1. Introduction

Whispering gallery mode (WGM) resonators confine light along the edge of a dielectric boundary via total internal reflection. Modes of these resonators have tight volumes and the highest Q factors of any other optical resonators of similar size, allowing for many applications [1, 2].

Recent research has been conducted on whispering gallery modes that are supported within a shaped protrusion of a crystalline disk, where researchers were able to show that the geometry of these protrusions controls the number of supported modes and their Q factors [3, 4]. It was also found that further microstructuring of the protrusion into a “photonic belt” enables geometrical dispersion engineering for both the multi-mode and single-mode regimes of resonator operation [5]. The microstructured photonic belt resonator (PBR) approach was used to generate a broadband Kerr frequency comb with record efficiency and may ultimately lead to the direct generation of octave spanning mode locked microcombs with repetition rates under 25 GHz.

In terms of geometrical approaches to dispersion engineering, photonic belt resonators of crystalline materials are similar to on-chip ring resonators that have been produced from silicon [6] and other materials [7–9], but are able to achieve much higher Q values. There is also similarity to dispersion engineered silica disks [10]. Even so, experimentally demonstrated single-mode resonators possess significantly lower Q values than those that have been achieved in whispering gallery mode resonators of the same materials with more typical geometries [11]. An ability to engineer the mode spectrum and dispersion seen in photonic belt resonators in combination with very high Q values consistent with well-polished crystalline resonators would be very useful for efficient broadband Kerr comb generation.

The purpose of this paper is to investigate the limitations of Q factors in crystalline single-mode resonators and compare the behavior of these specialized resonators to the behavior seen in other resonator geometries. It is currently an outstanding question as to why single-mode resonators have thus far possessed relatively lower Q values than other crystalline resonators. We study whether this is due to a limitation of fabrication techniques or due to some fundamental aspect of this unique geometry. At the same time, we have observed significant differences in the Q of TE and TM modes in photonic belt resonators. This difference was thus also investigated. Our primary focus of research has been on photonic belt resonators, but other resonator geometries are also considered.

2. Fundamental limit of Q due to radiative loss

The single modeness of a whispering gallery mode resonator is achieved by making the mode-confining structure smaller, such that only a lowest value of mode can be supported at a high Q. Smaller microresonators typically have very significant radiative losses in the radial direction, though these losses become negligible for resonators exceeding tens of wavelength in diameter. Nonetheless, given the unusual shape of single-mode resonators, it is necessary to re-evaluate the importance of radiative loss. The radiative loss in this context is not only limited to the bending loss. In a photonic belt resonator, light might also be leaking into the surrounding cylindrical substrate in the axial direction.

We studied radiative losses in whispering gallery mode resonators using a numerical modeling package based on the finite element method known as FreeFem++ [12]. This software is used with Maxwell’s equations in weak form using cylindrical coordinates and under the assumption of axisymmetry [13] to find the field distribution and associated eigenfrequencies of modes in a given resonator geometry. This tool has been used successfully in previous whispering gallery mode resonator studies [5, 14].

Perfectly matched layers (PML) [15, 16] are added to serve as reflectionless, absorbing boundary conditions. Any light radiated from the resonator is absorbed by the PML, leading to imaginary components in the computed eigenfrequencies. The quality factor due to radiative losses of a mode with complex frequency ν may then be calculated by Q = Re(ν)/(2Im(ν)).

Prior work studying single-mode resonators [4] used PML to simulate the specific case where energy was leaking into a nearby substrate. Though PML was primarily used in that paper to demonstrate single-mode behavior, it is an outstanding question as to whether there is any light leakage into the surrounding substrate that limits the Q factor of generalized single-mode resonators. We used the order of magnitude single-mode conditions derived in Ref. [4] for guidance when designing our resonator models.

Initial tests using FreeFem++ were performed by simulating the radiative losses of whispering gallery modes in silica microspheres. Throughout all simulations, the wavelength was 1.55 μm. Similar to the approach in [16], the PML thickness was kept to a quarter of this wavelength.

