This paper examines the opportunities existing for engineering dispersion in non-silica whispering gallery mode microbubble resonators, for applications such as optical frequency comb generation. More specifically, the zero dispersion wavelength is analyzed as a function of microbubble diameter and wall thickness for several different material groups such as highly-nonlinear soft glasses, polymers and crystalline materials. The zero dispersion wavelength is shown to be highly-tunable by changing the thickness of the shell. Using certain materials it is shown that dispersion equalization can be realized at interesting wavelengths such as deep within the visible or mid-infrared, opening up new possibilities for optical frequency comb generation. This study represents the first extensive analysis of the prospects of using non-silica microbubbles for nonlinear optics.
© 2016 Optical Society of America
Whispering gallery mode (WGM) resonators have gained considerable interest in recent years for various sensing applications [1–5]. Whispering gallery resonators can be thought of as mirror-less resonators, having at least one axis of revolution such as spheres and toroids, in which light is confined internally via continuous total internal reflection. If the circumference of the resonator is an integer multiple of the wavelength of light, resonance occurs due to phase matching. One particular class of whispering gallery resonators that has attracted significant interest in recent years are microbubble resonators [6–18]. Microbubble or microshell resonators are thin spherical shells which can support whispering gallery modes that have evanescent fields inside and outside the resonator. The thin walls can result in the evanescent fields extending significantly outside the cavity wall, resulting in a highly sensitive resonator .
Microbubble resonators are typically fabricated by locally heating an internally pressurized capillary which has been pre-tapered. When the capillary softens, the wall of the capillary expands, forming the bubble [6,8]. The heat required to keep the glass soft increases exponentially as the microbubble wall thickness decreases. Therefore for a given fixed heat source the convective heat loss from the surface will eventually exceed the heating rate and the glass rapidly cools. Therefore the size of the microbubble can be controlled by adjusting the heat applied . The flow-through capability of such capillary-based microbubbles also facilitates liquid delivery, hence allowing for upstream and downstream microfluidics . Some of the applications of microbubble resonators include refractive index sensing [9,13,15,16] and optomechanical and pressure sensing [9,12,16,19]. Microbubble resonators have also been used for temperature sensing and an order of magnitude increase in the thermal shift compared with solid microspheres has previously been demonstrated [9,10]. Microbubbles also allow for effective mechanical tuning of the resonance frequencies , and this could be used for tuning the resonator to the frequencies of atomic and molecular transitions for cavity quantum electrodynamics, and also as the basis of tunable microlasers and optical switches . Moreover, optofluidic ring resonator (OFRR) lasers based on microbubbles have also been demonstrated by filling the cavity with a liquid gain medium . Microbubbles are also of particular interest for biochemical sensing because they allow the samples to be placed inside the resonator, without disturbing the coupling mechanism with, for instance, a fiber-taper.
More recently microbubbles have also been used for optical frequency comb generation through hyperparametric excitation of the WGMs [17,20]. Optical frequency combs are regularly spaced coherent spectral lines, having a range of applications relating to metrology, spectroscopy and sensing [21–23]. Frequency comb generation in microresonators occurs due to four-wave mixing (FWM) which cascades along the resonances provided that dispersion is equalized [21,24–26]. The extremely small mode volumes (V) possible in microbubbles, is especially beneficial as it enhances the nonlinear interaction efficiency due to the increased energy density. The extra degree of freedom provided by the thickness of the microbubble walls when compared to solid spheres, also allows for much more effective tailoring of the dispersion. Frequency comb generation in microresonators requires high Q2/V values as well as dispersion equalization (see Section 2) [17,20,21,27]. In this paper these aspects are investigated as a function of microbubble diameter and shell thickness for various materials other than silica such as highly-nonlinear soft glasses, polymers and crystalline materials. The paper therefore provides the first extensive overview of the opportunities available for microbubble resonators consisting of materials other than silica for applications in nonlinear optics.
