## Abstract

Dual-coupled structure is typically used to actively change the local dispersion of microresonator through controllable avoided mode crossings (AMXs). In this paper, we investigate the switchability of dissipative cnoidal waves (DCWs) based on dual-coupled microresonators. The switching dynamics of DCWs are numerically simulated using two sets of nonlinear coupled-mode equations. It is found that the pulse number of DCWs can only be decreased (i.e. switched unidirectionally) when working as perfect soliton crystals and can either be decreased or increased (i.e. switched bidirectionally) when working as Turing rolls. Moreover, the stable regions of DCWs can be greatly expanded due to the existence of AMXs. The switchability of DCWs would further liberate the application potential of microcombs in a wide range of fields, including frequency metrology, optical communications, and signal-processing systems.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical frequency combs (OFCs) provide a coherent link between the optical and radio frequency domains and have brought great improvement to various fields [1,2]. The observation of Kerr frequency combs in optical microresonators, namely microcombs, makes it possible to generate broadband and mode-locked OFCs with chip-scale footprint and high repetition rate ranging from GHz to THz [3,4]. Microcombs have been proved to be a revolutionary fundamental technology in many important areas such as large-capacity optical communications [5–7], metrology [8–10], dual comb technology [11,12], and microwave photonics [13–15]. Moreover, optical microresonators provide an ideal test bench for studying nonlinear soliton physics such as soliton switching [16], bounding [17,18], crystallization [19–22] and breathing [23–25].

Among various microcombs states, Turing rolls (TRs) and perfect soliton crystals (PSCs) are two special kinds of coherent states which are both characterized by *N* equally spaced self-reinforcing pulses in time domain and correspondingly discrete comb lines separated by *N* times of free spectral ranges (FSRs) in frequency domain. In microresonator community, TRs and PSCs are usually distinguished by their effective pump-cavity detuning [20,26,27], i.e., TRs exist in the effectively blue detuned region while PSCs exist in the effectively red detuned region. In nonlinear waves community, as noted in Ref. [28], TRs and PSCs are both stable periodic solutions of the Lugiato-Lefever equations (LLEs) [29–31] and could be uniformly classified as dissipative cnoidal waves (DCWs). In contrast to single-soliton state, DCWs can be deterministically accessed because the comb line spacing (CLS) of TRs is determined by the modulational instability (MI) gain and the CLS of PSCs is determined by the period of the modulated intracavity background. Moreover, DCW microcombs have been applied in the field of coherent optical communications [32], microwave/THz generation [33–35] and microwave photonic filters [36]. It should be noted that realizing flexible switching of the CLS will further liberate the application potentials of DCW microcombs so that they can meet diverse application requirements.

The CLS of TRs is related to the maximum of MI gain which is determined by the pump power and the effective pump-cavity detuning [27], so it is possible to change the CLS of TRs by tuning the power and wavelength of the pump laser. However, it is shown [28] that the stable regions of adjacent TRs partially overlap with each other in the (frequency detuning)${\times}$ (pump amplitude) parameter space, which poses severe requirements on the tuning resolution of the pump laser to access a specific TRs state [33]. The CLS of PSCs is determined by the modulated *N*-period intracavity background which traps the soliton pulses [37] and leads them to distribute evenly on the microresonator circumference [20]. In a single microresonator, the modulation on the CW background is introduced either by the avoided mode crossings (AMXs) between different mode families [38,39] or by using a dichromatic pumping scheme [21]. However, coupling between mode families originates from inevitable fabrication errors [38] and is therefore an inherent property which is hard to control and modify manually [40]. Although some works show that the spectral position and strength of AMXs are sensitive to the temperature [20,41,42] and the switching phenomenon of PSCs has been observed [20,42], it is still unpredictable to some extent and cannot be controlled successively. The dichromatic pumping scheme has been demonstrated to be able to achieve deterministically switching of PSCs over a wide range, but it requires sophisticated control on two sets of tunable lasers and suffers from soliton vibration [21].

