Dynamics of soliton self-injection locking in optical microresonators

Soliton microcombs constitute chip-scale 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 self-injection 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 self-injection locking to a nonlinear microresonator, i.e., nonlinear self-injection locking, and construct state-of-the-art hybrid integrated soliton microcombs with electronically detectable repetition rate of 30 GHz and 35 GHz, consisting of a DFB laser butt-coupled 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 red-detuned. A novel technique to study the soliton formation dynamics as well as the repetition rate evolution in real-time uncover non-trivial features of the soliton self-injection 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 self-injection locking, key for future deployment of microcombs for system applications.

thinnest. It is completely hidden by the thick light red dashes indicating the locking region in that region. Should the dashes extend beyond the locking regime? Probably not . Similar, but not quite so pronounced in Fig. 1d. 4. Fig. 4d is quite unclear and not explained well. The thin grey lines have no legend. I do not think that the two straight pieces of thick solid black line is the theory as it seems to be indicated in the legend. Similarly, the experimental data need to be better identified.
Reviewer #3 (Remarks to the Author): The authors present theoretical investigations of self-injection locking of a DFB laser to a silicon nitride micresonator for Kerr comb generation and soliton formation. They investigate the soliton existence region for forward and backward tuning of the laser-cavity detuning. The paper is generally well written and would be of specific interest to the Kerr comb community. However, there has been many prior demonstrations of injection locking for Kerr comb generation and soliton formation in recent years, including the work done by the some of the authors of this manuscript, including Pavlov, et al., "Narrow -linewidth lasing and soliton Kerr microcombs with ordinary laser diodes," Nat. Photon. 12, 694 (2018); Raja, et al., "Electrically pumped photonic integrated soliton microcomb," Nat. Commun. 10, 680 (2019); Stern, et al., "Battery-operated integrated frequency comb generator," Nature 562, 401 (2018); Raja, et al., "Packaged photonic chip -based soliton microcomb using an ultralow-noise laser," arXiv:1906.03194; Raja, et al., "Chip-based soliton microcomb module using a hybrid semiconductor laser," Opt. Express 28, 2714 (2020); Lesko, et al., "Fully phase-stabilized 1 GHz turnkey frequency comb at 1.56 µm," arXiv:2005.03088; She n, et al., "Integrated turnkey soliton microcombs operated at CMOS frequencies," arXiv:1911.02636. While the experimental investigation of the noise performance is a nice addition, the overall performance of the comb source is similar to the previous demonstrations by the authors and do not see any notable advance. Especially with many prior experimental work being demonstrated using the self-injection locked scheme, the theoretical work will be more suitable in a specialized journal rather than a journal w ith broad readership such as Nature Communications. Specific comments are below .
First of all, we would like to express our deep appreciation to the three reviewers, for the time they took to carefully read our manuscript and the concerns that help us improve the quality of our manuscript. We appreciate that the reviewers read it carefully, made suggestions and gave overall positive evaluations:  "While there is room for improvement and providing more details, this is a rather good result with importance scientifically that may improve our understanding of soliton dynamics in nonlinear chip-based optical resonator" (Reviewer #1)  "The investigation and results seem to be also sound and well done." (Reviewer #2) In the following, we will respond in great detail (in black) to the reviewers' questions (in blue), point-by-point, as well as the action taken (in red):

Executive Summary:
In the revised manuscript, according to the Reviewer #1's request, we have added new experimental data to the SIthe FROG measurement confirming that the soliton self-injection locking provides generation of pulses in the temporal domain. Also, we measured the ultra-low phase noise of the photonic microwave signal generated by the soliton. We emphasize that the key novelty and scientific significance of our work is the comprehensive study of the dynamics of soliton self-injection locking corroborated by a novel quantitative theoretical model. We perform soliton beatnote spectroscopy and demonstrate real-time switching in the self-injection locking regime between different soliton states which have not been studied before. Our results could be applied to laser systems beyond Kerr microcombs, such as photonic circuits containing both nonlinear photonic elements and active media and microresonators made of active materials.

