The emergence of microresonator-based frequency combs (microcombs) has shaped the path toward miniaturized and scalable photonic devices [1–4]. In addition to their compactness, microcombs can be designed with comb spacing or free spectral ranges (FSR) from 2.6 GHz up to 1 THz [5–8] as opposed to fiber or solid-state mode-locked lasers where the comb fundamental repetition rate is often in the GHz range due to the bulk materials and cavity sizes [9–13]. Dissipative Kerr solitons (DKSs), where the temporal shape of the pulse is preserved due to achieving a balance between nonlinear gain and dispersion, can form in the microresonator [14–19]. The smooth spectrum and high coherence of the comb lines make DKS suitable for a wide variety of applications, such as optical frequency synthesis [20], optical clockworks [6], microwave generation [21,22], coherent optical communications [23], distance measurements [24,25], exoplanet searching [26], and optical coherence tomography [27].

Microwave signals derived from frequency combs feature high spectral purity because they are derived from a large division of the noise from the optical to the microwave frequencies via photodetection of the pulse train [28]. As a result, the small size, weight, and power (SWaP) and optical-to-microwave noise reduction have increased interest in microcombs, including those with low repetition rates [5,29]. Recently, DKS with microwave FSR has been demonstrated in different structures and materials such as crystalline microresonators [21,30,31], silica [32–35], silicon nitride (Si3N4) [7,8,14,22,36–39], and lithium niobate [40]. In particular, complementary metal–oxide semiconductor (CMOS) compatible planar integrated microresonators are of great interest because of their monolithic electronic and photonic integration capabilities. The lowest phase noise for a microwave carrier was achieved in crystalline microresonators [21,30]; however, they are not CMOS compatible for large-scale integration. Silica has demonstrated very low-noise microwave signal carriers due to their low thermal noise, but the low index of silica makes it difficult for monolithic integration, leading to the use of tapered fibers for optical coupling of the resonator [32,33]. Si3N4 microresonators show the most promising path toward full monolithic integration due to their integrated planar design and the simplicity of mode and dispersion engineering compared to the abovementioned materials; however, the generated microwave signals from these microresonators to date have shown higher phase noise compared to other materials [22,36].

The noise sources in the DKS-based microwave oscillators have been studied extensively [22,31,32,41–43]. The performance of these microwave oscillators still has not reached their fundamental limit given by the fundamental thermorefractive noise (TRN) of the microresonators [44,45] due to the presence of other higher noise sources such as laser phase noise transduction due to detuning dependent Raman self-frequency shift (RSFS) and soliton recoil induced by dispersive waves [31,42,46,47]. Such a phenomenon has been exploited to reduce the phase noise of the soliton repetition rate by operating on a narrow pump detuning region such that RSFS is balanced by soliton recoil where the repetition rate dependence on detuning is minimized, which is known as quiet point operation [41]. Furthermore, the use of an auxiliary mode has been demonstrated to provide passive thermal stabilization of the resonator and hence reduce the repetition rate phase noise [42,48,49]. The auxiliary mode provides a cooling effect such that the auxiliary mode pins the main resonance, which leads to fixed pump-cavity detuning. Also, it has been shown that power fluctuations within the microresonator can couple to the phase noise of the soliton repetition rate [50]. Dispersive waves induced by avoided mode crossing (AMX) [32] introduce additional spatiotemporal thermal noise to the soliton repetition rate through uncorrelated thermal fluctuations between different modes of the microresonator.

Here, we show that a soliton with a low phase noise over a large 4 GHz detuning range can be achieved by both minimizing the detuning-dependent RFSS and soliton recoil, and the use of a hybridized mode for self-thermal stabilization and a stable soliton generation. We achieve a measured single-sideband phase noise lower than −110dBc/Hz at 10 kHz offset from the 19.69 GHz carrier over the detuning range reaching a minimum of −117dBc/Hz (−123dBc/Hz-scaled to 10 GHz), which is, to the best of our knowledge, the lowest phase noise achieved in free-running microwave chip-based DKS without applying any active stabilization. Furthermore, we show that active stabilization of the DKS can overcome the TRN by locking the soliton repetition rate to an external reference derived from fiber-based frequency combs, with a resulting microwave phase noise lower than the microresonator TRN.

