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

Photonic integration of 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 Herr et al. (2014); Liu et al. (2021a); Jung et al. (2021), careful pump power manipulation Brasch et al. (2016); Yi et al. (2015), or microheater thermal tuning Joshi et al. (2016); Ji et al. (2021a). 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) Stern et al. (2018); Raja et al. (2019) has been proposed and exploited for turnkey soliton generation Shen et al. (2020) or even a remarkable octave-spanning soliton microcomb Briles et al. (2021), 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. Also, the likely accessible detuning reduction in SIL will limit the spectral bandwidth and dispersive waves (DWs) intensity Briles et al. (2021), as well as the SER and total comb power Voloshin et al. (2021), 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 deterministic generation and switching of the DKS Zheng et al. (2022); Zhang et al. (2019); Zhou et al. (2019); Lu et al. (2019). However, it drastically increases the system complexity and cost due to the use of another set of laser, amplifier, polarization controller, and fiber circulator Zheng et al. (2022); Zhou et al. (2019); Lu et al. (2019). Another dual-pumping scheme but activating a single resonance (2P1R) was also proposed, where the auxiliary pump is one modulation sideband away from the main pump Wildi et al. (2019); Nishimoto et al. (2022).

A simple and cost-effective octave-spanning Kerr soliton generator needs to be developed urgently to bring the microcombs towards 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) Weng et al. (2021a). Dual-mode but with mixed polarization (1P2R-2P) was also found to help the soliton stabilization process Li et al. (2017), 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.

This paper presents the production of octave-spanning Kerr solitons with improved performance through careful design of the microresonators while we also discuss the feasibility of fabrication with high yield. 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 Herr et al. (2014). Moreover, using the same resonance, octave-spanning soliton crystals at the telecommunication C-band are also demonstrated. Similar soliton behavior 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 towards 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 through photonic Damascene processes Liu et al. (2021b). To build a reliable layout design, we first conducted the simulations with finite element method. For the MRRs with a fixed radius of 23.25 $\mu$m 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 d$\lambda$/dRW is 0.012 and 0.094 for the TE₀₀ and TE₁₀ modes. Consequently, a 1 nm RW variation will lead to a $\sim$0.08 nm adjustment in the mode separation $\Delta\lambda$ ($\lambda_{10}$-$\lambda_{00}$), i.e., a 100 nm RW variation will lead to a $\sim$8 nm $\Delta\lambda$ change, which is almost one free spectra range (FSR). Specifically, the TE₀₀ mode with an angular number (m) of 164 and TE₁₀ mode (m=151) have a minimum $\Delta\lambda$ of 0.09 nm at $\sim$1563 nm when RW=1.68 $\mu$m. Then the two modes coincide again at $\sim$1572 nm with a separation of 0.12 nm when RW=1.78 $\mu$m.

Figure 2(b) displays the measured transmission spectra of seven neighbouring MRRs with RWs discretely increasing from 1.68 to 1.80 $\mu$m. 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 DUV lithography. The target dual-mode in MRR1 (denoted by a circle) has a $\Delta\lambda$ of 0.084 nm, i.e., $\sim$10.5 GHz, and is enlarged as an inset of Fig. 2(c). The TE₀₀ and TE₁₀ modes have an FSR of $\sim$1012 and $\sim$979 GHz, respectively. Their statistical Q factors are 1.1×10⁶ and 4.2×10⁵, respectively. The dual-mode position shifts to $\sim$1566.3 nm when RW=1.78 $\mu$m, while the microcombs cannot be tuned to the soliton regime due to a relatively large $\Delta\lambda$ 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) Brasch et al. (2016) 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 D₂/2$\pi$ is 37.2 and 19 MHz when RW is 1.68 and 1.78 $\mu$m, 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 2nd-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 Ramelow et al. (2014). The wider MRRs tend to have flatter spectral envelopes and stronger comb lines near the DWs because ofer the low 2nd-order dispersion.

