Monolithic Kerr and electro-optic hybrid microcombs

Optical frequency combs are precision spectral rulers that have revolutionized frequency and time measurements Diddams, Vahala, and Udem (2020). Since early demonstrations of frequency combs with solid-state mode-locked lasers, persistent efforts have been made to advance the comb technology from laboratory concepts to field deployment Fortier and Baumann (2019). An important progress is the soliton mode-locking in high-quality-factor Kerr microresonators Herr et al. (2014); Xue et al. (2015) which pushes comb systems towards integrated chips Xiang et al. (2021) for portable operation Stern et al. (2018). Due to the tight optical confinement in a microresonator, the Kerr ($\chi^{(3)}$) nonlinearity is enhanced, leading to sub-milliwatt comb initiation Chang et al. (2020); Ji et al. (2017) as well as new avenues for studying soliton physics Yang et al. (2016); Cole et al. (2017); Weng et al. (2020); Tikan et al. (2021). The versatile waveguide dispersion control further permits broadband comb operation Li et al. (2017); Pfeiffer et al. (2017); Gong et al. (2020a); Yu et al. (2016); Liu et al. (2021); He et al. (2021). To date, a multitude of nanophotonic material platforms Kippenberg et al. (2018) have been developed to support soliton microcomb sources which have already enabled new capabilities in optical clock Newman et al. (2019); Drake et al. (2019), optical frequency synthesis Spencer et al. (2018), spectroscopy Dutt et al. (2018); Suh et al. (2016) and etc Kippenberg et al. (2018).

To further the integration of microcomb systems for scalable production and high volume applications, efforts have been made to integrate soliton initiation circuitry onto chip Raja et al. (2019); Jin et al. (2021); Stern et al. (2018); Xiang et al. (2021). Another key element is the detection and control of the soliton repetition rates Del’Haye, Papp, and Diddams (2012); Yu et al. (2019); Drake et al. (2019); Wang et al. (2020). Despite that Kerr solitons with microwave repetition rates are achievable Liu et al. (2020); Yang et al. (2018), broadband self-referenceable solitons often exhibit much wider mode spacings in the terahertz range which are usually not amenable to direct photodetection Pfeiffer et al. (2017); Li et al. (2017); Zhang et al. (2019a). Previous works divided such high soliton repetition rates by the use of a separate Kerr microcomb device Spencer et al. (2018); Wang et al. (2020) or off-chip electro-optic (EO) modulators Del’Haye, Papp, and Diddams (2012); Yu et al. (2019); Drake et al. (2019). Under cascaded phase modulation, dense sidebands can be created between neighboring soliton comb lines to facilitate the electronic detection and stabilization of the soliton repetition rate Del’Haye, Papp, and Diddams (2012).

Meanwhile, a flurry of studies have been devoted to exploiting the additional $\chi^{(2)}$ nonlinearity existing in non-centrosymmetric Kerr microcomb materials Jung et al. (2014); Wilson et al. (2019); He et al. (2019). Not only do they unveil novel nonlinear dynamics Bruch et al. (2021); Guo et al. (2018), they also bring about opportunities to extend Kerr microcomb system functionalities, such as phase-matched frequency doubling Lu et al. (2019); Luo et al. (2018); Bruch et al. (2018), and EO modulation Wang et al. (2019); Zhang et al. (2019b, 2021).

Here, we demonstrate an integrated approach to detect and control the high repetition rate of Kerr solitons using a monolithic hybrid Kerr and EO microcombs circuit patterned on a single z-cut lithium niobate (LN) thin film. Owing to the coexistence of Kerr ($\chi^{(3)}$) and Pockels ($\chi^{(2)}$) nonlinearities on this emerging nanophotonic platform He et al. (2019); Gong et al. (2019); Okawachi et al. (2020); Yu et al. (2020); Wang et al. (2019); Zhang et al. (2019b); Gong et al. (2020b), Kerr soliton and EO combs can be produced simultaneously within a single photonic circuit. The fast solitons and slow EO pulses converge onto a common output waveguide and lead to optical-heterodyning between the two pulse trains. In this way, a soliton repetition rate of 334 GHz is divided down to a low frequency RF beatnote which can be processed electronically and monitored in real time. We then show the dependence of soliton repetition rate on pump frequency and pump power, and further stabilize it to an external microwave reference. Our work establishes an integrated interface connecting terahertz-rate Kerr solitons and readily detectable microwaves, providing a path for on-chip photonic frequency division as well as comb locking.

