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Abstract
Soliton crystals are a novel form of microcomb, with relatively high conversion efficiency, good thermal robustness, and simple initiation among the methods to generate them. Soliton crystals can be easily generated in microring resonators with an appropriate mode-crossing. However, fabrication defects can significantly affect the mode-crossing placement and strength in devices. To enable soliton crystal states to be harnessed for a broader range of microcomb applications, we need a better understanding of the link between mode-crossing properties and the desired soliton crystal properties. Here, we investigate how to generate the same soliton crystal state in two different microrings, how changes in microring temperature change the mode-crossing properties, and how mode-crossing properties affect the generation of our desired soliton crystal state. We find that temperature affects the mode-crossing position in these rings but without major changes in the mode-crossing strength. We find that our wanted state can be generated over a device temperature range of 25 $^\circ$C, with different mode-crossing properties, and is insensitive to the precise mode-crossing position between resonances.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Microcombs are powerful tools that can be used for many purposes, from spectroscopy [1], optical frequency synthesis [2], communications [3–5], ultra-fast distance measurement [6,7], optical clocks [8,9], astronomical spectrometer calibration, the search for exoplanets [10,11], sensor applications and more [12,13].
Dissipative Kerr solitons (DKSs) are self-reinforcing waves that can be generated in nonlinear optical microresonators. Bright DKSs can be generated in optical resonators with anomalous dispersion [12] and can produce coherent low-noise microcombs [12]. In single-ring devices, single DKS states require complex techniques such as fast wavelength sweeping, rapid thermal tuning [14] forward and backward pump frequency scans [15], self-injection locking [16] and power kicking [17]. These techniques are required to overcome fast resonance shift due to the sudden intracavity power drop [18] and change in cavity temperature [15].
Two alternative microcomb states to these are Dark Pulses or Platicons that can be generated in normal dispersion devices and Soliton Crystals(SC) in anomalous ones. Dark Pulses or Platicons are coherent pulses, however, require techniques such as self-injection locking [19,20], modulated pumping [21], or pumping on modes affected by an avoided mode crossing [22] to generate. In single-ring devices, Dark Pulses have a lower intracavity power drop during formation than single bright DKS states and can be easier to form, along with higher internal conversion efficiency. For Dark Pulses, the internal conversion efficiency is typically around 20-35${\%}$ [5,23] compared to $\sim 1{\%}$ for DKS states [3,24]. Dual ring devices offer higher efficiency for both states [25,26] as well as easier generation for single DKS states [26] however require a more complicated device architecture and hence increased fabrication difficulty.
Soliton crystals consist of a set of tightly packed solitons in an orderly crystal-like structure [27] and have a highly structured spectrum. In contrast to single DKS microcomb states, SCs have a low intracavity power drop and do not require the same complex pumping techniques needed to overcome the thermo-optic effect [27]. As a result, SCs can be generated via a simple slow laser frequency ramp, or from thermal tuning [28,29]. SC states form mainly at lower pump powers as transient chaos can prevent the formation of low-defect-number soliton states when using a simple frequency sweep [30]. SCs can have high internal conversion efficiency [27] with $\sim 40{\%}$ internal efficiency reported [4].
Of the many soliton crystals, there has been a particular focus on both perfect soliton crystal states, due to their high repetition rate and SC states with a single defect. Single defect soliton crystals provide the robust and minimal intracavity drop of a soliton crystal state but allow for 1-FSR line spacing between comb lines. In this paper, we look at generating the ‘Palm’ like state also called the ‘Scalloped’ soliton crystal state [27,28]. The ‘Palm’ like state is a single defect soliton crystal state, that is favourable due to the high line-to-line power within the main lobe and ease of generation. These ‘Palm’ like states have been used in previous works for; ultra-high data-carrying capacity from a single microcomb source on single mode fibre [4], radio frequency photonic signal processing [31,32], microwave integrator [32] and optical neuromorphic processing [33,34].
There are a few ways of generating SC states, including using an avoided mode crossing [27,28,35], harmonic modulation [36], bi-chromatic pumping [29], non-linear mode-coupling [37] in $\chi ^{(2)}$ material and use of a saturable absorber [38]. For the ‘Palm’ like state it has been shown to be produced using both thermal tuning and frequency sweeping on devices with an avoided mode crossing [27,28], as well as nonlinear mode-coupling [37] in a $\chi ^{(2)}$ device.
