1. Introduction

Plasmonic metal nanostructures have been utilized in a wide range of nanophotonics technologies and devices, mainly because they can support plasmon resonances and have the ability to concentrate light into subwavelength volumes and produce highly localized fields [1–3]. In some applications such as biosensors [4–6] and low threshold lasing [7–10], resonances with high quality (Q) factors or narrow linewidths are highly desirable. However, plasmonic nanostructures suffer non-radiative damping due to Ohmic loss, ultimately limiting the achievable Q-factors of the plasmon resonances to values on the order of magnitude of 10 [3,11,12]. Recently, combined photonic and plasmonic resonators supporting hybrid modes have been proposed as a promising way to further increase the Q-factors of plasmonic resonators. For example, hybrid plasmon-photon resonances with high Q-factors in the thousands, which is about two orders of magnitude larger than the previously reported achievable Q-factors of ~10 in the plasmonic nanostructures, have been theoretically predicted in combined structures such as 3.0-μm-diameter hollow silver tubes [13] and 10-μm-diameter silica toroidal microcavities coated with a silver film [14] or with a silver nanoring [15], and even experimentally verified in silver-coated 16-μm-diameter micro-disks [16], 3.2-μm-diameter micro-bottle resonators [17] or fibers with a diameter of 100-μm [18]. Furthermore, it has also been found in these studies that the Q-factors of the hybrid resonances in the combined photonic and plasmonic resonators are still smaller than that of the whispering-gallery modes supported by the corresponding dielectric microcavities [16,17]. This, however, is not always true. Our recent experimental studies have, for example, shown that the hybrid modes (cavity plasmon resonances) in silver-coated polystyrene spheres with a relatively small diameter of 1.0-μm could exhibit higher Q-factors than the whispering-gallery modes supported by polystyrene spheres without silver shells [19].

In this paper, we comparatively investigate the Q-factors of whispering-gallery modes in spherical dielectric resonators and hybrid resonances in dielectric-metal core-shell resonators (DMCSRs) formed by wrapping a thin silver shell layer around the same dielectric resonator. We theoretically demonstrate that the Q-factors of the hybrid resonances are mainly limited by the metal absorption loss, while the Q-factors of the whispering-gallery resonances are limited only by the radiation loss. Depending on the size of the resonators, the metal absorption loss of the DMCSRs could be smaller or larger than the radiation loss of the dielectric resonators, resulting in that in a certain wavelength range the Q-factors of hybrid modes could be correspondingly higher or lower than that of whispering-gallery modes. Experimentally, dye-doped polystyrene (PS) spheres themselves act as the dielectric resonators, and are wrapped with a nearly complete silver shell using a simple two-step approach we recently developed to form the DMCSRs [20]. By comparing the reshaped fluorescence spectra that arise from the coupling of the dye molecules to the whispering-gallery and hybrid modes, we show that for a 0.5-μm-radius PS sphere the hybrid modes exhibit higher Q-factors in the emission window of the dye molecules, while the Q-factors of the whispering-gallery modes become larger than that of the hybrid modes for a relatively large 1.6-μm-radius PS sphere, which thus provide direct experimental support to the theoretical predictions.

2. Results and discussions

As schematically shown in the bottom and top insets of Fig. 1, a PS sphere (radius: r; refractive index: n) itself and coated with a silver shell (thickness: t) act as the dielectric resonator and the DMCSR, respectively. Throughout this paper, the problem of scattering, extinction or absorption of a plane wave by the dielectric resonators and DMCSRs which are embedded in the air is solved analytically using Mie theory [21]. In the calculations, the refractive index of the PS sphere is taken to n = 1.59. The permittivity of silver is described by a Drude model ${\epsilon }_{Ag}=1-{\omega }_p^2/\left[\omega \left(\omega +i{\tau }^{-1}\right)\right]$, with ћω_p = 9.0 eV and ћτ⁻¹ = 0.1 eV [22]. Previous studies have shown that the Q-factors of the hybrid resonances in the DMCSRs could initially increase with increasing the metal shell thickness, and begin to converge as the shell thickness increases beyond the optical skin depth of metal [23,24]. Therefore, in the present study the silver shell thickness is fixed to t = 60 nm, which is larger than the optical skin depth of silver, so that the achievable Q-factors of the hybrid resonances are maximized.