Our hybrid Kerr and EO circuits are monolithically fabricated on a millimeter-sized z-cut LN thin film (see Methods), as shown in Fig. 1c. The circuit consists of a microring Kerr soliton generator and a microring EO comb generator that are serially coupled to a common bus waveguide. The bus waveguide couples separate pump sources into the two microrings and delivers the resulting interleaved hybrid combs for $f_{\mathrm{rep,Kerr}}$ detection. A separate drop-port waveguide is added alongside the Kerr soliton microring for direct access of the soliton spectrum. This convenient circuit configuration requires only one pair of optical input and output ports to produce and access the hybrid Kerr soliton and EO combs for $f_{\mathrm{rep,Kerr}}$ measurement. The use of the auxiliary laser to generate the EO comb can be eliminated, for example, by electro-optically aligning an EO microring resonance to a soliton comb line.

The EO comb, generated in a racetrack microring-based EO modulator, harnesses resonantly enhanced EO modulation strength Zhang et al. (2019b). Favorable phase-matching condition for the $\mathrm{TE}_{00}$ EO comb is achieved by tailoring the microring geometries (see Methods). Here, in-plane electric fields are applied across the microring to modulate the $\mathrm{TE}_{00}$ modes through the LN EO tensor component $r_{22}$ (5.4 pm/V Yariv, Yariv, and Yeh (2007)). The gap between the microring and the in-plane GSG gold electrodes is set to be 2 ${\mu}$m to provide strong RF drive fields while not attenuating the intracavity optical fields.

Similar to Kerr soliton generation, the auxiliary c.w. pump can be steadily launched into a $\mathrm{TE_{00}}$ resonance of the EO microring from the red-detuned region. In experiments, a 1.5-mW on-chip optical pump and an RF drive with an approximately 14-V peak amplitude were used to generate the EO combs (Fig. 2). The pump power was set below the thresholds of Kerr four-wave mixing and SRS which otherwise could disturb the EO comb formation. The RF modulation field was provided by a microwave synthesizer running at a frequency close to the microring FSR ($\sim$ 16 GHz). The measured microcomb spectra under different EO pump wavelengths are shown in Fig. 2b. At certain wavelengths, mode-crossings between the EO comb $\mathrm{TE_{00}}$ modes and other cavity modes are present and causes the asymmetry of the comb profile (see Supplementary Note 1). By selecting a suitable pump wavelength, well-developed EO comb can be obtained to bridge the gap of adjacent soliton lines. In the following $f_{\mathrm{rep,Kerr}}$ measurement, we chose to pump the $\mathrm{TE}_{00}$ resonance at 1556 nm.

The hybrid combs spectrum output from the bus waveguide (Fig. 2a) shows that the Kerr soliton comb spectral profile remains largely intact after passing through the EO microring. This is because the likelihood for soliton comb lines being aligned with EO microring resonances is low. Meanwhile, one can also obtain an unfiltered soliton spectrum from the drop-port of the soliton microring (see Supplementary Note 2). Here, the circuit allows both combs to exploit identical polarization state ($\mathrm{TE}_{00}$), facilitating the following heterodyne measurement.