Figures 1(a) and 1(b) show the logarithmic intensity distribution for a whispering gallery mode in a 10 μm silica microsphere with the absorbing PML placed respectively near and far from the mode. Unsurprisingly, the absorbing boundary placed near the mode leads to losses due to the evanescent field, yielding a low Q value. As the PML is moved farther away, the calculated Q initially rises and then approaches an asymptote [Fig. 1(c)], approaching the known analytic value for a whispering gallery mode in a microsphere [17].

 

Fig. 1 Simulations with 10 μm radius silica microspheres, examining the TE fundamental mode. PML layers are set along the top, bottom and right boundaries to allow determination of Q due to radiative losses. Window height is 32 μm. (a) and (b) show the simulated intensity distribution of the TE fundamental mode at a wavelength of 1.55 μm in a logarithmic scale (difference of 30 dB between contour lines with the highest value line located at half the maximum intensity). The location of the right border was adjusted to understand how PML affects simulation results, with the location varying from (a) the PML placed 1 μm away from the microsphere to (b) the PML being placed 20 μm away from the microsphere. The results of Q as determined for different PML locations are shown in (c).

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This asymptotic behavior can be replicated for the case of photonic belt resonators a few tens of micrometers in diameter for which the radiative losses into free space are significant. Figures 2(a) and 2(b) show the simulated logarithmic intensity distribution for a photonic belt resonator with a 10 μm radius and a 2 μm by 2 μm belt. This belt size was chosen because it shows single-mode behavior at this radius and for the simulated wavelength of 1.55 μm. The material properties were set to be consistent with MgF₂ as it was previously used with photonic belt resonators [5].

 

Fig. 2 Simulations with 10 μm radius MgF₂ photonic belt resonators with 2 by 2 μm belts. (a), (b), and (d) show the simulated intensity distribution of the TE fundamental mode for various window sizes at a wavelength of 1.55 μm in a logarithmic scale (difference of 30 dB between contour lines with the highest value line located at half the maximum intensity). (a) and (b) correspond to simulations where the window height was set to 60 μm while the location of the right PML boundary was varied from (a) 13 μm to (b) 1 μm from the right edge of the belt. The simulated Q values for the window width sweep between (a) and (b) are shown in (c). We also performed a window height sweep from (a) 60 μm height to (d) 2.25 μm height while the left and right window boundaries were kept unchanged. The simulated Q results for the sweep from (a) to (d) are shown in (e).

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PML was placed around the simulation window’s borders, both against air and against the MgF₂ cylinder itself, similar to the PML substrate used in [4]. As with Fig. 1, the right absorbing boundary was varied between the positions shown in Figs. 2(a) and 2(b), producing the plot shown in Fig. 2(c) with the asymptotic behavior analogous to the microsphere case. Simulated radiative losses approaching an asymptote are also observed when the top and bottom borders are moved by varying the height of the simulation window. These results are shown in Figs. 2(a), 2(d), and 2(e).

The results in Fig. 2 correspond to the TE fundamental mode. We performed these same simulations for the TM fundamental mode (not pictured) and obtained a similar structure in the intensity distribution and saw a lower simulated Q value. The Q factor of the TM mode was approximately 442,000 in contrast to the Q factor of roughly 719,000 seen for the TE mode plots given in Fig. 2. It is worth noting here that while the TM mode does possess a significantly smaller Q factor than the TE mode, this discrepancy is much smaller than experimentally observed in larger radius resonators, as we shall cover later. We also emphasize that both TE and TM mode simulation results for this small radius case possessed similar asymptotic behavior as the window borders were altered.

It is interesting to note that for a 10 μm PBR we clearly observe energy leaking into the surrounding cylinder [Fig. 2(a)] that limits Q at 10 times below that of a microsphere with similar size [Fig. 1(c)]. We saw significantly different results when we adjusted our models to study resonators with more realistic dimensions, implementing a PBR with a cylinder radius of 750 μm and a belt that is 7 μm wide and 5 μm high above the cylinder substrate [5]. The TE fundamental mode intensity distribution for this resonator is given in Figs. 3(a) and 3(b). At this radius, radiative losses into the air around the substrate for a sufficiently large simulation window proved to be negligible. Placing PML in contact with the substrate on the top and bottom borders of the simulation area does lead to a significant impact on the simulated Q value, but as shown in Fig. 3(e), the Q value returned by the simulation appears to be arbitrarily determined by the window height. This means that the mode energy is highly localized in the vicinity of the PBR belt for the TE fundamental mode and the Q values simulated in Fig. 3(e) are only being altered by the PML infringing upon the large mode area.