2. Analytical models
2.1 Cavity dispersion
The total dispersion of a microbubble resonator, consisting of both material and geometric components , can be determined from the resonance wavelengths of the transverse electric (TE) or transverse magnetic (TM) whispering gallery modes. The wavelengths of these modes can be solved using the general model of a microsphere with a high-index coating [29,30], in which the index of the sphere and surrounding medium is assumed to be that of air, n1 = n3 = 1.0. A schematic of the microbubble is shown in Fig. 1.
We assume that the microsphere has a radius ρ0 and index n1, the coating (or shell) has a thickness t and index n2 and the surrounding medium has index n3. The characteristic equation used to determine the resonance wavelengths for the TE (p = 1) and TM (p = −1) modes is [29,30],31],Eqs. (1) and (2) can be expressed as,Eq. (1) are considered. The wavelength-dependence of the refractive indices (i.e. material dispersion) can be incorporated using the relevant Sellmeier equations, allowing for the exact resonance positions to be solved using e.g. the fsolve function in MATLAB. The resonances determined from Eq. (1) are categorized by radial and azimuthal mode numbers q and l, which determine the number of nodes of the electric field in the radial and azimuthal directions, respectively. In this paper only the fundamental first-order modes (q = 1) are considered which are the solutions of Eq. (1) with the smallest value of kr for each mode number l. We note here that the radial mode order q provides an additional degree of freedom, and could be used to further tailor the geometric dispersion .
To ensure correct identification of the first-order mode for a given azimuthal mode number l, the MATLAB script implemented solves the characteristic equation for several initial wavelength guesses over an appropriate wavelength range centered on the rough estimate of 2πnρ/l. The real solutions kr are then ordered, and the smallest value corresponds to the first-order q = 1 mode. We note that the characteristic equation is convex around the fundamental first-order eigenvalue. If the starting guess for the wavelength is chosen larger than the expected value of the q = 1 eigenvalue it will converge to the q = 1 mode. Solving the characteristic equation for multiple initial wavelength guesses, however provides added robustness for the code and also allows higher radial order modes to be identified.
After the first-order resonances are identified for a range of l, the total cavity dispersion is then given by the change in free-spectral range (FSR) over the resonances, Δ(Δvl) = Δ(Δ(ckr,l/2πn)), where vl are the discrete resonance frequencies. The zero dispersion wavelength (ZDW, λZD) for the microbubble of given material, thickness and diameter is then determined from the kr value, corresponding to azimuthal mode l, nearest to the zero crossing of Δ(Δvl). Examples of detuning curves (blue traces) showing the zero crossings (with red line being Δ(Δvl) = 0) are shown in the insets of Fig. 2. This approach of calculating the cavity dispersion is exact, as it correctly incorporates the interplay between material and geometric dispersion. For microbubble resonators there are typically several ZDWs of which only the shortest wavelength solution is considered in this paper.
The figure of merit (FOM) most commonly used to quantify the efficiency of nonlinear interactions in microresonators is Q2/V [33,34]. The figure of merit can also be specified as nkerrQ2/V, where nkerr is the nonlinear index of the resonator material [27,35]. Therefore efficient nonlinear interactions require ultra-high Q-factors and small mode volumes. The total Q-factor of a microbubble resonator is determined from several contributing factors which add in parallel (i.e. Q−1 = Qgeo−1 + Qss−1 + Qmat−1) including intrinsic geometric losses (Qgeo), scattering losses due to surface in-homogeneities (Qss) and material losses (Qmat) . The geometric losses occur because total internal reflection is incomplete at a curved surface resulting in tunneling loss . The geometric Q-factor component could be determined from the complex solutions of Eq. (1) as Qgeo ~kr/2kim. Geometric losses however generally only have a significant influence on the overall Q-factor for small microbubbles of the order of 10s of microns or less, or those having subwavelength wall thicknesses [13,18]. The Qgeo component has a roughly exponential dependence on diameter and thus becomes insignificant for larger diameters [13,18]. Note that the Q-factor component Qgeo of microbubbles can differ significantly from that of microspheres as shown in . The total cavity Q-factor of a microbubble resonator is typically limited by the material or scattering losses. In the case of smooth glass-like surfaces assumed here, the material losses usually dominate the overall Q-factor through ,
