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This paper evaluates the opportunities for using materials other than silica for optical frequency comb generation in whispering gallery mode microsphere resonators. Different materials are shown to satisfy the requirement of dispersion compensation in interesting spectral regions such as the visible or mid-infrared and for smaller microspheres. This paper also analyses the prospects of comb generation in microspheres within aqueous solution for potential use in applications such as biosensing. It is predicted that to achieve comb generation with microspheres in aqueous solution the visible low-loss wavelength window of water needs to be exploited. This is because efficient comb generation necessitates ultra-high Q-factors, which are only possible for cavities with low absorption of the evanescent field outside the cavity. This paper explores the figure of merit for nonlinear interaction efficiency and the potential for dispersion compensation at unique wavelengths for a host of microsphere materials and dimensions and in different surroundings.
© 2015 Optical Society of America
1. Introduction
Often generated using mode-locked lasers, using a periodic train of ultrafast laser pulses, optical frequency combs consist of regularly spaced spectral lines in the ultraviolet, visible or infrared. Optical frequency combs have attracted considerable interest in recent years with applications as diverse as metrology, spectroscopy and sensing [1–3]. Another way to generate frequency combs is by hyperparametric excitation of the whispering gallery modes (WGM) of a microresonator such as a microtoroid or microsphere [1–3], with the latter forming the focus of this paper. Microresonators are an ideal platform for nonlinear interactions due to their capacity to store energy as quantified by their high quality factors. The successful realization of comb generation at novel wavelengths using microresonators, or within aqueous environments, has considerable potential to spawn further applications. For example, there may exist opportunities for spectral fingerprinting [3] and biosensing using microresonators in aqueous environments. The dependency of various comb features such as the bandwidth or spectral offset on the surrounding environment could for instance be exploited for various applications such as refractive index sensing. As an example, Silverstone et al., have shown that by exploiting a full WGM spectrum consisting of multiple modes using the Fourier Transform, higher resolution of the wavelength shifts occurring from changes in the surrounding refractive index can be achieved when compared to single mode tracking [4,5]. Similar improvements in the refractive index sensitivity could be expected when tracking frequency combs. Moreover, it is envisioned that binding events of e.g. proteins or molecules onto the surface of the microresonator could be detected by monitoring comb features such as the power threshold which would be affected by the spoiling of the Q-factor [6].
Frequency comb generation in microresonators is a nonlinear process in which four-wave mixing cascades along the resonances of the cavity. The main condition for frequency comb generation to occur is dispersion equalization (see Section 2.2), in which the material dispersion cancels the geometric component, ensuring that the whispering gallery modes of the resonator are equally spaced in frequency. This is necessary for the four-wave mixing process to spread among the resonances without breaking energy conservation conditions. If instead the mode spacing varies, energy conservation cannot be maintained and this drastically limits the bandwidth of the comb generated [7–9]. Dispersion compensation is however difficult to realize in small microresonators due to the often severe negative geometric dispersion. For the same reason dispersion compensation is also generally difficult to achieve at short wavelengths such as within the visible or near-infrared. The latter poses a significant problem if microresonators such as microspheres are to be used, for instance, in aqueous environments for comb generation since the high Q-factors necessary for efficient parametric oscillations may not be possible. This is because of the high absorption (see inset of Fig. 2(b)) of the cavity mode evanescent fields in water, at the longer wavelengths necessary for dispersion equalization. Therefore, if frequency combs are to be feasible in aqueous solution for applications such as biosensing, effective tailoring of the dispersion is needed to allow for operation at shorter wavelengths. Conversely, by exploiting the properties of high-index materials this paper shows that it is possible to achieve dispersion compensation in microspheres at longer wavelengths as would be required for mid-IR comb generation in applications such as molecular ‘finger printing’ [3].
