Abstract
Soliton microcombs constitute chipscale optical frequency combs, and have the potential to impact a myriad of applications from frequency synthesis and telecommunications to astronomy. The demonstration of soliton formation via selfinjection locking of the pump laser to the microresonator has significantly relaxed the requirement on the external driving lasers. Yet to date, the nonlinear dynamics of this process has not been fully understood. Here, we develop an original theoretical model of the laser selfinjection locking to a nonlinear microresonator, i.e., nonlinear selfinjection locking, and construct stateoftheart hybrid integrated soliton microcombs with electronically detectable repetition rate of 30 GHz and 35 GHz, consisting of a DFB laser buttcoupled to a silicon nitride microresonator chip. We reveal that the microresonator’s Kerr nonlinearity significantly modifies the laser diode behavior and the locking dynamics, forcing laser emission frequency to be reddetuned. A novel technique to study the soliton formation dynamics as well as the repetition rate evolution in realtime uncover nontrivial features of the soliton selfinjection locking, including soliton generation at both directions of the diode current sweep. Our findings provide the guidelines to build electrically driven integrated microcomb devices that employ full control of the rich dynamics of laser selfinjection locking, key for future deployment of microcombs for system applications.
Introduction
Recent advances in bridging integrated photonics and optical microresonators^{1,2,3,4} have highlighted the technological potential of solitonbased microresonator frequency combs (“soliton microcombs”)^{5,6,7,8,9} in a wide domain of applications, such as coherent communication^{10,11}, ultrafast optical ranging^{12,13}, dualcomb spectroscopy^{14}, astrophysical spectrometer calibration^{15,16}, lownoise microwave synthesis^{17}, and to build integrated frequency synthesizers^{18} and atomic clocks^{19}. Likewise, soliton microcombs also are a testbed for studying the rich nonlinear dynamics, arising from a nonequilibrium driven dissipative nonlinear system, governed by the Lugiato–Lefever equation or extensions thereof, that leads to the formation of “localized dissipative structure”^{8,20,21,22,23,24,25}. To generate soliton microcombs, commonly, the cavity is pumped with a frequency agile, highpower narrowlinewidth, continuouswave laser with an optical isolator to avoid back reflections. The fast tuning of the laser frequency^{26} is applied to access the soliton states, which are affected by the thermal resonator heating. Previously, laser selfinjection locking (SIL) to highQ crystalline microresonators has been used to demonstrate narrowlinewidth lasers^{17,27}, ultralownoise photonic microwave oscillators^{28}, and soliton microcomb generation^{29}, i.e., soliton SIL. Microresonators provide a high level of integration with the semiconductor devices, integrated InPSi_{3}N_{4} hybrid lasers have rapidly become the point of interest for narrowlinewidth onchip lasers^{30,31,32}. Moreover, 100 mW multifrequency Fabry–Perot lasers have recently been employed to demonstrate an electrically driven microcomb^{33}. Another approach was based on a Si_{3}N_{4} microresonator buttcoupled to a semiconductor optical amplifier (SOA) with onchip Vernier filters and heaters for soliton initiation and control^{34}. The integrated soliton microcomb based on the direct pumping of a Si_{3}N_{4} microresonator by a III–V distributed feedback (DFB) laser has been reported^{35,36}. Recent demonstrations of integrated packaging of DFB lasers and ultrahighQ Si_{3}N_{4} microresonators with low repetition rates^{37} made turnkey operation of such devices possible. Low power consumption of integrated microcombs^{33} will allow increasing the efficiency of datacenters, which use an estimated 200 TWh each year and are responsible for 0.3% of overall carbon emissions^{38}.
However, despite the inspiring and promising experimental results, the principles and dynamics of the soliton SIL have not been sufficiently studied yet. Only recently some aspects of the soliton generation effect were investigated^{37}, where static operation was considered, but comprehensive theoretical and experimental investigation is still necessary. The common SIL models consider either laser equations with frequencyindependent feedback^{39,40,41} or linearresonant feedback^{42,43,44}.