In this paper, a dual-coupled microresonator scheme is proposed to realize switchable DCWs taking advantage of its flexible control on the spectral position of AMX. The tuning procedure of AMXs is modeled by introducing time-variant resonance frequencies to the auxiliary microresonator. By solving two sets of nonlinear coupled-mode equations, the switching dynamics of DCWs (i.e. PSCs and TRs) are numerically investigated. It is found that the AMXs between dual-coupled microresonators can not only break the degeneracy of DCWs with different CLSs and lead to the switching, but also greatly expand the stable regions of DCWs compared to the single microresonator case. It is discovered that the pulse number of PSCs can only be decreased, while the pulse number of TRs can either be decreased or increased due to the presence of MI gain. The CLS of DCWs can thus be flexibly switched over a wide range which verifies the feasibility and advantages of the proposed dual-coupled microresonator scheme.

## 2. Numerical model

The schematic diagram of the proposed dual-coupled microresonators is shown in Fig. 1(a). Two microresonators with slightly different FSRs are coupled with each other and the main microresonator is pumped by a CW laser through the on-chip bus waveguide. Here, it is assumed that the auxiliary microresonator is of the same resonance linewidth and dispersion of the main microresonator.

The fields in dual-coupled microresonators obey the following nonlinear coupled-mode equations [43,44]:

*f*is the external pumping term and ${\delta _{0\mu }}$ is the Kronecker delta indicating that only mode $\mu = 0$ is pumped. It should be noted that a time-variant term $\omega _\mu ^b\textrm{(}\tau \textrm{)}$ is introduced to Eq. (2) which means that the resonance frequencies of the auxiliary microresonator are continuously tuned, this can be realized experimentally through microheater [45], piezoelectric control [46] or electro-optical Pockels effect [47].

The AMX between the two microresonators leads to a pair of symmetric and asymmetric modes whose eigenfrequencies are given as [48]:

As the FSRs of the two microresonators are slightly different, the spectral position of AMX can be successively tuned from one longitude mode to another (see Fig. 1(b)) by tuning the resonance frequencies $\omega _\mu ^b$ utilizing the Vernier effect [49]. It will be seen in the following that the tuning of AMX will lead to the switching of DCWs. Therefore, as shown in Fig. 1(a), the auxiliary microresonator works like a knob which can be used to control the CLS of DCW microcombs.

## 3. Simulation results and discussion

The switching dynamics of DCWs in dual-coupled microresonators are numerically simulated using the model described in Section 2. The measured parameters of typical Si_{3}N_{4} microresonators are adapted in the simulations, which include: ${\kappa / {2\pi }} = 200\textrm{ MHz}$, ${{{D_2}} / {2\pi }} = 0.87\textrm{ MHz}$, ${{D_1^a} / {2\pi }} = 100\textrm{ GHz}$, ${{D_1^b} / {2\pi }} = 101\textrm{ GHz}$. In each dynamic simulation, the resonance frequencies are tuned linearly with time until the coupling strength at $\mu = {\mu _0}$ reaches its maximum, i.e., $\omega _{{\mu _0}}^b = \omega _{{\mu _0}}^a$. In this case, the hybrid eigenmodes in the mode-splitting region exhibit a frequency separation of $\tilde{\omega }_\mu ^ +{-} \tilde{\omega }_\mu ^ -{=} |{\kappa {g_c}} |$. The simulations will be carried out in two tuning directions starting from an initial state $\omega _N^b = \omega _N^a\textrm{ }(N > 0)$ as shown in Fig. 1(b). In the forward tuning process, $\omega _\mu ^b$ are decreased from $\omega _N^b = \omega _N^a$ to $\omega _{N + 1}^b = \omega _{N + 1}^a$; In the backward tuning process, $\omega _\mu ^b$ are increased from $\omega _N^b = \omega _N^a$ to $\omega _{N - 1}^b = \omega _{N - 1}^a$. For simplicity, it is assumed that each $\omega _\mu ^b$ is tuned by the same amount.

First of all, a specific PSC or TR state is generated under a fixed AMX. For example, as shown in Fig. 2, the AMX is set at $\mu = 22$ (i.e., $\omega _{22}^b = \omega _{22}^a$) with the coupling strength ${g_c} = 1.2$ and the pump power ${f^2} = 3.5$. When initiated from random noise, the PSC_{22} (PSC with $\textrm{CLS} = 22 \times \textrm{FSR}$) state is generated by tuning $\zeta _0^a$ from −0.5 to 2.8 (see Fig. 2(b)), and the TR_{22} state can be generated at a fixed detuning $\zeta _0^a = 2.3$ due to the presence of MI gain (see Fig. 2(c)).

