d3=20nm, and nd1=nd2=nd3=nd4=1.4. The b1 term is enlarged 50 times for clarity. (c)–(h) The electric field intensity distributions of WGM1,1 (0.602 eV), WGM1,2 (0.873 eV), WGM1,3 (1.305 eV), TM1,1 (2.827 eV), WGM2,1 (1.021 eV), and TM2,1 (3.217 eV), respectively. Dashed circle lines show the silver/dielectric interfaces of the HMM cavity.

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In order to further understand these sharp resonant modes supported by the spherical HMM cavity, we perform the near-field profiles at the selected energies using an analytical Mie solution. Figures 1(c)–1(h) show the electric field intensity (|E/E0|2) distributions calculated at the resonances WGM1,1 (E0.602eV), WGM1,2 (E0.873eV), WGM1,3 (E1.305eV), TM1,1 (E2.827eV), WGM2,1 (E1.021eV), and TM2,1 (E3.217eV), respectively. It is clear in Figs. 1(c)–1(h) that the electric fields for all resonant modes are tightly confined in the different dielectric layers (WGMs) or the dielectric core (TM modes) of the HMM cavity, and their intensities are all greatly enhanced with factors from 101 to 2550. In addition, the electric field distributions of WGM1,1, WGM1,2, and WGM1,3 shown in Figs. 1(c)–1(e) are found to exhibit a similar twofold symmetry, which reveals that these three resonances correspond to the excitations of a dipolar WGM with an angular mode number of l=1. The difference in the field distribution is that the electric fields for WGM11, WGM1,2, and WGM1,3 are mainly concentrated in the first (d1), second (d2), and third dielectric layers (d3) of the HMM cavity, respectively. This confirms that WGM1,1, WGM1,2, and WGM1,3 are the first-order (m=1), second-order (m=2), and third-order modes (m=3), respectively, from the dipolar WGMs (l=1). For the resonance TM1,1 (E2.827eV), the electric fields are mainly localized within the dielectric core, showing a single-lobed feature [Fig. 1(f)], which corresponds to the excitations of electric dipolar cavity plasmons (l=1) in the HMM cavity. Similar to the analysis above, the electric field intensity distributions for both WGM2,1 and TM2,1 modes should have the same fourfold symmetry feature (l=2), and should mainly be concentrated within the first dielectric layer (d1) and the dielectric core, as displayed in Figs. 1(g) and 1(h), respectively. The above results clearly demonstrate that both the plasmonic WGMs and the cavity plasmon modes can be excited in the HMM cavity. In addition, the sharpness of the WGMs and TM modes in the absorption spectrum reveals a long lifetime of τ1484fs (Fig. 6 in Appendix A). As a result, the spherical HMM cavity can provide us with an ideal platform for investigating the coupling effect between two high-Q (long-lifetime) plasmonic modes.

B. Ultralarge Rabi Splitting via Strong Coupling between WGM1,3 and TM1,1 Modes

In the following, we mainly focus on the coupling effect between dipolar cavity plasmons (TM1,1) and dipolar WGMs (a1) in the spherical HMM cavity. In Fig. 1(b), there is a large energy difference (ΔE) between the WGMs and TM1,1 mode. To achieve coupling between these resonant modes, ΔE should be reduced by adjusting the structural (rcore, d) and material parameters (ncore, nd) of the HMM cavity. As demonstrated above, the electric field intensity distribution feature (WGMs mainly concentrated within the dielectric layers and TM1,1 mode mainly concentrated within the dielectric core) allows us to tune their resonant energies by independently changing the parameters rcore, ncore, d, and nd of the HMM cavity. Compared with changing the parameters rcore, d, and nd, adjusting ncore has been demonstrated to be the most effective method for reducing ΔE (Fig. 7 in Appendix A). The parameters d and nd are intentionally chosen to be 20 nm and 1.4 in the following discussion. The absorption efficiency spectra (a1) of the HMM cavity (rcore=50nm, s=15nm) are shown in Fig. 2(a) as ncore increases from 2.9 to 6.0. It is clear in Fig. 2(a) that the resonant energies of WGM1,1 and WGM1,2 are almost entirely unshifted as ncore increases from 2.9 to 6.0. The resonant energy of TM1,1 mode clearly decreases (longer wavelength) with increasing ncore, and displays apparent anti-crossing behavior as TM1,1 mode approaches WGM1,3.

 figure: Fig. 2.

Fig. 2. (a) Absorption efficiency spectra (a1) of a spherical HMM cavity (rcore=50nm, s=15nm, d=20nm, and nd=1.4) as a function of the dielectric core index (ncore). The two red lines with circles present the dispersions of two hybrid modes predicted by the CTM fitting. The dashed white horizontal and oblique lines represent the resonant energy (left y axis) and wavelength (right y axis) of the uncoupled WGM1,3 and TM1,1 mode, respectively. (b) The absorption efficiency spectrum of the spherical HMM cavity with a dielectric core index of ncore=4.3 (ETM1,1=EWGM1,3). The blue lines are the Fano fitting results for the calculated absorption peaks.

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To further reveal the physical origins of the observed anti-crossing effect between WGM1,3 and TM1,1 mode, we use a semiclassical coupled two-oscillator model (CTM) in the following expression [34,36]:

Ehybrid±=ETM+EWGM2±Δ2+(ETMEWGM)24,
where Δ is the coupling strength and EWGM and ETM are the resonant energies of the uncoupled WGM1,3 and TM1,1 mode, respectively. To accurately obtain the resonant energies of the uncoupled WGM1,3 and TM1,1 mode, the coupling effect between these two modes should be completely excluded. As has been demonstrated above, most of the electric fields of the WGMs are tightly concentrated within the dielectric layers of the HMM cavity (Fig. 1). This feature reveals that the resonant energies of the WGMs should be insensitive to the core materials, even for metals. In addition, if we set the dielectric core of the HMM cavity as a solid silver nanosphere, the cavity plasmons cannot be supported. As a result, the coupling effect between cavity plasmons and WGMs can be completely excluded. The resonant energy of the uncoupled WGM1,3 is extracted as 1.444 eV by changing the dielectric core of the HMM cavity to a same-sized solid silver nanosphere, which is displayed as the horizontal white dashed line in Fig. 2(a). Similarly, the resonant energy of the uncoupled TM1,1 mode can be obtained via a metallic nanoshell, which cannot support plasmonic WGMs. The cavity plasmons supported by the metallic shell are highly dependent on both the dielectric core and the metal layer thickness [45]. For a silver nanoshell with a thicker silver layer (more than 60 nm), the cavity plasmons cannot be efficiently excited by the visible and near-IR light because the silver thickness is much greater than the optical skin depth in the current spectral range [45]. For a thinner silver layer (less than 15 nm), the light confinement ability is weak, and thus it cannot effectively concentrate the resonant energy within the dielectric core of the silver nanoshell, which also results in a broad feature in the far-field spectrum due to the large radiative loss [45]. Therefore, the resonant energy of an uncoupled TM1,1 mode is extracted in a silver nanoshell by choosing an appropriate value for the silver thickness (30 nm) for the same dielectric core size (rcore=50nm). The resonant energies of uncoupled TM1,1 modes as a function of ncore are shown by the oblique white dashed line in Fig. 2(a).