Here, we first develop a theoretical model, taking into account nonlinear interactions of the counterpropagating waves in the microresonator, to describe nonlinear SIL, i.e., SIL to a nonlinear microresonator. Using this model, we show that the principles of the soliton generation in the selfinjection locked devices differ considerably from the conventional soliton generation techniques. We find that the emission frequency of the laser locked to the nonlinear microresonator is strongly reddetuned and located inside the soliton existence range. For experimental verification, we develop a technique to experimentally characterize the SIL dynamics and study it in a hybridintegrated soliton microcomb device with 30 GHz repetition rate, amenable to the direct electronic detection, using an InGaAsP DFB laser selfinjection locked to a highqualityfactor (Q_{0} > 10^{7}) integrated Si_{3}N_{4} microresonator. We demonstrate the presence of the nontrivial dynamics upon diode current sweep, predicted by the theoretical model, and perform the beatnote spectroscopy, i.e., study of the soliton repetition rate evolution under SIL.
Results
Principle of laser SIL
First, we introduce the general principles of the laser SIL to the microresonator (Fig. 1a, b) and clarify definitions of basic terms (Fig. 1c). The generation frequency of the freerunning DFB diode is determined by its laser cavity (LC) resonant frequency ω_{LC} and can be tuned by varying the diode injection current I_{inj} exhibiting practically linear dependence. When ω_{LC} is tuned into a highQ resonance of the Si_{3}N_{4} microresonator with frequency ω_{0}, laser SIL can happen. In that case, ω_{eff} is the actual or effective laser emission frequency, that differs from the ω_{LC} as the optical feedback from the microresonator affects the laser dynamics.
It is convenient to introduce the normalized laser cavity to microresonator detuning ξ = 2(ω_{0} − ω_{LC})/κ and the actual effective detuning ζ = 2(ω_{0} − ω_{eff})/κ, where ω_{0} is the frequency of the microresonator resonance and κ is its loaded linewidth. Note that ξ practically linearly depends on the injection current I_{inj}. The ζ is the actual detuning parameter that determines the dynamics of the nonlinear processes in microresonator^{45}. Here we define the “tuning curve” as the dependence of ω_{eff} on the injection current I_{inj}, or equivalently, the dependence of ζ on ξ (Fig. 1c).
When the laser frequency ω_{LC} is far detuned from the resonance ω_{0} (∣ξ∣ ≫ 0), the tuning curve first follows the line ζ = ξ (Fig. 1c) when the injection current I_{inj} changes. When ω_{LC} is tuned into ω_{0}, i.e., ξ → 0, the laser frequency becomes locked to the resonance due to the Rayleighbackscatteringinduced SIL, so that ζ ≈ 0 despite the variations of ξ within the locking range. The stabilization coefficient (inverse slope at ξ = 0) and the locking range Δω_{lock} are determined by the amplitude and phase of the backscattered light^{44}. When ξ increases further and finally moves out of the locking range, the laser becomes freerunning again, such that ζ = ξ. Note also, that the locked state region for continuous oneway scanning may not fully coincide with the locking range (see Fig. 1c) and even be different for different scan directions.
Such tuning curves (Fig. 1c and gray curve in Fig. 1d) are wellstudied for the case of the linear microresonator (or for small pump powers)^{44,46}. However, analyzing experimental results on the soliton formation in the linear SIL regime^{29,33}, we found that such linear model can’t predict soliton generation.