In the following simulations, we will start from a stable DCW state as generated above and then carry out the forward and backward tuning to investigate its switching dynamics. In Section 3.1, the switching dynamics of PSCs are investigated. In Section 3.2, the switching dynamics of TRs are investigated. Besides, the three mode MI gain of the dual-coupled system is analyzed in order to explain the bidirectional switchability. In Section 3.3, the stable regions of DCWs are mapped in the dual-coupled system using the evolutionary method.

#### 3.1 Unidirectional switching of PSCs

Starting from the PSC_{22} state under $(\zeta _0^a,{f^2}) = (2.8,3.5)$ as generated above, we try to decrease its soliton number by tuning the spectral position of AMX from mode number $\mu = 22$ to $\mu = 21$ (i.e. backward tuning). As shown in Fig. 3(a) (the first panel), the resonance frequencies of the auxiliary microresonators are increased by $\Delta \omega _\mu ^b = D_1^b - D_1^a = 10{\kappa / 2}$ and then settled down. Because the coupled microresonators possess the same dispersion, $\Delta \omega _\mu ^b = D_1^b - D_1^a$ means that a backward tuning process is completed once. Figure 3(a) (the third panel) shows the switching dynamics of the intracavity waveform, it is found that the switching process does not start immediately after the backward tuning because the new modulated background is not completely formed. From a spectral point of view, as shown in Fig. 3(a) (the fourth panel), the extra spectrums between original combs take time to grow up gradually from the noise floor and the modulation strength of the intracavity background reaches its maximum when the optical spectrum is filled up by extra comb lines. After that, the new intracavity background pushes the soliton pulses of PSC_{22} to rearrange. During the rearrangement, all solitons adjust their relative positions to match the new modulated background while two of them are pushed close to each other. Finally, two solitons merge into one and cause a wave splash as shown in the inset of Fig. 3(a) (the third panel) which has been experimentally observed in Ref. [50]. During the merging process, the overall intracavity power increases firstly and then drops rapidly due to the soliton merging, the merged soliton acts like a breather and experiences a damped oscillation before reaching its stable state as shown in Fig. 3(a) (the second panel). In frequency domain, the extra comb lines between $\mu = n \times 21\textrm{ }(n \in \textrm{{Z}})$ decay to noise floor after soliton merging and the PSC_{21} state forms eventually. The switching process could also be reflected in the stair-like pattern of the overall intracavity power which is similar with the switching process of traditional non-equidistant multiple-soliton states [16,51,52], but it should be noted that the switching of multiple-soliton states is caused by the transient chaos [20] rather than soliton merging.

After switching from PSC_{22} to PSC_{21}, we try to reverse the tuning direction and find out whether PSC_{22} state could be recovered. As shown in Fig. 3(b), during the backward tuning process, the hybrid mode $\tilde{\omega }_{21}^ -$ will sweep through the comb mode $\mu = 21$ which causes a resonance in the overall intracavity power trace. This resonance heavily perturbs the equidistant soliton pulses and results in soliton merging multiple times. Eventually, PSC_{21} state evolves to a defect soliton crystal state [19,22] which indicates that the defect-free switching of PSCs can only be realized unidirectionally (i.e. once the soliton number of PSC is decreased, the process is irreversible).

Furthermore, in order to verify the feasibility of successive switching, the numerical experiment is implemented using the same strategy as described above. Starting from PSC_{22} state, the spectral position of AMX is tuned from mode index $\mu = 22$ to $\mu = 11$ and realize successive switching from PSC_{22} to PSC_{11} as shown in Fig. 4. Because the switching process only occupies a small fraction of the whole tuning process (see Fig. 3), the evolution dynamics around each soliton-merging process are intercepted and spliced together for better displaying the switching details. It should be noted that each tuning process in Fig. 4 follows the same way as shown in Fig. 3(a), the AMX is fixed for a sufficiently long time to make sure that each PSC state can exist stably.