 

Fig. 1 The theoretical Q-factors of multipolar hybrid (circle symbols) and whispering-gallery resonances (square symbols) in a certain wavelength range of 550 nm - 655 nm supported by a dielectric-metal core-shell resonator and a same dielectric resonator, respectively, depending on the radii of the dielectric sphere. The bottom and top insets display the schematics of a dielectric resonator (radius: r; refractive index: n) and a dielectric-metal core-shell resonator (radius: r; refractive index: n; metal shell thickness: t), respectively. The red, blue and black dashed lines are guides to the eye.

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In the analytical Mie solutions, the total scattering efficiency Q_sca defined as scattering cross section divided by the cross section of the particle is $Q_{sca}=\frac{2}{k^2{r}^2}{{\displaystyle \sum _l(2l+1)(\left|a_l\right|}}^2+{\left|b_l\right|}^2)$, and the total absorption efficiency Q_abs defined as absorption cross section divied by the cross section of the partilce is $Q_{\text{abs}}=\frac{2}{k^2{r}^2}{\displaystyle \sum _l(2l+1)\left[\mathrm{Re}(a_l)-{\left|a_l\right|}^2+\mathrm{Re}(b_l)-{\left|b_l\right|}^2\right]}$, where k is the wavenumber, r is the outer radius of the particle, a_l is the l-th order transverse-magnetic (TM) Mie scattering coefficient, and b_l is the l-th transverse-electric (TE) Mie scattering coefficient [21]. One of the main advantages of the analytical Mie solution is its ability to decompose the total spectra into separate multipolar contributions [21]. The decomposed scattering efficiency spectra and absorption efficiency spectra are, respectively, calculated for the dielectric resonators and DMCSRs containing PS spheres with different radii. The Mie modes (TE or TM) in a sphere could be ordered using three indices (l, n, m), where l is the angular mode number, n is the polar mode number (due to spherical symmetry, n = 1 in all cases), and m is the radial mode number [21]. For clarity, the hybrid and whispering-gallery modes supported by the DMCSRs and dielectri resonators are denoted by HPP(TM/TE,l,m) and WG(TM/TE,l,m), respectively. For each whispering-gallery mode, its resonance wavelength (λ_res) and linewidth (Г) can be extracted from the Lorentz fitting of the corresponding scattering spectrum calculated for the dielectric resonator. Similarly, the resonance wavelength and linewidth of the hybrid resonance can be obtained from the Fano fitting of the absorption spectrum of the DMCSR. With the extracted λ_res and Γ, the Q-factor can be calculated as Q = λ_res/Γ. Figure 1 summarizes the Q-factors of all the whispering-gallery and bybrid modes locating within a certain wavelength range from 550 nm to 655 nm for the dielectric resonators and DMCSRs containing PS spheres with different radii of r = 0.5 μm, 0.9 μm, 1.25 μm, and 1.6 μm. For a PS sphere with a relatively small radius of r = 0.5 μm, it is seen that the Q-factors of the HPP resonances [open circles for HPP(TE,l,m) and solid circles for HPP(TM,l,m)] can reach the maximum value of ~520 in the DMCSR, which is much larger than the achievable maximum Q-factor of ~18 for the whispering-gallery modes [open squares for WG(TE,l,1) and solid squares for WG(TM,l,1)] supported by the dielectric resonator. As schematically guided by the dashed-lines in Fig. 1, with increasing the radius of the PS sphere, the Q-factors of the HPP(TM/TE,l,m) resonances are slowly increased (circle symbols), while the Q-factors of the WG(TM/TE,l,1) modes appearing within the same certain wavelength range are found to be increased very fast (square symbols). For example, when the radius of the PS sphere is increased to r = 1.6 μm, the Q-factors of the WG(TM/TE,l,1) modes and the HPP(TM/TE,l,m) resonances are increased from 5~18 to 2100~21180 and from 100~520 to 116~1850, respectively. In particular, at the PS sphere radius of r = 1.25 μm, the Q-factors (275~1660) of the WG(TM/TE,l,1) modes in the dielectric resonator are found to be on the same level as the Q-factors (116~1328) of the HPP(TM/TE,l,m) resonances in the DMCSR. It should be noted that for relatively large PS spheres (r = 1.25 μm and r = 1.6 μm in Fig. 1), the whispering-gallery modes with raidal mode number of m = 2, WG(TM/TE,l,2), can also appear within the specified wavelength range, and their Q-factors can reach relatively small values of 46~177, compared with the corresponding WG(TM/TE,l,1) modes.