SIL to a nonlinear microresonator
Previous works^{23,45} have shown that soliton generation occurs in a certain range of the normalized pump detunings ζ, with the lower boundary being above the bistability criterion and the upper boundary being the soliton existence criterion:
where \(f=\sqrt{8{\omega }_{0}c{n}_{2}\eta {P}_{{\rm{in}}}/({\kappa }^{2}{n}^{2}{V}_{{\rm{eff}}})}\) is the normalized pump amplitude, ω_{0} is the resonance frequency, c is the speed of light, n_{2} is the microresonator nonlinear index, P_{in} is the input pump power, n is the refractive index of the microresonator mode, V_{eff} is the effective mode volume, κ is the loaded resonance linewidth, and η is the coupling efficiency (η = 1/2 for critical coupling). Note, however, that the lower boundary (bistability) includes also a region of breather existence for high pump power^{47}, so actual soliton region starts at a bit higher detunings. The linear SIL model^{44} predicts that, in the SIL regime, the attainable detuning range of \(\zeta \in \left[0.7;0.7\right]\) in the locked state does not overlap with the soliton existence range despite the sufficient pump power.
We attribute this contradiction to the absence of the microresonator Kerr nonlinearity in the linear SIL model. The modified nonlinear SIL model including the Kerr nonlinearity^{48} is presented as follows.
Consider the microresonator coupled mode equations^{45} with backscattering^{49} for forward and backward (or clockwise and counterclockwise propagating) mode amplitudes a_{μ} and b_{μ}, which is analogous to the linear SIL model^{44} with additional nonlinear terms:
where Γ is the normalized coupling rate between forward and backward modes (mode splitting in units of mode linewidth), α_{x} is a coefficient derived from mode overlap integrals and ζ_{μ} = 2(ω_{μ} − μD_{1} − ω_{eff})/κ is the normalized detuning between the laser emission frequency ω_{eff} and the μth cold microresonator resonance ω_{μ} on the FSRgrid, with μ = 0 being the pumped mode and D_{1}/2π is the microresonator free spectral range (FSR). For numerical estimations we use α_{x} = 1 as for the modes with the same polarization. Equation (2) provides a nonlinear resonance curve and the soliton solution^{45,49}. For analysis of the SIL effect we combine Eq. (2) with the standard laser rate equations similar to the Lang–Kobayashi equations^{39}, but with resonant feedback^{44}. The pumped mode corresponding to μ = 0 is of main interest. We search for the stationary solution:
where we define a = a_{0}, b = b_{0} and ζ = ζ_{0} for simplicity. These equations define the complex reflection coefficient of the WGM microresonator which is used for SIL theory. To solve Eq. (3) and make resemblance to the linear case, we introduce the nonlinear detuning shift δζ_{nl}
and nonlinear coupling shift δΓ_{nl}
We further transform \(\bar{\zeta }=\zeta \delta {\zeta }_{{\rm{nl}}}\), \({\bar{\Gamma }}^{2}=\delta {\Gamma }_{{\rm{nl}}}^{2}+{\Gamma }^{2}\), in order to achieve Eq. (3) in the same form as in the linear SIL model^{44,50}. After redefinition \(\bar{\xi }=\xi \delta {\zeta }_{{\rm{nl}}}\), the nonlinear tuning curve in the new coordinates \(\bar{\xi }\)\(\bar{\zeta }\) becomes:
where \({K}_{0}=8\eta \Gamma {\kappa }_{\text{do}}\sqrt{1+{\alpha }_{g}^{2}}/(\kappa {R}_{\text{o}})\) is the SIL coefficient and \(\bar{\psi }={\psi }_{0}\kappa {\tau }_{s}\zeta /2\) is the selfinjection phase, κ_{do} is the laser diode output mirror coupling rate, α_{g} is the Henry factor^{51} and R_{o} is the amplitude reflection coefficient of the laser output facet. We also note that the laser cavity resonant frequency ω_{LC}, as well as ξ, are also assumed to include the Henry factor in its definition. The κτ_{s}/2 is usually considered to be small, i.e., κτ_{s}/2 ≪ 1, so the locking phase \(\bar{\psi }\approx {\psi }_{0}={\omega }_{0}{\tau }_{s}\arctan {\alpha }_{g}3\pi /2\) depends on both the resonance frequency ω_{0} and the roundtrip time τ_{s} from the laser output facet to the microresonator and back. The SIL coefficient K_{0} is analogous to the feedback parameter C used in the theory of the simple mirror feedback^{40,41}, where the SIL is achieved with the frequencyindependent reflector forming an additional FabryPerot cavity. However, in the resonant feedback setup the SIL coefficient