Whilst recent studies have explored the use of micro-shell resonators to more readily tailor dispersion, and in particular to reduce the zero dispersion wavelength (ZDW) [10], for simplicity the present study focuses purely on microsphere resonators. Microspheres are not only simpler to fabricate, but are also far less restrictive on the choice of material. The latter benefit provides flexibility for dispersion compensation. The geometric dispersion in both microsphere and micro-shell resonators is high and as suggested this is what makes compensation with material dispersion difficult [10]. This issue is explored in the present paper for microspheres of several different material groups, unlike previous studies which have focused almost exclusively on the use of silica microspheres [7,8,10]. As mentioned, the challenge lies in achieving dispersion compensation at short wavelengths and for small microsphere size. Small microspheres are preferred for sensing applications due to their enhanced sensitivity to the surrounding environment. In order to maintain strong confinement of the light, the smaller resonator would however have to be operated at shorter wavelengths, providing an additional incentive for reducing the ZDW to the visible.
2. Analytical models
2.1 Cavity dispersion
The total dispersion of a microsphere, which consists of both material and geometric contributions, can be determined by first solving the resonance frequencies for the TE or TM whispering gallery modes. The resonance conditions for the TE (p = 1) and TM (p = −1) modes are given by [11],
where n = n(k0) is the sphere index, nout = nout(k0) is the surrounding index, ψl(z) = zjl(z) and χl(z) = zηl(z) are the spherical Ricatti-Bessel and Ricatti-Neumann functions, jl(z) and ηl(z) are the spherical Bessel and Neumann functions, respectively, k0 is the resonance wavenumber, l is the angular mode number and p is the polarization coefficient [11]. By incorporating the wavelength-dependence of the refractive indices using the relevant Sellmeier equations, the exact resonance positions can be solved using e.g. the fsolve function in MATLAB®. The resonances determined from Eq. (1) are categorized by two quantum numbers q and l. The radial mode number q determines the number of nodes of the WGM in the radial direction, whereas the angular mode number l determines the number of nodes in the azimuthal direction. In this paper only the first-order modes (q = 1) are considered which are the solutions of Eq. (1) with the largest value of k0 for each mode number l. To ensure correct identification of the first-order mode for a given mode number l, the code implemented solves the equation for multiple initial wavelength guesses over an appropriate wavelength range centered on 2πρn/l. The solutions k0 are then ordered, with the largest value corresponding to the first-order q = 1 mode. Once the first-order resonances are determined 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(ck0,l/2πn)), where vl are the discrete resonance frequencies. The zero dispersion wavelength for the sphere of given material, diameter and surrounding medium is then determined from the first-order k0 value, corresponding to angular mode l, nearest to the zero crossing of Δ(Δvl). Note that this approach of calculating the cavity dispersion is exact as it correctly incorporates the dependency between material and geometric dispersion.2.2 Frequency combs
The basic process by which frequency comb generation occurs in high Q resonators is four-wave mixing (FWM) [12–15]. In this process two pump photons (2ωP) are converted into signal (ωS) and idler (ωI) photons. Frequency combs can be thought of as a propagation of this FWM process across the resonances that occurs provided that both energy and momentum are conserved. The latter is conserved intrinsically when the signal and idler angular mode numbers, are located symmetrically around the pump (i.e. lS,I = lP ± N) [7,8]. Energy conservation is not satisfied apriori, since the resonant frequencies in a microresonator, such as a microsphere, are usually irregularly spaced due to dispersion as is shown in Fig. 1 [7].
Fig. 1 The effect of normal (ΔFSR < 0) and anomalous (ΔFSR > 0) dispersion on the modes of a microresonator. In the case of zero dispersion, the resonances match the equidistant comb lines.