It is found that the switching time gets longer as the soliton number *N* decreases. This is due to that the angular separation of adjacent soliton pulses become larger after the backward switching, so it would take longer for these soliton pulses to rearrange, interact with each other and finally complete the switching process. In the numerical simulations, the soliton number *N* can only be switched to $N = 8$ within a reasonable simulation time, although the PSC* _{N}* states with $1 \le N \le 7$ are also stable in the dual-coupled system under $(\zeta _0^a,{f^2}) = (2.8,3.5)$. It is noted that the small laser-cavity detuning will enhance the modulated background induced by the AMX [37,53], which is essential for preventing the closely spaced soliton pulses from merging with each other. As a result, small laser-cavity detuning is preferred for maintaining PSCs with large

*N*while large laser-cavity detuning is preferred to generate PSCs with small

*N*.

#### 3.2 Bidirectional switching of TPs

In this section, we start from the TR_{22} state under $(\zeta _0^a,{f^2}) = (2.3,3.5)$ as generated above and execute the same numerical experiments as described in Section 3.1. As shown in Fig. 5(a), the backward tuning process of TR_{22} is similar with that of PSC_{22}, the TR_{22} state is successfully switched to the TR_{21} state through the merging of two pulses. When the tuning direction is reversed, the resonance between the hybrid mode $\tilde{\omega }_{21}^ -$ and the comb mode $\mu = 21$ also heavily perturbs the equidistant pulses and results in pulse merging. Interestingly, new pulses will burst from the intracavity background to fill the vacancies between adjacent pulses (see Fig. 5(b), the third panel). After pulse bursting and merging multiple times, the TR_{21} state is finally switched back to the TR_{22} state which indicates that the TRs can be switched bidirectionally in dual-coupled system.

The bidirectional switchability of TRs can be attributed to the presence of MI gain [27]. In dual-coupled system, the steady-state solution of ${b_\mu }$ can be derived as ${b_\mu } = {{ig_c^ \ast {a_\mu }} / {(1 + i\zeta _\mu ^b)}}$ by ignoring the nonlinear term in Eq. (2). Therefore, the linearized equations of ${a_{ + \mu }}$ and ${a_{ - \mu }}$ can be written as [43,44]:

The MI gain can be expressed as $G = \Re ({\lambda + 1} )$ where $\lambda$ are the eigenvalues of *M*. Figure 6 shows the MI gain under three sets of pump parameters when the AMX is set at $\mu = 22$ with the coupling strength ${g_c} = 1.2$. It is shown that there is no MI gain in PSC region when the system is worked at the lower branch [27]. Therefore, the collapsed solitons during the forward switching process cannot regenerate from the background (see Fig. 3(b)), which explains why PSCs can only be switched unidirectionally. As for TRs, the presence of MI gain gives them the bidirectional switchability. In principle, the modes $\mu$ that satisfy $G(\mu ) > 1$ can be directly excited by the pump, which leads to the overlap between different TR states in the $(\zeta _0^a,f)$ parameter space, i.e., there are multiple TR solutions under the same pump parameters [28]. However, as shown in Fig. 6(b), the AMX perturbs the MI gain profile and lift the degeneracy of different TR states. In conclusion, the bidirectional switchability of TRs is originated from the joint action of MI and AMX, i.e., MI provides the gain so that the lost pulses can burst from the background while the AMX forces the intracavity waveform to evolve to the specific TR state. Moreover, it can be seen from Fig. 6(b) that the switching range of TRs will be greatly expanded if the pump parameters get close to the PSC region, which coincides with the conclusion in Ref. [28] that the stable regions of multiple TRs closely overlap with each other near the PSC region. Therefore, it is preferred to operate the TRs near the PSC region to get the maximum switching range.

In Fig. 7, the bidirectional switching between the TR_{22} and TR_{17} state is realized under $(\zeta _0^a,{f^2}) = (2.3,3.5)$ (point B in Fig. 6). The inset in Fig. 7(a) indicates that the backward switching of TRs will sometimes go through a more complex process (i.e. pulse merging twice and bursting once).