does not depend on the lasertoreflector distance, depending on the parameters of the reflector instead. Though the system has qualitatively similar regimes as the simple one^{52}, their ranges and thresholds are different^{42,44,46}. The value of K_{0} > 4 is required for the pronounced locking with sharp transition, naturally becoming a locking criterion. For highQ microresonators this value can be no less than several hundred. We also note that in the linear regime (or in nonlinearly shifted coordinates \(\bar{\xi }\), \(\bar{\zeta }\)) the stabilization coefficient of the setup is close to K_{0}, full locking range is close to 0.65K_{0} × κ/2. The nonlinear detuning and coupling can be expressed as
Equations (6)–(8) can be solved numerically and plotted in \(\zeta =\bar{\zeta }+\delta {\zeta }_{{\rm{nl}}}\), \(\xi =\bar{\xi }+\delta {\zeta }_{{\rm{nl}}}\) coordinates^{48}. We observe that calculated tuning curve in the nonlinear case, when Kerr nonlinearity is present, differs drastically from the tuning curve predicted by the linear model (cf. red and gray lines in Fig. 1d). Also, we can see from Eq. (7) that the nonlinear detuning shift is positive, and allows for larger detuning ζ (proportional to the pump power). The proposed nonlinear SIL model is valid for both anomalous and normal group velocity dispersions.
We show the distinctions between conventional generation of the dissipative Kerr solitons and selfinjection locked soliton excitation in Fig. 2. Figure 2a, b shows the conventional case where the laser pumps the microresonator with an optical isolator between the laser and the microresonator, preventing SIL. In Fig. 2a, the solid black line corresponds to the linear tuning curve (ζ = ξ) of a freerunning laser. In Fig. 2b, the thick solid black curve corresponds to the solution of Eq. (2) in the ξ frame, which provides soliton solutions^{49}. Horizontal dashed green lines are the boundaries of the soliton existence range in the ζ frame, i.e., Eq. (1), highlighted also with the green area.
Next, we consider tuning curves corresponding to the nonlinear SIL, described by Eq. (6). They are plotted in Fig. 2c, e in the ζ–ξ frame with dashed red lines. Due to the multistability of the tuning curves, forward and backward laser scans within the same diode current range result in different ranges of the effective detuning ζ and different tuning curves (thick solid lines). Figure 2c, e shows the attainable values of detunings ζ, and Fig. 2d, f shows the intracavity power ∣a(ξ)∣^{2} for the forward and backward scans (thick dashed lines). We studied a set of the realworld parameters and found that the following key conclusions can be done: first, the effective detuning ζ predominantly locks to the reddetuned region, where the soliton microcomb formation is possible (Fig. 1e shows the union of the SIL regions for different locking phases together with the soliton existence region). Second, in the SIL regime soliton generation may be observed for both directions of the current sweep (see Fig. 2c, e) that is impossible for the freerunning laser. Also, it is possible to obtain larger values of the detuning ζ in the SIL regime using backward tuning (cf. Fig. 2c, e). At the same time, the locked state width can be shorter for the backward tuning than that for the forward tuning. Third, while decreasing the diode current (i.e., backward tuning, the freerunning laser frequency rising), the detuning ζ can grow (Fig. 2e) that is counterintuitive. Moreover, such nonmonotonic behavior of ζ may take place in the soliton existence domain and may affect soliton dynamics. As it was shown in^{53} decrease of the detuning value in the soliton regime may lead to the switching between different soliton states.
More study of the nonlinear tuning curves dependence on the locking phase ψ_{0}, pump power, and the modecoupling parameter Γ is presented in Supplementary Note 2.
Dynamics of soliton SIL
For experimental verification, the integrated soliton microcomb device consisting of a semiconductor laser diode and a highQ Si_{3}N_{4} microresonator chip is developed. In our experiment, we use a commercial DFB laser diode of 120 kHz linewidth and 120 mW output power, which is directly butt coupled to a Si_{3}N_{4} chip (Fig. 3a) without using an optical isolator (see “Butt coupling” in Methods).