In order to explain the above demonstrated size dependence of the Q-factors (total quality factors, Q_tot), the radiation Q-factors (Q_rad) and absorption Q-factors (Q_abs) are further investigated for the hybrid and whispering-gallery modes. Since the PS sphere is assumed to be lossless, the total Q-factors of the whispering-gallery resonances supported by the dielectric resonators are limited only by the radiation loss (Q_tot = Q_rad). For DMCSRs, the radiation Q-factors and the absorption Q-factors arising from the ohmic loss of metal have contributions to the total Q-factors, which can be described as 1/Q_tot = 1/Q_rad + 1/Q_abs [16]. By neglecting the metal loss in the DMCSRs, the radiation Q-factors (Q_rad) of the hybrid resonances can be directly obtained, and consequently the absorption Q-factors (Q_abs) are calculated according to the relation of 1/Q_abs = 1/Q_tot - 1/Q_rad.

Figure 2 mainly summarizes the Q_rad, Q_abs, Q_total of the WG(TM/TE,l,1) and HPP(TM/TE,l,1) resonances locating within a certain wavelength range from 550 nm to 655 nm for the dielectric resonators and DMCSRs containing PS spheres with different radii of r = 0.5 μm and 1.6 μm. As shown in Figs. 2(a) and 2(b), for a PS sphere with a relatively small radius of r = 0.5 μm, we find that the Q_total (open triangles) or equivalently the Q_rad (open circles) of WG(TM,5,1) and WG(TE,6,1) modes can only reach values as low as 5~18, revealing the huge radiation loss of the low-order (small l) whispering-gallery resonances. The field intensity distribution of WG(TM,5,1) is calculated and shown in Fig. 2(c), which clearly reveals that a large portion of the electric fields are distributed outside the dielectric resonator. On the other hand, the Q_rad of the HPP(TM/TE,l,1) resonances in a DMCSR with a radius of r = 0.5 μm locating within the same spectral range are found to reach the values of 1739~5971 [open circles in Figs. 2(a) and 2(b)], which are about one order of magnitude larger than the corresponding Q_abs of 107~570 [open squares in Figs. 2(a) and 2(b)], revealing that the total Q-factors of the hybrid resonances are mainly limited by the metal absorption loss [open triangles in Figs. 2(a) and 2(b)]. The field intensity distribution is also calculated for HPP(TM,7,1) and shown in Fig. 2(d). It is seen from Fig. 2(d) that the silver layer can efficiently concentrate the electric field inside the dielectric core, which is expected to result in a small radiation loss. In this case, although the fields are found to be closed to the inner surface of the silver shell [Fig. 2(d)], which could cause metal absorption loss, the absorption loss of the HPP(TM/TE,l,m) resonances in the DMCSR could be smaller than the radiation loss of the WG(TM/TE,l,1) resonances in the dielectric resonator. Therefore, the HPP(TM/TE,l,1) modes are expected to exhibit relatively large total quality factors.

 

Fig. 2 Total (Q_total), radiation (Q_rad), and absorption (Q_abs) quality factors of TM resonances (a) and TE resonances (b) supported by the 0.5-μm-radius dielectric resonator and DMCSR, and TM resonances (e) and TE resonances (f) supported by the 1.6-μm-radius dielectric resonator and DMCSR. Normalized electric field intensity distributions calculated for WG(TM,5,1) (c), HPP(TM,7,1) (d), WG(TM,20,1) (g), and HPP(TM,30,1) (h). The scale bar is 0.5 μm in (c) and (d), and 1.0 μm in (g) and (h).

