Universal Kerr-thermal dynamics of self-injection-locked microresonator dark pulses
自注入锁定微谐振器暗脉冲的普适克尔热动力学

Optical frequency combs (OFCs) Udem et al. (2002); Cundiff and Ye (2003); Fortier and Baumann (2019); Diddams et al. (2020) constitute broadband frequency rulers with equidistant lines of continuous-wave (CW) components, and have revolutionized timing, spectroscopy, precision measurement, and testing fundamental physics. While conventional OFCs are constructed with solid-state or fiber mode-locked lasers, microcombs – harnessing enhanced Kerr nonlinearity in ultrahigh-$Q$ optical microresonators driven by CW pumps – have allowed miniaturized OFCs with small size, weight and power consumption Del’Haye et al. (2007); Kippenberg et al. (2018). With the emergence and quick maturing of ultralow-loss photonic integrated circuits Moss et al. (2013); Gaeta et al. (2019); Zhang et al. (2017); Liu et al. (2021), as well as heterogeneous and hybrid integration with semiconductor lasers Stern et al. (2018); Xiang et al. (2021), today microcombs can be built entirely on-chip and manufactured in large volume with low cost, catalyzing wide deployment outside laboratories and in space.
光频率梳 (OFC) Udem 等人 ( 2002)；Cundiff 和 Ye ( 2003)；Fortier 和 Baumann ( 2019)；Diddams 等人 ( 2020) 构成具有等距连续波 (CW) 分量线的宽带频率标尺，并彻底改变了定时光谱学、精密测量和测试基础物理学传统 OFC 是用固态或光纤锁模激光器构造的，而微梳 - 利用由 CW 泵驱动的超高 $Q$ 光学微谐振器中增强的克尔非线性 - 允许小型化 OFC，具有小尺寸、重量和功耗 Del'Haye 等人 ( 2007)；Kippenberg 等人 ( 2018)随着超低损耗光子集成电路的出现和快速成熟 Moss 等人 ( 2013)；Gaeta 等人 ( 2019)；Zhang 等人（2017 年）；刘等人（2021 年）以及与半导体激光器的异质和混合集成 Stern 等人（2018 年）；Xiang 等人（2021 年）如今，微梳可以完全在芯片上构建，并以低成本进行大批量生产，从而在实验室外和太空中广泛部署

Depending on the microresonator’s group velocity dispersion (GVD), there are two types of coherent microcombs. The bright dissipative soliton microcombs Herr et al. (2013); Yi et al. (2015); Brasch et al. (2016a); Liang et al. (2015); Joshi et al. (2016); He et al. (2019) require anomalous GVD, while the dark-pulse (also termed “platicon”) microcombs require normal GVD Xue et al. (2015); Lobanov et al. (2015); Huang et al. (2015); Parra-Rivas et al. (2016); Nazemosadat et al. (2021); Wang et al. (2022). Compared to solitons, platicons exhibit remarkably higher CW-to-pulse power conversion efficiency Xue et al. (2017); Jang et al. (2021), thus are advantageous for coherent optical communication Fülöp et al. (2018) and photonic microwave generation Sun et al. (2024a). To initiate platicons, laser self-injection locking (SIL) Kondratiev et al. (2017, 2023) offers the most robust and effective form, and permits seamless integration of the pump laser and the microresonator Voloshin et al. (2021); Jin et al. (2021); Lihachev et al. (2022). However, while the current linear SIL theory and model Kondratiev et al. (2017) have addressed the linear coupling between the laser cavity and the external microresonator, it fails to manage the complicated nonlinear processes in the laser gain media and in platicon formation.
根据微谐振器的群速度色散 (GVD)，相干微梳有两种类型。亮耗散孤子微梳Herr等人( 2013 ); Yi等人( 2015 ); Brasch等人( 2016a ); Liang等人( 2015 ); Joshi等人( 2016 ); He等人( 2019 )需要异常 GVD，而暗脉冲 (也称为“platicon”) 微梳需要正常 GVD Xue等人( 2015 ); Lobanov等人( 2015 ); Huang等人( 2015 ); Parra-Rivas等人( 2016 ); Nazemosadat等人( 2021 ); Wang等人( 2022 ) 。与孤子相比，扁子表现出明显更高的连续波到脉冲功率转换效率Xue等（ 2017 ）；Jang等（ 2021 ） ，因此有利于相干光通信Fülöp等（ 2018 ）和光子微波产生Sun等（ 2024a ） 。 为了启动扁晶格，激光自注入锁定 (SIL) Kondratiev等人（ 2017 年、 2023 年）提供了最稳健、最有效的形式，并允许泵浦激光器和微谐振器Voloshin等人（ 2021 年）；Jin等人（ 2021 年）；Lihachev等人（ 2022 年）的无缝集成。然而，虽然当前的线性SIL 理论和模型Kondratiev等人（ 2017 年）已经解决了激光腔和外部微谐振器之间的线性耦合，但它无法管理激光增益介质和扁晶格形成中的复杂非线性过程。