However in the case of zero dispersion the modes are equidistant, allowing for them to match the equidistant comb lines [8]. For certain microsphere sizes a small mismatch between resonator modes and comb lines can be maintained even over broad bandwidths [8]. This is true despite the narrow cavity linewidths associated with high Q microspheres. The realization of broadband frequency combs is made possible by cross- and self-phase modulation that distorts the mode spectrum via the third order Kerr effect and it essentially adjusts the mode spectrum. The resonance frequency distortion based on Kerr nonlinearity induced index change is also known as ‘mode-pulling’ and is approximately given by [8]:
where v is the resonance frequency, n and n2 are the linear and nonlinear refractive indices of the microsphere resonator and I is the circulating intensity. Equation (2) demonstrates that enhanced mode-pulling may be possible using materials of high nonlinearity such as tellurite (n2 = 55 × 10−20 m2 /W) or germanate (n2 = 57 × 10−20 m2 /W) when compared to silica (n2 = 2.7 × 10−20 m2 /W). This could have implications for dispersion-flattening for broadband comb generation. However larger refractive index n or n2 values (which are related according to Miller’s Rule [16,17]), in general correlate with significant increases in the ZDW as is later shown in Section 3. Equation (2) also demonstrates that the mode-pulling effect is enhanced (up to a certain point) by using smaller microspheres because of the increase in circulating intensity that may be realized. To achieve an overall flat dispersion actually requires a slight wavelength dependent variation in the mode positions. This is realized by having anomalous dispersion (i.e. pump wavelength is slightly beyond the ZDW) as shown in Fig. 1, since nonlinear mode-pulling can naturally only compensate a positive Cavity-Comb frequency mismatch [8].2.3 Nonlinear figure of merit
The figure of merit (FOM) most commonly used to quantify the efficiency of nonlinear interactions in microresonators is Q2/V [7,18]. Efficient nonlinear interactions therefore require ultra-high Q-factors and small mode volumes. The latter has been the main motivation for recent research into microtoroidal resonators [13,14]. In the case of microspheres the mode volume can be decreased only by either reducing the diameter or increasing the index of the microsphere.
The effective volume of whispering gallery modes exhibits a near-quadratic dependence on mode number l, and assuming l >> 1 can be approximated by [19],
where λ is the free-space wavelength, m is the polar mode number (l = m for the fundamental modes modelled in this paper) and n is the refractive index of the microsphere.The calculation of Quality factor (Q−1 = Qgeo−1 + Qss−1 + Qmat−1) for whispering gallery modes is more complex and is determined by several factors which add in parallel, including intrinsic radiative losses (Qgeo), scattering losses due to surface in-homogeneities (Qss) and material losses (Qmat). The radiative or geometric losses occur because total internal reflection is incomplete at a curved surface resulting in tunneling losses [7]. The Q-factor contribution can be approximated in the limit of l >> 1, by expanding the characteristic equation (Eq. (1)) and allowing the wave-vector k0 to be complex [7],
where t0q is the qth airy function zero, f is the relative index of refraction (with respect to the surrounding medium), and,Whilst the radiative losses are large for spheres with sizes of the order of the wavelength of light, the total cavity quality factor (Q) is typically limited by the material or scattering losses for larger microspheres. The material and scattering losses can be approximated by [20],where α is the bulk material attenuation (dB/m), neff is the effective mode index, ρ is the microsphere radius and σ and L are the RMS size and correlation length of the surface in-homogeneities.3. Numerical results
The zero dispersion wavelength (ZDW) represents the boundary between the normal (ΔFSR = Δ(Δvl) < 0) and anomalous (ΔFSR = Δ(Δvl) > 0) dispersion regimes, and can be thought of as a lower limit on the comb wavelength. The ZDWs of microspheres as a function of diameter, are shown in the following set of figures for various different materials and material groups. The figures show that the ZDW is largely independent of polarization. Assuming a reasonable index contrast is maintained (e.g. Δn ≥ nsilica – nwater), the ZDW is also found to be insensitive to the surrounding medium. Therefore microspheres in aqueous solution will in general have nearly the same ZDW as those in air. This is especially true in the limit of large microspheres for which the discrepancy becomes negligible. It is also worth noting that as the microsphere size increases the ZDW tends towards that of bulk material (see Table 1) since the geometric dispersion contribution diminishes.