After we perform the CTM model, the resonant energies of the two hybrid modes are predicted and displayed with the red lines with circles in Fig. 2(a). Comparing the CTM model (red/circle lines) with the theoretical results (Mie theory) shows good agreement [Fig. 2(a)], which successfully reproduces the anti-crossing behavior of WGM1,3 and TM1,1 mode. As a result, the normal Rabi splitting energy is obtained at E(TM1,1)=E(WGM1,3) or ncore=4.3, with an ultralarge value of up to 629 meV (Ω=629meV) displayed in Fig. 2(b), which corresponds to a large coupling strength of Δ=0.198(eV)2 (Ω=2Δ). It should be pointed out that the refractive index of several semiconductor materials can reach a value of 4.3, such as Si, Ge, and GaAs [46,47]. We also note that Ω(ΓTM+ΓWGM)/2=28meV as required for SC. It is also clearly seen in Fig. 2(a) that the TM1,1 resonance can also result in anti-crossing behavior with WGM1,2 mode when shifting close to the resonant energy of WGM1,2 at a large ncore (6), while the Rabi splitting energy of TM1,1 and WGM1,2 is much smaller than that of TM1,1 and WGM1,3 due to the fact that the electric field spatial overlap between TM1,1 mode and WGM1,3 is much larger than that of WGM1,2 and WGM1,1 both at the dielectric core and in the dielectric layers (Fig. 8 in Appendix A). In Fig. 2(b), we further find that two hybrid modes formed by SC exhibit ultralong lifetimes of 71.9 fs (Ehybrid=1.1117eV) and 81.6 fs (Ehybrid+=1.7405eV), which are about 2 times larger than those of the uncoupled WGM1,3 and TM1,1 mode (Fig. 6 in Appendix A).

Next, we characterize the near-field distributions of the two hybrid modes formed by SC in the HMM cavity. Figures 3(a) and 3(c) display the electric field intensity distributions of the lower-energy hybrid mode (Hybrid) at 1.1117 eV and the higher-energy hybrid mode (Hybrid+) at 1.7405 eV, respectively. It is clear in Figs. 3(a) and 3(c) that the electric fields are showing similar features and are mostly concentrated within the third dielectric layer (d3), with the maximum enhancement factors up to 886 and 1545. The electric field intensities along the white dashed lines in Figs. 3(a) and 3(c) are also displayed in Figs. 3(b) and 3(d), which clearly show that both the dielectric core (97–170) and the third dielectric layer (d3) have large electric field enhancement, revealing the SC feature (strong energy change) between WGM1,3 and TM1,1 mode.

 figure: Fig. 3.

Fig. 3. (a), (c) Electric field intensity distributions of Hybrid (resonant energy: 1.1117 eV) and Hybrid+ (resonant energy: 1.7405 eV) at the kE plane, respectively; (b), (d) electric field intensities along the white dashed lines in (a) and (c), respectively. The light blue and turquoise vertical stripes in (b) and (d) are used to indicate the dielectric core and dielectric layers (d1, d2, and d3) of the HMM cavity, respectively. Dashed circle lines show the silver/dielectric interfaces of the HMM cavity.

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For a deeper understanding of the two hybrid resonances, the x component of the electric field (Ex) is further calculated by Mie theory [41]. Figures 4(a) and 4(c) display the Ex of the lower-energy hybrid mode (Hybrid) at 1.1117 eV and the higher-energy hybrid mode (Hybrid+) at 1.7405 eV, respectively. The Ex values along the white dashed lines in Figs. 4(a) and 4(c) are further shown in Figs. 4(b) and 4(d), respectively. It is clear in Fig. 4 that the Ex for two hybrid modes shows the same positive phase at the dielectric layers, but displays the opposite phases both at the dielectric core (negative phase for Hybrid and positive phase for Hybrid+) and in the silver layers (positive phase for Hybrid and negative phase for Hybrid+).

 figure: Fig. 4.

Fig. 4. (a), (c) x component of the electric field (Ex) for Hybrid (resonant energy: 1.1117 eV) and Hybrid+ (resonant energy: 1.7405 eV) at the kE plane, respectively; (b), (d) Ex along the white dashed line in (a) and (c), respectively. The light blue and turqoise vertical stripes in (b) and (d) are used to indicate the dielectric core and dielectric layers (d1, d2, and d3) of the HMM cavity, respectively.

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In the following, we further demonstrate that the normal Rabi splitting energy (Ω) or the coupling strength (Δ) is mainly determined by the inner silver thickness (s4). As s4 increases from 15 nm to 25 nm and to 35 nm, not only does the Rabi splitting energy (or coupling strength Δ) decrease quickly from 629 meV [Δ=0.198(eV)2] to 335 meV [Δ=0.056(eV)2] and to 185 meV [Δ=0.017(eV)2], but the ncore also increases rapidly from 4.3 to 5.0 and to 5.5, respectively. The values of 5.0 and 5.5 obviously exceed the refractive index of most materials in the visible to the near-IR spectral range (Fig. 9 in Appendix A).

C. Broadband Strong Coupling Produced by Varying the Dielectric Core Size

So far, we have theoretically demonstrated that the SC between WGM1,3 and TM1,1 mode in an HMM cavity with a dielectric core radius of rcore=50nm occurs in the near-IR spectral range from 1.1117 eV to 1.7405 eV (1115.4 nm to 712.4 nm). To further achieve the SC effect over a broad spectral range (from the visible to the near-IR spectral range), we change the dielectric core radius of the HMM cavity from 30 nm to 80 nm, and other parameters are the same as for the HMM cavity with rcore=50nm. The calculated absorption efficiency spectra for the HMM cavity (s=15nm, d=20nm, nd=1.4) with dielectric core radii of 30 nm, 40 nm, 60 nm, and 80 nm as a function of ncore are displayed in Fig. 5(a). It is obvious in Fig. 5(a) that all the absorption efficiency spectra show anti-crossing behavior between WGM1,3 and TM1,1 mode. In addition, the resonant energies of the hybrid modes (red lines with circles) predicted by the CTM fitting show excellent agreement with the theoretical values (Mie theory). What is more, it is especially interesting that the dielectric core indices (ncore) where the SC occurs at EWGM1,3=ETM1,1 maintain the same value of 4.3 for different radii of the HMM cavity.

 figure: Fig. 5.

Fig. 5. (a) Absorption efficiency spectra (a1) of the spherical HMM cavity (s=15nm, d=20nm, and nd=1.4) as a function of ncore for dielectric core radii of (bottom to top) 30 nm, 40 nm, 60 nm, and 80 nm. The two red lines with circles in each panel present the dispersions of two hybrid modes predicted by the CTM fitting. The dashed white horizontal and oblique lines in each panel represent the resonant energies of the uncoupled WGM1,3 and TM1,1 mode, respectively. The vertical dashed line indicates the dielectric core index ncore=4.3 where the resonant energies of two uncoupled modes are equal, ETM1,1=EWGM1,3. (b) The absorption efficiency spectra of the spherical HMM cavity at ncore=4.3 with dielectric core radii of 30 nm (black line), 40 nm (red line), 60 nm (blue line), and 80 nm (magenta line) (with absorption spectra offset vertically for clarity). (c) The Rabi splitting energy as a function of dielectric core radius (rcore).