Reviewer #1
This paper reports a theoretical and experimental investigation of soliton comb dynamics in a nonlinear integrated Si3N4 microresonator chip pumped by a self-injection locked diode laser. It is shown in particular that the effective emission frequency of the self-injection locked laser is reddetuned relative to the microresonator resonance due to the microresonator's Kerr nonlinearity. It is further shown that the self-and cross-phase modulation of the forward and backward light circulating in the microresonator enables soliton formation with accessible large detunings that are unreachable according to linear self-injection locking theory.
The paper is well written and provides high quality experimental results supported by simulations, although somewhat unclear and lacking in important details. The abstract is relevant, physically speaking. While there is room for improvement and providing more details, this is a rather good result with importance scientifically that may improve our understanding of soliton dynamics in the nonlinear chip-based optical resonator. However, I think the paper is not amenable for Nature Communications in its current form and with substantiate claims and thorough completion of all items on the checklist. My main reservation about the paper is that, despite the claim to have generated soliton, there is no plot of temporal soliton pulse including shape and width measurement, nor even any autocorrelation or FROG trace, which would show clear evidence of soliton related to the comb spectrum shown in Fig. 3b. This is a serious omission and in my view makes the paper less interesting and significant.

Our reply:
First of all, we thank the reviewer for the positive comments. We also appreciate that the reviewer shares his concern with us on the significance of our study. We clarify that the main focus of our work is to describe and characterize the dynamics of soliton formation via laser self-injection locking in Kerr nonlinear optical microresonators.
The first FROG measurements of dissipative Kerr solitons in optical microresonators was presented in Ref. [R1, R2]. It has been shown that the FROG measurement in microresonators distinguished a pulsed waveform from an non-pulsed, periodic, modulated waveform. However, FROG is not the only criterion in determining if the comb state is indeed a soliton pulse. It is now widely accepted in the microcomb community that two measurements [R3] could suffice to prove the existence of soliton pulses in microresonators: (1) the sech 2 profile of the optical spectrum (in the case of the single soliton state), and (2) low phase noise of the beatnote of the soliton repetition rate. Self-injection locked solitons meet both criteria. Particularly, in our original manuscript, we have shown the low-phase-noise beatnote signal of the soliton repetition rate, which is convincing evidence of soliton existence and demonstrates the mode-locked regime. For this reason, we did not include the FROG measurement in our initial submission. To respond to the reviewer's request, and to prove a pulsed waveform, we have performed additional experiment, and have measured the temporal profile of soliton pulses with FROG. The photonic chip-based microresonator used here has an FSR of 35.397 GHz. The data is presented in Fig. F1.
Before the FROG setup, the optical spectrum of self-injection locked soliton is filtered by a fiber Bragg grating (FBG) for pump suppression and amplified to ~100 mW by using two EDFAs. The FBG suppressed not only the central comb line but also neighbouring lines because the FBG's bandwidth (~100GHz) is larger than the comb lines spacing (35 GHz). Also, the optical amplifiers used here have a gain profile which is not flat. This causes the distortion of the soliton's optical spectrum. The reconstructed intracavity power from the FROG trace proves that the spectrum corresponds to a short pulse train. We confirmed the pulsed waveform of the soliton microcomb. A detailed study of soliton parameters is not the key point of our work, so we did not analyze the FROG trace fine structure.
Moreover, we would like to mention that our theoretical model of the self-injection locking is valid for microresonators with normal group velocity dispersion (GVD). In these microresonators, bright solitons do not exist, while dark pulses, i.e. platicons, can be observed [R4]. The principles of laser self-injection locking remain the same in this case of normal-GVD microresonators. Our approach was successfully used for the normal-GVD microresonators, and the preliminary result has been presented at CLEO US 2020 [R5]. The microcomb was, first, filtered by using fiber Bragg grating to suppress the central line and amplified to ~100 mW by using cascaded optical amplifiers (with non-flat gain). That is why the FROG trace does not correspond to the pure single soliton temporal profile, but still exhibits the optical pulse with a width of less than 1 picosecond. The reconstructed FROG trace is on the right panel.

