Dual-mode microresonators as straightforward access to octave-spanning dissipative Kerr solitons

The dissipative Kerr soliton (DKS), as a self-reinforcing wave packet that maintains its shape while circulating around a microresonator, has been demonstrated under a double balance between nonlinearity and dispersion, as well as parametric gain and cavity loss.^1,2 Due to the unprecedented compactness, low-noise, high power-efficiency, and broad spectral bandwidth, soliton Kerr combs (microcombs) have attracted considerable research interest and been extensively studied for spectroscopy,³ communications,⁴ frequency synthesizer,⁵ optical clock,⁶ microwave photonics,⁷ and sensor applications.⁸ Over the past several years, through the substantial exploration of the fundamental physics and microresonator fabrication, researchers have realized Kerr solitons in a growing number of platforms, including ultra-high Q MgF₂,⁹ silica,¹⁰ and monolithic integrated platforms such as Si₃N₄,^1,11–13 LiNbO₃,¹⁴ AlGaAs,¹⁵ and Ta₂O₅,¹⁶ as well as the wide-bandgap semiconductors AlN,^17,18 SiC,¹⁹ and GaN.²⁰ 

The photonic integration of the laser pump and passive resonators offers the possibility of achieving chip-scale operation, but there are significant challenges to overcome before the widespread deployment of these soliton comb systems. One key challenge comes from the thermo-optic instability in the microresonator when the pump enters into the red-detuned regime for soliton formation. To stably access the soliton state, a number of experimental techniques were developed including rapid laser frequency scanning,^9,16,18 careful pump power manipulation,^1,10 or microheater thermal tuning.^11,21 With the extra radio-frequency (RF) generator, modulator, or microheaters, these schemes can bring the short-lived soliton to a steady state. Self-injection locking (SIL)^22,23 has been proposed and exploited for turnkey soliton generation²⁴ or even a remarkable octave-spanning soliton microcomb,²⁵ by directly coupling a laser chip to a passive microresonator. This approach enables a miniaturized frequency comb source but demands challenging photonic integration and great caution to control the back-reflected Rayleigh scattering. In addition, the likely accessible detuning reduction in SIL will limit the spectral bandwidth and dispersive wave (DW) intensity,²⁵ as well as the soliton existence range (SER) and total comb power,²⁶ which are serious issues for the precision metrology and timing. Recently, dual-pumping for two resonances (2P2R) has been applied to mitigate the thermal effects, thus leading to the generation and switching of the DKS.^20,27–29 However, it drastically increases the system complexity and cost due to the use of another set of laser, amplifier, polarization controller, and fiber circulator.^20,28,29 Another dual-pumping scheme that activates a single resonance (2P1R) was also proposed, where the auxiliary pump is one modulation sideband away from the main pump.^30,31

A simple and cost-effective octave-spanning Kerr soliton generator needs to be developed urgently to bring the microcombs toward practical applications. Here, we present the straightforward access to octave-spanning DKSs by injecting a single pump to two close resonances (dual-mode) with the same polarization, which we called the 1P2R-1P scheme. The modes on the blue and red sides are used for parametric processes and thermal compensation, respectively. Figure 1 compares different thermal accessible soliton systems and sketches the 1P2R mechanism. In contrast to dual-pumping, our method leverages a much simplified setup once the front-end design is properly managed. The idea used in this paper was proposed by our group and successfully applied for octave-spanning DKS generation in an AlN microring resonator (MRR).¹⁷ Dual-mode but with mixed polarization (1P2R-2P) was also found to help the soliton stabilization process,³² which requires careful adjustment of the polarization. However, the real potential of the 1P2R scheme needs investigation due to the rigorous requirements in design and fabrication for ensuring the pump and auxiliary modes are in close proximity.

For the first time, this paper shows a systematic design to realize the desired dual-mode reliably and we discuss the feasibility of fabrication with high yield. The strong dependence of the carrier-envelope offset frequency (f_ceo) on the dimension is also revealed. The advanced single-soliton features a 17-GHz-wide soliton existence range (SER) and a 200-THz-wide spectral bandwidth. SER means an effective detuning range where a soliton state is maintained during the pump wavelength tuning.⁹ Moreover, using the same resonance, octave-spanning soliton crystals (SCs) at the telecommunication C-band are demonstrated. Similar soliton behaviors are also observed in multiple chips and thereby illustrate its universal nature. The presented results provide a solid strategy for broadband DKS generation, which is transferable to alternative materials with a tailored repetition rate (f_rep). From an application perspective, the 1P2R-1P scheme paves the way toward making reliable, dynamic, low-cost, and easy-to-operate soliton microcomb sources.