Here, we investigate – theoretically, numerically and experimentally – the Kerr-thermal dynamics of platicon formation using a semiconductor laser self-injection-locked to an integrated silicon nitride (Si₃N₄) microresonator. We unveil an intriguing platicon switching behaviour with discrete steps, allowing operation of platicons in noise-quenched states and immunity to laser noise. Figure 1a illustrates the principle of our study. When a laser-cavity mode is tuned into a resonance mode of the Si₃N₄ microresonator and light is coupled into the microresonator, SIL occurs that locks the laser frequency to the resonance Kondratiev et al. (2017, 2023) and suppresses the laser linewidth. The linewidth suppression ratio is proportional to $Q_{\text{r}}/Q_{\text{d}}$, where $Q_{\text{r/d}}$ is the quality factor of the microresonator/laser cavity. Meanwhile, photo-thermal effect Carmon et al. (2004); Brasch et al. (2016b); Yi et al. (2016); Gao et al. (2022) induces a global frequency shift of the microresonator’s resonance grid. With sufficient intracavity power, Kerr nonlinearity induces four-wave mixing (FWM) Kippenberg et al. (2004) that translates photons to other resonances. In microresonators of normal GVD, the synergy of SIL, photo-thermal and Kerr effects ultimately yield platicon formation, as shown in Fig. 1b.
在这里，我们从理论、数值和实验上研究了使用自注入锁定到集成_氮化硅( _Si3N4 ) 微谐振器的半导体激光器形成平晶格的克尔热动力学。我们揭示了一种有趣的平晶格开关行为，它具有离散步骤，允许平晶格在噪声猝灭状态下工作，并且不受激光噪声的影响。图1a说明了我们研究的原理。当激光腔模式调谐到_Si3N4微谐振_器的谐振模式并将光耦合到微谐振器中时，会发生 SIL，将激光频率锁定到谐振Kondratiev等人（ 2017 年， 2023 年）并抑制激光线宽。线宽抑制比与 $Q_{\text{r}}/Q_{\text{d}}$ 成正比，其中 $Q_{\text{r/d}}$ 是微谐振器/激光腔的品质因数。同时，光热效应Carmon等人（ 2004 年）；Brasch等人。 （ 2016b ）；Yi等人（ 2016 ）；Gao等人（ 2022 ）在微谐振器的谐振网格中引起全局频率偏移。在腔内功率足够的情况下，克尔非线性会引起四波混频 (FWM) Kippenberg等人（ 2004 ），从而将光子转换为其他谐振。在正常 GVD 的微谐振器中，SIL、光热和克尔效应的协同作用最终导致形成扁晶格，如图1b所示。

Figure 1d illustrates the schematic of SIL. Light $E_{\text{L}}$ in the laser cavity is emitted and coupled into the microresonator’s counter-clockwise direction $E^{+}$. Rayleigh scattering in the microresonator reflects a portion of light to the clockwise direction $E^{-}$ and to the laser cavity. To facilitate SIL, we optimize the edge coupling and the gap distance (thus the feedback phase $\phi$) between the two chips. In the SIL regime, we continuously tune the laser current, and monitor the output optical power and frequency from the Si₃N₄ chip. The output light is beaten against a reference laser, and the beat frequency is recorded. The forward (backward) tuning corresponds to increasing (decreasing) laser current, which decreases (increases) laser frequency.
图1d显示了 SIL 的示意图。激光腔中的光 $E_{\text{L}}$ 被发射并耦合到微谐振器的逆时针方向 $E^{+}$ 。微谐振器中的瑞利散射将一部分光反射到顺时针方向 $E^{-}$ 并反射到激光腔。为了促进 SIL，我们优化了两个芯片之间的边缘耦合和间隙距离（因此优化了反馈相位 $\phi$ ）。在 SIL 状态下，我们不断调整激光电流，并监测 Si ₃ N ₄芯片的输出光功率和频率。输出光与参考激光器进行拍频，并记录拍频。前向（后向）调整对应于增加（减少）激光电流，从而降低（增加）激光频率。