Action taken:
• We added the Supplementary Note 6 to the Supplementary Materials, where we explain how one can prove the soliton presence in microresonators, demonstrate original FROG trace, reconstructed FROG trace and pulse waveform. • We added the following text in the abstract (marked in red) on the 1st page in the revised manuscript: "….locking theories. We construct state-of-the-art integrated soliton microcombs with an electronically detectable repetition rate of 30 GHz and 35 GHz with as low phase noises as -96 dBc at 10 kHz frequency offset. These devices, consisting of a DFB laser self-injectionlocked to a Si 3 N 4 microresonator chip, allow us to implement a novel experimental technique and study the soliton formation dynamics as well as the repetition rate evolution in real-time. Conducted experiments..." • We added the following text to Section IV "Self-injection locking to a nonlinear microresonator" (page 4): "Having access to the stable operation of these soliton states, we prove experimentally (in addition to their ultra-low noise RF beatnote signal (Fig. 3d, inset), optical spectrum (Fig. 3b) and zero background noise) that such optical spectra correspond to the ultra-short pulses representing bright temporal solitons. We perform a frequency-resolved optical gating (FROG) experiment [45]. This corresponds to a second-harmonic generation autocorrelation experiment in which the frequency doubled light is resolved in spectral domain (see Supplementary Note 6 for details). Reconstructed optical field confirms that we study temporal solitons with width less 1 picosecond. Therefore, self-injection locking is a reliable [33,37] platform to substitute bulky narrow-linewidth lasers for pumping microresonators allowing the generation of ultra-short pulses and providing ultra-low noise RF spectral characteristics. But, as we stated above, this platform has much more complicated principles of operation."

Reviewer #2
The manuscript provides a detailed analysis of soliton self-injection locking, i.e. that the feedback of the microresonator in the pumping laser will affect the laser frequency, which in turn will influence the nonlinear dynamics of the microresonator. This can lead to the stabilization of a frequency comb relaxing needs on other resource demanding and complex active and passive stabilization schemes, or to a further destabilization of the frequency comb, if things are done wrongly. The problem is demanding as it evolves the coupled dynamics of two systems displaying already complex nonlinear dynamics. Due this challenge and the potential impact on the communities in metrology, integrated photonics and nonlinear dynamics I welcome the publication in a high profile journal. The investigation and results seem to be also sound and well done, although I have a few questions on the model. Most figures (in particular Fig. 2) are also done nicely to illustrate the complex relationships.
1. Section II: In the first paragraph omega_0 is introduced without a proper definition. In the second paragraph omega_0 is introduced as the "loaded" cavity resonance. I do not understand this phrasing as the model is linear. It should be explained explicitly, what is included in omega_0 and what not. In addition, no model is presented on the locking behaviour of the lasers diode. This should include the Henry factor, introduced only much later. I assume the corresponding equations are already used here. This needs to be rewritten.

Our reply:
The word "loaded" in the sentence was intended to refer only to the cavity linewidth, not the frequency. We meant that " 0 is the frequency of the microresonator resonance and is its loaded linewidth". As 0 is the frequency of the microresonator, the Henry factor is not related to it. However, it is, of course, included in the laser cavity frequency as it is by definition a frequency of the freerunning laser.
In this section we discuss only the general principle of the self-injection locking. In the following theoretical section we provide more clarifications that are cold microresonator resonances. We also indicate where the reader can find the locking behaviour of the laser diode. The derivation of the tuning curve equations is shown and explained in the text. We suppose that the detailed description of the methods of laser and microresonator eigenfrequency calculation will greatly complicate the article.

Action taken:
• We changed the wording to avoid the confusion: «...where 0 is the frequency of the microresonator resonance and is its loaded linewidth.» • We add the following sentence to the Section II "Principle of laser self-injection locking": «We also note that the laser cavity resonant frequency as well as are also assumed to include the Henry factor in its definition.» • To clarify the laser model definition, we add the following sentence: "For analysis of the SIL effect we combine the eqs.
(2) with the standard laser rate equations similar to the Lang-Kobayashi equations [39], but with resonant feedback [44]." The latter also classifies feedback regimes. Is the notion of regime V in the supplementary material linked to the classification or is it an own one? The equation for the self-injection coefficient after Eq. (6) would probably justify an original citation.