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The absorption efficiency spectra of the spherical HMM cavity at ncore=4.3 with dielectric core radii of 30 nm, 40 nm, 60 nm, and 80 nm are shown in Fig. 5(b). For the HMM cavity with a relatively small dielectric core radius of rcore=30nm, a record normal Rabi splitting energy as large as 788 meV [Δ=0.31(eV)2] arising from the SC between WGM1,3 and TM1,1 mode is observed in the visible spectral range from 2.23 eV (556.1 nm) to 1.441 eV (860.5 nm), as indicated by the black line in Fig. 5(b). As rcore increases from 30 nm to 40 nm, to 60 nm, and to 80 nm, the Rabi splitting energy decreases from 788 meV (black line) to 713 meV (red line), to 552 meV (blue line), and to 428 meV (magenta line), respectively. It should be noted that the Rabi splitting energy for the HMM cavity with a relatively large dielectric core radius of rcore=80nm still reaches a relatively high value of 428 meV, and a high-order cavity plasmon mode of TM1,2 appears at 2.813 eV in the current spectral range, which is shown by the magenta line in Fig. 5(b). In addition, the spectral ranges where the SC occurs are shifted from 2.23–1.441 eV (556.1–860.3 nm) to 1.965–1.252 eV (631–990.4 nm), to 1.554–1.003 eV (797.7–1236.4 nm), and to 1.273–0.844 eV (974.4–1469.1 nm) as rcore increases from 30 nm to 40 nm, to 60 nm, and to 80 nm [Fig. 5(b)], demonstrating that the SC between WGM1,3 and TM1,1 mode in the HMM cavity can be tuned across a broad spectral range by simply changing the dielectric core size.

Figure 5(c) displays the Rabi splitting energy in the HMM cavity as a function of rcore (from 10 nm to 80 nm), and presents a trend of first an increase and then a decrease, revealing a maximum Rabi splitting energy of 805 meV at rcore=20nm. As rcore is increased from 20 nm to 80 nm, the Rabi splitting energy linearly decreases from 805 meV to 428 meV, due to the fact that the longer-wavelength light has a smaller optical skin depth in silver, and the coupling strength (Rabi splitting) is thus reduced for the longer-wavelength modes supported by the larger rcore of the HMM cavity. For the smaller rcore, the coupling strength (Rabi splitting) between WGM1,3 and TM1,1 mode in the HMM cavity is also limited when the resonant modes blueshift to the vicinity of the interband transition of the silver [42,43]. It is shown in Fig. 5(c) that the Rabi splitting energy decreases quickly from 805 meV at rcore=20nm to 626 meV at rcore=10nm.

3. CONCLUSION

In summary, we have theoretically demonstrated a record Rabi splitting energy of up to 805 meV, arising from the SC between two high-Q (long-lifetime) plasmonic resonances (WGM1,3 and TM1,1 mode) in a spherical HMM cavity. The strong energy exchange between these two modes leads to the formation of new hybrid modes, which exhibit a longer lifetime (τ71.981.6fs) and great enhancement of the electric field intensity both at the dielectric core and in the dielectric layers in the HMM cavity. Importantly, by adjusting the dielectric core size, we have shown that the SC occurs at the same dielectric core index and can be tuned across a broad range from the visible to the near-IR spectral range (broadband nature). These results provide direction for achieving ultralong lifetimes of surface plasmons and may find potential applications in bright single-photon sources [48,49].

APPENDIX A: DETAILED NUMERICAL CALCULATION METHOD

1. Lifetime of the Resonant WGMs and TM Cavity Plasmon Modes

To calculate the lifetime (τ) of resonant WGMs and TM modes in a spherical HMM cavity, the sharp absorption peaks are first fitted using a Fano formula: F(ε)=σbg+σ0(ε+q)2/(1+ε2), where σ0 and σbg are the normalized and background absorption, q is the asymmetry parameter, and ε=2(EEres)/Γ, with Eres and Γ being the resonant energy and linewidth of the mode, respectively. The Fano fitting results for WGM1,1, WGM1,2, WGM1,3, TM1,1, WGM2,1, WGM2,2, WGM2,3, and TM2,1 are shown as olive lines in Figs. 6(a)–6(h), and show good agreement with the theoretical absorption efficiency spectra. With the extracted linewidth of Γ, the lifetime (τ) of the resonant mode is then calculated by the formula τ=2/Γ, where =6.582119514×1016eV·ns.

 figure: Fig. 6.

Fig. 6. Fano fitting (olive lines) for the multiple absorption peaks (hollow red circles) of a spherical HMM cavity (rcore=50nm, ncore=2.0, s=15nm, d=20nm, and nd=1.4): (a) WGM1,1, (b) WGM1,2, (c) WGM1,3, (d) TM1,1, (e) WGM2,1, (f) WGM2,2, (g) WGM2,3, (h) TM2,1.

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2. Resonant Energy and Energy Difference (ΔE) between TM1,1 Mode and WGMs

The resonant energies of the TM1,1 mode (magenta triangles), WGM1,3 (blue triangles), WGM1,2 (red circles), and WGM1,1 (black squares) in the spherical HMM cavity as a function of dielectric layer thickness (d), dielectric layer index (nd), dielectric core radius (rcore), and dielectric core index (ncore) are shown in Figs. 7(a)–7(d), respectively. Figures 7(e)–7(h) display the corresponding energy differences (ΔE) between the TM1,1 mode and WGM1,3/WGM1,2/WGM1,1[ΔE(TM1,1WGM1,3), blue trangles; ΔE(TM1,1WGM1,2), red circles; ΔE(TM1,1WGM1,1), black squares] as a function of d, nd, rcore, and ncore, respectively. For the dielectric layer thickness of d values smaller than 20 nm, the resonant energies of WGM1,1, WGM1,2, and WGM1,3 are obviously reduced (longer wavelength) due to the strong layer-by-layer coupling [Fig. 7(a)].

 figure: Fig. 7.

Fig. 7. (a)–(d) Resonant energies of TM1,1 (magenta triangles), WGM1,3 (blue triangles), WGM1,2 (red circles), and WGM1,1 (black squares) as a function of dielectric layer thickness (d), dielectric layer index (nd), dielectric core radius (rcore), and dielectric core index (ncore), respectively; (e)–(h) energy differences (ΔE) between TM1,1 mode and WGM1,3/WGM1,2/WGM1,1 [ΔE(TM1,1WGM1,3), blue trangles; ΔE(TM1,1WGM1,2), red circles; ΔE(TM1,1WGM1,1), black squares] as a function of d, nd, rcore, and ncore, respectively.

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 figure: Fig. 8.

Fig. 8. Radial (vertical direction) electric field intensity distributions of (a) WGM1,1, (b) WGM1,2, (c) WGM1,3, and (d) TM1,1. The light blue and turquoise vertical stripes are used to indicate the dielectric core and dielectric layers (d1, d2, and d3), respectively, of the HMM cavity (rcore=50nm, ncore=2.0, s=15nm, d=20nm, and nd=1.4).

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 figure: Fig. 9.

