Bottle resonators can support high Q-factor whispering gallery modes (WGMs) and demonstrate a rich mode spectrum. Resonators were fabricated using a fiber tapering apparatus and were coated with a thin, smooth layer of luminescent silicon nanocrystals. The photoluminescence spectrum showed WGM peaks with Q-factors near 2,500; however, evanescent measurements showed that these modes are a composite of many modes with Q-factors exceeding 106, the highest yet seen for a silicon-nanocrystal-coated microresonators. The mode structure showed strong polarization and sensitivity to position within the bottle resonator. An analysis of loss mechanisms establishes surface roughness scattering as the limiting factor in these nanocrystal-coated bottle resonators in the absence of excited carriers.
© 2010 Optical Society of America
Silicon nanocrystals (Si-NCs) hold high promise as light emitters in future microphotonic devices . Ensembles of oxide-embedded silicon nanocrystals display a broad, spectrally featureless luminescence band centered near 800 nm, with the possibility of quantum efficiencies above 50% . One of the greatest hopes in microphotonics is to develop a Si-NC laser; numerous research groups are currently pursuing this objective and there have been several reports of optical gain or stimulated emission [3–5], although it is not entirely clear what conditions are required to produce it [5, 6]. Coupling Si-NC luminescence into high-quality microcavities can provide fine control over its spectrum, may be one of the best avenues toward lasing, and could also lead to new applications including silicon-nanocrystal-based luminescence sensors . There is one report of stimulated emission in a nanocrystal-coated microsphere, although there was no clear evidence of cavity modes in the emission spectrum. The photoluminescence peak reported as possible stimulated emission was weak, blueshifted from the normal emission peak near 800 nm, and broad (full-width half-maximum ≈ 40 nm) . For this article we fabricate Si-NC-coated high-Q bottle resonators, explore their mode structure in photoluminescence (PL) and transmission configurations, analyze possible loss mechanisms to estimate an upper bound on the achievable Q-factors, and study the time-resolved behavior of the PL.
In order to form a reasonably high-Q microcavity, the nanocrystals should be placed onto the exterior surface of a cylindrical or spherical structure. The luminescence can then couple into the low-radial-order whispering gallery modes (WGMs). Several methods have been developed in recent years for coupling the luminescence of Si-NCs into the resonant WGMs of different structures. These methods include: (i) using physical vapor deposition and annealing to coat arrays of silica microspheres with Si-NCs that luminesce with Q-factors up to 1,200  (ii) CVD deposition of single microspheres using a rotatable specimen holder produced evanescent Q-factors up to 105, although PL was not reported , (iii) lithographically defining silicon-rich-oxide microdisks with PL Q-factors up to 3,000 [11,12], and (iv) a dip-coating method that showed PL Q-factors close to 3,000 . In addition to demonstrating fairly narrow luminescent modes, these structures can be used as temperature sensors , or as refractometric dip-type PL fiber sensors . In microdisks, surface roughness and free carrier absorption are the dominating loss mechanisms that so far prevented the PL Q-factor from exceeding ≈ 103 [10, 11].
In the present work, we investigate bottle resonators coated with a layer of oxide-embedded silicon NCs. Bottle resonators are attracting interest [13, 14] for several reasons: they support high Q-factors and small mode volumes; the guided modes (a major source of loss in cylindrical cavities) can be strongly suppressed; bottle resonators combine azimuthal and axial modes that can provide enhanced tunability of the resonant frequencies ; and they are not especially difficult to fabricate or handle. For example, bottle resonators can be made by “pinching” adjacent regions of an optical fiber to form a central bulged region  [Fig. 1(a)].
The resonators were made from standard optical fibers using a hydrogen torch to heat the fibers and a motorized apparatus to pull them into the required prolate spheroidal shape. They were subsequently dipped into a solution of hydrogen silsesquioxane (HSQ) dissolved in methyl isobutyl keytone that formed a thin layer on the fiber surface . Subsequent thermal processing at 1100°C for 1 hour under 95% N2 +5% H2 drove off excess solvent, collapsed the HSQ cage structures, and induced disproportionation that results in silicon nanocrystals ≈ 3 nm in diameter embedded in a silicon oxide matrix . The nanocrystal containing film is thin (on the order of 100 nm ), and smoother than typical results obtained using lithography and etching  or evaporated thin films . We used an atomic force microscope (AFM) in contact mode to estimate a root-mean-square surface roughness. The dimensions of the resonator were measured from an image taken with an optical microscope.