To access $f_{\mathrm{rep,Kerr}}$, we then mix two of the RF beat notes from different paths and record their frequency difference, $\delta=|\delta_{1}-\delta_{3}|$ $\approx$ 2.2 GHz. Fig. 3c shows a typical RF spectrum at the output of the RF mixer, whose input RF/LO frequency range is from 3.7 to 8.5 GHz. Due to the limited conversion efficiency and finite isolation of the mixer, residual input RF beat notes can still been seen on the mixer output spectrum. They are shown here for illustration purpose, and can be further suppressed with RF filters. At last, we arrive at $f_{\mathrm{rep,Kerr}}=21f_{\mathrm{rep,EO}}-\delta$, based on the measurement shown in Fig. 3. Note that the sign before $\delta$ should be reversed in the calculation if $\delta_{1}<\delta_{3}$.

The electronic access of $f_{\mathrm{rep,Kerr}}$ allows us to investigate and quantify the tunability of $f_{\mathrm{rep,Kerr}}$ in LN microcombs. We vary the Kerr pump frequency and pump power separately while monitoring $\delta$ to infer the change in $f_{\mathrm{rep,Kerr}}$. Fig. 4a shows the measured $\delta$ beat note spectra as pump frequency is modulated. The $f_{\mathrm{rep,Kerr}}$-to-pump-frequency tuning rate is extracted to be 4.6 MHz/GHz under an on-chip pump power of 50mW (Fig. 4b). Here, the pump laser frequency is piezo controlled by an arbitrary function generator and calibrated with a wavelength meter. On the other hand, one can also tune $f_{\mathrm{rep,Kerr}}$ by varying the pump power, e.g., via an external acousto-optic modulator (Fig. 4c). The $f_{\mathrm{rep,Kerr}}$-to-pump-power tuning rate is found to be -0.18 MHz/mW around the 41.5-mW operating point (Fig. 4d).

We move on to characterize the frequency stability of the free-running $f_{\mathrm{rep,Kerr}}$. The drift of $f_{\mathrm{rep,Kerr}}$ is reflected in $\delta$, since $f_{\mathrm{rep,EO}}$ is set by a reference RF synthesizer and has negligible frequency drift within the measurement time span (see Methods). Due to the limited amplitude of the $\delta$ beat note, we chose to use an electronic spectrum analyzer instead of a frequency counter Drake et al. (2019); Zhang et al. (2019a); Del’Haye, Papp, and Diddams (2012) to capture $\delta$ (see Methods). A time series measurement of $\delta$ is presented in Fig 4e, where the Kerr soliton comb is free-running. The corresponding Allan deviations (Fig 4f) reveal a fractional-frequency stability of 1.6 $\times$ $10^{-7}$ for 1-s measurements, which translates to a 53.4-kHz fluctuation in $f_{\mathrm{rep,Kerr}}$. The observed 1-s Allan deviation is 36 times higher than the previously reported value from a silica microtoroid of similar FSR Zhang et al. (2019a). Several factors may contribute to the larger instability. First, LN is known to exhibit photorefactive effect which may introduce cavity resonance oscillations Sun et al. (2017). Second, the Kerr soliton pump laser used here (TSL-710, Santec) has $>$ 10 times broader short-term linewidth than the laser used in Zhang et al. (2019a). Furthermore, our fiber-to-chip edge coupling tends to be more sensitive to the environmental perturbations than the fiber-taper setup. The mitigation of these drifts will be subject of future investigations, for example, by minimizing the technical noise sources mentioned above and also by operating the Kerr soliton at a quiet point, as demonstrated in Yi et al. (2017); Lucas et al. (2020).