Fig. 9. Calculated absorption efficiency spectra (a1) of the spherical HMM cavity (rcore=50nm, s=15nm, d=20nm, and nd=1.4) as a function of ncore with an inner silver thickness (s4) of (a) 15 nm, (b) 25 nm, and (c) 35 nm. The two red lines with circles in each panel show the resonant energies of two hybrid modes predicted by the CTM fitting.

Download Full Size | PPT Slide | PDF

However, for d values larger than 20 nm, the resonant energies begin to converge as the layer-by-layer coupling becomes weak [Fig. 7(a)]. The resonant energy of TM1,1 mode shows a slightly linear decrease as d increases from 2 nm to 50 nm [Fig. 7(a)]. It should be noted that the strong layer-by-layer coupling leads to a larger ΔE (2.5eV) between TM1,1 mode and WGM1,1/WGM1,2/WGM1,3 for a 2 nm dielectric layer thickness, and ΔE decreases as d increases from 2 nm to 50 nm (decreases quickly before 20 nm and decreases slowly after 20 nm) [Fig. 7(e)]. Although ΔE can be more smaller for a larger d, a larger d leads to a lower excitation efficiency of the resonant modes and a relatively large HMM cavity. As a result, when we comprehensively consider ΔE, the exicitation efficiency, and the cavity size, the dielectric layer thickness (d) of the sperical HMM cavity is choosen to be 20 nm in this study. In Fig. 7(b), the resonant energies of both TM1,1 mode and WGM1,1/WGM1,2/WGM1,3 all decrease linearly as the dielectric layer index (nd) increases from 1.4 to 2.0. It is also shown in Fig. 7(b) that the slope of the TM1,1 mode [0.276eV/RIU (refractive index unit)] is smaller than that of WGM1,1 (0.294eV/RIU), WGM1,2 (0.423eV/RIU), or WGM1,3 (0.611eV/RIU), resulting in ΔE increasing as nd increases from 1.4 to 2.0 [Fig. 7(f)]. Therefore, nd is choosen and fixed to be 1.4 in this paper, which corresponds to the refractive index of MgF2 material. The resonant energies of both TM1,1 mode and WGM1,1/WGM1,2/WGM1,3 are also decreasing near-linearly as the dielectric core radius (rcore) increases from 30 nm to 100 nm [Fig. 7(c)]. However, the slope of TM1,1 mode (–0.019 eV/nm) is obviously larger than that of WGM1,1 (–0.0037 eV/nm), WGM1,2 (0.0066eV/nm), or the WGM1,3 (0.012eV/nm), resuling in ΔE decreasing as rcore increases from 30 nm to 100 nm, as shown in Fig. 7(g). As the dielectric core index (ncore) increases from 1.8 to 2.4, the resonant energy of TM1,1 mode decreases linearly, while the resonant energies of WGM1,1, WGM1,2, and WGM1,3 all remain at the same value [Fig. 7(d)]. It is clear in Fig. 7(h) that ΔE decreases linealy as ncore increases from 1.8 to 2.4, demonstrating that adjusting the dielectric core index (ncore) is the most effective method of reducing ΔE while keeping the HMM cavity size unchanged.

3. Electric Field Spatial Overlap between TM1,1 Resonance and WGM1,3/WGM1,2/WGM1,1

Figure 8 displays the spatial overlap of electric field between TM1,1 resonance and WGM1,3/WGM1,2/WGM1,1 in the HMM cavity (rcore = 50 nm, ncore = 2.0, s = 15 nm, d =20 nm, and nd = 1.4), clearly revealing that the TM1,1 mode [Fig. 8(d)] has the largest electric field spatial overlap (both at the dielectric core and in the dielectric layers) with the WGM1,3 resonance [Fig. 8(c)] as compared to the WGM1,2 [Fig. 8(b)] and WGM1,1 [Fig. 8(a)].

4. Rabi Splitting versus the Inner Silver Thickness

Figure 9 shows the normal Rabi splitting energy (ℏΩ) or coupling strength (∆) in the HMM for different inner silver thickness (s4). It is clear in Fig. 9 that the Rabi splitting energy (or coupling strength) decreases quickly from 629 meV [∆ = 0.198 (eV)2] to 335 meV [∆ = 0.056 (eV)2] and to 185 meV [∆ = 0.017 (eV)2] when the s4 increases from 15 nm [Fig. 9(a)] to 25 nm [Fig. 9(b)] and to 35 nm [Fig. 9(c)]. In addition, the ncore where the SC occurs at EWGM1,3=ETM1,1 increases rapidly from 4.3 to 5.0 and to 5.5 (Fig. 9).

Funding

National Natural Science Foundation of China (11704183, 11974188, 11704184, 51701176); Double Innovation Project of Jiangsu Province (CZ106SC19010); NUPTSF (NY219015).

Acknowledgment

G. P. thanks the NUPTSF and the Double Innovation Project of Jiangsu Province.

Disclosures

The authors declare no conflicts of interest.