Our reply:
We appreciate the reviewer for providing us with these classical works. All mentioned papers are about frequency-independent non-resonant feedback and their derivations are generally not applicable to the highly selective backscattering of the high-Q microresonator. We added references to the papers in several places in the revised manuscript.
To answer "The latter also classifies feedback regimes. Is the notion of regime V in the supplementary material linked to the classification or is it an own one": If we understand the question correctly, we are speaking about the regimes of the diode emission, according to Fig. 4. The regime of multi-frequency (regime V on the spectrogram) is characterized by the rise of suppressed laser diode modes, being suppressed in the normal regime. That is why the beatnote signal at the frequency corresponding to the laser diode cavity length is observed. We do not suppose that this regime is similar to one described in Ref. [52], because the laser emission frequency remains narrow and locked to the microresonator. Tkach's classification does not work for the narrow-band feedback systems. One can argue that the high-Q cavity can be viewed as a long arm with effective distance derived from the Q-factor thus making the dynamics of such system to be inside the Tkach's V regime, but this seems to be unnecessarily complicated. If we refer to the reviewer question to SUPPLEMENTARY NOTE 2 and Fig.3 in the SI, then the different regimes correspond to different optical phases and do not correspond to the regimes from [52] and are just five exemplary traces without any classification purposes.
To answer "The literature list is very heavy on the papers of the authors": we've deleted some citations of the authors.

Action taken:
• The last paragraph of the introduction was modified as follows: "However, despite the inspiring and promising experimental results the principles and dynamics of the soliton self-injection locking have never been thoroughly studied. Only recently some aspects of the soliton generation effect were investigated [37], where a static operation was considered, but a comprehensive theoretical and experimental investigation is still necessary. The common SIL models consider either laser equations with frequencyindependent feedback [39][40][41], or linear-resonant feedback [42][43][44]. Here, we first develop an original 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." • We add the following sentence to the Section II "Principle of laser self-injection locking": «We also note that the laser cavity resonant frequency as well as are also assumed to include the Henry factor in its definition.» • To clarify the laser model definition, we add the following sentence: "For analysis of the SIL effect we combine Eqs.
(2) with the standard laser rate equations similar to the Lang-Kobayashi equations [39], but with resonant feedback [44]." • We add the following after the equation (6): "The self-injection locking coefficient 0 is analogous to the feedback parameter C used in the theory of the simple mirror feedback [40,41], where the self-injection is achieved with the frequency-independent reflector forming an additional Fabry-Perot cavity. However, in the resonant feedback setup, the self-injection locking coefficient does not depend on the laser-to-reflector distance, depending on the parameters of the reflector instead. Though the system has qualitatively similar regimes as a simple one [52], their ranges and thresholds are different [42,44,46]. The value of 0 > 4 is required for pronounced locking with the sharp transition, naturally becoming a locking criterion. For high-Q microresonators, this value can be no less than several thousand. We also note that in linear regime (or in nonlinearly shifted coordinates , ) the stabilization coefficient of the setup is close to 0 , full locking range is close to 0.65 0 ." • Also, we have deleted the following citations of authors: https://doi.org/10.1109/CLEOE-EQEC.2019.8873388, https://doi.org/10.1051/epjconf/201922002006, http://www.osapublishing.org/abstract.cfm?URI=ASSL-2019-JTu3A.32 that region. Should the dashes extend beyond the locking regime? Probably not. Similar, but not quite so pronounced in Fig. 1d.

Our reply:
We thank the reviewer for the suggestion to make this important figure better.
To answer "Should the dashes extend beyond the locking regime?": We would like to highlight the range of in the locked state in the revised manuscript.

Action taken:
• We revised the Fig. 1 according to the reviewer's comment. We changed the dashed region so that it corresponds to the locking regime only.
4. Fig. 4d is quite unclear and not explained well. The thin grey lines have no legend. I do not think that the two straight pieces of thick solid black line is the theory as it seems to be indicated in the legend. Similarly, the experimental data need to be better identified.

Our reply:
We thank the reviewer for the suggestion. There were some graphical artifacts which have been deleted in our revised manuscript. Also, we improved the visual representation of the experimental data. We added the legend for thin grey lines.

Action taken:
• We revised the Fig. 4d.
The authors present theoretical investigations of self-injection locking of a DFB laser to a silicon nitride micresonator for Kerr comb generation and soliton formation. They investigate the soliton existence region for forward and backward tuning of the laser-cavity detuning. The paper is generally well written and would be of specific interest to the Kerr comb community.
However, there has been many prior demonstrations of injection locking for Kerr comb generation and soliton formation in recent years, including the work done by the some of the authors of this manuscript, including Pavlov, et al., "Narrow-linewidth lasing and soliton Kerr microcombs with ordinary laser diodes," Nat. Photon While the experimental investigation of the noise performance is a nice addition, the overall performance of the comb source is similar to the previous demonstrations by the authors and do not see any notable advance. Especially with many prior experimental work being demonstrated using the self-injection locked scheme, the theoretical work will be more suitable in a specialized journal rather than a journal with broad readership such as Nature Communications.