Figure 2 shows the design and resonance characteristics of the dual-mode MRRs. The 800-nm-thick Si₃N₄ MRRs that we are using were fabricated by LIGENTEC.³³ To build a reliable layout design, we first conducted the simulations with a finite element method. For the MRRs with a fixed radius of 23.3 µm and varied ring widths (RWs), the simulated resonant wavelengths are plotted in Fig. 2(a). As the RW increases, an obvious mode redshift is observed, while the slope dλ/dRW is 0.012 and 0.094 for the TE₀₀ and TE₁₀ modes. Consequently, a 1 nm RW variation will lead to an ∼0.08 nm adjustment in the mode separation Δλ (λ₁₀ − λ₀₀), i.e., a 100 nm RW variation will lead to an ∼8 nm Δλ change, which is almost one free spectra range (FSR). Specifically, the TE₀₀ mode with an angular number (m) of 164 and the TE₁₀ mode (m = 151) have a minimum Δλ of 0.09 nm at ∼1563 nm when RW = 1.68 µm. Then, the two modes coincide again at ∼1572 nm with a separation of 0.12 nm when RW = 1.78 µm.

Figure 2(b) displays the measured transmission spectra of seven neighboring MRRs with RWs discretely increasing from 1.68 to 1.80 µm. The 600-nm-wide straight waveguides are employed for the coupling, with a 650 nm gap apart from the resonators. As the RW increases, both modes show the redshift (denoted by arrows) with the speeds almost the same as simulations, illustrating that a 20 nm variation in the MRR dimension is achievable even using the 248 nm deep ultraviolet (DUV) lithography. The target dual-mode in MRR1, denoted by a circle and enlarged as an inset of Fig. 2(c), has a center wavelength of ∼1565.15 nm and a Δλ of 0.084 nm (∼10.5 GHz). The TE₀₀ and TE₁₀ modes have an FSR of ∼1012 and ∼979 GHz, respectively. Their statistical Q factors are 1.1 × 10⁶ and 4.2 × 10⁵, respectively. The dual-mode position shifts to ∼1566.3 nm when RW = 1.78 µm, while the microcombs cannot be tuned to the soliton regime due to a relatively large Δλ of 0.262 nm. Overall, the experimental results are consistent with simulations, suggesting that dual-mode resonators in the C-band can be reliably attained by tailoring the MRR dimensions.

Near-zero anomalous dispersion is crucial for broadband microcomb generation. Figure 2(c) presents the calculated integrated dispersion (D_int)¹ of the TE₀₀ mode family. All the MRRs can support dual-DW, where the D_int equals zero. With the increase of RW, the DW position at low-frequency has a small blue shift, while the high-frequency DW dramatically drops from 311 to 254 THz. The simulated second-order dispersion D₂/2π is 37.2 and 19 MHz when RW is 1.68 and 1.78 µm, respectively. Figure 2(d) summarizes the measured modulation instability (MI) microcombs when pumping the MRRs with an on-chip power of P_in = 400 mW. All spectra exceed an octave span, thanks to the dual-DW. The groups of dense lines below 130 THz result from the second-order diffraction of the optical spectrum analyzer (OSA) and are thus artifacts. The peculiar comb lines with enhanced or reduced power (denoted by dotted lines) are caused by mode interactions.³⁴ The wider MRRs tend to have flatter spectral envelopes and stronger comb lines near the DWs because of the lower second-order dispersion.

In this section, we present the octave-spanning soliton results using the first 1.68-µm-wide device MRR1 as discussed in Fig. 2. All transmission curves and optical spectra presented in this work are obtained with a forward pump tuning speed of 1 nm/s, which allows for adiabatic tuning within the cavity. The experimental setup is similar to the one reported in Refs. 17 and 35 and can be found in the supplementary material.