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For a relatively large PS sphere with a radius of r = 1.6 μm, both the dielectric resonator and the DMCSR can support resonances with much higher angular momentum indexes (l ≈19-30). It is seen from Figs. 2(e) and 2(f) that the Q_total or the Q_rad of the WG(TM/TE,l,1) resonances can reach large values as high as 2100~21180, demonstrating the relatively low radiation loss of the high-order whispering-gallery resonances in the dielectric resonator. As an example, the field intensity distribution of WG(TM,20,1) is calculated and shown in Fig. 2(g), which clearly reveals that the large dielectric resonator could provide tight field confinement, indicating the low radiation loss. The Q_rad of the HPP(TM/TE,l,1) resonances in a DMCSR with such a large PS sphere can reach the ultra-high values of Q_rad ≈4.8 × 10⁵~1.1 × 10⁹ (open circles in Figs. 2(c) and 2(d)), which are several orders of magnitude larger than the Q_rad (Q_rad ≈2100~21180) of the WG(TM/TE,l,1) resonances in the dielectric resonator. The field intensity distribution calculated for the hybrid HPP(TM,30,1) resonance, as shown in Fig. 2(h), confirms that the light confinement ability of high-order hybrid resonances are much better than the high-order whispering-gallery modes. However, the total quality factors of the HPP(TM/TE,l,1) resonances are still limited by the metal absorption loss in the DMCSR. Since the radiation loss of the high-order WG(TM/TE,l,1) resonances in the dielectric resonator is smaller than the metal absorption loss of the high-order HPP(TM/TE,l,1) modes in the DMCSR, it is found that for large PS sphere the total quality factors of HPP(TM/TE,l,m) modes are lower than that of WG(TM/TE,l,1).

To experimentally compare the total Q-factors of whispering-gallery and hybrid resonances locating within the same spectral range, the dielectric resonators and DMCSRs are first prepared by using our recently developed approach [20]. In brief, a monolayer of self-supporting dye-doped PS spheres (Thermo-Scientific; dye-tye: Firefli Fluorescent Red; central emission wavelength at 612 nm) is formed onto a substrate with tens of micrometer-sized through-holes. Each sphere within the through-hole area can thus act as a dielectric resonator. Thin silver films with a thickness of ~60 nm are successively deposited onto the lower and upper half-surfaces of the as-prepared self-supporting dye-doped PS spheres by plasma sputtering to wrap a nearly-complete thin silver layer around each colloid, and thus forming the dye-doped DMCSR. With the aid of the coupling of the dye molecules to the whispering-gallery and hybrid modes, the re-shaped fluorescence spectra instead of the scattering or extinction spectra of the dye-doped dielectric resonators and DMCSRs are directly measured using a homebuilt photoluminescence (PL) setup. Light from a 514 nm wavelength diode laser with ~5 μW power is focused by an objective (50 × , 0.75 N. A.) to a ~1-μm-diamter spot onto the samples to pump the dye molecules. The same objective is also used to collect the fluorescence signals, which are then guided into a grating spectrometer, outputting the fluorescence emission spectra. Corresponding to the theoretical results demonstrated in Fig. 2, dye-doped PS spheres with two different radii of r = 0.5 μm and 1.6 μm are intentionally chosen in our experiments to access the low-order and high-order whispering-gallery and hybrid modes within the spectral range of the emission of the dye molecules (550 nm < λ < 655 nm). The representative scanning electron microscopy (SEM) images of the DMCSRs containing the dye-doped PS spheres with diameters of 1.0 μm and 3.2 μm are shown in Figs. 3(a) and 3(b), respectively.

 

Fig. 3 SEM images of the DMCSRs containing the dye-doped PS spheres with diameters of 1.0 μm (a) and 3.2 μm (b). The dashed-line boxes indicate the free-standing areas.