Table 1. Exponential Fits of ZDW as a function of Diameter.
Our calculations are in agreement with previous reports that silica microspheres operating in the low-loss C-band (1530-1565 nm) have anomalous dispersion for diameters exceeding approximately 160 µm [8,10]. For smaller microspheres, exhibiting higher refractive index sensitivity [21], the ZDW shifts to even higher wavelengths as is shown in Fig. 2(a). Since the low-loss wavelength window for water (see inset of Fig. 2(b)) is in the visible, this poses a significant problem for the use of silica microspheres for frequency comb generation in aqueous applications such as biosensing. This is because strong evanescent field absorption will drastically lower the Q-factor [22], rendering comb generation impractical. The total cavity dispersion could however be further tailored to lower the ZDW closer towards the visible by instead using a hollow microsphere, also known as a micro-shell [10,23].
Fig. 2 (a) The zero dispersion wavelength as a function of diameter for silica microspheres in air and water. (b) The spectral absorption of silica, and (inset) water [40,41].
In the case of soft glasses, the ZDWs of the microspheres are found to be higher as shown in Fig. 3(a) and typically range from ~2-3 microns. The low-loss wavelength windows for such glasses however tend to be at shorter wavelengths as is shown in Fig. 3(b). In fact the ZDWs coincide with relatively high-loss spectral regions (i.e. >> 1 dB/m) which means that comb generation would be challenging to achieve in soft glass microspheres. The types of glass considered here are lead-germanate (GPL5) [24], Na-Zn tellurite (TZNL [24]) and the Schott soft glasses: SF57 (Dense Flint), F2 (Flint), LLF1 (Very Light Flint) and NBK7 (Borosilicate Crown) [25]. The zero dispersion wavelengths for microspheres of diameter 70 µm are approximately 2844, 2678, 2440, 2150, 2066 and 1866 nm, respectively. Note that soft glasses such as tellurite and lead-silicate (e.g., F2) tend to have low chemical stability in water, which poses a problem if they were to be used for applications such as biosensing [26]. One possibility would be to use suitable coatings to protect the microsphere surfaces from deteriorating.
Fig. 3 (a) The zero dispersion wavelength as a function of diameter for various soft glass microspheres in air and water and (b) the absorption spectra of the glasses [24,25].
The ZDWs of various polymer microspheres are shown in Fig. 4(a), and the values range from around 870-1050 nm. The lower refractive indices of the polymers compared with the soft glasses correlate with reduced ZDWs, but also lower n2 values. The reduction in n2 values potentially decreases the comb bandwidth according to Eq. (2). For the case of 70 µm microspheres, the ZDWs for polystyrene, polycarbonate, cyclic-olefin polymer (Zeonex E48R), poly(methyl methacrylate), and polylactide are 1022, 987, 932, 912, and 893 nm, respectively. As shown by Fig. 4(b), the ZDWs of polymer microspheres in general occur within the respective low-loss (< 1 dB/m) wavelength windows. By further tailoring the dispersion (e.g. using a micro-shell geometry) comb generation well within the visible may be possible. Polymers are also typically immune to degradation in water, and therefore represent ideal candidates for realizing nonlinear phenomena such as comb generation in aqueous settings.
Fig. 4 (a) The zero dispersion wavelength as a function of diameter for various polymer microspheres in air and water and (b) the absorption spectra [42,43]. The vertical axes are identical for all plots in (b).
It is clear from the previous figures that the dependency of the ZDW on microsphere diameter can be approximated by an exponential decrease of the form:
where d is the microsphere diameter and A, B and C are constants given in Table 1 which are determined by the material used. Note that for large diameters the ZDWs do not converge exactly to the bulk material ZDWs (i.e. zero Group Velocity Dispersion (GVD) in Table 1), since the exponential decrease is accurate only in the range of diameters considered.† Nonlinear indices calculated using Miller’s Rule and measured values in brackets [ 17 ]
‡ Ordinary index of refraction assumed.