Figure 3(a) shows the simultaneously measured transmission of all output and pump alone, as well as their difference when P_in = 150 mW. The striking steps related to the single-soliton formation are observed, with a width of 0.08 nm (∼10 GHz), which is close to Δλ. The soliton step is always accessible among tens of sweeping attempts, indicating the deterministic generation of the single-soliton. The soliton state can be easily reached even by tuning the wavelength manually with step mode, which takes a couple of minutes. By sweeping the laser to different wavelengths, we record the microcomb spectra and plot the evolution map in Fig. 3(b), which confirms the aforementioned wide soliton existence window conclusively. Two dashed lines (i) and (ii) indicate the state of MI comb and single-soliton, whose spectra are plotted in Figs. 3(c)(i) and 3(c)(ii), respectively. The spectrum ranges from 136 to 240 THz at MI state but is conspicuously widened to 125–322 THz for the single-soliton. The soliton microcomb covers 1.5-octaves and is state of the art.^32,36,37 The dual-DW at the frequency of 132 and 311 THz agree with the simulation results very well. The transition from MI to soliton state can also be verified by the drastic reduction of the RF intensity noise, as shown in Fig. 3(d). For comparison, we also pump the TE₀₀ mode at 1549.1 nm, which is far from the auxiliary mode. Only an MI comb ranging from 136 to 240 THz appears as a final state when P_in = 150 mW (see the supplementary material). These results illustrate that the dual-mode scheme could also decrease the pump power required to reach the soliton state.

The soliton crystal (SC) is an extraordinary state with regularly distributed soliton pulses and enhanced comb line power spaced by multiples of the cavity FSR.^38,39 For example, N-SC exhibits comb lines separated by N×FSR. Such SCs are typically formed in the presence of avoided mode crossing.^40,41 In our scheme, a weak mode coupling occurs between the two neighboring resonances (see the supplementary material), which results in several GHz changes in the FSRs and the generation of 2-SC and 3-SC when setting a slightly higher pump power, as shown in Figs. 3(c)(iii) and 3(c)(iv), respectively. Both spectra exceed an octave-spanning range (127–270 THz) and exhibit stronger comb lines near the pump. To the best of our knowledge, this is the first report of octave-spanning dissipative SCs centered on the C-band. All the soliton states are sufficiently stable that enables us to record the spectra with a high resolution of 0.05 nm. As with the single-soliton, the SCs are free of low-frequency RF noise.

As regards 3-SC, there are 8 and 15 comb lines with powers greater than 1 mW and 100 µW, respectively. By adding up the intensities of all lines except that of the pump, the collected comb powers in the lensed fiber are 6.4, 12.8, and 17.4 mW for the single-soliton, 2-SC, and 3-SC, respectively. Based on a fiber-coupling efficiency of 56% (measured at C-band) per facet, the in-waveguide comb powers can be conservatively estimated as 11.5, 22.8, and 31 mW, corresponding to a conversion efficiency (CE) of 7.7%, 11.4%, and 13.5%, respectively. Compared with conventional single DKSs (a few percent CE),⁴² the CE of SC is greatly enhanced and we believe it can be further improved by refining the external coupling rate.⁴³ SCs offer regular temporal profiles and strong individual line intensity, which would be useful for telecommunication and radio-frequency filter applications. Given the octave-spanning ranges here, they are also promising for frequency synthesizer or optical clock applications, while the detection and locking of the high f_rep might be challenging. To counteract this, SCs generated with larger resonators would be helpful.

Figure 3(e) shows the experimental pulse traces carried out by an autocorrelator, where the periods of single-soliton, 2-SC, and 3-SC are ∼1, ∼0.5, and ∼0.33 ps, respectively, inversely proportional to the f_rep of ∼1, ∼2, and ∼3 THz. With sech-squared fitting, the pulse width of the single-soliton is deduced to be 35 fs from the autocorrelation trace, while an 18 fs width is obtained from the spectrum by assuming a transform-limited pulse. This discrepancy can be mainly attributed to the phase variation across the pulse spectrum, which can be more precisely determined through a characterization technique such as frequency resolved optical gating (FROG). In the supplementary material, we also present the 3-SC and 4-SC from other dual-mode devices.