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Figures 4(a) and 4(b) show the fluorescence spectra measured for the 0.5-μm-radius dye-doped PS sphere and DMCSR, respectively. It is clearly seen from Figs. 4(a) and 4(b) that only broad peaks are present in the fluorescence spectrum measured for the dye-doped PS sphere, while for the dye-doped DMCSR the most remarkable spectral features are several sharp emission peaks. Our previous experimental studies have already demonstrated that the plasmonic properties of the prepared DMCSRs are dominated by multipolar hybrid modes with their optical fields being tightly confined within the dielectric cores [20]. Such highly localized nature prevents the hybrid resonances in adjacent DMCSRs from interacting with each other. Therefore, even though the DMCSRs are closely packed like in this study [Fig. 3], they are allowed to be analyzed using Mie theory [20, 24]. In general, when the dielectric resonators are closely packed and illuminated by a plane wave, the whispering-gallery modes supported by the adjacent dielectric resonators may interact with each other. To make sure that the dielectric resonators located in a periodic format can also be analyzed using Mie theory, the excitation laser is thus tightly focused to a ~1-μm-diamter spot onto the samples to selectively excite only a single dye-doped PS sphere. The total and decomposed scattering efficiency spectra calculated for the individual dielectric resonator and the absorption efficiency spectra calculated for the individual DMCSR are shown in upper and lower panels of Figs. 4(c) and 4(d), respectively, in which the radius of the dye-doped PS sphere is r₁ = 508 nm and all the other parameters are the same as those used in the above theoretical calculations. It is seen from Fig. 4(c) that the total scattering efficiency spectrum presents two broad scattering peaks at the wavelengths of λ ≈580 nm and 620 nm, corresponding to the excitations of low-Q WG(TM,5,1) and WG(TE,6,1) modes, which are consistent with the experimental observations [Fig. 4(a)]. The linewidths of the experimentally observed whispering-gallery modes are estimated to be ~50 nm [Fig. 4(a)], giving rise to the Q-factors lower than 15. The total absorption efficiency spectrum of the DMCSR shown in Fig. 4(d) clearly presents several narrow absorption peaks, corresponding to the excitations of HPP(TM,3,2), HPP(TM,1,3), HPP(TE,4,1), HPP(TE,2,2) and HPP(TE,5,1) resonances. By comparing the Fig. 4(b) and Fig. 4(d), it is directly seen that there is a very good one-to-one correspondence between experimentally observed spectral features and theoretically predicted absorption peaks, providing strong evidence that the dramatic modification of the fluorescence emission spectrum is due to the coupling of the dyes to multipolar hybrid resonances. Although experimentally achieved Q-factors (80~150) of the hybrid resonances supported by the DMCSR [Fig. 4(b)] are lower than the predicted values [350~520, Fig. 4(d)], which are mostly due to the dissipation mechanisms induced by electron scattering in the thin silver shell, the roughness of the deposited silver shell layer and the six nano-windows existed in the equator region where the original PS beads touch each other [20], they are still much higher than the Q-factors (~15) of the whispering-gallery modes supported by the dielectric resonator [Fig. 4(a)], which confirms that for a relatively small PS sphere the hybrid modes exhibit higher Q-factors in the emission window of the dye molecules, in other words, the achievable Q-factors can be improved by coating the dielectric resonator with a thin metal shell to form the DMCSR.

 

Fig. 4 Measured PL spectra for the dielectric resonator (a) and DMCSR (b) containing a dye-doped PS sphere with radius of r₁ = 0.5 μm. (c) Total and decomposed scattering efficiency spectra of a dielectric resonator (r₁ = 0.508 μm). (d) Total and decomposed absorption efficiency spectra of a DMCSR (r₁ = 0.508 μm and t = 60 nm).