Figure 2(a, b) shows the experimental data. When SIL occurs, the output power experiences a sudden drop, and the output frequency is shifted. In the SIL regime, we observe decreasing (increasing) output power and frequency with backward (forward) tuning, due to the photo-thermal effect Carmon et al. (2004); Brasch et al. (2016b); Yi et al. (2016); Gao et al. (2022). For example, the decreasing output power corresponds to decreasing transmitted power and increasing intra-cavity power. The latter enhances the photo-thermal effect and causes increasing resonance red-shift (towards longer wavelength) Carmon et al. (2004). As the laser frequency is locked to the resonance, the red-shifted resonance drags the laser frequency, leading to decreasing output optical frequency.
图2 （a，b）显示了实验数据。当发生SIL时，输出功率会突然下降，并且输出频率会发生偏移。在SIL范围内，我们观察到由于光热效应导致输出功率和频率随着后向（前向）调谐而降低（增加） Carmon等人（ 2004 ）；Brasch等人（ 2016b ）；Yi等人（ 2016 ）；Gao等人（ 2022 ） 。例如，输出功率的降低对应于透射功率的降低和腔内功率的增加。后者增强了光热效应并导致共振红移增加（朝向更长的波长） Carmon等人（ 2004 ） 。由于激光频率锁定在共振频率上，红移的共振会拖动激光频率，导致输出光频率降低。

Besides, it is apparent that the output optical power and frequency exhibit discrete step features. This is contrary to the conventional, linear SIL model Kondratiev et al. (2017) showing continuous, nearly anchored tuning curves of power and frequency. Here, the appearance of these steps are attributed to the Kerr nonlinearity in the microresonator with sufficient intra-cavity power. Note that similar discontinuous curves have been observed and characterized in self-injection-locked solitons Voloshin et al. (2021), while here we observe and characterize these curves for the first time for self-injection-locked platicons. Figure 2c shows the zoom-in profile of the gray-shaded zoom in Fig. 2b. Steps in backward tuning with 320.4 to 321.3 mA laser current are marked with 1 to 4. Typical optical spectra within each step are shown in Fig. 3a, evidencing formation and switching of different platicon states. Distinct fringes are marked with arrows on the spectral envelopes. Numerical simulation of time-domain pulse shapes in Fig. 3c (discussed later) confirms that all these platicon states are “single platicon” comprising only one dark pulse in the microresonator.
此外，很明显输出光功率和频率呈现离散阶跃特征。这与传统的线性 SIL 模型Kondratiev等人（ 2017 年）相反，该模型显示了连续的、几乎锚定的功率和频率调谐曲线。在这里，这些步骤的出现归因于具有足够腔内功率的微谐振器中的克尔非线性。请注意， Voloshin等人（ 2021 年）在自注入锁定孤子中已经观察并表征了类似的不连续曲线，而在这里我们第一次观察和表征了自注入锁定扁晶格的这些曲线。图2c显示了图2b中灰色阴影缩放的放大轮廓。使用 320.4 至 321.3 mA 激光电流进行后向调谐的步骤用 1 到 4 标记。图3a显示了每个步骤内的典型光谱，证明不同扁晶格状态的形成和切换。光谱包络上用箭头标记了不同的条纹。图3c中时域脉冲形状的数值模拟（稍后讨论）证实，所有这些平面状态都是“单个平面”，微谐振器中仅包含一个暗脉冲。

Moreover, identical Kerr-thermal platicon dynamics has also been observed in other two independent SIL setups, where different DFB lasers and 21.3-GHz-FSR Si₃N₄ microresonators are used. Details are found in Supplementary Information Note 1 and 2. These parallel experiments suggest that, our observation of the Kerr-thermal dynamics and discrete steps is universal and independent of the particular lasers or microresonators.
此外，在其他两个独立的 SIL 装置中也观察到了相同的克尔热平台动力学，其中使用了不同的 DFB 激光器和 21.3-GHz-FSR Si ₃ N ₄微谐振器。详细信息请参阅补充信息注释 1 和 2。这些平行实验表明，我们对克尔热动力学和离散步骤的观察是通用的，并且与特定的激光器或微谐振器无关。