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 Cole et al. (2017); Wang et al. (2018). For example, N-SC exhibits comb lines separated by N×FSR. Such SCs are typically formed in the presence of avoided mode crossing Karpov et al. (2019); Weng et al. (2021b). In our scheme, a weak mode coupling occurs between the two neighboring resonances (see supplementary material) and result in the observation of 2-SC and 3-SC when adjusting the power slightly, as shown in Fig. 3(c)-(iii) and -(iv), respectively. Both spectra exceed an octave-spanning range (127-270 THz) and exhibit stronger comb lines near the pump. To our knowledge, this is the first report of octave-spanning dissipative SCs centered on the C-band. As regards 3-SC, there are 8 and 15 comb lines with powers greater than 1 mW and 100 $\mu$W, respectively. The in-waveguide comb powers of the single-soliton, 2-SC, and 3-SC are estimated to be 11.5, 22.8, and 31 mW, corresponding to a conversion efficiency (CE) of 7.7$\%$, 11.4$\%$, and 13.5$\%$. Compared with conventional single DKSs (a few percent CE) Bao et al. (2014), the CE of SC is greatly enhanced and we believe it can be further improved by refining the external coupling rate Jang et al. (2021). 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 $\sim$1, $\sim$0.5 and $\sim$0.33 ps, inversely proportional to the f_rep of $\sim$1, $\sim$2 and $\sim$3 THz. With sech-squared fitting, the pulse width of single-soliton is deduced to be 35 fs from the autocorrelator trace, while an 18 fs width will be resulted 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 like frequency resolved optical gating (FROG). In the supplementary material, we also present the 3-SC and 4-SC from other dual-mode devices.

Figure 5(c) shows examples of transmission spectra measured at high power, where the relevant soliton number is labelled at the top. Specifically, the TSM with a notable SER of $\sim$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 SER of $\sim$17 GHz, slightly wider than the 0.105-nm-wide $\Delta\lambda$ ($\sim$13 GHz). The single-soliton spectra acquired at 310, 320, and 330 K are plotted in Fig. 5(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 1st-order and 2nd-order comb lines near 125.5 THz. At 310, 320, and 330 K, the f_ceo is calculated to be $\sim$105, $\sim$108 and $\sim$109 GHz, respectively, which is about half of the f_ceo shown in Fig. 3(d) from MRR1.

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. 5(e)], which has a similar profile as Fig. 3(c)-(ii) (MRR1) and Fig. 5(d) (MRR2). In this case, the $\Delta\lambda$, SER and f_ceo are $\sim$0.1 nm ($\sim$12.5 GHz), $\sim$11 GHz and $\sim$100 GHz, respectively. It should be mentioned that the MRR2 and MRR3, with almost identical mode separation and f_ceo, are in the same chip, which strongly suggests the fabrication uniformity. These results indicate that temperature control will be crucial for the deterministic creation of single-solitons.

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, according to Fig. 2, a fine (a few nm) ring dimension scan is necessary when designing the layout to ensure the accuracy of relative variation in fabrication. Considering the reliability, uniformity, and ability to fabricate a high density of MRRs in the commercial foundry, a reasonable variation in MRR dimensions is possible to provide more samples featuring both the desired dual-mode and low f_ceo. Second, post-fabrication such as etching Moille et al. (2021) can be used to tune the resonance characteristics and the mode spacing. Finally, as investigated with MRR2 and MRR3, temperature control can effectively modify the mode separation and change the soliton state. 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 Xue et al. (2016). In practice, we can also tune pump wavelength or change the laser source if the real dual-mode region deviates from the designed position.

The proposed straightforward and low-cost soliton generation system 1P2R-1P is universally feasible for appropriately designed microresonators. We have demonstrated this first in an AlN MRR and expanded on this in the present study. It should also be possible across other platforms for soliton microcomb generation in LiNbO₃ and SiC as examples. It is foreseeable that, by using a passive dual-mode microresonator with an upgraded Q such as $\sim$5×10⁶ Ji et al. (2021b), 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. We also note that a recent work Lei et al. (2022) shows the positive effects of 1P2R-2P in improving the timing jitter and effective linewidth of the soliton microcomb lines. We believe such a simple system and the versatile soliton states will not only accelerate the achievement of commercial, portable and affordable soliton microcomb sources but also contribute to the extension of their applications.

Figure S1 shows the experimental setup for soliton microcomb generation and characterization. The light source used in the experiment is a tunable laser ranging from 1480 to 1640 nm (Santec TSL-710). An erbium-doped fiber amplifier (EDFA) is adopted to amplify the laser power for pumping the resonator through a lensed fiber with a spot size of 2.5 $\mu$m. A fiber polarization controller (FPC) is utilized to adjust the polarization. The generated soliton microcombs are collected with another lensed fiber for characterization. The optical spectra are recorded by two optical spectrum analyzers (1200-2400 nm and 600-1700 nm range). By injecting the output into an electrical spectrum analyzer (ESA) after a photodiode (PD), the radio-frequency (RF) intensity noise of the combs can be detected. The autocorrelation measurement shown in Fig. 3(e) is carried out by the autocorrelator (AC). For the transmission measurement, the output is divided into two parts, one is for all-transmission and the other one is for the pump-transmission only after a band-pass filter. By calculating the difference between the two branches, we can easily identify the soliton steps in the transmitted curves, as shown in Figs. 3(a) and 4(a) in the main text.