The f_ceo is an important parameter for microcomb in the applications of metrology and timekeeping, which can be detected via the f-2f self-referencing technique.⁴⁴ A near-zero f_ceo is ideal for electronic detection and phase-locking.³⁷ Figure 4(a) shows the single-soliton spectrum at low-frequency, where the circles and triangles indicate the first-order (i.e., realistic) and second-order comb lines, respectively. The latter has a spacing of half FSR and an intensity increasing trend with the frequency decrease. The f_ceo can be calculated via

where f_n and f_2n/2 are frequencies of the adjacent first-order and second-order comb lines, respectively. Consequently, an f_ceo of ∼200 GHz is extracted. Figure 4(b) illustrates the f_ceo dependence on the RW. For the MI combs shown in Fig. 2(d), the f_ceo declines from 141 to −460 GHz when the RW increases from 1.68 to 1.8 μm, with a fitted linear slope of −5 GHz/nm. In particular, for the 1.68-μm-wide device, the f_ceo of the soliton comb is 60 GHz higher than that of the MI comb. The data are missed at RW = 1.7 μm due to the weak and unrecognized comb lines. The value is fitted to be as small as 20 GHz well within the range for RF measurements. Different from the conventional f-2f beatnote detection via frequency doubling, our calculation accuracy is only at the level of 1 GHz limited by the OSA resolution. Nevertheless, this simple measurement could help to achieve a low f_ceo in the further. A preliminary simulation study reveals that the ring width and radius are negatively and positively correlated with the f_ceo, respectively. Therefore, a modest optimization based on the current design will help to further reduce f_ceo to near zero, while the dual-mode scheme is maintained.

By repeatedly scanning the laser over the dual-mode in MRR1 at 200 mW, four typical transmission spectra (excluding pump) are observed and recorded, as shown in Fig. 5(a). Besides the single-soliton, we also obtain the multi-solitons with soliton number (N) of 2 and 3 as well as the switching from N = 3 to N = 2. Particularly, the two-soliton microcomb (TSM) has a soliton existence range of 0.12 nm (∼15 GHz). We clarify that these curves are not real comb power considering that some of the pump power, even if lower than the parametric threshold, will be absorbed by the auxiliary TE₁₀ mode. Despite this, the soliton number is easily identified through the OSA and power meter measurements. Figure 5(b) is a diagram showing the soliton number and effective detuning range dependence on P_in, which indicates that the single-soliton can be definitely generated at 140 and 150 mW. As P_in increases, both the access probability and existence range of single-soliton operation are decreasing, while the multi-solitons appear with higher possibility. The TSM attained at 180 mW has a 0.15-nm-wide (∼18.8 GHz) soliton step, near twice that of Δλ. The 1P2R-1P scheme is, therefore, shown to allow the creation of two-soliton states with ease.

Figure 5(c) shows several spectra of the TSMs when P_in is around 200 mW. These soliton combs can be reproduced easily and sustained for long periods without noise. The relative azimuthal angles of 37.6°, 60.5°, 120°, 172.4°, 175.1°, and 178.2° are retrieved by fitting the spectral envelope with

where ψ is the relative azimuthal angle between the two pulses, μ is the comb mode index relative to the pump position, and S⁽¹⁾ (μ) is the spectral of a single-soliton following a sech-squared shape fitted from the experimental data.¹ The TSMs with ψ of 37.6°, 175.1°, and 178.2° are reproducible in the other two 1.68-µm-wide devices (i.e., MRR2 and MRR3), as shown in Figs. 5(d) and 5(e), respectively. All resonators are over-coupled at the pump wavelength, while MRR1, MRR2, and MRR3 have a coupling gap of 650, 650, and 550 nm, respectively. Thus, for MRR3, the pump power to access solitons is higher and the powers of individual lines are stronger at both high and low frequencies. These TSMs have a generally higher CE compared with the single-soliton, especially for the ψ = 178.2° case, which has a CE beyond 10%. Such diverse soliton states with improved CE are of interest in applications such as optical arbitrary waveform generation⁴⁵ and microwave photonic filters (with larger resonators).⁴⁶ The results of three-soliton microcombs are shown in the supplementary material.