The achievable tuning range of TRs is preliminarily determined by $G(\mu ) > 1$ while the maximum value of *N* is further limited by the condition that the adjacent pulses cannot interact with each other. We check the stability of different TR states numerically and find that ${N_{\min }} = 16$ (which is close to the MI gain analysis result ${N_{\min }} = 15$) and ${N_{\max }} = 22$. However, as shown in Fig. 8, the backward switching from TR_{17} to TR_{16} cannot be realized in the dynamic simulations. Although the intracavity position of 17 pulses are rearranged at 16 equidistant potential ‘sites’ introduced by the new background modulation, one of the 17 pulses never merge with others but undergo periodic pulse collisions [54] and the intracavity power exhibits a chaotic oscillation. This phenomenon always happens when the pulse number *N* approaches its lower limit and may be attributed to that the MI gain $G(N)$ is too close to 1. The underlying physical mechanism will be further studied in the future. It is worth noting that the conclusions in this section may be used to explain the experimental phenomenon in Ref. [42] where the bidirectional switching and chaotic oscillation are observed.

#### 3.3 Stable regions of DCWs

In single microresonator case, it is shown [55] that if a pair of solitons are separated by an angular distance $\Delta \Phi \ge 8\sqrt {{{{D_2}} / {\kappa {\zeta _0}}}}$, they will not interact with each other. The maximum soliton number of the PSC can thus be calculated as ${N_{\max }} = {{2\pi } / {\Delta \Phi }} \le 0.25\pi \sqrt {{{\kappa {\zeta _0}} / {{D_2}}}}$. Under the parameters used for PSC switching (i.e. ${\kappa / {2\pi }} = 200\textrm{ MHz}$, ${{{D_2}} / {2\pi }} = 0.87\textrm{ MHz}$, ${\zeta _0} = 2.8$), it is found that ${N_{\max }} = 19$ which is smaller than the soliton number of PSC_{22} generated above. This indicates that the existence of AMX increases the maximum achievable soliton number of DCW. In other words, the stable regions of DCWs can be very different when the mode coupling effects are concerned.

Here, the evolutionary method [56] is used to map the stable regions of DCWs in the $(\zeta _0^a,\textrm{ }f)$ parameter space. It should be noted that, in single microresonator case (or ${g_c} = 0$), the dynamical method described in Ref. [28,57] provides a powerful mathematical tool to track the stability boundaries of DCWs precisely. But in dual-coupled system (or ${g_c} \ne 0$), the AMXs between two microresonators lead to the group velocity shifts of DCWs relative to the moving framework (see Fig. 2). The group velocity shift in dual-coupled system is hard to be expressed analytically and can only be numerically calculated using the evolutionary method. This increases the computational complexity of the dynamical method. Besides, it is not the point of this work to precisely map the stable boundary of DCWs. Therefore, the stable regions of DCWs under ${g_c} = 0$ (calculated using dynamical method) are taken as references to estimate the simulation region, then the evolutionary method is used to map the stable regions of DCWs under AMX.

In each simulation, we set the AMX at mode index $\mu = N$ (i.e. $\omega _N^b = \omega _N^a$) and initiate the dual-coupled system with a period-*N* solution, the stable region of DCW* _{N}* is then mapped in the $(\zeta _0^a,\textrm{ }f)$ parameter space with a resolution of 0.05 using evolutionary method.

Figure 9 shows the stable regions of DCWs under ${g_c} = 1.2$ (used for previous simulations). It can be seen that the AMX moves the stable region towards the lower right in the $(\zeta _0^a,\textrm{ }f)$ parameter space (e.g. DCW_{28}), open up a new stable region in the PSC region (e.g. DCW_{26}), or extend the stable region all the way from the TR region to the PSC region (e.g. DCW_{22}). As for DCW* _{N}* that already extend into the PSC region (e.g. DCW

_{15}), the AMX will shrink its stable region a little. In conclusion, the AMX-induced background modulation traps the pulses of DCW at the potential ‘sites’ and thus prevents them from moving towards each other. As a result, the presence of AMX greatly expands the stable regions of DCWs that possessing relatively large

*N*, which explains why the PSCs generated in experiments often have a larger soliton number

*N*than the theoretical predicted value [20].