The Si_{3}N_{4} chip is fabricated using the photonic Damascene reflow process^{54,55} and features intrinsic quality factor Q_{0} exceeds 10 × 10^{6} (see “Silicon nitride chip information” in Methods). We study two different photonic chips, containing microresonators with FSR of 30.6 GHz and 35.4 GHz. Tuning of the injection current makes the laser diode selfinjection locked to the microresonator. The Lorentzian linewidth of the SIL laser is 1.1 kHz. Laser phase noise in the freerunning and SIL regimes presented in Fig. 3c demonstrates the laser linewidth reduction by more than 100 times (see Supplementary Note 1 for details). At some particular currents soliton states are generated (see “Comb generation in the SIL regime” in Methods). Figure 3b shows the single soliton spectrum with 30.6 GHz repetition rate. Soliton repetition beatnote signal^{56,57,58,59,60} and corresponding phase noise are shown in Fig. 3d (see Supplementary Note 5 for details). The phase noise for the 35.4 GHz soliton is shown in Fig. 3d as well. This soliton state provides ultralow noise beatnote signal of −96 dBc at 10 kHz frequency offset, that is comparable to some breadboard implementation without any complicated feedback schemes.
Having access to the stable operation of these soliton states, we prove experimentally (in addition to their ultralow noise RF beatnote signal (Fig. 3d, inset), optical spectrum (Fig. 3b) and zero background noise) that such optical spectra correspond to the ultrashort pulses representing bright temporal solitons. We perform a frequencyresolved optical gating (FROG) experiment^{45}. This corresponds to a secondharmonic generation autocorrelation experiment in which the frequencydoubled light is resolved in spectral domain (see Supplementary Note 6 for details). Reconstructed optical field confirms that we observe temporal solitons with the width less than 1 picosecond. Therefore, SIL is a reliable^{33,37} platform to substitute bulky narrowlinewidth lasers for pumping microresonators allowing the generation of ultrashort pulses and providing ultralownoise RF spectral characteristics. But, as we stated above, this platform has much more complicated principles of operation.
To clarify these principles and corroborate our findings from the theoretical model of nonlinear SIL, we develop a technique to experimentally investigate the soliton dynamics via laser SIL, based on a spectrogram measurement of the beatnote signal between the SIL laser and the reference laser. This approach allows the experimental characterization of the nonlinear tuning curve ζ(ξ), which can be compared to the theoretical model.
First, we measure the microresonator transmission trace by applying 30 Hz triangle diode current modulation from 372 to 392 mA, such that the laser scans over a nonlinear microresonator resonance. As shown in Fig. 4a, the resonance shape is prominently different from the typical triangle shape with soliton steps, characteristic for the conventional generation method, which uses a laser with an isolator, and tunes the laser from the bluedetuned to the reddetuned state (i.e., forward tuning)^{45}. However, in the case of the nonlinear SIL, soliton states are observed for both tuning directions, as illustrated below (see Fig. 4) and as it was predicted by the developed model.
Then we measure the tuning curve inside this transmission trace. The reference laser’s frequency is set higher than the freerunning DFB laser frequency, such that the heterodyne beatnote signal is observed near 15 GHz. The laser diode current is swept rather slow at 10 mHz rate, such that the laser scans across the resonance in both directions. The spectrogram data in the range of 0 to 25 GHz is collected by ESA (Fig. 4b), and the soliton beatnote signal is measured at 30.696 GHz (Fig. 4c).