The transmission spectrum of MRR1 measured in a wide range is shown in Fig. S2(a), where the TE₀₀ and TE₁₀ modes are marked by circles and crosses, respectively. The dual-mode near 1565 nm used for the soliton generation is labelled with a relative mode number $\mu$=0. Figure S2(b) plots the experimental loaded Q factors, where the values of TE₀₀ modes have an abrupt change between $\mu$=0 and $\mu$=-1 due to the mode coupling. The extinction ratio of TE₀₀ mode also changes significantly at $\mu$=-1. Figure S2(c) shows the free spectra ranges (FSRs) of the two mode families. Clearly, the fundamental and first-order modes have a statistical FSR of $\sim$1012 and $\sim$979 GHz, respectively. However, because of the mode coupling, the obvious FSR changes are observed for the TE₀₀ mode at $\mu$=0 as well as TE₁₀ modes at $\mu$=-1 and -2. Such doublet resonances with avoided mode crossing are pretty general especially in the resonators with relatively small FSR [1-3]. It has also been exploited for dispersive wave formation [4] and soliton crystal (SC) generation [5].

For comparison, we also pump the TE₀₀ mode at $\mu$=-2, which has a separation of 0.62 nm with the adjacent TE₁₀ mode. Figure S3(a) shows the measured transmission spectra at an on-chip pump power (P_in) of 150 mW. There is no overlap between the two resonances, indicating that the modes will be pumped independently. By sweeping the pump wavelength to 1549.11 nm, just before dropping out of the TE₀₀ resonance, we can successfully access the modulation instability (MI) comb. As shown in Fig. S3(b), the MI comb ranges from 135 to 242 THz, below an octave span. It has a similar envelope like the one obtained from the fundamental mode at $\mu$=0 [see Fig.3(c)-(i)]. Thus the dual-mode scheme could decrease the required power to reach the soliton state.

In this section, we will introduce the other two 1.68-$\mu$m-wide dual-mode MRRs firstly and then present the versatile soliton states. Figures S4(a) and S4(c) show the transmission spectra of the dual-mode in MRR4 and MRR5, which have a separation of 0.067 and 0.019 nm, respectively. The transmitted power curves at 200 mW excluding the pump of MRR4 are plotted in Fig. S4(b), where the visible steps correspond to the formation of two-soliton (0.02-nm-wide) and three-soliton (0.11-nm-wide) states. For MRR5, the extinction ratio of the TE₁₀ resonance is even higher than the TE₀₀ resonance, suggesting a strong mode coupling between the two modes. 3- and 4-SC are realized with MRR5 using the slow pump sweeping technique. There is no single-soliton observation for these two resonators when increasing the TEC temperature, which we attribute to the relatively small mode separations. Nevertheless, deterministic generation of the multi-solitons with pronounced soliton existence window manifests the reliability and practicality of our approach.

Figure S4(d) shows the octave-spanning three-soliton combs generated with MRR1, MRR3, and MRR4 under the pump power of 210, 320, and 200 mW, respectively. Similar three-soliton states are reproducible among different chips. In addition, the three-solitons obtained from MRR3 have stronger intensities due to the narrower coupling gap between resonator and waveguide. Figure S4(e) summarizes the spectra of SCs. The 3-SC spectra originating from MRR4 exceeds an octave-spanning (130-270 THz) with an estimated conversion efficiency of 12.8$\%$. However, because of the smaller mode separation and enhanced mode coupling in MRR5, the 3-SC and 4-SC have a relatively narrow bandwidth (140-230 THz).

Overall, it is easier to access the single-soliton state in the dual-mode resonators (MRR1, MRR2, and MRR3) with a relatively large mode separation (e.g., 10-13 GHz). However, we perceive that the mode separation is a critical but not the only factor that affects the soliton behavior. More intensive work needs to be implemented in the future to sort out the influence of mode coupling on thermal compensation and soliton behavior.