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In the following, the fluorescence spectra measured for the dielectric resonator and the DMCSR containing a relatively large dye-doped sphere with a radius of 1.6 μm are shown in Figs. 5(a) and 5(b), respectively. Figure 5(a) clearly shows that ultra-sharp emission peaks are present in the fluorescence spectrum measured for the dye-doped PS sphere. In addition to the sharp peaks, relatively broad peaks with weak intensities can also be found in Fig. 5(a). To identify the observed fluorescence emission peaks, the scattering spectra are calculated for the individual dielectric resonator with a radius of r₂ = 1.6 μm and shown in Fig. 5(c). The calculated result faithfully reproduces the main features of the measured spectrum, revealing that the observed narrow peaks correspond to the excitations of WG(TM,20,1), WG(TM,21,1), WG(TM,22,1), WG(TM,23,1), WG(TE,20,1), WG(TE,21,1), WG(TE,22,1), WG(TE,23,1) and WG(TE,24,1) modes, and the broad peaks should arise from the excitations of WG(TM,16,2), WG(TM,17,2), WG(TM,18,2), WG(TM,19,2), WG(TE,16,2), WG(TE,17,2), WG(TE,18,2), WG(TE,19,2) and WG(TE,20,2) modes [marked by the blue arrows in the total scattering spectrum in Fig. 5(c)]. The linewidths of the experimentally observed WG(TM/TE,l,2) resonances are estimated to be about 10 nm, while the linewidths of the WG(TM/TE,l,1) modes can reach the value as narrow as ~0.41 nm. Therefore, the maximum achievable Q-factor of the WG(TM/TE,l,1) modes supported by the dielectric resonator with a radius of 1.6 μm can reach a high value of ~1400 within the spectral range of the emission of the dye molecules (550 nm < λ < 655 nm). As shown in Fig. 5(b), the fluorescence emission spectrum measured for the dye-doped DMCSR is also strongly modulated. The absorption spectra of the DMCSR are calculated and shown in Fig. 5(d). In this case, a series of the absorption peaks are clearly present [Fig. 5(d)], and the number of the hybrid resonances supported by the DMCSR within the spectral range of the emission window is much larger than that in the dielectric resonator case [Fig. 5(c)]. By comparing Fig. 5(b) and Fig. 5(d), it is seen that some theoretically predicted hybrid resonances, such as HPP(TE,l,2) and HPP(TM,l,3) [the first and second series of the lower panel in Fig. 5(d)], match the experimentally observed emission peaks, as indicated by blue dashed lines in Figs. 5(b) and 5(d). In addition, as indicated by green dashed lines, the hybrid resonances, such as HPP(TM,l,4) [the third series of the lower panel in Fig. 5(d)], are found to cause several shoulders in the fluorescence spectrum as shown in Fig. 5(b), from which it is hard to extract their linewidths. Nevertheless, the minimum linewidth of those well-formed emission peaks are estimated to be ~1.4 nm, from which the maximum Q-factors are then estimated to be ~440. Therefore, for a relatively large PS core with a radius of 1.6 μm, the maximum achievable Q-factor of the hybrid resonances supported by the DMCSR is smaller than that of the whispering-gallery modes in the dielectric resonator. However, it should be noted that associated with the excitations of the hybrid resonances the DMCSR is expected to provide a plasmon-type field enhancement at the outer metal surface [17].

 

Fig. 5 Measured PL spectra for the dielectric resonator (a) and DMCSR (b) containing a dye-doped PS sphere with radius of r₂ = 1.6 μm. (c) Total and decomposed scattering efficiency spectra of a dielectric resonator (r₂ = 1.6 μm). (d) Total and decomposed absorption efficiency spectra of a DMCSR (r₂ = 1.6 μm and t = 60 nm).

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3. Conclusions

In summary, we have comparatively studied the Q-factors of whispering-gallery modes supported by a dielectric resonator and multipolar hybrid resonances in a DMCSR formed by wrapping a thin silver shell around the same dielectric resonator. We theoretically and experimentally demonstrate that within a certain wavelength range, which in our case is the spectral range of the emission of the dye molecules (550 nm < λ < 655 nm), the achievable maximum Q-factors of hybrid modes are larger than that of whispering-gallery modes for a small sized (0.5 μm in radius) dielectric core sphere, while for a relatively large sized (1.6 μm in radius) core sphere, the achievable maximum Q-factors of hybrid modes locating within the same spectral range are smaller than that of whispering-gallery modes. Our findings thus provide guidance for appropriately choosing plasmonic core-shell (hybrid modes) or dielectric sphere resonators (whispering-gallery modes) in applications such as ultrasensitive bio-sensors [3,6], low-threshold lasing [8–10], slow-light and nonlinear optical devices [3].