The luminescence was excited in one of two ways: (i) bringing the beam of an Ar+ laser directly onto the resonator (free-space pumping), or (ii) by coupling the laser into a tapered fiber that was brought into contact with the bottler resonator (evanescent pumping). The mode structure was also measured in two ways: by using a microscope objective to collect the luminescence and deliver it to a grating spectrometer, or by coupling a tunable diode laser (centered around 828 nm, bandwidth 4 MHz) and measuring the transmission through the tapered fiber as a function of wavelength. When evanescently pumping the resonator with the the Ar+ laser, we tuned the resonant mode (to enhance the pumping efficiency) using a mounting fork with a piezoelectric stack to apply small strains to the bottle resonator, in a way similar to that in Ref. .
For the time-resolved studies, the excitation Ar+ laser was chopped into 500 μs square pulses using an acousto-optic modulator and delivered either through free-space or evanescently. The PL was collected by the microscope after filtering to remove the scattered excitation laser light and was finally routed to a photomultiplier tube (PMT). The PMT was connected to a multi-scale analyzer (MSA) with a maximum temporal resolution of 5 ns.
Upon excitation of the coated resonator with the Ar+ laser through the evanescently coupled fiber (the power coupled into the resonator was estimated to be 37 μW) a bright reddish-orange PL, localized on the bottle edges, could be readily observed [Fig. 1(b)]. By adjusting the position of the image with respect to the spectrometer slit (which is 18 μm wide), we measured spectra at different points in the resonator, marked in Fig. 1(b). A polarizer in the collection path permitted two orthogonal polarizations to be measured: one where the electric field is oriented along the resonator axis (TM) and one where it is perpendicular (TE). Spectra taken from a resonator with a 76 μm diameter are shown in Figs. 1(c) (TM) and 1(d) (TE) after a constant background removal. The TM spectra are more intense than the TE ones (this has been observed in cylindrical resonators by ourselves and other groups citeKippPRL06, RakovichAFM07 and is likely related to interference of spiraling modes ). Overall both polarizations show a similar behavior as a function of the position on the resonator from where the light is collected: At the outer positions (1,8) there are no distinguishable features, but for positions closer to the center (4) equispaced peaks become more noticeable in the spectra. The individual peaks show evidence of fine structure, indicating that they are not single-moded but rather a composite of features from different modes [as can be clearly seen in the close-up in Fig. 2(a)]. The PL Q-factors for these composite modes can be obtained by fitting the peaks to a sum of lorentzian functions, resulting in values between 500 and 2,500. Spectral beating can be seen for both polarizations which is consistent with interference between the different axial modes composing the peaks, as discussed in in Section 4.
The measured PL Q-factors are consistent with previous reports [7, 11, 21], but it is clear that PL measurements do not fully resolve the mode structure. We performed transmission measurements in the evanescently coupled configuration to probe the mode structure with high resolution (Fig. 2). The spectrum shows a very rich structure, which confirms that the individual PL peaks are indeed composed of substructures with much smaller linewidth. There is a large number of resonant features present in the scanned wavelength window. In this particular luminescent resonator several of the modes showed Q-factors on the order of 105, with some peaks slightly exceeding 106 [see Fig. 2(c)], setting an new record for the highest Q-factors observed in Si-NC coated microresonators. As a comparison, the Q-factors in uncoated bottle resonators approach 108.
4. Mode structure
The prolate geometry of the bottle resonators results in a distinctive mode structure. Using a geometrical optics description, it can be seen that these resonators support standard WGMs (where the ray path closes on itself after just one round-trip around the resonator axis or, equivalently with kz = 0) and spiral WGMs (with kz ≠ 0 and a ray trajectory that not only bounces just below the resonator surface but also moves back and forth along the axis between two turning points). Figure 3(a) shows a pictorial representation of one of these spiral trajectories.
The resonator profile is approximated (far from the pinched regions) by a parabola:
where R0 = 38.1μm is the maximum radius (where is arbitrarily located at z = 0) and Δk = 0.002μm-1 is the profile curvature. These values were obtained by capturing an image using an optical microscope and a color charge-coupled-device (CCD) camera, tracing the resonator profile from the image and fitting Eq. (1) to the trace.