 

Fig. 3 Simulations of the TE and TM fundamental modes at a wavelength of 1.55 μm for 750 μm radius MgF₂ photonic belt resonators. The belt is 7 μm wide and extends 5 μm above the cylinder. The window width is 48 μm. (a) and (b) show the simulated intensity distribution (difference of 30 dB between contour lines with the highest value line located at half the maximum intensity) for the TE fundamental mode, while (c) and (d) correspond to the logarithmic intensity distribution of the TM fundamental mode. The window height was varied from 120 μm for (a) and (c) to 10 μm for (b) and (d). The simulated Q values are shown in (e) and (f), corresponding to the TE and TM modes respectively. Note that, due to precision limitations in calculating Q with FreeFem++, (e) is limited to plotting simulation results with window heights no higher than 60 μm.

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In contrast to the small radius case, no asymptotic behavior is observed for the TE mode when the window is increased with a 750 μm radius photonic belt resonator.. In fact, although we simulated resonator intensity distributions with window heights up to 120 μm, we reached the FreeFem++ precision limitations for calculating Q when the window height exceeded 60 μm. Within the precision limits, we simulated Q values in excess of 10¹⁵. This is far larger than experimentally observed Q values for these resonator geometries. We can conclude that there is no radiative loss within the substrate for the TE fundamental modes. The Q obtained in the TE fundamental mode studied in [5] was not limited by radiative losses.

The TE fundamental mode results shown in Figs. 3(a), 3(b), and 3(e) stand in contrast to the TM fundamental mode results given in Figs. 3(c), 3(d), and 3(f). For the TM fundamental mode, light is apparently still leaking upwards and downwards along the cylinder edge, similar to the TE and TM mode results seen for a 10 μm radius cylinder. Once the PML is placed far enough from the mode center, the simulated Q values shown in Fig. 3(f) oscillate depending on the window height, varying with the modal structure of the intensity shown in Fig. 3(c). As such, while radiative losses into the surrounding air are not significant for the TM mode, leakage into the surrounding cylindrical substrate might set a fundamental limit to the quality factor of the TM fundamental mode of PBRs.

3. Field distributions on the surface of WGM resonators

In addition to geometry-related limitations to Q, we also investigated fabrication limitations for photonic belt resonators. It is well known that the surface roughness often limits the Q of practical crystalline resonators, where resonators with the highest quality factor require a very small surface roughness [11]. Single-mode resonators, especially photonic belt resonators, have a very specific field distribution profile and it is worth studying how the surface roughness limits these resonators.

Scattering by the surface of WGM resonators has been analyzed for the case of fused silica, where surface roughness is homogeneous in all directions [18] and is not correlated at distances exceeding the wavelength of light. In this case the surface roughness can be characterized only by variance and correlation length. The approximations then lead to simple estimates of scattering loss. Surface scattering in disk and ring microresonators has been analyzed using the volume current approach [19–22] and a unified approach based on Green functions [23]. However, in all cases it is assumed that the surface perturbation is one-dimensional, such that the surface height remains constant along the disk axis. Such a choice is justified by the characteristic features of the disk etching process. Even in such a simplified case a detailed derivation of scattering is rather involved [20].

We have used an optical interferometric profilometer Veeco NY9300 to obtain the measurements of PBRs used in this work as shown in Fig. 4. The profiler images reveal that the surface of PBRs has polishing traces such that correlation length strongly depends on the direction in which it is measured. Roughness also depends on the measurement direction and varies on different surfaces of the resonator. For example, the top surfaces of the structures in Fig. 4 have the same polish quality as the best demonstrated WGM resonators. While the side walls of the structures could not be measured, their roughness is expected to be similar to that of the area adjacent to the belts. Thus we can’t rely on one-dimensional surface roughness models and a more detailed model is needed.