For proof-of-concept stabilization of $f_{\mathrm{rep,Kerr}}$, we build a simple computer-aided feedback control loop (see Methods) to improve $f_{\mathrm{rep,Kerr}}$ frequency stability. We digitally sample $\delta$ and compare it to a frequency setpoint, and use the generated error signal to adjust the Kerr soliton pump laser frequency to compensate for the $f_{\mathrm{rep,Kerr}}$ drift. Due to the latency (60 ms) of this digital approach, the effective feedback control bandwdith is limited to 17 Hz. Fig. 4g shows the stabilized $f_{\mathrm{rep,Kerr}}$ with much reduced long-term drift compared with the free-running case (Fig. 4e). Allan deviations of the in-loop $f_{\mathrm{rep,Kerr}}$ also reveal better long-term stability, e.g., a two order of magnitude improvement is achieved for 100-s measurements (Fig. 4f). Additionally, the in-loop $f_{\mathrm{rep,Kerr}}$ frequency drift distribution is shown in Fig. 4h, exhibiting a standard deviation of 79 kHz. For future improvement, we believe that a higher $\delta$ beat note SNR enabled by a stronger EO comb will permit the use of a faster frequency locker to improve both the long-term and short-term stability of $f_{\mathrm{rep,Kerr}}$ (see Methods).

In summary, we demonstrate a monolithic, hybrid Kerr and EO combs platform based on a LN thin film which permits the electronic access and control of terahertz Kerr soliton repetition rates by leveraging the simultaneous Kerr and Pockels nonlinearities inherent to LN. The monolithic circuit integrates Kerr soliton and EO microcombs on a single chip, and provides an on-chip interface between fast Kerr solitons and readily detectable microwaves. We also point out opportunities for further improvement of this chip platform. For example, EO microring modulation efficiency can be enhanced by either accessing a larger LN EO tensor component $r_{33}$ which is 6 times higher than $r_{22}$ used here or incorporating a microwave resonator Zhang et al. (2019b); Rueda et al. (2019). The need for the auxiliary pump laser could also be relaxed by aligning a soliton mode to the target EO microring mode. The demonstrated monolithic scheme thus holds great promise for the future development of fully integrated optical frequency divider on the LN thin-film platform for optical clock application, as well as study of soliton dynamics in hybrid Kerr and EO systems. It should be also noted that the hybrid comb architecture demonstrated here can be extended to other materials platforms that possess simultaneous Kerr and Pockels nonlinearities, such as AlN Liu et al. (2021), AlGaAs Chang et al. (2020), and GaP Wilson et al. (2019).

The Kerr soliton microring is constructed in an add-drop configuration. The drop-port waveguide directly guides solitons out of the chip, while the through-port waveguide couples to the following EO microring and delivers the hybrid combs for $f_{\mathrm{rep,Kerr}}$ detection.

To favor Kerr soliton formation over SRS, the intracavity pump mode threshold energy for SRS ($\mathrm{\varepsilon_{th,R}}$) needs to be raised above the threshold for soliton generation ($\mathrm{\varepsilon_{th,S}}$) Gong et al. (2020b, a). In practice, we fix the drop-port coupling rate ($\mathrm{\kappa_{ex1,Kerr}}$) and adjust the through-port coupling rate ($\mathrm{\kappa_{ex2,Kerr}}$) to raise $\mathrm{\varepsilon_{th,R}}$ over $\mathrm{\varepsilon_{th,S}}$. The Kerr soliton microring $\mathrm{TE_{00}}$ modes exhibit a typical intrinsic decay rate of $\mathrm{\kappa_{in,Kerr}}/(2\pi)$ $=$ 106 MHz and $\mathrm{\kappa_{ex1,Kerr}}/(2\pi)$ $=$ 13 MHz (900-nm coupling gap) in the telecom band. For example, in order to generate a Kerr soliton with a 5.6-THz FWHM, $\mathrm{\kappa_{ex2,Kerr}/(2\pi)}$ should be made smaller than 72 MHz. Here, we assume that $\mathrm{\kappa_{in,Kerr}}$, $\mathrm{\kappa_{ex1,Kerr}}$ and $\mathrm{\kappa_{ex2,Kerr}}$ are constant across the modes of interest, and the dominant Raman gain ($\mathrm{E(LO)_{8}}$) center Gong et al. (2020b, a) overlaps a local soliton forming mode. In the experiment, we adopted a 850-nm through-port-Kerr-microring coupling gap, which yields a $\mathrm{\kappa_{ex2,Kerr}/(2\pi)}$ $=$ 58 MHz and leads to successful soliton generation (Fig. 2). SRS is also suppressed in the EO microring, since the intracavity pump mode energy is far below $\mathrm{\varepsilon_{th,R}}$. The EO microring exhibits a typical intrinsic decay rate of $\mathrm{\kappa_{in,EO}}/(2\pi)$ $=$ 60 MHz and a waveguide coupling rate of $\mathrm{\kappa_{ex,EO}/(2\pi)}$ $=$ 20 MHz (600-nm coupling gap). The measured $\mathrm{TE_{00}}$ resonance spectra of both microrings are shown in Supplementary Note 1.