Initially, the DFB laser is freerunning (Frame I). The diode current and ξ is decreasing, the laser is tuned into a nonlinear microresonator resonance (Frame II). Further decreasing of the diode current locks ζ to the microresonator inside the soliton existence range (Frame III). In this case, two effects can be observed: the appearance of the 20 GHz beatnote signal (Frame IV) between the reference laser and the first comb line, and the narrowlinewidth and powerful signal of the 30.6 GHz soliton repetition rate, as shown in Fig. 4c (Frame X). The spectrogram regions IV and VII are locally enhanced for better visual representation. Further reducing of the diode current can lead to the switching of (multi)soliton states. Such switching is caused by the decrease of the effective pump detuning ζ^{53}, that was predicted by the nonlinear SIL model and is observed in the experiment (see Fig. 4d).
In our experiment, the DFB laser remains locked when the current scan direction reverses from backward to forward (Frame V). This region V, which represents the multifrequency regime of the DFB (see Supplementary Note 1 for details), is truncated for better visual representation.
The DFB laser then changes its regime to a singlefrequency locked emission again (Frame VI), and generates a chaotic Kerr comb with a wide beatnote signal (Frame XI). As the effective detuning ζ rises with increasing ξ, switching between different soliton states and the first comb line beatnote (Frame VII) is observed, as well as the appearance of the breather soliton states (Frame VIII). Note that the soliton repetition rate reduces with increasing ζ (Frame X), and vice versa (Frame XII). Further diode current tuning causes the DFB laser to jump out of the SIL regime, its frequency returns to the freerunning regime, and thus ζ = ξ again (Frame IX).
Based on the spectrogram data shown in Fig. 4b, the nonlinear tuning curve ζ(ξ) is easily extracted, as shown in Fig. 4d, e. These experimentally measured nonlinear tuning curves are fitted using Eq. (6)–(8). In our experiments DFB diode provides 7 < f < 15. The fitted optical phase is ψ_{0} = −0.27π, and the fitted normalized pump is f = 13.1. One may see nearly excellent agreement of theoretical and experimental results. For example, transitions to and from the locked state (Frames II and IX) is perfectly fitted with loops of the theoretical curves in Fig. 4d, e.
Thus, we can conclude that conducted experiments confirm the presence of the nontrivial soliton formation dynamics predicted by the theoretical model. We demonstrate several predicted effects: first, soliton generation at both directions of the diode current sweep; second, switching of soliton states when the effective detuning ζ is decreasing^{53}. Third, we observe the effect of repetition rate decrease with the growth of the effective detuning ζ (reported in ref. ^{57,61}). Measured nonlinear tuning curves are in good agreement with theoretical curves.
Discussion
In summary, we have studied theoretically and experimentally the effect of the soliton generation by the diode laser selfinjection locked to an integrated microresonator. We have developed a theoretical model to describe SIL to a nonlinear microresonator (“nonlinear SIL”) and have shown that the complicated dependence of the emission frequency on the injection current leads to the nontrivial dynamics of nonlinear processes. It has been shown that the effective emission frequency of the selfinjection locked laser is reddetuned relative to the microresonator resonance and is located inside the soliton existence region for most combinations of parameters. We have checked theoretical results experimentally and we have demonstrated singlesoliton generation enabled by a DFB laser selfinjection locked to an integrated Si_{3}N_{4} microresonator of 30 GHz and 35 GHz FSRs. Also, it has been predicted that in contrast with the freerunning laser, in the SIL regime soliton generation is possible for both forward and backward scans of the laser diode injection current. We have developed a spectrogram technique that allows to measure accessed soliton detuning range, and to observe features of nonlinear SIL. Obtained experimental results are in good agreement with theoretical predictions.
Some deviations from the predicted behavior can be attributed to the nonlinear generation of sidebands not included in our theory, which depletes the power in the pumped mode and changes the nonlinear detuning shift (7). Also, the pump power slightly depends on the injection current. Nevertheless, many important theoryderived conclusions have been observed experimentally. Some more important features, predicted by the theory, are yet to study. For example, there are regions of tuning curves with dζ/dξ = 0, where noise characteristics of the stabilized laser may be significantly improved.
The radiophotonic signal of the soliton repetition rate provides better phase fluctuations at high offsets (>10 kHz) than the commercial microwave analog signal generator (Keisight N5183B). Note that different soliton states correspond to different phase fluctuations, i.e., optimization of the SIL soliton state may lead to the further decreasing of the phase noise.