Using the avoided mode crossing (AMX) to generate the SC state requires the least complexity and equipment, however, it requires a ring to have a weak mode crossing at a suitable location [27]. This is difficult due to fabrication differences affecting the mode crossing placement and strength in rings. AMXs can both aid and hinder the generation of DKS states, as they give DKS extended oscillatory tails [39]. Weak oscillatory tails can allow solitons to lock on to each other when forming soliton crystals and soliton molecules [40] or to reduce the number of solitons when the loss due to the AMX is large [41]. Unlike Dark Solitons, soliton crystals require the AMX to be a distance from the pump location, and this distance can affect the type and breadth of potential soliton crystal states [27].
AMXs are caused by the interaction of different mode families and are common in anomalous dispersion rings. Waveguides with anomalous dispersion tend to have a larger cross-section and are often able to support multiple modes, where inter-modal crossings can occur. AMX’s cause significant changes in the individual dispersion profiles of the crossing modes families near the AMX location. For no cladding devices, the dispersion and AMX may be able to be tuned after fabrication [42], however, this is not the case with integrated devices. In dual ring devices, mode crossings can be induced [25,26,43,44] as the auxiliary ring can be used as a source of a second mode family. The mode crossing placement can be shifted over a larger range compared to single ring devices, particularly with the inclusion of separate integrated heaters as the two rings’ temperature and hence mode placement can be controlled independently.
In single-ring devices increasing the intra-cavity power by use of an auxiliary laser has been shown to tune the AMX position and strength, for a $Si_3N_4$ MRR [45]. Another way to thermally tune devices is to heat change the entire MRR device’s temperature, which can be done through a temperature-controlled holder.
In this paper, we show that the temperature of the MRR chip can be used to fine-tune the AMX placement in two 49-GHz, high-index doped silica, anomalous micro-rings. These two rings can both generate the same ‘Palm’ like soliton crystal state. By measuring the dispersion profiles at different temperatures, the temperature was shown to only have a minor effect on the mode crossing strength. Changing the device temperature can shift the minute AMX position, however, the same ‘Palm’ like state can still then generated on both devices. Showing that the precise AMX is not a requirement for generating these states. This was done for an operating range of 10-40 $^\circ$C and 15-40 $^\circ$C which suggests the ‘Palm’ like state has a large temperature operating range. The temperature variation was shown to also shift the resonance’s position, with the entire comb shifting 1.05 and 1.02 FSR over the temperature range, allowing for control of the comb placement within the FSR grid spacing.
2. Thermal effect on dispersion profile
To see the thermal effect on the AMX, dispersion measurements were undertaken for two high-index doped silica micro-rings; Ring 1 and Ring 2. The two micro-ring resonators (MRR) were fabricated based on a complementary metal–oxide semiconductor (CMOS) compatible doped silica glass platform [46]. These rings are 4-port rings with a radius of around 592 $\mu$m, with a free spectral range (FSR) of $\sim$49 GHz, and anomalous group velocity dispersion (GVD) in the C-band.
Ring 1 has an average Q factor of $\sim$1.9 million near 1550 nm as seen in Fig. 1(bi) and a coupling loss of 1.5 dB per facet, from single-mode fibre to waveguide. Similarly, Ring 2 has an average Q factor of $\sim$1.5 million in Fig. 1(bii) and a coupling loss of 2 dB. The rings were situated in a temperature-stable housing with a Peltier module controlled via a commercial PID temperature controller.
Fig. 1. a) Experimental setup for the dispersion measurement. PC polarisation controller, MZI unbalanced fibre Mach-Zehnder interferometer, MRR micro-ring resonator, HCN H$_{13}$C$_{14}$N calibration cell, OSC oscilloscope b) Quality factor for each resonance, taken from the linewidth of the resonance for bi) Ring 1 and bii) Ring 2, centred around the closest mode to 1550 nm c) Integrated dispersion profiles for the two rings taken at $25 ^\circ C$ centred around the same modes as b). The blue dots are the experimental data and the orange line the theoretical fitting, for ci) Ring 1 and cii) Ring 2
The dispersion was characterised using a frequency-swept laser where the setup is shown in Fig. 1(a) similar to the setup used in [47], with a low power fast sweep, so that the thermal effects on the MRR resonances were negligible [48]. The laser was also swept through a fibre Mach-Zehnder interferometer (MZI) with a $\sim$50 m unbalanced arm providing a resolution of $\sim$4.1 MHz, and a H$_{13}$C$_{14}$N (HCN) calibration cell. The HCN cell was used for the frequency calibration [47,49] and to compensate for the dispersion of the fibre MZI. Each peak from the MRR absorption was fitted with a Lorentzian profile, which was used to find the centre frequency and linewidth of the resonance. From this integrated dispersion and AMX can then be found for each ring. In each device, only one polarisation was of interest, as it had a weak mode crossing in a good location relative to C-band.