To allow for efficient interaction between the applied in-plane RF field and EO microring $\mathrm{TE_{00}}$ modes, we place 200-nm-thick GSG gold electrodes closely along the microring, keeping a 2 $\mu$m gap between them (Fig. 1c). The $\mathrm{TE_{00}}$ resonance EO tuning efficiency (DC) is estimated to be 1.5 pm/V based on COMSOL simulations and perturbation theory Teraoka, Arnold, and Vollmer (2003). The simulation of EO microcomb generation and EO microring DC tuning characterization are detailed in Supplementary Notes 3 and 4 respectively.

The proof-of-concept slow-speed feedback loop for stabilizing $f_{\mathrm{rep,Kerr}}$ is shown in Fig. 5a. We used an electronic spectrum analyzer (N9020a, Agilent) to capture $\delta$ with the built-in peak-search function. The recorded $\delta$ was then compared to a frequency setpoint on a computer. The frequency errors were converted to a voltage waveform by an arbitrary function generator (NI USB-6251). We fed the error signal to a proportional-integral-derivative controller (DigiLock 110, Toptica) to generate the feedback signal for adjusting the Kerr pump laser frequency via its internal piezoelectric transducer.

Each feedback loop took approximately 60 ms to complete, translating to an effective feedback bandwidth of 17 Hz. While it took 9 ms for the ESA to scan over a 8-MHz frequency span (1000 data points, 40-kHz RBW) and find $\delta$, most latency came from the computer executing the LabVIEW program (on Windows 10 operating system) and communication between the computer and instruments. Fig. 5c shows one example of the generated error signal and the corresponding feedback signal. The error signal was proportional to the frequency error by a ratio of 0.1 MHz/mV. The feedback signal drove the Kerr pump laser piezoelectric transducer, which has a frequency tunability of -3.3 MHz/mV and a 1-kHz actuation bandwidth, to reduce $f_{\mathrm{rep,Kerr}}$ drifts. The on-chip Kerr pump power is fixed at 50 mW during the process.

In the current system, the $\delta$ beat note exhibits an optimal SNR of 30 dB at a 40-kHz RBW with the peak power reaching -17 dBm (Fig. 5b). The SNR is currently limited by the available EO comb sideband power (Fig. 3b). Since the EO comb is free-running during the measurement, cavity-pump detuning could drift and thus cause the EO comb line power to fluctuate. These factors combined prevent the fast locking of $f_{\mathrm{rep,Kerr}}$. We anticipate that the use of a microwave resonator Zhang et al. (2019b); Rueda et al. (2019) and Pound-Drever-Hall stabilization technique Stone et al. (2018) will improve the beat note SNR. A larger SNR, e.g., 53 dB at a 40-kHz RBW, could allow using a frequency counter to directly read $\delta$, and a fast frequency locker (D2-135, Vescent Photonics, $>$500 kHz locking bandwidth) to enhance both the short-term and long-term stability of $f_{\mathrm{rep,Kerr}}$.

The add-drop configuration of the Kerr soliton microring allows for direct access of the soliton microcomb from the drop-port waveguide, where the soliton is not filtered by the EO microring. Supplementary Figure 8 shows one example of the outgoing Kerr soliton spectrum from the drop-port waveguide. The soliton spectrum exhibits a FWHM of 5.9 THz under a 60-mW on-chip pump power. Here, a weak EO pump signal appears on the spectrum. This may be due to residual reflection in between facets and couplers.