Therefore, the soliton SIL provides, first, laser diode stabilization, second, microcomb generation, third, ultralow noise photonic microwave generation. The main problem of this technique is the limitation of achievable effective detunings: a singlesoliton state with a large detuning and broad bandwidth may be hard to obtain in the SIL regime. Further careful parameter optimization is needed for the comb bandwidth enhancement. One possible solution may be based on the fact that backscattering plays a major role and different schemes with increased backscattering may extend the range of effective detunings in the locked state.
Another important question is the increase in the total comb power (see Supplementary Note 7). First, the optimization of the laser diode mode and the photonic chip bus waveguide mode matching is an essential task to decrease the total insertion loss of 7 dB and increase the pump power. Optimization of the parameters of the photonic chip waveguide, particularly speaking, the secondorder dispersion D_{2} and the comb coupling rate to the bus waveguide will lead to the better generation and extraction of the comb lines out of the microresonator^{10}. Moreover, bright dissipative Kerr solitons are wellstudied structures exhibiting fundamental limitation of the pumptocomb conversion efficiency. Utilization of the dark soliton pulses, which formation is possible in microresonators with normal GVD, provides Kerr microcombs with high power per each line^{62,63}. Moreover, our recent numerical studies suggest that the SIL allows solitonic pulse generation in the normal GVD regime without any additional efforts like mode interaction or pump modulation^{64}.
Our results provide insight into the soliton formation dynamics via laser SIL, which has received wide interest recently from the fields of integrated photonics, microresonators, and frequency metrology. Also, they may be used for the determination of the optimal regimes of the soliton generation and for the efficiency enhancement of integrated microcomb devices. We believe, that our findings, in combination with recent demonstrations of industrial packaging of DFB lasers and Si_{3}N_{4} waveguides, pave the way for highly compact microcomb devices operated at microwave repetition rates, built on commercially available CMOScompatible components and amenable to integration. This device is a promising candidate for highvolume applications in datacenters, as scientific instrumentation, and even as wearable technology in healthcare. This result is significant for laser systems with strong optical feedback (such as lownoise IIIV/Si hybrid lasers and modelocked lasers), oscillator synchronization, and other laser systems beyond integrated microcombs. A related example microrings are made of quantum cascade active media^{65}. Therefore, our findings are relevant not only for integrated photonics community but for a wide range of specialists.
Methods
Silicon nitride chip information
The Si_{3}N_{4} integrated microresonator chips were fabricated using the photonic Damascene process^{54,66}. The pumped microresonator resonance is measured and fitted including backscattering^{50} to obtain the intrinsic loss κ_{0}/2π = 20.7 MHz, the external coupling rate κ_{ex}/2π = 48.6 MHz, and the backwardcoupling rate γ/2π = 11.8 MHz (mode splitting). These correspond to the full resonance linewidth κ/2π = κ_{0}/2π + κ_{ex}/2π = 69.3 MHz, the pump coupling efficiency η = κ_{ex}/κ = 0.70, the normalized modecoupling parameter Γ = γ/κ = 0.17, and the amplitude resonant reflection coefficient from the passive microresonator r ≈ 2ηΓ/(1 + Γ^{2}) = 0.23.
Butt coupling
We do not use any optical wire bonding techniques in our work. The DFB laser diode and the Si_{3}N_{4} chip are directly butt coupled and are mounted on precise optomechanical stages. The distance between the diode and the chip can be varied with an accuracy better than 100 nm, thus enabling the control of the accumulated optical phase from the Si_{3}N_{4} microresonator to the diode (i.e., the locking phase). The output light from the Si_{3}N_{4} chip is collected using a lensed fiber. The total insertion loss, i.e., the output free space power of the freerunning laser diode divided by the collected power in the lensed fiber, is 7 dB. Note that matching the polarization of the DFB laser radiation and the polarization of the microcavity highQ modes is critically important to achieve maximum pump power either via rotation of laser diode or by “polarization engineering” of highQ modes.