 

Fig. 4 (a) Optical profilometer scan of the MgF₂ cylinder with three photonic belt structures. Polishing traces and varying surface roughness are visible across different parts of the structures. (b) Profile of the middle structure from (a).

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The Q values of the resonators shown in Fig. 4 were measured by observing the resonators’ absorption spectra using a tunable laser. Q is calculated from the Lorentzian resonance linewidth. A phase modulator is used to produce sidebands that allow for frequency calibration and a direct measurement of resonance linewidth. The experimentally measured Q for TE modes (electric field is along the resonator axis) at a wavelength of 1561 nm for the middle and right structures shown in Fig. 4 are 400 and 490 million respectively in critical coupling condition with around 70% coupling efficiency. The Q of TM modes, however, were found to be 75 times lower for the same structures. We shall give a qualitative explanation of this difference by looking at the field strength at the surface near the belt, as well as the direction of the field.

Figure 5 shows the optical field intensity distribution for a photonic belt resonators with a 10 μm by 10 μm belt structure. Due to significant differences in the experimentally observed Q between TE and TM modes in these resonators, we plotted both the TE and TM mode intensity following the arc of the resonator edge and examined the ratio between these two modes. Potentially consistent with much lower Q for TM modes than TE modes, there are regions where the TM mode is significantly more intense along the resonator edge than the TE mode and therefore more likely to suffer scattering losses within these regions depending on the surface roughness. Specifically, the TM mode is more intense along the side wall of the belt and much of the edge of the cylinder past point C in Fig. 5(a). The ratio of TM to TE intensity given in Fig. 5 is closely related to the difference in power scattered between the two modes in that region. These results are qualitatively consistent with our previous simulations using PML from Fig. 3, as well as our experimental observations.

 

Fig. 5 Simulation of the TE and TM fundamental mode at a wavelength of 1.55 μm for a 1337 μm radius MgF₂ photonic belt resonator. The belt was set to be 10 by 10 μm. (a) shows the simulated TE mode intensity distribution (linear scale). (b) shows the intensity along the resonator edge for both the TE and TM modes. The labels A, B, and C on figures (a) and (b) correspond to the same points of the arc of the resonator edge, located at 0 μm, 5 μm, and 10 μm from the belt center along the arc. (c) shows the ratio of the TM fundamental mode intensity over the TE fundamental mode intensity along the resonator edge.

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If surface roughness is assumed to be uniform, we cannot explain the large difference between TE and TM mode Q values for the model results shown in Fig. 5 as the difference between the TE and TM modes is not substantial in the region where the fields are most intense. If, however, surface roughness is assumed to be significantly higher for the regions where the TM mode is substantially more intense than the TE mode on the surface, such as beyond 20 μm in Fig. 5(c), then it is plausible that Q values consistent with experimental results may be due to surface scattering.

To give further insight, Fig. 6 shows the absolute value of the components of the simulated electric field. In the mode naming convention chosen here, the TE mode is the one with the strongest E_Z-component. Thus, the Z-component is responsible for most of the scattering energy loss for this mode type. For the TM mode, in the immediate vicinity of the belt, the E_R-component has the strongest intensity, but as shown in Figs. 6(g) and 6(h), the TM modes we’ve simulated using a PBR geometry consistently show a significant hybridized cylindrical mode structure for the E_z and E_Phi component. Further from the vicinity of the belt, this Z-component becomes the most significant and would be the primary contribution to scattering. For the modeled ideal case of a perfectly straight and smooth cylinder, these hybrid cylindrical modes appear to extend along the resonator edge without decaying (corresponding to the intensity distribution seen in Fig. 3(c). This can also be interpreted as coupling between the PBR and cylinder modes, which happens only for the TM modes and mostly to the E_Z component.

 

Fig. 6 Simulations showing lines of equivalent magnitude for the electric field components of the TE and TM fundamental modes at a wavelength of 1.55 μm for a 750 μm radius MgF₂ photonic belt resonator. The belt was set to be 7 by 5 μm. (a)–(c) correspond to components of the TE fundamental mode, while (e)–(g) correspond to the TM fundamental mode. (a) and (e) show the magnitude of the E_R-components. (b) and (f) show the E_ϕ -components. (c) and (g) show the E_Z-components. (d) and (h) plot the magnitude of these components along the resonator edge, starting from the belt center and following the resonator arc upwards.