In the EO microring, the normalized electric field of a $\mathrm{TE_{00}}$ mode (by the sqaure root of its photon number) can be written as

$\displaystyle\vec{E}_{m}\left(\vec{r},t\right)$

$\displaystyle=$

$\displaystyle\vec{\phi}_{m}(\vec{r},\theta)e^{i\omega_{m}t}+\vec{\phi}_{m}^{*}(\vec{r},\theta)e^{-i\omega_{m}t},$

(2)

where $\phi_{j}(\vec{r},\theta)=\phi_{j}(\vec{r})e^{im\theta}$, $\omega_{m}$ and $m$ denote the spatial distribution, angular frequency and azimuthal number of the mode respectively. Optical modes are mutually orthogonal and their electric field distributions satisfy

$\displaystyle\begin{split}\int\int drd\theta\varepsilon_{0}{\varepsilon_{r,m}}\left(\vec{r},\theta\right)\vec{\phi}_{m}(\vec{r},\theta)\cdot\vec{\phi}_{n}^{*}(\vec{r},\theta)\\ =\frac{1}{2}\hbar\omega_{m}\delta\left(m-n\right),\end{split}$

(3)

with $\varepsilon_{0(r)}$ referring to the vacuum (relative) permittivity. Now, consider that the EO microring is modulated by a strong in-plane RF electric field with an angular frequency of $\Omega=\omega_{m}-\omega_{n}$ and an amplitude distribution of $\vec{E}_{eo}(\vec{r})f(\theta)$ ($\vec{E}_{eo}$ is in units of V/m, representing the RF drive strength) respectively, the EO interaction Hamiltonian between the optical modes $m$ and $n$ can be given by

	$\displaystyle H_{nl}$	$\displaystyle=\frac{1}{2}\epsilon_{0}\int\int{\vec{E}}\cdot(\chi^{(2)}:\vec{E}\vec{E})drd\theta$
		$\displaystyle\approx 3\epsilon_{0}\int\int{r_{22}}E_{eo}\left(\vec{r}\right)f(\theta)\phi_{m}(\vec{r})\phi_{n}^{*}(\vec{r})e^{i(m-n)\theta}drd\theta a_{m}a_{n}^{\dagger}$
		$\displaystyle+h.c.$		(4)

where $\vec{E}$ is the total electric field of the two optical modes and the RF drive. $a_{m}$ and $a_{n}$ are the Bosonic operators of the two optical modes. Here, we neglect high-frequency rotating terms $a_{m}a_{n}+h.c.$ and only consider the dominant electric field components of the $\mathrm{TE_{00}}$ modes and the RF drive (along LN Y-axis). Based on Eq. 3 and 4, we introduce the EO coupling strength ($g_{\mathrm{EO}}$) as

	$\displaystyle\hbar g_{\mathrm{EO}}=$	$\displaystyle 3\epsilon_{0}\int\int{r_{22}}E_{eo}\left(\vec{r}\right)f(\theta)\phi_{m}(\vec{r})\phi_{n}^{*}(\vec{r})e^{i(m-n)\theta}drd\theta$
	$\displaystyle=$	$\displaystyle 3\epsilon_{0}{r_{22}}\sqrt{\frac{\hbar\omega_{m}\hbar\omega_{n}}{4\epsilon_{0}\epsilon_{r,m}\epsilon_{0}\epsilon_{r,n}}}\times\frac{\int\int{E_{eo}\left(\vec{r}\right)}f(\theta)\phi_{m}(\vec{r})\phi_{n}^{*}(\vec{r})e^{i(m-n)\theta}drd\theta}{\sqrt{\int\int|\phi_{m}(\vec{r})|^{2}d\vec{r}d{\theta}}\sqrt{\int\int|\phi_{n}(\vec{r})|^{2}d\vec{r}d{\theta}}}.$		(5)

Note that in order to achieve effective coupling between the two optical modes, the RF field azimuthal distribution $f(\theta)$ needs to compensate the phase mismatch between optical modes, i.e., $e^{i(m-n)\theta}$.