The frequency of the cavity modes $\omega _\mu$, can be represented as $\omega _\mu =\omega _0 +\mu D_1+D_{int}$ where $D_1/(2\pi )$ is the FSR of the cavity, $\omega _0$ is the centre frequency and $D_{int}$ is the integrated dispersion of the MRR relative to the mode number $\mu$. In the absence of the mode crossing the integrated dispersion can be described using the Taylor expansion of the resonance frequencies around the pumped mode, $D_{int}=\mu ^2D_2/2+\mu ^3D_3/6+\mu ^4D_4/24\cdots$ .
For the dispersion measurements the centre mode, $\mu =0$, was set to be the nearest to 1550nm at 25 $^\circ$C. The FSR and integrated dispersion are then found from the resonance frequencies. A polynomial was fitted to the integrated dispersion excluding the perturbed modes near the AMXs to get the dispersion curve and $D_2$ value.
In Fig. 1(c), at 25 $^\circ$C, Ring 1 has a weak spatial AMX centred at 1580.3 nm of around 202 MHz from maximum to minimum, and an FSR 48.911 GHz with a second-order dispersion $D_2/(2\pi )$ of 131 kHz$/\mu ^2$. Ring 2 has a weaker spatial AMX centred around 1581.5 nm of around 60 MHz an FSR of 48.78 GHz and a $D_2/(2\pi )$ of 118 kHz$/\mu ^2$, these values are summarised in Table 1.
Table 1. Device properties including; FSR, second-order dispersion $D_2/(2\pi )$, mode crossing strength, mode crossing location and average quality factor, values taken for a temperature of 25 $^\circ$C.
The same dispersion measurement was undertaken for a temperature range of 10-40 $^\circ$C for Ring 1 and 15-40 $^\circ$C for Ring 2. The MRRs were mounted in separate holders, and the temperature range chosen was based on the physical capacity of the temperature holder and the maximum chosen to avoid the risk of damaging the components through overheating. Ring 2’s holder had a reduced cooling capacity accounting for the 5 $^\circ$C difference in the two ranges.
In Fig. 2, the temperature dependence frequency shift of the centre mode for Ring 1 was -1.696 GHz/$^\circ$C and Ring 2 the modes shifted by -1.693 GHz/$^\circ$C. The FSR decreased by 0.45 MHz/$^\circ$C and 0.44 MHz/$^\circ$C for Ring 1 and Ring 2, respectively, which might be due to the temperature dependent expansion of the ring. The dispersion values from this fit were found not to significantly change over this temperature range.
Fig. 2. Change in passive characteristics of the two rings with Ring 1 in blue and Ring 2 in orange, a) Shift of the centre mode frequency which was selected as the mode closest to 1550 nm at 25$^\circ$ b) FSR of the rings at each temperature c) $D_2/(2\pi )$ for each temperature
The AMX maximum and minimum strength were taken as the largest deviation values from the fitted dispersion curve as seen in Fig. 3. As seen in Fig. 4, the AMX strength is not strongly affected by the temperature with a maximum variation of 10 MHz for Ring 1 and 6 MHz for Ring 2. This small variation in AMX strength is potentially due to the shallow gradient around the extrema of deviation caused by the mode crossing and the relatively small FSR spacing. This was seen for both the maxima and minima mode crossing strengths in both rings.
Fig. 3. Deviation from the dispersion curve taken from the difference between the measured value and the fitted curve, showing the mode crossing strength for all temperatures. Ring 1 is on the left and Ring 2 on the right. In a), deviation against wavelength. A single mode shifting with temperature is shown in a blue dotted line. The AMX strength for 40 $^\circ$C is shown with the arrows and is found from deviation extrema. The wavelength where a fit crosses zero deviation does not change significantly against temperature. In b), deviation against mode number. The similarly coloured connecting lines from a cubic interpolation, with arrows pointing to the AMX position for 10 $^\circ$C on Ring 1 which is found from the approximate zero crossing.