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References

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    [Crossref]
  2. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
    [Crossref]
  3. P. Song, J. H. Wang, M. Zhang, F. Yang, H. J. Lu, B. Kang, J. J. Xu, and H. Y. Chen, “Three-level spaser for next-generation luminescent nanoprobe,” Sci. Adv. 4, eaat0292 (2018).
    [Crossref]
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    [Crossref]
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    [Crossref]
  6. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3, 654–657 (2009).
    [Crossref]
  7. G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8, 835–840 (2014).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  12. F. Kato, H. Minamimoto, F. Nagasawa, Y. S. Yamamoto, T. Itoh, and K. Murakoshi, “Active tuning of strong coupling states between dye excitons and localized surface plasmons via electrochemical potential control,” ACS Photon. 5, 788–796 (2018).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  20. L. Liu, L. Y. M. Tobing, X. Yu, J. Tong, and B. Qiang, “Strong plasmon–exciton interactions on nanoantenna array–monolayer WS2 hybrid system,” Adv. Opt. Mater. 8, 1901002 (2019).
    [Crossref]
  21. X. Yan and H. Wei, “Strong plasmon–exciton coupling between lithographically defined single metal nanoparticles and monolayer WSe2,” Nanoscale 12, 9708–9716 (2020).
    [Crossref]
  22. M. Geisler, X. Cui, J. Wang, T. Rindzevicius, L. Gammelgaard, B. S. Jessen, P. A. D. Goncalves, F. Todisco, P. Bøggild, A. Boisen, M. Wubs, N. A. Mortensen, and S. Xiao, “Single-crystalline gold nanodisks on WS2 mono- and multilayers for strong coupling at room temperature,” ACS Photon. 6, 994–1001 (2019).
    [Crossref]
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    [Crossref]
  24. Y. Zhang, Q. S. Meng, L. Zhang, Y. Luo, Y. J. Yu, B. Yang, Y. Zhang, R. Esteban, J. Aizpurua, Y. Luo, J. K. Yang, Z. C. Dong, and J. G. Hou, “Sub-nanometre control of the coherent interaction between a single molecule and a plasmonic nanocavity,” Nat. Commun. 8, 15225 (2017).
    [Crossref]
  25. H. Groß, J. M. Hamm, T. Tufarelli, O. Hess, and B. Hecht, “Near-field strong coupling of single quantum dots,” Sci. Adv. 4, eaar4906 (2018).
    [Crossref]
  26. H. Leng, B. Szychowski, M. C. Daniel, and M. Pelton, “Strong coupling and induced transparency at room temperature with single quantum dots and gap plasmons,” Nat. Commun. 9, 4012 (2018).
    [Crossref]
  27. P. Zeng, J. Cadusch, D. Chakraborty, T. A. Smith, A. Roberts, J. E. Sader, and T. J. Davis, “Photoinduced electron transfer in strong coupling regime: waveguide–plasmon polaritons,” Nano Lett. 16, 2651–2656 (2016).
    [Crossref]
  28. Z. Xi, Y. Lu, W. Yu, P. Yao, P. Wang, and H. Ming, “Strong coupling between plasmonic Fabry-Pérot cavity mode and magnetic plasmon,” Opt. Lett. 38, 1591–1593 (2013).
    [Crossref]
  29. C. Hägglund, G. Zeltzer, R. Ruiz, A. Wangperawong, K. Roelofs, and S. F. Bent, “Strong coupling of plasmon and nanocavity modes for dual band, near-perfect absorbers and ultrathin photovoltaics,” ACS Photon. 3, 456–463 (2016).
    [Crossref]
  30. X. Shi, K. Ueno, T. Oshikiri, Q. Sun, K. Sasaki, and H. Misawa, “Enhanced water splitting under modal strong coupling conditions,” Nat. Nanotechnol. 13, 953–958 (2018).
    [Crossref]
  31. Y. Chu and K. B. Crozier, “Experimental study of the interaction between localized and propagating surface plasmons,” Opt. Lett. 34, 244–246 (2009).
    [Crossref]
  32. C. Zhang, J. Fang, W. Yang, Q. Song, and S. Xiao, “Enhancing the magnetic resonance via strong coupling in optical metamaterials,” Adv. Opt. Mater. 5, 1700469 (2017).
    [Crossref]
  33. H. Shan, Y. Yu, X. Wang, Y. Luo, S. Zu, B. Du, T. Han, B. Li, Y. Li, J. Wu, F. Lin, K. Shi, B. K. Tay, Z. Liu, X. Zhu, and Z. Fang, “Direct observation of ultrafast plasmonic hot electron transfer in the strong coupling regime,” Light Sci. Appl. 8, 9 (2019).
    [Crossref]
  34. J. Yang, Q. Sun, K. Ueno, X. Shi, T. Oshikiri, H. Misawa, and Q. Gong, “Manipulation of the dephasing time by strong coupling between localized and propagating surface plasmon modes,” Nat. Commun. 9, 4858 (2018).
    [Crossref]
  35. W. Ren, Y. Dai, H. Cai, H. Ding, N. Pan, and X. Wang, “Tailoring the coupling between localized and propagating surface plasmons: realizing Fano-like interference and high-performance sensor,” Opt. Express 21, 10251–10258 (2013).
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2020 (1)

X. Yan and H. Wei, “Strong plasmon–exciton coupling between lithographically defined single metal nanoparticles and monolayer WSe2,” Nanoscale 12, 9708–9716 (2020).
[Crossref]

2019 (7)

M. Geisler, X. Cui, J. Wang, T. Rindzevicius, L. Gammelgaard, B. S. Jessen, P. A. D. Goncalves, F. Todisco, P. Bøggild, A. Boisen, M. Wubs, N. A. Mortensen, and S. Xiao, “Single-crystalline gold nanodisks on WS2 mono- and multilayers for strong coupling at room temperature,” ACS Photon. 6, 994–1001 (2019).
[Crossref]

L. Liu, L. Y. M. Tobing, X. Yu, J. Tong, and B. Qiang, “Strong plasmon–exciton interactions on nanoantenna array–monolayer WS2 hybrid system,” Adv. Opt. Mater. 8, 1901002 (2019).
[Crossref]

H. Shan, Y. Yu, X. Wang, Y. Luo, S. Zu, B. Du, T. Han, B. Li, Y. Li, J. Wu, F. Lin, K. Shi, B. K. Tay, Z. Liu, X. Zhu, and Z. Fang, “Direct observation of ultrafast plasmonic hot electron transfer in the strong coupling regime,” Light Sci. Appl. 8, 9 (2019).
[Crossref]

Y. Ji, C. Tang, N. Xie, J. Chen, P. Gu, C. Peng, and B. Liu, “High-performance metamaterial sensors based on strong coupling between surface plasmon polaritons and magnetic plasmon resonances,” Results Phys. 14, 102397 (2019).
[Crossref]

T. Xue, W. Liang, Y. Li, Y. Sun, Y. Xiang, Y. Zhang, Z. Dai, Y. Duo, L. Wu, K. Qi, B. N. Shivananju, L. Zhang, X. Cui, H. Zhang, and Q. Bao, “Ultrasensitive detection of miRNA with an antimonene-based surface plasmon resonance,” Nat. Commun. 10, 28 (2019).
[Crossref]

O. S. Ojambati, R. Chikkaraddy, W. D. Deacon, M. Horton, D. Kos, V. A. Turek, U. F. Keyser, and J. J. Baumberg, “Quantum electrodynamics at room temperature coupling a single vibrating molecule with a plasmonic nanocavity,” Nat. Commun. 10, 1049 (2019).
[Crossref]

K. S. Menghrajani, H. A. Fernandez, G. R. Nash, and W. L. Barnes, “Hybridization of multiple vibrational modes via strong coupling using confined light fields,” Adv. Opt. Mater. 7, 1900403 (2019).
[Crossref]

2018 (7)

A. J. Moilanen, T. K. Hakala, and P. Törmä, “Active control of surface plasmon−emitter strong coupling,” ACS Photon. 5, 54–64 (2018).
[Crossref]

F. Kato, H. Minamimoto, F. Nagasawa, Y. S. Yamamoto, T. Itoh, and K. Murakoshi, “Active tuning of strong coupling states between dye excitons and localized surface plasmons via electrochemical potential control,” ACS Photon. 5, 788–796 (2018).
[Crossref]

P. Song, J. H. Wang, M. Zhang, F. Yang, H. J. Lu, B. Kang, J. J. Xu, and H. Y. Chen, “Three-level spaser for next-generation luminescent nanoprobe,” Sci. Adv. 4, eaat0292 (2018).
[Crossref]

J. Yang, Q. Sun, K. Ueno, X. Shi, T. Oshikiri, H. Misawa, and Q. Gong, “Manipulation of the dephasing time by strong coupling between localized and propagating surface plasmon modes,” Nat. Commun. 9, 4858 (2018).
[Crossref]

X. Shi, K. Ueno, T. Oshikiri, Q. Sun, K. Sasaki, and H. Misawa, “Enhanced water splitting under modal strong coupling conditions,” Nat. Nanotechnol. 13, 953–958 (2018).
[Crossref]

H. Groß, J. M. Hamm, T. Tufarelli, O. Hess, and B. Hecht, “Near-field strong coupling of single quantum dots,” Sci. Adv. 4, eaar4906 (2018).
[Crossref]

H. Leng, B. Szychowski, M. C. Daniel, and M. Pelton, “Strong coupling and induced transparency at room temperature with single quantum dots and gap plasmons,” Nat. Commun. 9, 4012 (2018).
[Crossref]

2017 (6)