Due to the optical confinement on the resonator boundary, there is a discrete set of allowed eigenmodes , with wavenumbers
where m is the usual azimuthal order number (corresponding to the number of nodes in the field along the azimuthal direction), q is an axial order number representing the number of nodes along the resonator axis and ΔE = 2mΔk/R0 (only the lowest radial mode is considered in the analysis, so there is no radial order number). The axial number q will also influence the z positions of the peak field since there are two symmetric positions, the caustics, on the axis where the field peaks: the larger q is, these peaks will be farther away from the center.
Using these equation with n = 1.453 for silica at room temperature, we can quantify the mode spacings (while the model does not account for the thin nanocrystal film, we can assume that the latter will cause a minor red shift of the mode resonant wavelengths ). For example, there is an upper limit of m = 420 for the mode wavelengths to fall within the scanned range (827.95–828.07 nm). The spacing between two modes with the same q number and consecutive m numbers is Δvm ≈ c/2πnR0 ≈ 862 GHz ≈ 2 nm, as can be appreciated in Fig. 2(a). The spacing between two modes with the same m number but consecutive q numbers, would then be Δvq ≈ cΔk/2π ≈ 95 MHz ≈ 0.2 nm, which is larger than the scanned range of approximately 0.11 nm. Thus we can safely infer that the multitude of modes observed in the transmission spectrum in Fig. 2(b) possess different m and q numbers. Since the tapered fiber was located close to the center of the resonator, modes with lower q numbers will be more efficiently excited so the measured modes should have values near the upper limit in m. A comparison showing qualitative agreement between the calculated spectra and the measured PL at a given position is given in Fig. 3(b).
The eigenmode distribution from Eq. (2) also explains why the resonant peaks in Fig. 1 seem to shift when measured at different axial positions. Mode PL emission will be strongest at the caustics, so the higher q numbers imply that the strongest emission will be located farther away from the resonator midpoint. As we collect emission from locations away from the center, the predominant PL comes from modes with higher q values (and lower m values) which will certainly show a different resonant wavelength. This also provides an explanation for the fine structure observed in Fig. 2(a), as the spectrometer slit (1.8 μm accounting for the 10X magnification of the microscope) collects emission from modes with different values for q.
5. Loss mechanisms
The addition of the coating decreases the Q-factor in the resonators by one or two orders of magnitude. The ability to maintain high Q-factors in coated resonators matters in different contexts: it is critical to minimize the threshold in a laser and it enhances the interaction between light and matter in a way that could make cavity phenomena like the Purcell effect or strong coupling easier to achieve. It is also important for low-power operation of resonator-based optical filters. We can analyze the contribution of different loss mechanisms to the degradation of Q-factors to see if it would be possible in principle to improve them still further. The main loss mechanisms in Si-NC-containing resonators when the excitation power is kept low and the wavelength is in the near-infrared range are radiation losses in the modes, surface roughness scattering, Mie scattering and inter-band absorption by the nanocrystals . The final Q-factor is then given by
When the resonator is excited by the blue light emitted by the Ar+ laser, a population of carriers builds up in the Si-NCs. These excited carriers can absorb the light circulating within the cavity , and thus introduce a new loss channel. This additional loss mechanism is normally called confined-carrier-absorption (CCA), and we will consider it separately.
In all the following calculations, we have used the modal properties (field profile, effective index, confinement factor) calculated for a coated cylindrical resonator , which should be similar to those of modes which don’t travel far from the center of the bottle resonator.
5.1. Radiation loss
Since we have measured Q-factors close to 108 in uncoated cavities, we can assume QRad ≥ 108, and the loss due to radiation is negligible. This is not surprising, given the relatively large diameter of 76 μm.
5.2. Surface roughness
We can estimate the loss coefficient due to scattering from the roughness at the surface adapting the result from Ref.  for the case of a whispering gallery mode, approximating the field distribution and effective index by those calculated for a coated cylinder :
where φ(r) is the field distribution at the resonator surface, with the normalization ∫0∞ φ2(r)dr = 1, nfilm = 1.68  is the refractive index of the nanocrystal film, neff = 1.41 is the effective refractive index of the whispering gallery mode, k0 = 2π/λ is the vacuum wave vector and σ = 3.40 nm is the AFM-measured root-mean-square roughness of the surface. We take the wavelength to be λ = 829.9 nm.