The ZDWs of microspheres consisting of some other material candidates for comb generation are shown in Fig. 5(a). These crystalline materials include ZnO, CaF2, MgF2 and Al2O3, which have anomalous dispersion beyond 4305, 4132, 3338 and 1768 nm operating wavelengths for the case of 70 µm spheres. Of these the only microspheres that exhibit anomalous dispersion within the low-loss wavelength window of the respective material, and for which these losses are of a reasonable level, is MgF2 which is a material that has attracted considerable attention for mid-IR comb generation in microtoroidal resonators [3]. The use of such crystalline materials may however present considerable fabrication challenges especially for the case of small microspheres.
Fig. 5 (a) The zero dispersion wavelength as a function of diameter for several microspheres of crystalline materials, in air and water and (b) the absorption spectra [44]. These materials present opportunities for mid-infrared comb generation.
As shown in Fig. 6, the zero dispersion wavelength has a near linear dependence on the refractive index of the microsphere, and this is true for all diameters considered. The precise relationship depends on the specific material group, and shown here are Glass and Polymer. The normal and anomalous dispersion regimes occur either side of the ZDW. By extrapolation of Fig. 6 it seems that anomalous dispersion well within the visible is not possible for glasses of typical or even low refractive index. On the other hand, it appears that low-index polymers might be suitable candidates for comb generation in the visible. We note that by further tailoring the dispersion using a hollow-core structure [10,23] comb generation in the visible may be possible without necessitating the use of a polymer of ultra-low index. This would allow for a greater index contrast with water, facilitating its use in an aqueous setting.
Fig. 6 (a) The zero dispersion wavelength as a function of refractive index for the (a) Glasses and (b) Polymers. This is shown for several different microsphere diameters. The nonlinear indices as given by Miller’s Rule are shown in the insets [17].
As mentioned, microspheres in air will in general have nearly the same ZDW as those in aqueous solution. The Q-factors are however in general far more sensitive to the surrounding environment [19,20,27]. As mentioned, the Q-factor is made up of several individual components. For the large microspheres considered here the geometric Qgeo component is excessive, and the limiting factor on the cavity Q-factor is either material absorption or scattering loss. Scattering losses at the surface however only contribute significantly for small to intermediate-size spheres [20]. The RMS size and correlation length of the surface in-homogeneities modelled were σ = 0.3 nm and L = 3 nm, respectively, which are typical of very smooth glass-like surfaces [20].
The figure of merit, Q2/V [1/µm3] for nonlinear interaction efficiency is shown in Fig. 7 as a function of wavelength and microsphere diameter for a selection of materials. With the benchmark of silica, material absorption limited Q-factors approaching 1010 are possible forlarge microspheres [20,28]. For smaller diameters, the Q-factor is slightly reduced due to increased scattering and hence a reduction of Qss. The mode volume is however also lower (~104 µm3) for smaller spheres, such that the Q2/V value is actually enhanced as is shown in Fig. 7(a). Within the anomalous dispersion regime, Q2/V values upwards of 1017 µm−3 are possible in the near-infrared [19,20,28].
Fig. 7 (a) The dependence of the nonlinear figure of merit Q2/V [1/µm3] on diameter and wavelength for (a) silica, (b) F2 glass, (c) tellurite (TZNL), (d) MgF2, (e) polystyrene and (f) PMMA microspheres in air. The RMS size and correlation length of the surface in-homogeneities modelled were σ = 0.3 nm and L = 3 nm, respectively. The absorption data shown in previous figures was used to determine Qabs.