Fig. 4. a) Mode crossing strength for the two ring; for ai) Ring 1 on top, aii) Ring 2 below, at all temperatures. Full strength in blue, maximum positive deviation in green and maximum negative deviation in red. b) Mode crossing position for both rings in regards to wavelength and mode number, bi) Ring 1 and bii) Ring 2 below, showing the mode crossing shift respective to the wavelength in orange and the mode crossing shift respective to the mode number in blue with $\mu =0$ being defined as the nearest mode to 1550nm at $25 ^\circ$C,
As seen by the circled mode in Fig. 3 the modes travel over the AMX structure in a continuous manner, and the multiple modes lie between the two extrema of the AMX. We then define the AMX position the point between the two extrema of the AMX perturbation, at the point where the perturbation should be zero. The fractional AMX position serves as a means to characterise the position of the modes in the AMX structure. This mode crossing position was found by a cubic interpolation of the deviation, or the difference between the measured integrated dispersion as seen in Fig. 3, and the zero-crossing taken as approximate AMX position, this is done for each temperature and ring.
In Fig. 4 the mode crossing wavelength shifts 0.06 nm in total or at a rate of -0.1 GHz/$^\circ$C for Ring 1 and 0.13 nm in total or -0.5 GHz/$^\circ$C for Ring 2. The mode crossing shifts at a significantly slower rate than the resonance modes, and in the same direction as the resonance shift. This slight change in the AMX frequency could be due to different thermal tuning rates of the two-mode families crossing. For Ring 1 AMX location shifts 0.88 resonances over the 10-40 $^\circ$C temperature range and 0.55 for Ring 2 over the smaller 15-40 $^\circ$C temperature range. This means the precise location of the frequency grid can be shifted in regard to the mode crossing, and temperature can be used to fine-tune the mode crossing position relative to the modes. We can then use this to shift the relative AMX position during SC formation.
3. Generating the same spectra on two different rings
Having characterised the dispersion profiles of the two rings, which both have a weak AMX located near 1580 nm, we show that they can be used to generate the desired SC state. The same ‘Palm’ like state can be generated on the two devices, the state consisting of a 73 FSR spacing between the pump line and the line with excess power caused by the mode crossing, as seen in Fig. 5. For SC generation, continuous wave (CW) pump light was amplified and aligned to the desired MRR polarisation using an inline fibre polarization controller, the pump power was held steady and the CW frequency was swept from blue to red by hand-tuning the frequency. Slow hand tuning allows time for the thermal drift to reach a constant value ensuring the end state is thermally stable.
Fig. 5. a) Set up for soliton crystal generation including a frequency swept laser, an erbium-doped fibre amplifier (EDFA), a polarisation controller (PC), the MRR, and the optical spectrum analyser (OSA) at 0.1 nm resolution. b) In blue, the measured ‘Palm’ like spectrum generated on both the rings, both having the same 73 FSR spacing between the pump line and the line with excess power near the mode crossing. In orange the simulation results for each ring.
Ring 1 was pumped with 1.5 W at a specific resonance that sits around $\sim$1551 nm and Ring 2 was pumped with 2.2 W at a resonance around $\sim$1553 nm. Both produced the same ‘Palm’ like state as seen in 5. Ring 1 produced a smoother spectrum, likely due to a smoother dispersion curve and a more consistent quality factor between resonances. The internal conversion efficiency can be taken as the power ratio between the pump line and the comb lines [3,4,13]. Ring 1 has an internal conversion efficiency of 41${\%}$ and Ring 2 of 39${\%}$.
It was also observed that it is easier to generate the desired state on Ring 1, with Ring 1 being able to generate a ‘Palm’ like spectrum on multiple resonances. Including the 69, 71, 73, 75 and 79 FSR ‘Palm’ like states by pumping on different resonances around 1549nm-1553nm. Ring 1 has been used in previous works [4,31–34] with some of the above listed ‘Palm’ states. The mode crossing position was similar for both of these rings, the AMX strength between the two is significantly different with Ring 2 having a very weak mode crossing strength of around 60 MHz compared to Ring 1 which was 210 MHz. This indicates that there is a large range the AMX strength can be and still generate the same ‘Palm’ like state.
Simulation results in the background for Fig. 5, using the normalised Lugiato-Lefever Equation (LLE) [12] solved numerically.
Both rings can be used to generate the ‘Palm’ like spectra as seen in Fig. 5. The differences in the dispersion curve account for a small amount of the spectral differences between the two rings. However, the normalised LLE does not take into account the difference in quality and coupling loss between each resonance, which could also contribute to the spectral differences.