Using these numbers in Eq. (4), we obtain αRough = 0.05 cm-1, and from that we estimate QRough as follows:
5.3. Inter-band absorption
The estimated losses caused by inter-band absorption by Si-NC can be estimated by following the procedure outlined in Ref.  from the absorption cross-section, the nanocrystal size distribution and the mode field profile. We obtained the nanocrystal size distribution ρ(R) from energy-filtered transmission electron microscopy (see the appendix for details). Then, the absorption is given by 
where σabs(R) is the absorption cross-section (as a function of the nanocrystal radius), R(λ) is the radius for a nanocrystal with an emission of wavelength λ and Rmax is the radius of the largest nanocrystal in the sample and Γ(λ) ≈ 0.01 is the confinement factor calculated from the mode profile. Using the values reported in Ref. , we find αIB = 2.24 × 10-3 cm-1 and QIB = 4.7 × 107.
5.4. Mie scattering
Finally, the Mie scattering losses can be estimated by adding the contribution for each nanocrystal size weighted by the size distribution,
with the scattering cross-section given by the Rayleigh formula 
Taking nSiO2 = 1.46 and nSi = 3.69, we find σMie = 9.25 × 10−4 cm−1 and QMie = 1.2 × 108.
The final estimated Q-factor results Qf= 1.9 × 106, comparable to what we observed in the transmission spectra. From this analysis, surface roughness scattering is the dominant loss factor, followed by interband absorption. Mie scattering does not seem to contribute significantly to the scattering, just like radiation losses. The films produced by the dip-coating method are already very smooth (as compared, for instance, to lithographically fabricated microdisks which have a much higher root-mean-square roughness  or evaporated thin films. However, the experimental Q-factors could be increased (up to a factor of 6) by further improving their surface properties. It is worthwhile to note that Mie scattering and interband absorption are greatly reduced because of the thinness of the nanocrystal-containing film (expressed by the small confinement factor Γ). Thicker films would increase the losses due to these intrinsic mechanisms, and correspondingly decrease the maximum achievable Q-factors. Our fabrication method then becomes attractive for applications requiring resonators with high Q-factors.
5.5. Confined carrier absorption
To estimate the degree of confined carrier absorption, we measured the linewidth of a resonant mode under free-space excitation of the resonator with the Ar+ laser. Then, comparing this linewidth with the one in the absence of an external excitation we can obtain an estimate of the extra loss induced by the free carrier absorption. At an excitation intensity of 34 kW/cm2 (the maximum we could safely route through our microscope), the linewidth increased by 0.42 pm. Following Ref. , this results in αCCA = 2πneff/(λ2Γ) = 0.054 cm-1 and QCCA = 2.0 × 106. Combined with the near-infrared Q-factor calculated above, the Ar+ -excited Q-factor would be
The evolution of the transmission spectra as the pump power increases [Fig. 4(a)] shows the gradual increase in the full-width half-maximum (FWHM) of the modes as the pump intensity increases. There is also a red-shift associated with the pumping, which is due to thermal effects on the resonator (we verified this by corroborating that the time scale of the shift is in the order of a tens of milliseconds). Figure 4(b) displays the fitted mode FWHM as a function of incident power. At the lower intensities the mode linewidth stays roughly constant while it increases for pump intensities larger than 19 kW/cm2, the point where the confined carrier absorption becomes the dominant loss mechanism.
5.6. Losses and PL linewidth
The previous analysis established Q-factors in the order of 106 taking into account different loss mechanisms that apply to resonators coated with a thin film containing Si-NCs. However, the Q-factors measured in the PL are much lower (between 500 to 2,500). These values are not limited by the resolution of the spectrometer (the nominal 0.25 nm resolution would give us a Q-factor close to 3,000), so there must be some other mechanism reducing the PL Q-factors. One issue to consider is the spatial resolution of the PL measurement, which will cause emission from many modes to be collected from a single position. Since we have shown there is a multitude of modes with slightly different resonant wavelength in a short span, is is possible that emission from all those modes is being convolved together into the wider measured peaks. Another contributing factor (which would be present even in the case of unlimited spatial resolution) comes from the homogeneous linewidth of the Si-NCs which would broaden significantly the PL emission. The detailed theroy about this broadening mechanism, arising from a full integration and theoretical analysis of Fermi’s Golden Rule for coupled quantum dot-cavity systems, will be published in a separate paper .