Experimental setup
The DFB laser diode’ temperature and injection current are controlled by an SRS LD501 controller and an external function generator (Tektronics AFG3102C). The output optical signal of the soliton microcomb is divided by optical fiber couplers and sent to an optical spectrum analyzer (Yokogawa AQ6370D), a fast photodetector (NewFocus 1014), an oscilloscope, and an electrical signal analyzer (ESA Rohde&Schwarz FSW26). The heterodyne beatnote measurement of various comb lines is implemented with a narrowlinewidth reference laser (IDPhotonics DX2 or TOptica CTL). A passive doublebalanced MMIC radiofrequency (RF) mixer (Marki MM11140H) is utilized to downconvert and study the RF signal above 26 GHz in combination with a local oscillator (Keysight N5183B). The phase noises of 35.4 GHz repetition rate signal are measured directly by highRF electronics, without RF mixer.
Comb generation in SIL regime
Besides meeting the soliton power budget, a key requirement for soliton generation in the SIL regime is to reach sufficient detuning ζ = 2(ω_{0} − ω_{eff})/κ. When the laser is tuned into resonance from the reddetuned side, SIL can occur so that the laser frequency ω_{eff} becomes different from ω_{LC} and is close to ω_{0}. In the conventional case where soliton initiation and switching are achieved using frequency tunable lasers with isolators, the soliton switching, e.g., from a multisoliton state to a singlesoliton state, can lead to the intracavity power drop which causes the resonance frequency ω_{0} shift due to the thermal effect, and ultimately the annihilation of solitons. However, in the case of laser SIL, the optical feedback via Rayleigh backscattering is much faster (instantaneous) than the thermal relaxation time (at millisecond order), therefore the laser frequency can follow the resonance shift instantaneously such that the soliton state is maintained. The slope of the tuning curve dζ/dξ in the SIL regime allows to control the effective detuning ζ by varying ξ, realized by increasing the laser injection current for forward tuning^{45}, or decreasing the current for backward tuning^{53}. However, note that the entire soliton existence range may not be fully accessible at certain locking phases ψ_{0} and locking coefficient K_{0}. Therefore, a single soliton state with a large detuning and broad bandwidth may be hard to obtain in the SIL regime and further careful parameter optimization is needed. In our experiments, the subsequent switching from chaotic combs to breather solitons and multisolitons in forward and backward scans is observed (see Supplementary Note 4).
Data availability
The code and data used to produce the plots within this paper are available at 10.5281/zenodo.4079515. All other data used in this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
The authors thank Miles H. Anderson, Joseph Briggs, and Vitaly V. Vassiliev for the fruitful discussion. The Si_{3}N_{4} microresonator samples were fabricated in the EPFL center of MicroNanoTechnology (CMi). DFB diode characterization was made at VNIIOFI center of the shared use (VNIIOFI CSU). This work was supported by the Russian Science Foundation (grant 171201413Π), Contract HR001115C0055 (DODOS) from the Defense Advanced Research Projects Agency (DARPA), Microsystems Technology Office (MTO), by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF under Award No. FA95501910250, and by the Swiss National Science Foundation under grant agreement No. 176563 (BRIDGE) and NCCRQSIT grant agreement No. 51NF40185902.
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A.S.V., G.V.L., N.Y.D. conducted the experiment. W.W. conducted the FROG experiment. N.M.K.,V.E.L. developed a theoretical model and performed numerical simulations. J.L. designed and fabricated the Si_{3}N_{4} chip devices. All authors analyzed the data and prepared the manuscript. I.A.B. and T.J.K initiated the collaboration and supervised the project.
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Voloshin, A.S., Kondratiev, N.M., Lihachev, G.V. et al. Dynamics of soliton selfinjection locking in optical microresonators. Nat Commun 12, 235 (2021). https://doi.org/10.1038/s4146702020196y
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DOI: https://doi.org/10.1038/s4146702020196y
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