Equation (2) describes the laser dynamics in the laser cavity, with $\alpha_{\text{L}}=2\delta\omega_{\text{L}}/\kappa$, $\varphi=\omega_{0}t_{\text{s}}$, $\tilde{\kappa}_{\text{L}}=2\sqrt{\kappa_{\text{ex}}\tau_{\text{R}}T_{\text{L}}T_{\text{c}}n_{\text{L}}V_{\text{L}}}/(\kappa\tau_{\text{L}}\sqrt{n_{0}V_{\text{eff}}})$. Here, $\alpha_{g}$ is the linewidth enhancement factor, $\delta\omega_{\text{L}}$ is the frequency difference between the cold-cavity resonance mode and the microresonator resonance mode, $t_{\text{s}}$ is the time delay between the laser cavity and the microresonator, $\kappa_{\text{ex}}$ is the external coupling rate of the microresonator, $\tau_{\text{R}}$ is the light round-trip time in the microresonator, $T_{\text{L}}$ is the light transmission rate at the laser cavity’s output surface, $T_{\text{c}}$ is the light coupling rate into the laser cavity from outside, $\tau_{\text{L}}$ is the light round-trip time in the laser cavity, and $b_{0}$ is the back-scattered light in the microresonator at the $0^{\text{th}}$ resonance mode.
方程（2）用 $\alpha_{\text{L}}=2\delta\omega_{\text{L}}/\kappa$ ， $\varphi=\omega_{0}t_{\text{s}}$ ， $\tilde{\kappa}_{\text{L}}=2\sqrt{\kappa_{\text{ex}}\tau_{\text{R}}T_{\text{L}}T_{\text{c}}n_{\text{L}}V_{\text{L}}}/(\kappa\tau_{\text{L}}\sqrt{n_{0}V_{\text{eff}}})$ 描述激光腔内的激光动力学，其中 $\alpha_{g}$ 为线宽增强因子， $\delta\omega_{\text{L}}$ 为冷腔谐振模式与微谐振器谐振模式之间的频率差， $t_{\text{s}}$ 为激光腔与微谐振器之间的时间延迟， $\kappa_{\text{ex}}$ 为微谐振器的外部耦合率， $\tau_{\text{R}}$ 为光在微谐振器中的往返时间， $T_{\text{L}}$ 为光在激光腔输出表面的传输速率， $T_{\text{c}}$ 为从外部进入激光腔的光耦合率， $\tau_{\text{L}}$ 为光在激光腔内的往返时间， $b_{0}$ 为 $0^{\text{th}}$ 谐振模式下微谐振器中的背向散射光