Y. Zhang, Q. S. Meng, L. Zhang, Y. Luo, Y. J. Yu, B. Yang, Y. Zhang, R. Esteban, J. Aizpurua, Y. Luo, J. K. Yang, Z. C. Dong, and J. G. Hou, “Sub-nanometre control of the coherent interaction between a single molecule and a plasmonic nanocavity,” Nat. Commun. 8, 15225 (2017).
[Crossref]

M. E. Kleemann, R. Chikkaraddy, E. M. Alexeev, D. Kos, C. Carnegie, W. Deacon, A. C. Pury, C. Große, B. Nijs, J. Mertens, A. I. Tartakovskii, and J. J. Baumberg, “Strong-coupling of WSe2 in ultra-compact plasmonic nanocavities at room temperature,” Nat. Commun. 8, 1296 (2017).
[Crossref]

D. Zheng, S. Zhang, Q. Deng, M. Kang, P. Nordlander, and H. Xu, “Manipulating coherent plasmon-exciton interaction in a single silver nanorod on monolayer WSe2,” Nano Lett. 17, 3809–3814 (2017).
[Crossref]

C. Zhang, J. Fang, W. Yang, Q. Song, and S. Xiao, “Enhancing the magnetic resonance via strong coupling in optical metamaterials,” Adv. Opt. Mater. 5, 1700469 (2017).
[Crossref]

M. Wan, P. Gu, W. Liu, Z. Chen, and Z. Wang, “Low threhold spaser based on deep-subwavelength spherical hyperbolic metamaterial cavities,” Appl. Phys. Lett. 110, 031103 (2017).
[Crossref]

P. Senellart, G. Solomon, and A. White, “High-performance semiconductor quantum-dot single-photon sources,” Nat. Nanotechnol. 12, 1026–1039 (2017).
[Crossref]

2016 (7)

Q. Sun, H. Yu, K. Ueno, A. Kubo, Y. Matsuo, and H. Misawa, “Dissecting the few-femtosecond dephasing time of dipole and quadrupole modes in gold nanoparticles using polarized photoemission electron microscopy,” ACS Nano 10, 3835–3842 (2016).
[Crossref]

E. M. Roller, C. Argyropoulos, A. Högele, T. Liedl, and M. Pilo-Pais, “Plasmon-exciton coupling using DNA templates,” Nano Lett. 16, 5962–5966 (2016).
[Crossref]

K. Santhosh, O. Bitton, L. Chuntonov, and G. Haran, “Vacuum Rabi splitting in a plasmonic cavity at the single quantum emitter limit,” Nat. Commun. 7, 11823 (2016).
[Crossref]

C. Hägglund, G. Zeltzer, R. Ruiz, A. Wangperawong, K. Roelofs, and S. F. Bent, “Strong coupling of plasmon and nanocavity modes for dual band, near-perfect absorbers and ultrathin photovoltaics,” ACS Photon. 3, 456–463 (2016).
[Crossref]

P. Zeng, J. Cadusch, D. Chakraborty, T. A. Smith, A. Roberts, J. E. Sader, and T. J. Davis, “Photoinduced electron transfer in strong coupling regime: waveguide–plasmon polaritons,” Nano Lett. 16, 2651–2656 (2016).
[Crossref]

D. Melnikau, R. Esteban, D. Savateeva, A. Sánchez-Iglesias, M. Grzelczak, M. K. Schmidt, L. M. Liz-Marzán, J. Aizpurua, and Y. P. Rakovich, “Rabi splitting in photoluminescence spectra of hybrid systems of gold nanorods and J-aggregates,” J. Phys. Chem. Lett. 7, 354–362 (2016).
[Crossref]

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

2015 (2)

G. Zengin, M. Wersäll, S. Nilsson, T. J. Antosiewicz, M. Käll, and T. Shegai, “Realizing strong light-matter interactions between single-nanoparticle plasmons and molecular excitons at ambient conditions,” Phys. Rev. Lett. 114, 157401 (2015).
[Crossref]

B. Ji, E. Giovanelli, B. Habert, P. Spinicelli, M. Nasilowski, X. Xu, N. Lequeux, J. P. Hugonin, F. Marquier, J. J. Greffet, and B. Dubertret, “Non-blinking quantum dot with a plasmonic nanoshell resonator,” Nat. Nanotechnol. 10, 170–175 (2015).
[Crossref]

2014 (3)

C. Wu, A. Salandrino, X. Ni, and X. Zhang, “Electrodynamical light trapping using whispering-gallery resonances in hyperbolic cavities,” Phys. Rev. X 4, 021015 (2014).
[Crossref]

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8, 835–840 (2014).
[Crossref]

L. Shi, T. K. Hakala, H. T. Rekola, J. P. Martikainen, R. J. Moerland, and P. Trömä, “Spatial coherence properties of organic molecules coupled to plasmonic surface lattice resonances in the weak and strong coupling regimes,” Phys. Rev. Lett. 112, 153002 (2014).
[Crossref]

2013 (3)

2011 (1)

T. Schwartz, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Reversible switching of ultrastrong light-molecule coupling,” Phys. Rev. Lett. 106, 196405 (2011).
[Crossref]

2009 (4)

Y. Chu and K. B. Crozier, “Experimental study of the interaction between localized and propagating surface plasmons,” Opt. Lett. 34, 244–246 (2009).
[Crossref]

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3, 654–657 (2009).
[Crossref]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2009).
[Crossref]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[Crossref]

2008 (1)

J. J. Penninkhof, L. A. Sweatlock, A. Moroz, H. A. Atwater, A. van Blaaderen, and A. Polman, “Optical cavity modes in gold shell colloids,” J. Appl. Phys. 103, 123105 (2008).
[Crossref]

2007 (1)

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1, 641–648 (2007).
[Crossref]

2002 (1)

C. Sönnichsen, T. Wilk, G. Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002).
[Crossref]

1998 (1)

T. Klar, M. Perner, S. Grosse, G. Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
[Crossref]

1986 (1)

D. E. Aspnes, S. M. Kelso, R. A. Logan, and R. Bhat, “Optical properties of AlxGa1-xAs,” J. Appl. Phys. 60, 754–767 (1986).
[Crossref]

1983 (1)

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0  eV,” Phys. Rev. B 27, 985–1009 (1983).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Aizpurua, J.

Y. Zhang, Q. S. Meng, L. Zhang, Y. Luo, Y. J. Yu, B. Yang, Y. Zhang, R. Esteban, J. Aizpurua, Y. Luo, J. K. Yang, Z. C. Dong, and J. G. Hou, “Sub-nanometre control of the coherent interaction between a single molecule and a plasmonic nanocavity,” Nat. Commun. 8, 15225 (2017).
[Crossref]

D. Melnikau, R. Esteban, D. Savateeva, A. Sánchez-Iglesias, M. Grzelczak, M. K. Schmidt, L. M. Liz-Marzán, J. Aizpurua, and Y. P. Rakovich, “Rabi splitting in photoluminescence spectra of hybrid systems of gold nanorods and J-aggregates,” J. Phys. Chem. Lett. 7, 354–362 (2016).
[Crossref]

Akselrod, G. M.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8, 835–840 (2014).
[Crossref]

Alexeev, E. M.