In the case of the soft glasses, material absorption tends to be significantly higher at the relevant ZDW boundaries (e.g. > 6 dB/m for TZNL [24], > 10 dB/m for F2 [25], and > 2 dB/m for NBK7 [25]), which reduces the Q-factors (Q ≈ Qmat) to ~105-106. These values are close to previous Q-factor measurements of e.g. 3 × 106 for tellurite [29] and 9 × 106 for lead-silicate [30]. It is worth noting, however, that the chemical compositions of the soft glasses could be tailored to minimize the losses in the relevant spectral regions and possibly to well below 1 dB/m. The scattering losses (Qss ~ 1010) are expected to have a very limited influence on the overall Q-factor for the soft glasses considered, and this is true even for microspheres of small diameters with enhanced scattering. For the examples of soft glasses considered the mode volumes decreased significantly (i.e. ~20-60%) compared with silica, but not nearly enough to offset the large reductions in Q-factor. For F2 and tellurite glasses, for instance, Q2/V values of the order of 107 and 108 µm−3 are predicted close to the ZDW boundary as shown in Figs. 7(b) and 7(c). If the ZDW boundary can be lowered to within the low-loss visible or near-infrared wavelength windows (i.e. << 1 dB/m) of the soft glasses by further tailoring the dispersion, several orders of magnitude improvement in the Q2/V values can be anticipated. As mentioned, there also exists scope for modifying the chemical composition of the glasses to reduce the losses in the relevant spectral regions. Note that tellurite has the advantage of a particularly high n2 value of 55 × 10−20 m2/W, which is twenty times that of silica. This contributes to lowering the minimum power threshold for comb generation by marginally offsetting the reduction in Q2/V since [1,15],
For MgF2 the absorption-limited Q2/V value is around 107 µm−3 close to the dispersion boundary as shown in Fig. 7(d), with the higher material losses again lowering the Q-factor considerably compared with that of silica.Polymer microspheres consisting of, for example, polystyrene or poly(methyl methacrylate), tend to have lower intrinsic losses close to their ZDW boundaries down to ~1 dB/m [31,32], when compared with the soft glasses, allowing for higher Q-factors of around ~107 in the wavelength ranges considered. These values are in close agreement with previously reported Q-factors of e.g. 4 × 106 for polystyrene [33] and 107 for PMMA [34]. As shown in Figs. 7(e) and 7(f), the Q2/V values reach ~1012 µm−3 for small microspheres (40-70 µm) at visible/near-infrared wavelengths. In practice these values might be very close to those of silica, given the typical challenges of realizing the high absorption-limited Q-factors of silica [20,28]. Moreover, increasing the RMS size and correlation length of the surface in-homogeneities by an order of magnitude in the simulations, for instance, result in comparable Q2/V values between silica and the polymers mentioned. In this case the Q-factors are limited by scattering and little incentive remains for using silica.
As mentioned, polymer microspheres have the advantage of ZDWs that are closer to the low-loss region of water, which means that evanescent field absorption is lower compared with the soft glasses at their respective ZDWs, when used in aqueous solution. This would allow for the preservation of high Q-factors [22] if such microspheres were to be used, for example, in biosensing applications such as refractive index sensing. As shown in Fig. 6 the reduced ZDW however also correlates with a reduced nonlinear index, which for example decreases the mode-pulling effect and thus has implications for the potential comb bandwidth. Note that in terms of minimizing the power threshold given in Eq. (8), maintaining a high Q-factor by reducing the ZDW close to the visible is critical even if it comes at the expense of the aforementioned reduction in nonlinear index, n2 (see Fig. 6). When microspheres are used in aqueous solution, the proportion of the modal field outside the microsphere and the mode volume itself are found to increase by typically only several percent for the diameters considered. This is true for all materials considered, except MgF2 in which case the index contrast becomes very small. Given that the losses in water are as low as those of the polymers for the visible spectrum, high Q2/V values could therefore still be maintained despite a significant proportion of the mode field residing outside the sphere. To realize optimal Q2/V values would, however, require further tailoring of the dispersion to lower the ZDW to well within the visible spectrum.