All the above results are obtained by maintaining the substrate temperature at 290 K (17 °C) with the aid of a thermoelectric cooler (TEC). However, only multi-solitons can be triggered in MRR2 and MRR3, which possess a relatively large Δλ of 0.125 and 0.127 nm. Next, we will show the effective control of the mode separation and soliton state by changing TEC temperature T. Figure 6(a) shows the measured dual-mode transmission spectra of MRR2 at different temperatures and a low injection power. As the temperature increases, a thermally induced redshift of the resonant wavelengths are observed with a dλ/dT of ∼0.02 nm/K, corresponding to a thermo-optic coefficient of ∼$2.3×10−5$/K, which is consistent with the result reported in Ref. 47. The Δλ declines from 0.125 to 0.086 nm when T increases from 290 to 350 K, indicating that the resonant wavelength of the TE₀₀ mode is more sensitive to the temperature variation. We also note that the coupling between the two modes is strengthened when they approach each other, leading to the increase in the extinction ratio of the TE₁₀ mode. Figure 6(b) depicts the relation between on-chip power and soliton number at various temperatures. It can be seen that only TSMs can be reached at the temperature of 290 and 300 K, while the single-solitons arise when T ≥ 310 K. When 310 ≤ T ≤ 330 K, low power can trigger the single-soliton only, while conversely, TSMs tend to be formed at high power, which is similar to the trend shown in Fig. 5(b).

Figure 6(c) shows examples of transmission spectra measured at high power, where the relevant soliton number is labeled at the top. Specifically, the TSM with a notable soliton step of ∼23 GHz is observed when T = 300 K and P_in = 190 mW. At 320 K, a soliton switching from N = 2 to N = 1 is observed when P_in = 170 mW. The deterministic access to single-soliton state is also realized at 320 K, accompanied by a maximum soliton existence range (SER) of ∼17 GHz, slightly wider than the 0.105-nm-wide Δλ (∼13 GHz). The single-soliton spectra acquired at 310, 320, and 330 K are plotted in Fig. 6(d) with pump wavelengths of 1566.550, 1566.750, and 1566.950 nm, respectively. The spectra have similar profiles and range from 125 to 320 THz, well beyond an octave span. As with the results originating from MRR1, dual-DW at the frequency of 130 and 312 THz are observed. The inset exhibits the two adjacent first-order and second-order comb lines near 125.5 THz. At 310, 320, and 330 K, f_ceo is calculated to be ∼105, ∼108, and ∼109 GHz, respectively, showing a variation slope of 100–300 MHz/K, which is similar to the previous study.⁴⁹ Because of the fabrication variation (even minor), here, f_ceo is about half of the one from MRR1, confirming the strong dependence of f_ceo on the dimension as explained in Fig. 4(b).

By setting the temperature of MRR3 at 330 K and tuning the pump wavelength to 1566.885 nm, the single-soliton can be stably accessed [see Fig. 6(e)], which has a similar profile as Fig. 3(c)(ii) (MRR1) and Fig. 6(d) (MRR2). In this case, Δλ, the soliton existence range, and f_ceo are ∼0.1 nm (∼12.5 GHz), ∼11 GHz and ∼100 GHz, respectively. It should be mentioned that the MRR2 and MRR3, with almost identical mode separation and f_ceo, are at the same area, suggesting the excellent local uniformity. These results indicate that temperature control will be crucial for the deterministic creation of single-solitons.

The results demonstrate that the 1P2R-1P scheme is applicable to our 23-µm-radius Si₃N₄ MRRs with the proper mode separation (e.g., 10–13 GHz). The pump detuning range enabling single-soliton state is generally equivalent to Δλ, while the detuning range of multi-soliton is up to almost twice Δλ. Table I compares the reported octave-spanning DKSs realized with various platforms. Clearly, the soliton existence range (SER) is much expanded with an auxiliary resonance in our 1P2R-1P scheme. More importantly, the proposed strategy enables access to the soliton state via straightforward pump frequency control instead of rapid frequency scanning or more complicated control. The present octave-spanning single-solitons are generated by the Si₃N₄ MRRs with Q of ∼1.1 × 10⁶, but the ongoing experiments suggest that a 2.7 × 10⁶ Q will reduce the required on-chip power to <40 mW,²⁵ which paves a way toward a miniaturized soliton system that is integrated with a laser diode. We also demonstrate the octave-spanning soliton crystals generation.