Our reply:
First of all, we thank the reviewer for reviewing our work and share the comments. We kindly but firmly disagree with the statement that in our work "the overall performance of the comb source is similar to the previous demonstrations by the authors and do not see any notable advance." First, we would like to explain and highlight the significance of our work, compared with previous works mentioned by the reviewer: 1. Our work is the first report presenting the study of the soliton self-injection locking dynamics via theoretical model, numerical simulations and experimental characterization. Prior works, including Ref. [R1-R3], studied only static comb states and only mentioned that switching of the soliton combs is possible. We simulate, for the first time, the self-injection locking dynamics taking into account the microresonator's Kerr nonlinearity, so-called "nonlinear laser self-injection locking". These results will become the point of interest not only for the Kerr comb community but for a wide community of photonics researchers because the integration of active structures with passive photonic integrated circuits and production of micro-rings from active material (for example, QCL) is in great demand. 2. We perform soliton beatnote spectroscopy and demonstrate switching in the self-injection locking regime between different soliton states, which has not been studied before. The measurement of the laser emission frequency depending on the laser injection current (Fig. 4 in the main text) has not been performed in previous works despite its importance to explain the locking dynamics. There were only attempts to measure such curves (Ref. [R4]). The researchers mistakenly expected that frequency tuning curves in the nonlinear case were similar to linear ones. 3. In this sense, our manuscript should be compared with Ref. [R5], because in both works the switching of soliton combs is investigated. Reference [R5] studied the conventional method of soliton initiation in the presence of an optical isolator (which is not amenable for photonic integration), while our work studies the case of fully integrated microcombs operating with self-injection locking (in the absence of an optical isolator). 4. Our work [R6] was done concurrently and independently with the paper Shen, et al., "Integrated turnkey soliton microcombs operated at CMOS frequencies," arXiv:1911.02636 (placed on arXiv back to back in November and December, respectively). Ref. [R5] presents a key step in photonic integration and packaging of DFB lasers to Si 3 N 4 photonic chips, but it contains only a qualitative analysis of laser self-injection locking in the static model in contrast with ours. That paper has been published in Nature recently.
Also, we strongly disagree that our work is "the theoretical work" and "will be more suitable in a specialized journal rather than a journal with broad readership such as Nature Communications." Our work includes thorough experimental results explained by the proposed novel theoretical model. Understanding of the soliton self-injection locking helps us to generate the single-soliton state with the lowest repetition rate of 30 GHz, that is the record to our knowledge. The community quickly adopted our approach to investigate self-injection locking. Please, let me show how our work is described in recent papers:  "Further theoretical investigation of the self-injection dynamics based on an adapted model like the one proposed by A. S. Voloshin and coworkers [R6] may explain the behavior of our system." -the published paper [R7] of Sylvain Boust (the group of Frédéric van Dijk, III-V lab in Thales Research and Technology, France).  "However, a recent theoretical and experimental demonstration of self-injection locking shows that the dynamics of self-injection locking is sensitive to various parameters, e.g., strength and phase of the backscattering and pump power [R6]" -preprint [R8].  "So here's one of these tuning curves for F^2 = 10, and again this is based on a very nice paper [R6]" -CLEO 2020 presentation of Travis Briles (the group of Scott Papp in NIST) [R9]. Our approach was chosen by NIST to investigate the integrated octave-spanning frequency comb, which they developed.
1. What is the required strength of the backscattered light for injection locking to occur? What is the value in the experiment?