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Investigating this further, we adjusted the height and width of the protrusion and simulated how this affected the hybridized TM mode component. Fig. 7 shows the simulated TM fundamental mode intensity following the resonator edge when the protrusion aspect ratio [Fig. 7(a)] and area A [Fig. 7(b)] are altered. While changing both the area and aspect ratio of the protrusion can clearly alter the hybridized intensity by multiple orders of magnitude, these results do not yet reveal a clear pattern and suggest an area for further research.

 

Fig. 7 Simulation results where the resonator protrusion dimensions were varied. The optical field intensity is plotted along the arc of the resonator surface profile, starting from the belt center. We used a wavelength of 1.55 μm for a 750 μm radius MgF₂ PBR. We denote the belt width by w, the belt height by h, and the belt area by A = w × h. (a) and (b) plot the TM mode intensities along the resonator edge for different protrusion dimensions. For (a), A was kept constant while w/h was varied. For (b), w/h was kept constant while A was varied.

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For comparison, we also performed simulations of resonators with Gaussian protrusions using a surface profile similar to that used in [4], with the results shown in Fig. 8. For these simulations, the TE mode is more intense than the TM mode throughout the protrusion, in contrast to the PBR geometry where the TM mode is more intense along the protrusion sidewall. The Gaussian structure shares the property with the PBR structure that the ratio between the TM mode and TE mode intensities along the resonator edge becomes substantial at large enough distances from the protrusion center. This suggests that higher surface roughness in regions beyond 17 μm along the resonator arc from the center of the Gaussian protrusion would lead to major discrepancies between the Q of the TE and TM modes for resonators possessing the Gaussian structure we simulated. Additionally, we observed a similar mode hybridization for Gaussian single-mode resonator geometry, seen only in the E_Z-component of the TM fundamental mode and shown in Fig. 8(b).

 

Fig. 8 Simulation of the TM fundamental mode with comparisons to the TE fundamental mode. The simulated wavelength was 1.55 μm with a 1337 μm radius MgF₂ resonator. The resonator had a Gaussian protrusion with a full width at half maximum of 10 μm and a height of 5 μm. The simulated TM mode intensity distribution (linear scale) is given in (a) and its E_Z-component is shown in (b). The intensity along the arc of the resonator edge is given in (c) for both the TE and TM modes. The plot in (d) shows the ratio of the intensity of the TM fundamental mode over the intensity of the TE fundamental mode along the resonator edge.

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We note that the large differences in TE and TM mode intensities observed in regions of the Gaussian and PBR geometries are in contrast to the case where the resonator edge is a shallow spherical curve without any protrusion. For a resonator with a shallow spherical edge, our simulations showed the TM to TE mode intensity ratio was relatively flat.

Lastly, we ran simulations where we relaxed the assumption that the belt was fabricated at the edge of a perfect cylinder, and instead added a tilt to the surrounding edges. We focused particularly on the electric field Z-component of the TM fundamental mode, which we show in simulation results in Fig. 9. Higher angles of tilt were seen to correspond to a more tightly confined Z-component.

 

Fig. 9 Simulation results where the resonator cylinder angle was varied. We used a wavelength of 1.55 μm for a 750 μm radius MgF₂ resonator. The belt was 7 by 5 μm. (a)–(d) show the electric field Z-component of the TM fundamental modes, at edge angles relative to the Z-axis of (a) 0 degrees, (b) 5 degrees, (c) 10 degrees, and (d) 15 degrees. (e) and (f) respectively plot the TE and TM mode intensities along the resonator edge for different edge angles.