We believe that the proposed 1P2R-1P approach for deterministic access to DKS could have profound significance on the microcomb field if a design could be reproduced in fabrication with a reasonable yield. Some solutions can be adopted to further control the mode separation and improve the yield. First, based on the variability below 20 nm in the fabrication, during layout designing, scanning the RW from 1.66 to 1.70 µm with step of a few nm while keeping the radius at 23.3 µm would be helpful. Considering the reproducibility, uniformity, and ability to fabricate a high density of MRRs in the commercial foundry, a reasonable variation in MRR dimensions (both radius and width) is possible to provide more samples featuring both the desired dual-mode and low f_ceo. Second, post-fabrication such as etching⁵⁰ can be used to tune the resonance characteristics and the mode spacing. Finally, as investigated with MRR2 and MRR3, temperature control can be used to effectively modify the mode separation, thus affecting the soliton behavior. The control can be implemented by a substrate TEC or surface microheaters, which has been demonstrated for the thermal tuning of f_rep and f_ceo.⁴⁹ In practice, we can also tune pump wavelength or change the laser source if the real dual-mode region deviates from the designed position.

Our dual-mode scheme allows for soliton creation with a broad existence range, while the mode coupling will change the mode Q factors thus affecting the thermal behavior, e.g., the absorption power of each mode and their thermally induced redshift, and the related soliton dynamics. Therefore, there is still work to be done on optimizing the designs for dual-mode creation with small mode coupling, such as using the resonators with an intra-cavity taper or adiabatic bend.^21,51,52

In summary, we have demonstrated the accessing of an octave-spanning single-soliton, soliton crystals, and multi-solitons in the dual-mode microresonators via simply slow pump tuning. In addition to rich soliton states, the conventional inaccessible soliton step is stabilized now accompanied by an expanded detuning range. Compared with the results achieved by the 2P2R method using an independent auxiliary laser,²⁸ the demonstrated 17-GHz-wide single-soliton step here has been significantly enhanced by two orders of magnitude. Such a broad soliton existence window will greatly enhance the potential for microcomb use in applications such as a parallel frequency-modulated continuous wave (FMCW) LIDAR.⁵³ Temperature control as an effective tool to tune the soliton dynamics for the dual-mode scheme is also demonstrated. This work paves the way toward a route for a reliable design and simple configuration to attain the stable soliton operation, while also providing guidance on how to achieve a low f_ceo. We believe these results will not only accelerate the achievement of commercial, portable, and affordable soliton microcomb sources but also contribute to the extension of their applications.

Recently, we also experimentally achieved the robust soliton generation in a Si₃N₄ MRR (TM₀₀–TM₁₀ as the dual-modes) with a 380 GHz-FSR. The soliton existence range can reach as high as 30 GHz, and the spectral bandwidth is nearly octave-spanning. Due to the weaker thermal effect in the bigger resonators, the ideal mode separation is then smaller than 50 pm. Therefore, the proposed straightforward and low-cost soliton generation system 1P2R-1P will be universally feasible for appropriately designed microresonators across all material platforms, including the LiNbO₃ and SiC.⁵⁴ For larger MRRs, the two mode families will have approximately the same FSR, thus allowing a larger dimension tolerance and easily achieving the dual-mode scheme in the desired pump region. In addition, the temperature control will be more effective to tune the mode separation in this case. We note recent two papers that used the 1P2R-2P scheme in Si₃N₄ MRRs to access the soliton state with an f_rep of 100 GHz⁵⁵ and to improve the timing jitter and effective linewidth of the microcomb lines.⁵⁶ It is foreseeable that, by using a passive dual-mode microresonator with an upgraded Q such as ∼$5×106$⁠,⁵⁷ the photonic integrated octave-spanning coherent microcomb source will be delivered soon, which is driven by a laser diode with a power of about 100 mW but without amplifier or optical feedback.

See the supplementary material for additional information.

This work was supported by the Science Foundation Ireland (Grant No. 17/NSFC/4918) and the National Natural Science Foundation of China (Grant No. 61861136001).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.