6. High-power excitation
Steady-state spectral measurements at increasing pump powers (up to an estimated 380 μW transferred from the waveguide into the resonator, corresponding to 9 mW of circulating power for a Q-factor of 105) did not show qualitative changes beyond an increase in PL intensity (in particular, there was no clear trend of an increasing PL linewidth as the pump power increased). The upper inset in Fig. 5 shows a representative spectrum, together with ones from the free-space configuration and a flat film. The WGMs are not observable in this spectra because light was collected from the whole resonator so as not to miss possible lasing modes. The free-space-excited resonator PL peak was blue shifted compared to that from the flat film, while the evanescently excited PL no longer had a well defined peak, instead gradually rising into the blue part of the spectrum and was eventually cut off by the longpass filter that blocked the laser. This blue shift is consistent with the high excitation power density inside the resonators, especially for evanescent pumping, which increases the contribution of fast processes.
7. Time-resolved behavior
Since there have been reports of short-lived gain in Si-NCs , we also performed time-resolved studies of the PL. As a baseline, we measured the time-resolved PL from a flat film of oxide-embedded Si-NCs (Fig. 5). This showed a typical stretched exponential decay, with τ = 21 μs and β = 0.77. The emission from a coated resonator using free-space excitation gave a similar curve (τ = 16 μs, β = 0.55). The trace from the evanescently excited resonator showed a fast rise (rise time ≈ 200 ns, close to the AOM modulator limit) followed by a slower decay to a steady state. The decay of this steady state was also on the order of 200 ns and was followed by a much weaker decay with a time constant on the order of microseconds.
The lifetime decreases and the spectrum blueshifts on going from a flat film to a free-space-pumped cavity and finally to an evanescently pumped cavity (Fig. 5). In fact, the evanescently coupled resonator is somewhat reminiscent of the gain reported in Si-NCs  but with a much longer decay to the steady state, which is inconsistent with Auger-induced gain quenching . No spectral narrowing or threshold behavior was observed when changing the pump power (see the lower inset of Fig. 5), only saturation at high pump powers. The fast (nanosecond) component of the rise and decay times are consistent with the so-called F-band emission , likely due to luminescent defects in silicon-rich oxide  which are not saturated in the evanescently-pumped configuration (the nanocrystals themselves contributing only to the low-amplitude slow dynamics). The initial decay to a steady state just after the pulse onset was unexpected, and it may be due to a reversible photobleaching of luminescent centers caused by the high excitation power density coupled into the cavity. It was only observed for evanescent pumping. Despite the small time constants and bluer emission, the lack of sharp peaks in the spectrum coupled with the absence of a threshold behavior leads us to conclude the Si-NCs do not show stimulated emission in these excitation and measurement configurations.
In summary, we studied Si-NC-coated bottle resonators under evanescent excitation. The tuning range and high Q- factors of bottle resonators allow for efficient pumping of the Si-NCs, leading to the emission of luminescence coupled to resonator modes. The PL modes approach 2,500; however, they were found to be composed by many modes with Q-factors on the order of 106, the highest yet reported in Si-NC-coated resonators. The measured spectra can be qualitatively explained by means of a mode analysis of the resonator geometry. We found no clear-cut evidence of either steady-state or short-time (5 ns time resolution) stimulated emission when pumping the Si-NC-coated resonator either in free space or evanescently despite the high Q-factors. These Q-factors correspond to the ones estimated when considering the most likely loss mechanisms. Even though the films as fabricated are very smooth (root-mean-square roughness σ = 3.40 nm) surface roughness scattering is still the dominant loss mechanism in the absence of excited carriers; when carriers are excited into the nanocrystals by a suitable excitation source, absorption by these confined carriers becomes the main limiting loss mechanism and the transition point between these two regimes has been found at an pump excitation power of 19 kW/cm2). Thus, it should be possible to decrease the losses and increase the Q-factors by careful optimization of the film surface properties and avoiding the buildup of large carrier populations. If such high Q-factors could be measured in PL using, for example, a chip-scale tapered slab spectrometer , the resolution of refractometric fiber sensors such as those developed in Ref.  would increase by several orders of magnitude. Considering a Q-factor of 106 and a shift of 33 nm/RIU the estimated detection limit would be 2.5×10-5 RIU, competitive with other sensing technologies such as surface plamon resonance , although the emitter bandwidth may play a limiting role in the case of Si-NCs . Future work will investigate the gain dynamics at short timescales (ps) where Auger-induced quenching may not limit the gain mechanisms in high-Q nanocrystal-coated resonators.