The resonator sizes modelled throughout this paper (in the hundreds of micron range), are typical of microspheres fabricated by melting the tip of a fiber. It must be noted, however, that many real-world resonator systems are in the millimetre range where the FSR is in the tens of GHz scale. This allows for direct photodetection of the emitted radiation, making the frequency comb more usable for most applications. The small resonators modelled here, however, have advantages in terms of e.g. smaller mode volume, and lower Q-factor requirements for achieving a given level of resonant power enhancement [13]. Moreover, if frequency combs were to be used for e.g. biosensing applications by the monitoring of changes in the comb features, smaller microspheres would be desirable because of their increased refractive index sensitivity. Frequency combs with higher repetition rates (e.g. 100 to 1000 GHz) are also suited to a number of specific applications, and they would be difficult to generate with mode-locked lasers because of the requirement of a short cavity length. The microspheres modelled in this paper have FSRs in the range of approximately 100 to 2000 GHz, most of which could still be directly measured with high-speed photodetectors. Note that the larger diameter and higher refractive index (e.g. Tellurite, ZnO) microspheres modelled have lower repetition rates. The emerging applications for high repetition rate frequency combs include astronomical spectrograph calibration, in which minute Doppler shifts could be measured as would be necessary for detecting Earth-like planets within the habitable zone of a distant star, or the direct observation of the accelerated expansion of the universe [35]. Microresonator optical frequency combs also have potential to be used in high-capacity telecommunications to generate hundreds of data channels, replacing the need for a large number of individual lasers [13]. Frequency combs could also be used for optical and microwave waveform synthesis, using e.g. Fourier synthesis of arbitrary waveforms by controlling the amplitude and phase of the comb lines [13]. High repetition rate frequency combs also have potential for spectral fingerprinting in which each of the comb lines could be used to probe atoms or molecules yielding e.g. absorption spectra in the mid-IR [3]. The exploration of novel materials with unique transmission windows and dispersion properties is expected to further broaden the potential applications.
This paper has shown that comb generation may also be possible in aqueous solution for the case of polymer microspheres, although the increased absorption at the longer wavelengths (i.e. far-visible/near-infrared) necessary for dispersion equalization, may render comb generation inefficient. The challenge remains to further tailor dispersion such that the zero dispersion wavelength region can more closely align with the low-loss wavelength window of water. This is likely to be possible using micro-bubbles or micro-shells, which provide greater freedom for modifying the geometric dispersion.
4. Conclusions
In conclusion, this paper has explored the opportunities for frequency comb generation in microspheres of various different sizes and materials. Frequency comb generation in microspheres requires dispersion compensation as well as high Q2/V values for enhanced nonlinear interaction efficiency. To satisfy the first condition, a host of different materials were considered, and it has been shown for the first time that the zero dispersion wavelength depends approximately linearly on the refractive index of the microspheres. Therefore to have dispersion compensation at longer wavelengths, permitting comb generation in e.g. the mid-IR, high-index materials such as tellurite or germanate can be exploited. Conversely, for comb generation in the visible or near-infrared wavelength windows, low-index materials such as polymers are required. It has been shown that by increasing the microsphere size the zero dispersion wavelength can in general be decreased further due to the reduced geometric dispersion. Reducing the zero dispersion wavelength to within the visible spectrum however remains challenging.
Operating microspheres in the visible would be crucial if frequency comb generation were to be exploited for aqueous applications. This is because of the significantly reduced evanescent field absorption in water of the cavity modes in the visible, which would be necessary for achieving the high Q2/V values for efficient nonlinear interaction. The low-index materials that are required for reducing the zero dispersion wavelength however generally have lower nonlinear indices according to Miller’s Rule, which affects some aspects of comb generation. We note that the use of micro-shell or micro-bubble cavities instead, may allow for both short zero dispersion wavelengths and higher refractive indices by virtue of the increased freedom to tailor geometric dispersion.
Acknowledgments
The authors acknowledge the support of an Australian Research Council Georgina Sweet Laureate Fellowship awarded to T. M. Monro, FL130100044. The authors thank Y. K. Chembo, H. Ebendorff-Heidepriem and M. Henderson for insightful discussions.
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