2.3 Mode volume & power fraction
As previously mentioned, efficient nonlinear interaction in a microresonator requires a small mode volume as this allows for a high peak intensity inside the shell. In a microbubble resonator the mode volume can be decreased, to a certain point, by reducing the wall thickness or diameter, or by increasing the index of the shell. The mode volume can be determined from the cross-sectional electric field distributions of the WGMs which can be obtained using a finite-element (FEM) simulation in which axial symmetry is used to reduce the 3D problem to 2D . The mode volume V, as well as the power fraction inside the shell P, is then determined by integrating over the volume of the resonator as follows,
3. Numerical results
In the case of silica microbubbles, the zero dispersion wavelength (ZDW) varies dramatically as the diameter and wall thickness is changed . As previously reported, the ZDW ranges from ~600 nm at small diameters and small wall thicknesses to ~1900 nm at small diameters and large wall thicknesses . This amounts to over three octave tuning of the ZDW by changing the geometry. Note also that as the diameter and wall thickness increase the ZDW approaches that of bulk material because of the decreasing geometric dispersion. In this paper we investigate the ZDWs of microbubble resonators consisting of various other materials including highly-nonlinear glasses. High Q-factor microbubble resonators having high nonlinear indices nkerr and small mode volumes V, hold considerable promise for realizing nonlinear phenomena with low power thresholds . For example, soft glasses such as tellurite (n = 1.98, nkerr = 55 × 10−20 m2 /W) or lead-germanate (n = 1.91, nkerr = 57 × 10−20 m2 /W) have nonlinear indices (at e.g. λ = 1.55 µm) that are twenty-times greater than silica (n = 1.44, nkerr = 2.7 × 10−20 m2 /W), and this lowers the nonlinear power threshold, provided that there is no compromise in Q-factor . Note however that the larger refractive index n or nkerr values (which are related according to Miller’s Rule ) of materials such as tellurite and lead-germanate tends to correlate with an increased ZDW .
The ZDWs of Na-Zn tellurite (TZNL) [42–44] and lead-germanate (GPL5) [42,44] soft glass microbubbles are shown in Figs. 2(a) and 2(b) as functions of diameter and shell thickness. The ZDW of tellurite microbubbles ranges from ~1100–2600 nm as the diameter (80–380 µm) and wall thickness (1-5 µm) is varied. As the diameter and wall thickness increase the ZDW converges to that of the bulk material as might be expected . Part v of Fig. 2(a) shows the absorption spectrum of tellurite [42,44] overlaid with the ZDW range for microsphere resonators (shaded green) as modelled in . The ZDW range for the microbubbles modelled here is also shown (shaded blue). The much broader ZDW range for the microbubbles arises because of the additional degree of freedom provided by the shell thickness for adjusting the geometric component of dispersion. This allows the ZDW to be made to coincide with the low-loss wavelength window of the glass, whilst this is not possible using microspheres . This would allow for the realization of higher absorption-limited Q-factors Eq. (6), in addition to smaller mode volumes compared with microspheres, hence lowering the threshold for comb generation. A similar scenario applies for lead-germanate microbubbles in which the ZDW range is between ~1100–2700 nm over the same parameter space, allowing for dispersion equalization in the low-loss wavelength window [42,44] of the material, not possible using microspheres.
Furthermore, the mode volumes for both tellurite and lead-germanate microbubbles (i.e. 100s to 10,000s µm3 over the parameter space) shown in part iii of Figs. 2(a) and 2(b) tend to be slightly smaller compared to those of silica microbubbles , whereas the fraction of power confined to the shell is nearly identical. For the first-order fundamental modes modelled here the fraction of power in the shell is close to 100% as shown in part iv of Fig. 2, whereas we note that for higher-order modes a large proportion of the light will reside outside the shell .