Our reply:
The strength of backscattering can be defined in several ways. First, it may mean the coupling rate described by the normalized mode-coupling parameter (see Si 3 N 4 chip information in Methods). This parameter can be calculated by fitting the microresonator's resonance and directly measured if the resonance splitting is detectable. It is easy to do because this parameter characterizes the microresonator, not the whole system "diode-microresonator", and one can do it for any microresonator in the external setup by coupling the light into the bus waveguide by the lensed fiber. For values < 0.16, it approximately equals the amplitude reflection coefficient in resonance. One can measure the backscattered light using beam-splitter or by using the circulator.
We measured = 0.17 by fitting the resonance profile (see Methods). Also, we measured the resonant backscattered light power to be ~5-10% of input power depending on particular mode by using the circulator.
Usually, a small value of is sufficient to trigger self-injection locking. We saw the selfinjection locking for high-Q crystalline microresonators with as low as 0.01.
The linewidth reduction depends on ( / ) 2 , so the influence of the microresonator's Q-factor is much higher than the influence of the normalized mode-coupling, because ∼ 10 4 and > 10 6 . In other words, one can obtain ultranarrow linewidth in the self-injection locking regime even with very low backscattering.
Second, another option to describe quantitatively the strength of the backscattering is the selfinjection locking coefficient 0 , which is similar to the classical feedback parameter C [40]. In some approximations the linewidth reduction Action taken: • The legend is changed to "Ref. laser phase noise" 3. In Figure 4, can the authors clarify the tuning path for backward scan? How is the resonance initially captured? In general the figure caption is incomplete. The authors should address what the Roman numerals are for each of the regions described in b. In general the figure caption is incomplete. The authors should address what the Roman numerals are for each of the regions described in b.

Our reply:
We have measured the nonlinear resonance (the microresonator transmission trace) on the oscilloscope directly. A 30 Hz triangle modulation was applied on the laser diode current from 372 to 392 mA, such that the laser frequency scans over the resonance in backward and then in forward directions. Then the frequency tuning curve inside this diode current range was measured. The reference laser's frequency is set higher than the free-running DFB laser frequency, such that the heterodyne beatnote signal is observed near 15 GHz. Then we slow the triangle modulation down to 10 mHz and observe the heterodyne beatnote signal in the spectrogram regime.
First, the laser diode is free-running (frame I). Then the laser diode is locked to the microresonator but the locking is weak and the Kerr comb does not form (frame II). In frame III we observed pure soliton self-injection locking. In frame V the laser diode operated in a multi-frequency regime and this region can not be analyzed. If the laser doesn't switch to this regime, it would go out of the locked state and start to be free-running again. Then the current sweep will change its direction and the laser scans over the same resonance in the forward direction.
Action taken: • We revised the caption of the Fig. 4 and gave the description of all Roman numbers.
4. The generated comb spectra is rather narrowband. There have been many demonstrations of broadband soliton microcomb generation in silicon nitride microresonators. Is this possible in an injection locked scheme? What, if any, are the limitations for achieving broad bandwidth?
Our reply: First, the width of the comb spectrum depends on the microresonator group velocity dispersion (GVD, D 2 ). In our case, the GVD D 2 is large, which ultimately limits our soliton bandwidth. We can optimize this parameter in the future to obtain lower D 2 and wider comb. Nevertheless, the width of the presented soliton microcombs in the manuscript is comparable with conventional microcombs with low rep.rates [55].
Secondly, the comb width depends on the effective detuning = 2 ( 0 − )⁄ for a specific D 2 . In conventional soliton initiation setup the external laser with isolator is used and it's easy to change the laser detuning. The self-injection locking makes the laser generation frequency ( ) locked to the microresonator ( 0 ), so is changing very slightly.
The main goal of our works was to understand how we change the effective detuning by changing the injection current. We have shown that not all values of the effective detunings are achievable (Fig. 1e). Our study shows ways to increase the effective detuning range and what parameters should be optimized. To get the broadest comb one should optimize Q-factor, the backscattering, and coupling.
Recently, broadband (near octave-spanning) soliton microcomb with 1-THz-FSR via selfinjection locking has been demonstrated in NIST [36], where 1-THz-FSR microresonator is used. Conclusions of our work were used to explain the behavior of the device presented in this work.
Action taken: • We added the following text to the "V. Conclusion and discussion" (page 9): "Therefore, the soliton self-injection locking provides, first, laser diode stabilization, second, microcomb generation, third, ultra-low noise photonic microwave generation. The main problem of this technique is the limitation of achievable effective detunings: a single-soliton 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 the effective detunings in the locked state." • We added the following to the Methods "Comb generation in SIL regime" (page 9): "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."