For each ring, the 73 FSR ‘Palm’ spectrum was a steady-state solution of the LLE. This was also true for all temperatures using the different dispersion profiles, with only minor differences in power between the spectra. Suggesting that these SC states have some in-variance to the AMX position.
4. Operating temperature
As the temperature was shown to shift the AMX placement with regard to the resonance number, and the simulations suggested there was some invariance to this. The temperature of the micro-ring resonator holder was changed to see if the exact mode position has an effect on generating the desired ‘Palm’ like state. The temperature range was chosen to match the temperature range used in the dispersion measurements above.
For each temperature, the same ‘Palm’ like state was regenerated by re-sweeping into the same resonance each time as seen in Fig. 6(a). Ring 1 can generate the 73-FSR ‘Palm’ like spectrum for the temperature range of 10 to 40 $^\circ$C. The same spectrum could be generated in Ring 2 for the temperature range of 15 to 40 $^\circ$C.
Fig. 6. a) The main lobe of the ‘Palm’ spectra generated at different temperatures on power per mode bases for the inner spectrum part with Ring 1 above and Ring 2 below mode number here is centred around the pump resonance, with Ring 1 a) small inset showing the corresponding spectrum in the blue line and the main lobe peaks in orange dots b) the difference in power level in each resonance compared to the spectrum at 25 $^\circ$C
Comparing each spectrum to the one generated at 25 $^\circ$C, in Fig. 6(b) there are small differences between each spectrum with little discernible pattern. Similar power fluctuations can be seen for a single comb held at a single temperature, as well as from re-generating the same microcomb state [4] at a single temperature.
The centre pump line and spectral resonances shifted with the change in temperature, with the entire spectrum shifting at a rate of -1.71 GHz/$^\circ$C for Ring 1 and -1.99 GHz/$^\circ$C for Ring 2. This allows the spectrum to be shifted up to 51.6 GHz in total or 1.05 FSR and for Ring 2, 49.7 GHz or 1.02 FSR, for the two respective rings as seen in Fig. 7.
Fig. 7. a) The ‘Palm’ spectra generated at different temperatures on power for each wavelength Ring 1 left and Ring 2 right b) the centre pump location for each temperature and neighbouring comb line $\mu =1$ c) the comb line with excess power associated with the AMX, 73 FSRs away from the pump location
5. Discussion
Repeated dispersion measurements for a temperature range of 10-40 $^\circ$C and 15-40 $^\circ$C was shown to change the mode crossing frequency at a rate of -0.10 GHz/$^\circ$C and -0.5 GHz/$^\circ$C for each ring respectively. This was in the same direction but smaller in magnitude than the change in frequency of the modes which was -1.696 GHz/$^\circ$C and -1.693 GHz/$^\circ$C resulting in the AMX position being tuned slightly with regard to the resonance position.
The AMX shape seen in Fig. 3 is typical of a spatial AMX in our devices and is accompanied by a large drop in quality factor and changes in the extinction ratio of the resonance affected by the mode interaction similar to [50]. Here we found the AMX strength only minorly fluctuated with temperature by $\sim$10 MHz and $\sim$6 MHz for Rings 1 and 2 respectively. The small AMX strength fluctuations could be due to the span or shape of the mode crossing and the FSR of the ring. As seen in Fig. 3 as one mode leaves the extrema, the neighbouring one approaches reducing the degree of fluctuation.
This is different to the results seen in [46]. In [46] an auxiliary laser was used to suppress different AMXs, allowing the generation of different perfect SC states [46]. The auxiliary laser power was used to change the intracavity power in the MRR. Heating and cooling the external device housing, and changes in the intracavity power would have a similar effect, both changing the on-chip temperature. In [45] they see significant AMX strength change and we do not. The difference between the two results could be due to the FSR of the MRR. The device in [45] has an FSR of 231 GHz, while our devices have an FSR of 48.9 GHz. As our devices have $\sim$4.7$\times$ more modes in an equivalent frequency interval and we see more modes affected by the AMX. The higher mode density could be partially responsible for why we do not see similar strong AMX strength change or suppression. If we pick every 5th mode we see a stronger change in AMX strength like [45], which can be inferred from Fig. 3 b). Other potential factors like the difference in crossing modes’ FSRs, and the degree of mode coupling could also contribute to the difference in results.