At this point it is interesting to note that the ZDWs of the TE modes (i) are generally higher than those of the TM modes (ii). This differs from microspheres for which the ZDW is largely independent of polarization . Indeed, as the wall thickness increases, and the resonator geometry approaches that of a microsphere, the discrepancy in ZDWs for the two polarizations diminishes. The polarization dependence is due to the shell-air interfaces, which result in large discontinuities in the perpendicular electric fields. The effective index of the TM modes, whose polarization is mainly perpendicular to the wall, is more sensitive to the wall thickness compared with the TE modes. This results in greater geometric dispersion for the TM modes, and thus accounts for the polarization dependence of the ZDW. This is also why the mode volume of the TM modes tends to be marginally smaller than for the TE modes, for the present case where the wall thickness is greater than the wavelength .
We note that one issue with high-index cavities is that coupling to the fundamental modes using tapered silica fibers is challenging, as higher-order radial modes (or higher-order Hermite-Gaussian modes) with effective index closer to the taper modes are more efficiently excited . Note however that modes with large polar mode number m can have reduced effective index, and hence could be efficiently coupled to . Alternatively, high-index tapers made of soft glasses such as tellurite could be used for effective phase-matching to the fundamental microbubble modes.
The ZDWs of microbubbles consisting of several other soft glasses were also modelled including F2 (Flint)  and LLF1 (Very Light Flint)  in Figs. 3(a) and 3(b), and NBK7 (Borosilicate Crown)  and SF57 (Dense Flint)  in Figs. 4(a) and 4(b), respectively. In the case of F2 microbubbles the ZDW ranges from ~900–2300 nm over the range of diameters (60–380 µm) and wall thicknesses (1–5 µm) considered. For the LLF1 microbubbles the ZDW range is similar spanning ~600–2300 nm over a comparable parameter space. This similarity is unsurprising given the closeness of the material compositions of these lead-silicate soft glasses. As before, the ZDW tends to be marginally smaller for TM modes compared with the TE modes. Part v of Figs. 3(a) and 3(b) clearly demonstrates that whilst dispersion equalization in the low-loss wavelength windows of the respective materials is not possible for microsphere resonators (green-shaded) , the extra degree of freedom of shell thickness available for microbubble resonators, allows for the ZDW to be tailored to coincide with the low-loss wavelength windows for the two materials (blue-shaded). This implies that the absorption-limited Q-factors, Eq. (6), of the soft glass microbubbles would be relatively high at zero dispersion. For these reasons, optical frequency comb generation is likely to be possible with a low power threshold for these, and other similar, soft glass microbubbles, whilst this is not the case for microspheres . In fact for the case of soft glass microspheres the ZDW ranges from ~2–3 µm which tends to coincide with relatively high-loss spectral regions of such glasses .
As shown in Fig. 4(a) the ZDW for NBK7 (Borosilicate Crown) microbubbles ranges from ~800–2600 nm over the range of diameters (40–380 µm) and wall thicknesses (1–5 µm) modelled. For the case of SF57 (Dense Flint) soft glass microbubbles the ZDW ranges from 1000 – 2600 nm over a similar parameter space, as is shown in Fig. 4(b). Part iii of the figures show that the ZDW space of NBK7 and SF57 microspheres coincide with high-loss spectral regions of the glasses, whereas for the case of microbubbles considered here, the thin walls allow for the ZDW to be reduced to well within the low-loss wavelength windows of these materials. This dramatically improves the absorption-limited Q-factors possible according to Eq. (6), and hence the prospects of realizing frequency comb generation.