M. E. Kleemann, R. Chikkaraddy, E. M. Alexeev, D. Kos, C. Carnegie, W. Deacon, A. C. Pury, C. Große, B. Nijs, J. Mertens, A. I. Tartakovskii, and J. J. Baumberg, “Strong-coupling of WSe2 in ultra-compact plasmonic nanocavities at room temperature,” Nat. Commun. 8, 1296 (2017).
[Crossref]

Antosiewicz, T. J.

G. Zengin, M. Wersäll, S. Nilsson, T. J. Antosiewicz, M. Käll, and T. Shegai, “Realizing strong light-matter interactions between single-nanoparticle plasmons and molecular excitons at ambient conditions,” Phys. Rev. Lett. 114, 157401 (2015).
[Crossref]

Argyropoulos, C.

E. M. Roller, C. Argyropoulos, A. Högele, T. Liedl, and M. Pilo-Pais, “Plasmon-exciton coupling using DNA templates,” Nano Lett. 16, 5962–5966 (2016).
[Crossref]

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Ciracì, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8, 835–840 (2014).
[Crossref]

Aspnes, D. E.

D. E. Aspnes, S. M. Kelso, R. A. Logan, and R. Bhat, “Optical properties of AlxGa1-xAs,” J. Appl. Phys. 60, 754–767 (1986).
[Crossref]

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0  eV,” Phys. Rev. B 27, 985–1009 (1983).
[Crossref]

Atwater, H. A.

J. J. Penninkhof, L. A. Sweatlock, A. Moroz, H. A. Atwater, A. van Blaaderen, and A. Polman, “Optical cavity modes in gold shell colloids,” J. Appl. Phys. 103, 123105 (2008).
[Crossref]

Avlasevich, Y.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3, 654–657 (2009).
[Crossref]

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[Crossref]

Bao, Q.

T. Xue, W. Liang, Y. Li, Y. Sun, Y. Xiang, Y. Zhang, Z. Dai, Y. Duo, L. Wu, K. Qi, B. N. Shivananju, L. Zhang, X. Cui, H. Zhang, and Q. Bao, “Ultrasensitive detection of miRNA with an antimonene-based surface plasmon resonance,” Nat. Commun. 10, 28 (2019).
[Crossref]

Barnes, W. L.

K. S. Menghrajani, H. A. Fernandez, G. R. Nash, and W. L. Barnes, “Hybridization of multiple vibrational modes via strong coupling using confined light fields,” Adv. Opt. Mater. 7, 1900403 (2019).
[Crossref]

Barrow, S. J.

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Baumberg, J. J.

O. S. Ojambati, R. Chikkaraddy, W. D. Deacon, M. Horton, D. Kos, V. A. Turek, U. F. Keyser, and J. J. Baumberg, “Quantum electrodynamics at room temperature coupling a single vibrating molecule with a plasmonic nanocavity,” Nat. Commun. 10, 1049 (2019).
[Crossref]

M. E. Kleemann, R. Chikkaraddy, E. M. Alexeev, D. Kos, C. Carnegie, W. Deacon, A. C. Pury, C. Große, B. Nijs, J. Mertens, A. I. Tartakovskii, and J. J. Baumberg, “Strong-coupling of WSe2 in ultra-compact plasmonic nanocavities at room temperature,” Nat. Commun. 8, 1296 (2017).
[Crossref]

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[Crossref]

Bent, S. F.

C. Hägglund, G. Zeltzer, R. Ruiz, A. Wangperawong, K. Roelofs, and S. F. Bent, “Strong coupling of plasmon and nanocavity modes for dual band, near-perfect absorbers and ultrathin photovoltaics,” ACS Photon. 3, 456–463 (2016).
[Crossref]

Benz, F.

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Bhat, R.

D. E. Aspnes, S. M. Kelso, R. A. Logan, and R. Bhat, “Optical properties of AlxGa1-xAs,” J. Appl. Phys. 60, 754–767 (1986).
[Crossref]

Bitton, O.

K. Santhosh, O. Bitton, L. Chuntonov, and G. Haran, “Vacuum Rabi splitting in a plasmonic cavity at the single quantum emitter limit,” Nat. Commun. 7, 11823 (2016).
[Crossref]

Bøggild, P.

M. Geisler, X. Cui, J. Wang, T. Rindzevicius, L. Gammelgaard, B. S. Jessen, P. A. D. Goncalves, F. Todisco, P. Bøggild, A. Boisen, M. Wubs, N. A. Mortensen, and S. Xiao, “Single-crystalline gold nanodisks on WS2 mono- and multilayers for strong coupling at room temperature,” ACS Photon. 6, 994–1001 (2019).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Boisen, A.

M. Geisler, X. Cui, J. Wang, T. Rindzevicius, L. Gammelgaard, B. S. Jessen, P. A. D. Goncalves, F. Todisco, P. Bøggild, A. Boisen, M. Wubs, N. A. Mortensen, and S. Xiao, “Single-crystalline gold nanodisks on WS2 mono- and multilayers for strong coupling at room temperature,” ACS Photon. 6, 994–1001 (2019).
[Crossref]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2009).
[Crossref]

Cadusch, J.

P. Zeng, J. Cadusch, D. Chakraborty, T. A. Smith, A. Roberts, J. E. Sader, and T. J. Davis, “Photoinduced electron transfer in strong coupling regime: waveguide–plasmon polaritons,” Nano Lett. 16, 2651–2656 (2016).
[Crossref]

Cai, H.

Carnegie, C.

M. E. Kleemann, R. Chikkaraddy, E. M. Alexeev, D. Kos, C. Carnegie, W. Deacon, A. C. Pury, C. Große, B. Nijs, J. Mertens, A. I. Tartakovskii, and J. J. Baumberg, “Strong-coupling of WSe2 in ultra-compact plasmonic nanocavities at room temperature,” Nat. Commun. 8, 1296 (2017).
[Crossref]

Cerullo, G.

P. Vasa, W. Wang, R. Pomraenke, M. Lammers, M. Maiuri, C. Manzoni, G. Cerullo, and C. Lienau, “Real-time observation of ultrafast Rabi oscillations between excitons and plasmons in metal nanostructures with J-aggregates,” Nat. Photonics 7, 128–132 (2013).
[Crossref]

Chakraborty, D.

P. Zeng, J. Cadusch, D. Chakraborty, T. A. Smith, A. Roberts, J. E. Sader, and T. J. Davis, “Photoinduced electron transfer in strong coupling regime: waveguide–plasmon polaritons,” Nano Lett. 16, 2651–2656 (2016).
[Crossref]

Chen, H. Y.

P. Song, J. H. Wang, M. Zhang, F. Yang, H. J. Lu, B. Kang, J. J. Xu, and H. Y. Chen, “Three-level spaser for next-generation luminescent nanoprobe,” Sci. Adv. 4, eaat0292 (2018).
[Crossref]

Chen, J.

Y. Ji, C. Tang, N. Xie, J. Chen, P. Gu, C. Peng, and B. Liu, “High-performance metamaterial sensors based on strong coupling between surface plasmon polaritons and magnetic plasmon resonances,” Results Phys. 14, 102397 (2019).
[Crossref]

Chen, Z.