In the experiment, the RF drive is strong and its frequency is set close to the EO microring FSR, leading to an effective linear coupling between adjacent $\mathrm{TE_{00}}$ modes under non-depletion approximation Rueda et al. (2019). The linearised interaction Hamiltonian among $\mathrm{TE_{00}}$ modes can be written as

$$H_{\mathrm{EO}}=\frac{1}{2}g_{\mathrm{EO}}\sum_{j}(a_{j}a_{j+1}^{\dagger}+a_{j}a_{j-1}^{\dagger})+h.c..$$

(6)

Assuming that $\int|\phi_{j}(\vec{r})|^{2}d\vec{r}$ is constant across the $\mathrm{TE_{00}}$ modes of interest and the RF drive has a 14-V peak amplitude, $g_{\mathrm{EO}}/(2\pi)$ can be estimated as 280 MHz via COMSOL simulations.

Based on the system Hamiltonian $H$ $=$ $H_{0}$ $+$ $H_{EO}$ ($H_{0}$ $=$ $\sum_{j}\hbar\omega_{j}a^{\dagger}_{j}a_{j}$, ignoring Kerr and Raman effects), the mean field dynamics of each $\mathrm{TE_{00}}$ mode can be derived as follows

	$\displaystyle\frac{d}{dt}a_{\mu}=$	$\displaystyle-(\kappa_{\mu}/2+i\Delta_{\mu})a_{\mu}-\frac{i}{2}g_{\mathrm{EO}}\sum_{k}[a_{k}\delta\left(k-\mu-1\right)$
		$\displaystyle+a_{k}\delta\left(k-\mu+1\right)].$		(7)

where $\Delta_{\mu}=\mathrm{D}_{\mathrm{int},\mu}+\omega_{0}-\omega_{\mathrm{P}}+\mu(D_{1}-\Omega)$, with $\omega_{0(\mathrm{P})}$ denoting the pump resonance (pump field) angular frequency, and $D_{\mathrm{1}}/(2\pi)$ representing the EO microring FSR. $\kappa_{\mu}$ is the total decay rate for the $\mu^{\mathrm{th}}$ $\mathrm{TE_{00}}$ mode, consisting of intrinsic cavity decay rate ($\kappa_{\mathrm{i},\mu}$) and coupling rate to the external bus waveguide ($\kappa_{\mathrm{e},\mu}$). The $\delta$ functions in Eq. 7 accounts for the phase-matching condition.

We move on to simulate EO microcomb formation by numerically solving the coupled mode equations (Eq. 7) using experimental parameters. The simulated spectrum is shown in Supplementary Figure 9, which agrees well with measured spectrum.

Finally, we experimentally and numerically characterized the modulator DC tuning efficiency. To avoid cancellation of EO modulation (a GSG probe was used here), we applied DC voltages across one racetrack arm and measured $\mathrm{TE_{00}}$ resonance spectra. The shifted $\mathrm{TE_{00}}$ resonance spectra under different DC offsets are plotted in Supplementary Figure 10a. The resonance tuning efficiency is extracted to be 1 pm/V for modulating one racetrack arm, which infers a $\sim$ 2-pm/V tuning efficiency for modulating the entire microring in a push-pull manner Wang et al. (2018). The inferred tuning efficiency is a bit higher than the simulated one (1.5 pm/V). The difference may be due to the mode crossing Xue et al. (2015) and uncertainties in material parameters. In simulations, LN refractive indices along Y(X)-axis and Z-axis are set as 2.21 and 2.14 respectively; LN DC dielectric constants along Y(X)-axis and Z-axis are set as 80 and 30 respectively.