The ZDWs of microbubbles consisting of certain crystalline materials were also considered, as is shown in Figs. 5(a)–5(c). The crystalline materials were CaF2 , MgF2  and ZnO . Note that the ordinary index of refraction was assumed for the latter two birefringent materials. For the case of CaF2 the ZDW ranges from ~600–4000 nm over the range of diameters (40–380 µm) and wall thicknesses (0.75–5 µm) considered, far exceeding the ZDW range possible in microsphere resonators . For instance CaF2 microspheres of diameter 70 µm have λZD = 4130 nm, whereas for a microbubble of equal diameter, reducing the shell thickness allows this value to be reduced down to 600 nm or less. The MgF2 microbubbles were found to have ZDWs spanning several octaves from ~700–3600 nm over a similar parameter space. For the case of MgF2 microspheres of diameter 70 µm, λZD = 3340 nm, whereas for an analogous microbubble resonator, the zero dispersion wavelength ranges from λZD = 800–3400 nm, providing far greater control over the dispersion.
Furthermore, as shown in Fig. 5(c), ZnO microbubbles were found to have ZDWs ranging from ~1100–4000 nm over a parameter space similar to before. Of the crystalline materials considered, the only materials that exhibit reasonably low loss within the relevant ZDW ranges are CaF2 and MgF2, which are materials that have attracted considerable interest in recent years for mid-infrared comb generation [22,51]. The fabrication of microbubble resonators using such crystalline materials is however challenging .
Microbubbles consisting of various polymers were also investigated. Polymers have the advantage of having high chemical stability in water and therefore represent appropriate candidates for realizing nonlinear phenomena such as comb generation in aqueous settings. The polymers considered included polymethyl-methacrylate (PMMA) , polystyrene , polycarbonate  and cyclic-olefin polymer (Zeonex E48R) . As shown in the inset of Fig. 6(a) part i, the dispersion curve of PMMA has two peaks. This implies that there are four zero dispersion wavelengths, and it means that a flat dispersion profile is possible over a large wavelength range. For PMMA we consider only the shortest zero dispersion wavelength solution which is nearly unchanged at 500 nm for the entire range of diameters (40–240 µm) and wall thicknesses (0.75–5 µm) considered. We note that the second zero dispersion wavelength solution, associated with the second dispersion peak, ranges from 800–1000 nm across this parameter space. Now, since 500 nm is within the low-loss (< 1 dB/m)  wavelength window of the polymer (see Fig. 6(a), part iii) , this implies that comb generation in the visible may well be possible, which is not the case for PMMA microsphere resonators which have a single ZDW centered at ~900 nm . Also, since the ZDW reaches deep into the visible, the confinement of light is greater at this operating wavelength, which allows for a smaller mode volume and an enhanced Q-factor, contributing to a reduction in the nonlinear power threshold for comb generation [27,35]. The fraction of power confined to the shell was found to reach 99.9% for the larger diameters and wall thicknesses considered. We also note that a reduction of the ZDW into the UV may be possible by reducing the diameter and thickness of the microbubble, although it must be noted that material absorption increases dramatically below 400 nm as seen in part iii of Fig. 6(a) .
The results of Fig. 6(b) show that the ZDW of polystyrene microbubbles ranges from 850 to 1050 nm over a similar parameter space as before, and this wavelength range is considerably broader than that possible for microspheres as suggested in part iii . For the polystyrene microbubbles the first dispersion peak occurring at shorter wavelengths does not have a zero crossing, resulting in higher ZDW values for this polymer compared with PMMA. However as before low dispersion over a broad wavelength range is still possible given the presence of two dispersion peaks which flattens the detuning even if zero dispersion is not strictly maintained over the entire wavelength range.