Equations (3, 4) describe the dynamics of the forward-propagating light $a_{\mu}$ and the back-scattered light $b_{\mu}$ at the $\mu^{\text{th}}$ resonance mode, with $a_{\mu}(b_{\mu})=\sqrt{g\epsilon_{0}n_{0}^{2}V_{\text{eff}}/(\hbar\omega_{0}\kappa)}E_{\mu}^{+}(E_{\mu}^{-})$, $d_{n}=2D_{n}/\kappa$, $\tilde{\beta}=2\beta/\kappa$, and $\tilde{\kappa}_{\text{R}}=2\sqrt{\kappa_{\text{ex}}T_{\text{L}}T_{\text{c}}n_{0}V_{\text{eff}}}/(\kappa\sqrt{\tau_{\text{R}}n_{\text{L}}V_{\text{L}}})$. Here, $E_{\mu}^{+}(E_{\mu}^{-})$ is the laser amplitude of the forward-propagating (back-scattered) light at the $\mu^{\text{th}}$ resonance mode, $D_{n}$ is the $n^{\text{th}}$-order dispersion of the microresonator, $\beta$ is the back-scattering ratio between the back-scattered light and the forward-propagating light, and $\mathcal{F}[|a|^{2}a]$ ($\mathcal{F}[|b|^{2}b])$ represents the Kerr interaction.
方程 (34) 描述了 $\mu^{\text{th}}$ 谐振模式下前向传播光 $a_{\mu}$ 和反向散射光 $b_{\mu}$ 的动力学，其中 $a_{\mu}(b_{\mu})=\sqrt{g\epsilon_{0}n_{0}^{2}V_{\text{eff}}/(\hbar\omega_{0}\kappa)}E_{\mu}^{+}(E_{\mu}^{-})$ 、 $d_{n}=2D_{n}/\kappa$ 、 $\tilde{\beta}=2\beta/\kappa$ 和 $\tilde{\kappa}_{\text{R}}=2\sqrt{\kappa_{\text{ex}}T_{\text{L}}T_{\text{c}}n_{0}V_{\text{eff}}}/(\kappa\sqrt{\tau_{\text{R}}n_{\text{L}}V_{\text{L}}})$ 其中 $E_{\mu}^{+}(E_{\mu}^{-})$ 是 $\mu^{\text{th}}$ 谐振模式下前向传播（反向散射）光的激光振幅， $D_{n}$ 是微谐振器的 $n^{\text{th}}$ 阶色散， $\beta$ 是反向散射光与前向传播光之间的反向散射比，并且 $\mathcal{F}[|a|^{2}a]$ （ $\mathcal{F}[|b|^{2}b])$ 表示克尔相互作用

Equation (5) describes the evolution of the normalized frequency shift $T$ induced by the temperature in the microresonator, with $k_{T}=K_{T}/g$, and $t_{\text{h}}=\kappa\tau_{\text{r}}/2$. Here, $K_{T}$ is the thermal-induced resonance shift coefficient, and $\tau_{\text{r}}$ is the thermal relaxation time.
方程 (5) 描述了微谐振器中温度引起的归一化频率偏移 $T$ 的演变，其中 $k_{T}=K_{T}/g$ 和 $t_{\text{h}}=\kappa\tau_{\text{r}}/2$ 。其中， $K_{T}$ 是热致谐振偏移系数， $\tau_{\text{r}}$ 是热弛豫时间。

We further perform numerical simulation of Eq. (1–5) using experimentally obtained or realistic values for each parameter. Details are found in Supplementary Information Note 3. Figure 2(d,e) presents the simulated output optical power and frequency, which agree with the experimental data on both the overall trend (due to the photo-thermal effect) and the platicon steps (due to the Kerr nonlinearity). Figure 2f shows the zoom-in profile of the gray-shaded zoom in Fig. 2(d,e), and highlights the full detuning range of platicon steps with gray curves. Steps corresponding to those in Fig. 2c are marked. We emphasize that, previous efforts Kondratiev et al. (2020); Lihachev et al. (2022); Wang et al. (2022) to model platicon formation in the SIL regime fail to investigate and analyze the transient behaviour, and thus fail to reveal the step features. Meanwhile, they have also omitted the photo-thermal effect.
我们进一步使用实验获得的每个参数的实际值对公式 (1-5) 进行数值模拟。详细信息见补充信息注释 3。图2 (d,e) 展示了模拟的输出光功率和频率，它们与实验数据在总体趋势（由于光热效应）和平板台阶（由于克尔非线性）上都一致。图2 f 显示了图2 (d,e) 中灰色阴影缩放的放大轮廓，并用灰色曲线突出显示了平板台阶的完整失谐范围。标记了与图2 c 中的台阶相对应的台阶。我们强调， Kondratiev等人（ 2020 年）；Lihachev等人（ 2022 年）；Wang等人（ 2022 年）在 SIL 范围内模拟平板形成的努力未能研究和分析瞬态行为，因此未能揭示台阶特征。同时，他们也忽略了光热效应。

Figure 3b presents the simulated platicon spectra for each labelled steps in Fig. 2f. They conform the experimental data in Fig. 3a, not only on the spectral envelopes but also on the number of fringes (marked with arrows). Figure 3c presents the simulated time-domain pulse shapes, revealing that the fringes are related to the oscillating tails at the bottom of the dark pulse (marked with arrows). Unlike the steps of bright dissipative solitons Guo et al. (2017); Voloshin et al. (2021), we demonstrate here that platicon steps can occur even in the single-platicon state, i.e. only one dark pulse in the microresonator.
图3b展示了图2f中每个标记台阶的模拟平板光谱。它们不仅在光谱包络上，而且在条纹数量（用箭头标记）上都与图3a中的实验数据相符。图3c展示了模拟的时间域脉冲形状，表明条纹与暗脉冲底部的振荡尾部有关（用箭头标记）。与 Guo等人（ 2017 年）；Voloshin等人（ 2021 年）的亮耗散孤子台阶不同，我们在此证明，即使在单平板状态下，即微谐振器中只有一个暗脉冲，也可以发生平板台阶。