A. Measuring the surface roughness
We measured the surface roughness of a coated resonator using an atomic force microscope. The captured AFM image of the resonator surface can be seen in Fig.6). We used the Roughness analysis feature of the software package Nanoscope 6.11r1 to calculate the root-mean-squared surface roughness from the captured data, with a resulting σ = 3.40 nm.
B. Calculating the nanocrystal size-dependent density
In order to estimate the losses due to some of the analyzed mechanisms (Mie scattering, interband absorption) we need the nanocrystal size-dependent density ρ(R), defined as the number of nanocrystals per unit volume with radius between R and R+ dR and measured in units of cm-3 nm-1 . We obtain this function from transmission electron microscope (TEM) analysis of a flat film prepared under the same conditions as the bottle resonators with proper normalization.
B.1. Measurement of the nanocrystal size distribution
Imaging Si-NCs embedded in an oxide matrix by conventional TEM methods is somewhat difficult due to the low contrast between Si and SiO2. Although high resolution TEM (HRTEM) is able to image Si-NCs [Fig. 7(a)], the appearance of the nanocrystals in the HRTEM image requires a zone-axis orientation, limiting the application of the technique for determining size distributions (while not directly relevant to our sample, it is worth noting that amorphous Si-NCs can’t be imaged by HRTEM either). Instead, we used energy filtered TEM (EFTEM) to detect Si-NCs. This method works because the plasmon losses for Si and SiO2 are different (centered at 17 eV and 23 eV respectively). In this work, the spectrometer slit was centered at the silicon plasmon loss peak at 17 eV, with a slit width of 4 eV, permitting the Si-NCs to be imaged without significant interference from the oxide matrix [Fig. 7(b)]. We can then measure the diameter of the Si-NCs which appear bright in the image. Sampling a number of different nanocrystals (200 in our case), we can reconstruct the the size distribution of the Si-NCs in our sample as shown in Fig. 8. The nanocrystal diameters were estimated by visual inspection of the intensity profile of each nanocrystal in the image (the software package Digital Micrograph was used to obtain the intensity profiles). It should be pointed out that delocalization of the plasmon scattering process can result in a size distribution that represents a slight overestimate of the true particle sizes . Finally, we fit a log-normal distribution,
to the experimental data, obtaining a good agreement with R0 = 1.57 nm and σ= 0.18.
Once we have the size distribution as measured from the TEM (which we call ρTEM(R)) we can compute ρ(R), following closely the procedure in Ref. . We begin by assuming that:
We can calculate the number of Si atoms in the nanocrystals by integrating the volume of a spherical nanocrystal times the nanocrystal density distribution, and dividing by the volume occupied by 1 atom (≈ 1/(8a3), where a is the lattice constant of silicon):
On the other hand, if we assume that all the excess silicon is in the form of nanocrystals and then take a unit volume, we can see that
using that NSi = ρSi VSi/ASi, where ρSi is the density of silicon and ASi its atomic mass.
Moreover, we can define a filling fraction f as
where ρSiO2 and ASiO2 are the density and molecular mass of SiO2, respectively.
We can then insert Eq. (15) into Eq. (13) to obtain then the number of silicon atoms as a function of the filling fraction f. Equating that to the one calculated from Eq. (12), we can calculate the value of the proportionality constant A. In our case, the filling fraction (measured using TEM) is f = 0.12, resulting in A = 1.94×1018cm-3nm-1. Integrating this density distribution over all possible nanocrystal sizes results in a number density NNC = 2.4×1018 cm-3.
The authors would like to acknowledge financial support by NSERC, iCORE and CIPI. The transmission electron microscopy was done at the National Institute for Nanotechnology Electron Microscopy Facility. The peak fitting was done using Fityk 0.8.8 (http://fityk.sourceforge.net). The pictorial representation of a resonator in Fig. 3(a) was drawn by M. Wilson. We would also like to acknowledge M. Sumetsky, who in a 2004 paper introduced theoretically the concept of whispering-gallery bottle resonators.
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