M. Wan, P. Gu, W. Liu, Z. Chen, and Z. Wang, “Low threhold spaser based on deep-subwavelength spherical hyperbolic metamaterial cavities,” Appl. Phys. Lett. 110, 031103 (2017).
[Crossref]

Chikkaraddy, R.

O. S. Ojambati, R. Chikkaraddy, W. D. Deacon, M. Horton, D. Kos, V. A. Turek, U. F. Keyser, and J. J. Baumberg, “Quantum electrodynamics at room temperature coupling a single vibrating molecule with a plasmonic nanocavity,” Nat. Commun. 10, 1049 (2019).
[Crossref]

M. E. Kleemann, R. Chikkaraddy, E. M. Alexeev, D. Kos, C. Carnegie, W. Deacon, A. C. Pury, C. Große, B. Nijs, J. Mertens, A. I. Tartakovskii, and J. J. Baumberg, “Strong-coupling of WSe2 in ultra-compact plasmonic nanocavities at room temperature,” Nat. Commun. 8, 1296 (2017).
[Crossref]

R. Chikkaraddy, B. Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
[Crossref]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

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ACS Nano (1)

Q. Sun, H. Yu, K. Ueno, A. Kubo, Y. Matsuo, and H. Misawa, “Dissecting the few-femtosecond dephasing time of dipole and quadrupole modes in gold nanoparticles using polarized photoemission electron microscopy,” ACS Nano 10, 3835–3842 (2016).
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Figures (9)

Fig. 1.
Fig. 1. (a) Schematic of a spherical HMM cavity composed of a dielectric nanosphere core (radius rcore; refractive index ncore) and seven alternating layers of silver (thickness: s1, s2, s3, and s4) and dielectric layers (thickness d1, d2, d3, and d4; refractive index nd1, nd2, nd3, and nd4). (b) Calculated decomposed absorption efficiency spectrum for the first two electric terms (a1, a2) and the first magnetic term (b1) of the Mie expansion for an HMM cavity with the parameters rcore=50nm, ncore=2.0, s1=s2=s3=s4=15nm, d1=d2=d3=20nm, and nd1=nd2=nd3=nd4=1.4. The b1 term is enlarged 50 times for clarity. (c)–(h) The electric field intensity distributions of WGM1,1 (0.602 eV), WGM1,2 (0.873 eV), WGM1,3 (1.305 eV), TM1,1 (2.827 eV), WGM2,1 (1.021 eV), and TM2,1 (3.217 eV), respectively. Dashed circle lines show the silver/dielectric interfaces of the HMM cavity.
Fig. 2.
Fig. 2. (a) Absorption efficiency spectra (a1) of a spherical HMM cavity (rcore=50nm, s=15nm, d=20nm, and nd=1.4) as a function of the dielectric core index (ncore). The two red lines with circles present the dispersions of two hybrid modes predicted by the CTM fitting. The dashed white horizontal and oblique lines represent the resonant energy (left y axis) and wavelength (right y axis) of the uncoupled WGM1,3 and TM1,1 mode, respectively. (b) The absorption efficiency spectrum of the spherical HMM cavity with a dielectric core index of ncore=4.3 (ETM1,1=EWGM1,3). The blue lines are the Fano fitting results for the calculated absorption peaks.
Fig. 3.
Fig. 3. (a), (c) Electric field intensity distributions of Hybrid (resonant energy: 1.1117 eV) and Hybrid+ (resonant energy: 1.7405 eV) at the kE plane, respectively; (b), (d) electric field intensities along the white dashed lines in (a) and (c), respectively. The light blue and turquoise vertical stripes in (b) and (d) are used to indicate the dielectric core and dielectric layers (d1, d2, and d3) of the HMM cavity, respectively. Dashed circle lines show the silver/dielectric interfaces of the HMM cavity.
Fig. 4.
Fig. 4. (a), (c) x component of the electric field (Ex) for Hybrid (resonant energy: 1.1117 eV) and Hybrid+ (resonant energy: 1.7405 eV) at the kE plane, respectively; (b), (d) Ex along the white dashed line in (a) and (c), respectively. The light blue and turqoise vertical stripes in (b) and (d) are used to indicate the dielectric core and dielectric layers (d1, d2, and d3) of the HMM cavity, respectively.
Fig. 5.
Fig. 5. (a) Absorption efficiency spectra (a1) of the spherical HMM cavity (s=15nm, d=20nm, and nd=1.4) as a function of ncore for dielectric core radii of (bottom to top) 30 nm, 40 nm, 60 nm, and 80 nm. The two red lines with circles in each panel present the dispersions of two hybrid modes predicted by the CTM fitting. The dashed white horizontal and oblique lines in each panel represent the resonant energies of the uncoupled WGM1,3 and TM1,1 mode, respectively. The vertical dashed line indicates the dielectric core index ncore=4.3 where the resonant energies of two uncoupled modes are equal, ETM1,1=EWGM1,3. (b) The absorption efficiency spectra of the spherical HMM cavity at ncore=4.3 with dielectric core radii of 30 nm (black line), 40 nm (red line), 60 nm (blue line), and 80 nm (magenta line) (with absorption spectra offset vertically for clarity). (c) The Rabi splitting energy as a function of dielectric core radius (rcore).
Fig. 6.
Fig. 6. Fano fitting (olive lines) for the multiple absorption peaks (hollow red circles) of a spherical HMM cavity (rcore=50nm, ncore=2.0, s=15nm, d=20nm, and nd=1.4): (a) WGM1,1, (b) WGM1,2, (c) WGM1,3, (d) TM1,1, (e) WGM2,1, (f) WGM2,2, (g) WGM2,3, (h) TM2,1.
Fig. 7.
Fig. 7. (a)–(d) Resonant energies of TM1,1 (magenta triangles), WGM1,3 (blue triangles), WGM1,2 (red circles), and WGM1,1 (black squares) as a function of dielectric layer thickness (d), dielectric layer index (nd), dielectric core radius (rcore), and dielectric core index (ncore), respectively; (e)–(h) energy differences (ΔE) between TM1,1 mode and WGM1,3/WGM1,2/WGM1,1 [ΔE(TM1,1WGM1,3), blue trangles; ΔE(TM1,1WGM1,2), red circles; ΔE(TM1,1WGM1,1), black squares] as a function of d, nd, rcore, and ncore, respectively.
Fig. 8.
Fig. 8. Radial (vertical direction) electric field intensity distributions of (a) WGM1,1, (b) WGM1,2, (c) WGM1,3, and (d) TM1,1. The light blue and turquoise vertical stripes are used to indicate the dielectric core and dielectric layers (d1, d2, and d3), respectively, of the HMM cavity (rcore=50nm, ncore=2.0, s=15nm, d=20nm, and nd=1.4).
Fig. 9.
Fig. 9. Calculated absorption efficiency spectra (a1) of the spherical HMM cavity (rcore=50nm, s=15nm, d=20nm, and nd=1.4) as a function of ncore with an inner silver thickness (s4) of (a) 15 nm, (b) 25 nm, and (c) 35 nm. The two red lines with circles in each panel show the resonant energies of two hybrid modes predicted by the CTM fitting.

Equations (2)

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εAg=εωp2/[ω(ω+iω)],
Ehybrid±=ETM+EWGM2±Δ2+(ETMEWGM)24,