Here we harness the platicon switching dynamics, and demonstrate a unique noise-quenching effect allowing suppression of microwave’s phase noise. In contrast to methods mentioned above, our system is free-running, all passive (i.e. without any active locking), and elegantly simple (i.e.without extra RF or laser sources). Experimentally, we collect the output light from the microresonator with a commercial photodetector (PD). The PD converts the dark pulse stream of $f_{\text{rep}}=10.7$ GHz to a 10.7-GHz microwave in the critical X-band which is dedicated for radar, wireless networks, and satellite communication.
在这里，我们利用 platicon 开关动力学，并展示出独特的噪声抑制效果，从而抑制微波的相位噪声。 与上述方法相比，我们的系统是自由运行的、完全被动的（即没有任何主动锁定）并且非常简单（即没有额外的射频或激光源）。 实验中，我们利用商用光电探测器（PD）收集微谐振器的输出光。 PD 将 $f_{\text{rep}}=10.7$ GHz 的暗脉冲流转换为关键 X 波段的 10.7 GHz 微波，该波段专用于雷达、无线网络和卫星通信。

By varying the laser current $I$ within the SIL regime, we monitor the platicon’s $f_{\text{rep}}$ and phase noise $S_{\phi}$ at 10 kHz Fourier offset frequency of the 10.7-GHz microwave, as shown in Fig. 3d. For each step, a local minimum of $S_{\phi}$ is observed. Meanwhile, it coincides with the local maximum of $f_{\text{rep}}$, as highlighted with vertical dashed lines in Fig. 3d. This coincidence is due to that, at the local maximum of $f_{\text{rep}}$, we have $\text{d}f_{\text{rep}}/\text{d}I=0$, as highlighted with horizontal dashed lines in Fig. 3d. Therefore, $f_{\text{rep}}$ becomes insensitive to current-noise-induced laser frequency jitter, resulting in the lowest $S_{\phi}$. More analysis of this coincidence is found in Supplementary Information Note 4. In fact, this effect is similar to the “quiet point” effect observed in bright dissipative solitons without SIL Yi et al. (2017); Triscari et al. (2023). The underlying mechanism is attributed to the multi-mode coupling between the laser cavity and the Si₃N₄ microresonator, which causes asymmetric comb-line enhancement or suppression, as shown in Fig. 3a. As a result, $f_{\text{rep}}$ depends on the laser frequency (be more specifically, the detuning of the free-running laser frequency to the cold resonance frequency). In our theoretical model, only the pump resonance is considered coupled, i.e. $b_{0}$ in Eq. (2). A more precise simulation on the optical spectra should include multi-mode coupling by substituting $b_{0}$ with $\sum_{\mu}b_{\mu}e^{i\mu d_{1}(\tau-\tau_{\text{s}})}$.
通过在 SIL 范围内改变激光电流 $I$ ，我们监测了 10.7 GHz 微波 10 kHz 傅里叶偏移频率处的平面波 $f_{\text{rep}}$ 和相位噪声 $S_{\phi}$ ，如图3 d 所示。对于每一步，都可以观察到局部最小值 $S_{\phi}$ 。同时，它与局部最大值 $f_{\text{rep}}$ 重合，如图3 d 中垂直虚线突出显示的那样。这种巧合是由于在局部最大值 $f_{\text{rep}}$ 处，我们有 $\text{d}f_{\text{rep}}/\text{d}I=0$ ，如图3 d 中水平虚线突出显示的那样。因此， $f_{\text{rep}}$ 对电流噪声引起的激光频率抖动不敏感，从而导致最低的 $S_{\phi}$ 。关于这种巧合的更多分析见补充信息注释 4。事实上，这种效应类似于在没有 SIL 的亮耗散孤子中观察到的“安静点”效应Yi等人（ 2017 年）；Triscari等人（ 2023 年） 。其根本机制归因于激光腔和 Si ₃ N ₄微谐振器之间的多模耦合，这会导致不对称梳状线增强或抑制，如图3a所示。因此， $f_{\text{rep}}$ 取决于激光频率（更具体地说，是自由运行激光频率与冷谐振频率的失谐）。在我们的理论模型中，仅考虑泵浦谐振耦合，即方程 (2) 中的 $b_{0}$ 。更精确的光谱模拟应包括多模耦合，方法是用 $\sum_{\mu}b_{\mu}e^{i\mu d_{1}(\tau-\tau_{\text{s}})}$ 代替 $b_{0}$ 。