The 73-FSR ‘Palm’ like SC state was able to be generated in two different rings with different mode crossing strengths, dispersion and quality factors. The state could also be generated by manually sweeping the laser on the same resonance for both the rings for the temperature range of 10-40 $^\circ$C and 15-40 $^\circ$C without significant change in shape or line-to-line power. Considering when the ring was passive, the AMX position shifted with regards to a fixed mode $\sim$0.88 resonances and $\sim$0.55 resonances over the same temperature range above for the two rings respectively. Assuming that there is a similar AMX shift when generating the SC state, then the precise AMX position does not have a large effect on the SC state generated. The temperature range was selected based on the physical limitations of the heater and the equipment, and the ‘Palm’ like SC state could potentially be generated at temperatures outside the range looked at here. The AMX shift during formation should also be small and is likely why switching dynamics were not observed [51]. This shows for the ‘Palm’ like SC state, dispersion, quality, precise AMX location and AMX strength can vary slightly and produce the same result.
Temperature is able to shift the comb position with both the pump line and the line with excess power associated with the AMX shifted at a rate of -1.71 GHz/$^\circ$C for Ring 1 and -1.99 GHz/$^\circ$C for Ring 2. Over the temperature range, the centre comb frequency shifted slightly over one FSR. Temperature can then be used to set the comb position within an FSR spacing and give full access to the wavelength space. Which is useful for applications where the comb lines need to be aligned to a particular frequency or grid.
The SC location and breadth are determined by the pump location and coarse AMX location, here the AMX lies around 73 FSR away from the pump, giving the 73-FSR Palm ’like’ state. The AMX location and shift would depend on the difference in the thermal coefficient between the two crossing mode families. In these devices, the AMX location is only able to shift $\sim$0.06 nm for the first ring and $\sim$0.13 nm for the second. So for devices such as these, the wavelength the AMX occurs at is still largely determined by the fabrication process and can only be slightly tuned with temperature. In other rings where there are greater differences in the thermal drift between the two crossing modes, for example in dual ring devices or for polarisation AMXs, the AMX might be able to be shifted more with temperature. This could potentially allow more flexibility in the SC placement but could also reduce the operating temperature range.
The SC state is consistent in shape and type with temperature suggesting that if the detuning value or frequency difference between the pump and the resonance is kept constant the state can be maintained with temperature shifts. A feedback loop, like fine thermal control [52,53] or pump control, may allow for active frequency shifting of the microcomb to tune it during operation. Additionally, this may remove the need for specific temperature control of the ring for applications where the frequency placement of comb lines is less important.
6. Conclusion
In two 49-GHz FSR high-index doped silica microrings, the first with AMX of $\sim$210 MHz located at 1580 nm and the second at 1582 nm with a strength of $\sim$60 MHz. By heating and cooling the devices, repeated dispersion measurements were shown to change the mode crossing frequency at a rate of -0.1 GHz/$^\circ$C and -0.5 GHz/$^\circ$C for each ring respectively. This was less than the change in frequency of the modes, resulting in the mode crossing position being able to be tuned in regard to the modes. The AMX strength was only minorly affected changing by a total of 10 MHz and 6 MHz for the two rings respectively.
Both these rings could generate the same ’Palm’ like soliton crystal state by frequency sweeping a particular resonance. This was done for a temperature range of 25 $^\circ$C. The thermal shift of the spectrum was -1.71 GHz/$^\circ$C and -1.99 GHz/$^\circ$C allowing for temperature control to adjust the placement of the comb within the resonance spacing giving full access to the frequency space.
As the same state can be generated over the temperature range, and the mode crossing position shifts with temperature, it means the ‘Palm’ like soliton crystal is insensitive to the minute mode crossing position. Combined with two different rings with different quality, mode crossing and dispersion, generating the same state signifies that the ‘Palm’ like soliton crystal state is robust to some fabrication variations. It also means soliton crystals of this type are unlikely to require a specific temperature to generate, removing temperature as a potential variable when searching for these states on devices with a similar spatial AMX.
Our work shows, in a more systematic study than previously presented in the literature, that one can obtain a wanted soliton crystal state without needing extremely fine control over the AMX properties. This shows that there is potential for these complex states to be made predictable. Opening up the possibility for others to take advantage of the simplified microcomb generation that is enabled by soliton crystals, in devices that exhibit spatial mode crossings.
Funding
Australian Research Council (FT220100835).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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