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When simulating the tilted edge resonators, we also examined how the intensity of both the TE and TM modes along the resonator edge changes with different edge angles in Fig. 9. Both see more rapid intensity decay along the surface profile for higher edge tilts, though the effect on the TM mode intensity is far more pronounced. For these higher angle tilts, so long as the region around the belt is well fabricated, we anticipate that scattering effects would be mitigated, especially for surface profile distances in excess of 30 μm from the belt. Moreover, repeating the PML simulations of the previous section with a tilted substrate edge yields TM mode Q values beyond the precision limits of FreeFem++ for sufficiently large window heights. These results demonstrate that the TM mode is localized in this tilted-edge geometry.

4. Discussion

Our simulations of single-mode resonators with both belt-shaped protrusions [Fig. 6] and Gaussian-shaped protrusions [Fig. 8] show how the TM mode of single-mode resonators can see coupling to cylindrical modes, in contrast to the well contained TE modes. The presence of this cylindrical mode coupling is, to our knowledge, a new observation and offers a likely explanation for the lower Qs observed in the TM mode of single-mode resonators relative to the TM modes.

The actual quality factor of the TM mode is likely heavily influenced by both leaking into the surrounding substrate [Fig. 3] as well as scattering along the surface of the resonator. TE fundamental mode simulations show no evidence of significant radiative losses, so surface scattering alone is likely to be the primary limiting factor for the achievable Q for TE modes of single-mode resonators. The observed photonic belt resonator Q of these modes is presumably lower than standard resonator geometries due to fabrication limitations. Still, TE modes see their field strength consistently decay for distances further from the mode center [Fig. 6], which limits the severity of scattering losses caused by surface irregularities in those regions. In contrast, simulations suggest that the TM mode field strength can persist for distances far from the mode center, suggesting that surface irregularities relatively far from the resonator protrusion might still significantly contribute to surface scattering losses. Stronger scattering losses and leakage into the substrate offer a plausible explanation for lower Q values of TM modes relative to TE modes in single-mode resonators.

Our later simulations offer some insight into how we might mitigate TM mode losses during the fabrication process. Figure 7 shows how the persisting TM mode strength along the surface outside of the protrusion can be reduced for different belt dimensions, where lower field strengths should in turn reduce losses due to light leakage and surface scattering. This suggests that the process of optimizing the quality factor of single-mode resonators should include careful selection and simulation of the protrusion dimensions prior to fabrication, with emphasis given to selecting dimensions that minimize the TM mode intensity spread.

Another approach to mitigating losses can be inferred from Fig. 9. By adding a tilt to the resonator surface profile, we see that the simulated TM mode is finitely confined. This confinement improves for steeper tilts. As such, adding an inwards tilt to the resonator profile during fabrication may allow achievement of higher quality factors. Our simulations suggest that this will potentially increase observed Q factors of both the TE and TM modes, though with stronger increases for the TM mode Q factors.

It’s plausible that resonator shapes besides a tilted line might see even less intensity spread along the edge. These results suggest that a carefully engineered surface structure in the vicinity of the single-mode guiding region might substantially reduce the impact of surface scattering. It may be possible to develop a special, micro-structured resonator geometry that optimizes quality factors and mode structure for a particular application, as will be reported elsewhere.

5. Conclusion

We performed finite element method modeling to investigate the quality factor of single-mode resonators, focusing on photonic belt resonators in particular. These simulations are compared to experimental results, where we observed a quality factor of 490 million for the TE mode of a photonic belt resonator, while the corresponding TM mode had a quality factor 75 times smaller. These are high quality factors compared to other approaches for fabricating resonators, but it is an outstanding challenge with single-mode resonators to achieve Q values as high as seen in standard crystalline resonator geometries. We have found that radiative losses are not a primary limiting factor on the Q of the TE fundamental modes of experimentally demonstrated photonic belt resonators, though the geometry might set fundamental limits to the Q of the TM fundamental modes. Our results indicate that surface scattering and mode hybridization effects offer a plausible explanation for the differences in Q between TE and TM modes, and we presume that surface scattering due to fabrication limitations leads to a lower currently achievable Q for photonic belt resonators relative to standard-geometry crystalline whispering gallery mode resonators. Improvements in fabrication techniques will lead to higher-Q crystalline resonators for improved Kerr comb generation and other applications. At the same time, proper surface structure design might allow for single-mode guiding while minimizing losses due to scattering and leakage into the substrate.