The ZDWs of polycarbonate and cyclic-olefin polymer (Zeonex E48R)  microbubbles were also considered as shown in Figs. 7(a) and 7(b), respectively. The ZDW for polycarbonate microbubbles was found to range from 800 to 1050 nm over the range of diameters (20–200 µm) and wall thicknesses (0.75–5 µm) considered. We note that the first dispersion peak might eventually reach zero dispersion in the UV at around 300 nm, however due to limits imposed by the Sellmeier equation  used this could not be confirmed. Nonetheless even if there are solutions for zero dispersion in the UV this is of limited interest given that it coincides with a very high-loss spectral region for the polymer. For the case of cyclic-olefin (Zeonex E48R) microbubbles modelled in Fig. 7(b), the second dispersion peak was found to reach zero dispersion only for small wall thicknesses as shown in the first inset of part i, resulting in a ZDW as low as 500 nm. However as the wall thickness increases the second peak no longer crosses zero dispersion as shown in the second inset of Fig. 7(b) part i. Plotting the shortest ZDW solution therefore yields a discontinuity as observed in Fig. 7(b). Beyond this boundary the ZDW ranges from about 850–1000 nm for wall thicknesses spanning ~1.5–5 µm. Again the ZDW range is far greater than that possible for equivalently-sized spheres, whilst the presence of two dispersion peaks allows for relatively flat dispersion between 500 and 1000 nm as suggested by the first inset of Fig. 7(b). At this point we note that in general a flatter dispersion profile occurs for the larger microbubbles corresponding to longer ZDWs, making them more suitable candidates for broadband comb generation in the anomalous regime.
The smallest ZDW possible for the microbubbles considered over the relevant parameter space is summarized in Table 1 for the various materials modelled in this paper. Also given are the linear n and nonlinear nkerr refractive indices at λ = 1550 nm for the respective materials. Note that further tailoring of the dispersion is likely to be possible by using various liquids inside and outside of the microbubble resonators. Further work could also involve investigating the potential span of frequency combs in microbubble resonators, by analysing the detuning (i.e. flatness of the dispersion) eitherside of the ZDW [32,35].
From the previous results it is apparent that for microbubbles of higher refractive index, the modal confinement is enhanced, as might be expected, and hence the mode volume tends to be smaller. As mentioned this can assist in increasing the nonlinear interaction efficiency. The results of Fig. 8 provide a visual demonstration of the difference in modal confinement between low-index (e.g. polystyrene, n ~1.5) and high-index (e.g. tellurite, n ~2) microbubbles. The discrepancy becomes particularly pronounced at smaller wall thicknesses. The figure shows the normalized radial TE field distributions at λ = 632 nm for the first-order modes, as the shell thickness is varied (0.3–2 µm) for a fixed diameter of 100 µm. The figure also illustrates how the field distribution and thus mode volume varies dramatically as the shell thickness is increased.
At this point it is also worth mentioning that the ZDWs of some of the microbubbles considered in this paper are likely to extend further into the visible for diameters smaller than those considered here. Provided that the wall thickness is greater than the wavelength of light, high Q-factors can still expected for smaller microbubbles , implying that frequency comb generation within the visible is likely to be possible. This is especially true given that, as mentioned, the small mode volumes of microbubble resonators enhance the nonlinear interaction efficiency as quantified by the figure of merit (Q2/V). The short operating wavelength also contributes to enhancing the mode confinement.
Also, provided a strong index contrast is maintained and the wall thickness is not subwavelength, the ZDW is found to be relatively insensitive to whether the surrounding medium is air or water. Therefore dispersion equalization could still be realized in the visible for microbubbles in aqueous solution. Given that the low-loss wavelength window of water is in the visible , the possibility may exist for realizing the high absorption-limited Q-factors  necessary for comb generation in aqueous solution for various applications including biosensing.
In conclusion, this paper has explored for the first time the scope for engineering dispersion in microbubble resonators by varying the shell material, thickness and diameter. Dispersion equalization is shown to be possible in the visible for microbubbles of small diameters (< 100 µm) for various different materials. Moreover, dispersion equalization in the mid-infrared is shown to be possible for large microbubbles (>100 µm) for the case of high-index materials such as tellurite and lead-germanate. The possibility of realizing dispersion equalization within these interesting wavelength windows, in addition to the small mode volumes possible for thin-walled microbubbles, could provide novel opportunities for nonlinear optics. One such opportunity might be visible frequency comb generation in microbubbles within aqueous solution for biosensing applications.
The authors acknowledge the support of an Australian Research Council Georgina Sweet Laureate Fellowship awarded to T. M. Monro, FL130100044. The authors thank S. A. Vahid, A. François and J. M. M. Hall for insightful discussions.
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