For comparison, the coupling strength is increased to ${g_c} = 2$ and the corresponding stable regions are mapped in Fig. 10. It is found that large coupling strength benefits the DCWs with large *N*, as it can open up a stable region in the PSC region (e.g. DCW_{28}) and connect the separated stable regions to form a closed region (e.g. DCW_{26}). However, for DCWs with smaller *N*, larger coupling strength will further shrink their stable regions (e.g. DCW_{15}). When the AMX is close to the pumped mode (e.g. DCW_{1}), the stable regions move up because the auxiliary microresonator consumes a part of pump power. Moreover, we try to increase the coupling strength to ${g_c} = 3$ and find that the area of stable regions is greatly reduced. When working as PSCs, the soliton pulses tend to annihilate and evolve to single soliton state when the coupling strength is further increased, this is consistent with the conclusion in Ref. [52] that the soliton number is reduced to one in order to lower the nonlinear loss into mode-interaction induced Cherenkov radiation when the coupling strength is strong enough. Concluded from our cases, ${g_c} \sim {1}$ is an appropriate choice for executing the switching.

## 4. Qualitative experiment

Here, a qualitative experiment is carried out using a single Si_{3}N_{4} microresonator. The coupling between the fundamental transverse electric (TE_{00}) and transverse magnetic (TM_{00}) mode families can be regarded as an analogue of dual-coupled system. In the experiment, the frequency of a tunable laser is scanned back and forth around the pumped resonance by applying a periodic triangular signal while the generated comb power is monitored after filtering the pump by a fiber Bragg grating. It is known that the spectral position of the mode crossing is sensitive to the temperature of the system [20,42], and the essential reason for this phenomenon is that the thermal-refractive coefficients of different mode families are different. Therefore, it could be expected that the spectral position of the mode crossing will change dynamically with the pump-cavity detuning.

As shown in Fig. 11(b) (the upper panel), when the pump power is set as ${\sim} 200\textrm{ mW}$, the comb power trace shows two high power ‘steps’ in the both tuning directions. The corresponding optical spectra (see Fig. 11(b), the lower panel) indicate that bidirectional switching between the TR_{21} and TR_{20} state is realized. In order to further verify that the switching is related to the AMX, the integrated dispersion is measured under two TR states using another probe laser through the Mach-Zehnder interferometer (MZI)-based optical sampling technique [34]. As shown in Fig. 11(c), the spectral positions of AMXs correspond well to the TR states. In addition, the interference between the probe laser and the pump laser can be used to determine whether the pump is blue detuned (i.e. in TR region) or red detuned (i.e. in PSC region) [58].

As shown in Fig. 11(d), when the pump power is increased to ${\sim} 250\textrm{ mW}$, the unidirectional switching from the PSC_{20} to PSC_{19} state can be realized along with the decrease of the laser frequency. Different from TRs, when the scan direction is reversed, the PSC_{20} state cannot recover from the PSC_{19} state but transits to the TR_{23} state.

This qualitative experiment partially verified the conclusion in previous sections, i.e., TRs can be switched bidirectionally while PSCs cannot. The difference between the simulation and the experiment could be contributed to three factors: firstly, the detuning is fixed in the simulation but changed in the experiment; secondly, the real microresonator suffers from the high-order mode coupling; thirdly, the thermal effect is not considered in the simulations. It should be noted that, in order to achieve reliable switching as described in the numerical simulations, single mode microresonators [59,60] are preferred to build the dual-coupled system and the offset Pound-Drever-Hall technique [13,61] can be used to stabilize the pump-cavity detuning at a preset value.

## 5. Conclusion

In conclusion, a dual-coupled microresonator structure is proposed and theoretically modeled for realizing feasible switching of DCWs. The switching dynamics of DCWs are numerically simulated in the TR region and the PSC region, respectively. It is found that the PSCs can only be switched unidirectionally while TRs can be switched bidirectionally due to the presence MI gain. Moreover, we map the stable regions of DCWs in the dual-coupled system and find that the presence of AMXs can greatly expand the stable regions of DCWs from the TR region to the PSC region. A qualitative experiment is carried out in a single Si_{3}N_{4} microresonator utilizing the coupling between TE and TM mode families. From the perspective of theory, the presented simulation results could give some inspiration for nonlinear wave dynamic research. From the perspective of application, the flexibly switchable microcombs can meet different demands of applications such as optical communication, signal process and metrology. It is worth noting that including thermal effect and considering coupled system composed of more microresonators could be important topics of the future work.

## Funding

National Natural Science Foundation of China (61625104, 61971065); State Key Laboratory of Information Photonics and Optical Communications (IPOC2020ZT03).

## Acknowledgments

The authors acknowledge Ligentec for device fabrication.

## 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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