The $S_{\phi}$ spectra of the local maximum and minimum points, marked with green and black dots in Fig. 3d, are measured and compared in Fig. 3e, showing 23.5 dB noise reduction. The lowest $S_{\phi}$ reaches $-45/-75/-105$ dBc/Hz at 0.1/1/10 kHz Fourier offset frequency, and is limited by the shot noise floor at higher frequency offset Savchenkov et al. (2008). Figure 3e inset shows the beatnote of the lowest $S_{\phi}$ (black data) with 10 Hz resolution bandwidth (RBW). Supplementary Information Note 5 illustrates that, such a noise quenching phenomenon has been also observed in parallel experiments with different Si₃N₄ microresonators and DFB lasers, thus is universal.
在图3e中测量并比较了图3d中用绿点和黑点标记的局部最大值和最小值点的 $S_{\phi}$ 光谱，显示噪声降低了 23.5 dB。最低的 $S_{\phi}$ 在 0.1/1/10 kHz 傅里叶偏移频率时达到 $-45/-75/-105$ dBc/Hz，并在更高频率偏移处受到散粒噪声基底的限制Savchenkov等人（ 2008 年） 。图3e插图显示了最低 $S_{\phi}$ （黑色数据）的拍音，分辨率带宽（RBW）为 10 Hz。补充信息注释 5 表明，这种噪声猝灭现象也在使用不同的 Si ₃ N ₄微谐振器和 DFB 激光器的平行实验中观察到，因此具有普遍性。

In conclusion, we unveil an intriguing yet universal Kerr-thermal dynamics of a semiconductor laser self-injection-locked to a Si₃N₄ microresonator where platicon microcombs are formed. We experimentally observe and characterize the platicon switching dynamics with discrete steps. We further establish a comprehensive theoretical model cooperating the laser-cavity dynamics, the platicon formation dynamics, the photo-thermal dynamics, and the mutual coupling between them. Numerical simulation confirms the experimental result, and illuminates that the platicon switching phenomenon is originated from the synergy of SIL, Kerr nonlinearity and the photo-thermal effect. Exploiting this finding, we showcase low-noise microcomb-based microwave generation. Via operation of platicons with specific laser current, we achieve 23.5 dB phase noise quenching of the 10.7-GHz microwave carrier. Our study not only add critical insight of pulse formation in linear-and-nonlinear-coupled laser-microresonator systems, but also offer a neat solution for photonic-chip-based microwave oscillators with high spectral purity, ideal for microwave photonics, coherent optical communication, analog-to-digital conversion, wireless links, and radar.
总之，我们揭示了一种有趣而又具有普适性的克尔热动力学，该动力学自注入锁定在 Si ₃ N ₄微谐振器上，其中形成了扁梳微梳。我们通过实验观察并以离散步骤表征了扁梳开关动力学。我们进一步建立了一个综合的理论模型，该模型结合了激光腔动力学、扁梳形成动力学、光热动力学以及它们之间的相互耦合。数值模拟证实了实验结果，并阐明了扁梳开关现象源于 SIL、克尔非线性和光热效应的协同作用。利用这一发现，我们展示了基于低噪声微梳的微波产生。通过以特定激光电流操作扁梳，我们实现了 10.7 GHz 微波载波的 23.5 dB 相位噪声抑制。我们的研究不仅增加了线性和非线性耦合激光微谐振器系统中脉冲形成的关键见解，而且还为基于光子芯片的高光谱纯度微波振荡器提供了一个完美的解决方案，非常适合微波光子学、相干光通信、模数转换、无线链路和雷达。