To generate a self-stabilized soliton microcomb, a tunable external cavity diode laser (ECDL) is amplified by an L-band erbium-doped fiber amplifier (EDFA), sent to a polarization beam splitter (PBS) to select the TM mode, and then coupled to a strongly coupled dispersion-managed paw-shaped compact Si3N4 microresonator with a single facet coupling loss of 3 dB resulting in ≈800mW on-chip power. This is illustrated in Fig. 1(a). A single soliton is generated via manual forward piezoelectric laser wavelength tuning into the resonance, followed by a small backward detuning to fall into the soliton state [18,51,52]. The soliton has an existence range of several GHz due to the auxiliary mode; once the soliton is generated, it stays for hours without any active stabilization. At the output, a fiber Bragg grating (FBG) is used to filter the pump from the spectrum before amplifying the comb power to ≈20mW. The amplified comb is subsequently sent into a fast high-power photodiode and the detected radio frequency (RF) beat is filtered by a bandpass filter and amplified to ≈7.9dBm by a low-phase noise RF amplifier.

The free spectral range (FSR) and loaded quality factor (Q) of the fundamental TE mode are ≈19.69GHz and ≈1.4 million, respectively. The optical spectrum of the 19.69 GHz soliton is shown in Fig. 1(c) via an optical spectrum analyzer (OSA; Yokogawa) with a 50 pm resolution. The measured local second-order dispersion near the AMX is found to be D2/2π=30.1±4.6kHz (see Appendix A). The presence of third-order dispersion leads to the formation of a dispersive wave near 1870 nm, which leads to asymmetry in the optical spectrum. Such asymmetry and the presence of an AMX signature at the pump result in the spectrum differing from a baseline sech2 fit, as illustrated in Fig. 1(c). By using the comb lines around the pump, an example sech2 fit is shown in orange, where there is no OSA-discernible soliton self-frequency and recoil shift from the pump.

Figure 2(a) shows the detected microwave signal at 19.69 GHz and phase noise characteristics in the free-running state. The detected microwave beat RF spectrum is shown in Fig. 2(b) with 1 kHz RBW. The absolute single-sideband (SSB) phase noise power spectral density of the 19.69 GHz microwave carrier is measured using a phase noise analyzer (Rohde & Schwarz FSWP26) and reaches −85dBc/Hz at 1 kHz offset, −117dBc/Hz at 10 kHz, and −137dBc/Hz at 100 kHz. At higher offset frequencies, the microresonator noise is limited at −147dBc/Hz, which is about 3 dB higher than the estimated shot noise. This higher noise floor is caused by amplified spontaneous emission (ASE) from EDFA [53].

The soliton repetition rate is expressed as frep=12π[D1+D2D1Ω(Δ)],(1)where D1/2π is the FSR and D2/2π is the second-order dispersion of the microresonator, Ω(Δ) is the detuning dependent total spectrum shift due to the Raman-induced soliton self-frequency shift (SSFS) ΩRaman and the dispersive-wave-induced frequency recoil of the soliton ΩRecoil. Δ is the difference between the pump frequency and the cold cavity frequency. The soliton repetition rate phase noise originates from different sources. It can be seen that the contribution of the fluctuations of Ω(Δ) to frep can be reduced by minimizing D2 or Ω(Δ), or both. Besides, the different contributions of soliton repetition rate noise Sfrepϕ can be expressed as [22,32] Sfrepϕ=β2Sδf,p+α2f2SRIN,p+γ2ωp2SAMX+D12STRN+SQ,(2)where Sδf,p is the pump phase noise, β2=∂frep∂δfp is the optical-to-microwave phase transduction (PM-to-PM) coefficient of the laser noise, SRIN,p is the pump intensity noise that is related to the soliton repetition rate phase noise by applying a 1/f2 factor, and α2 is the transduction factor of the intensity noise of the pump to the phase noise of soliton repetition rate through the Kerr effect [22,31,50]. SAMX is the AMX-induced spatiotemporal thermal noise of the hybridized mode [32], ωp is the frequency of the pump laser, γ2 is the transduction factor of the spatiotemporal thermal fluctuations to the phase noise of soliton repetition rate, STRN is the thermorefractive noise of the soliton mode [44], and SQ is the quantum jitter. First, we measure the frequency noise of our ECDL by heterodyne beating with a 1 Hz linewidth stable laser to estimate the influence of the frequency noise of our ECDL in the generated microwave signal. The noise transfer from our laser to microwave noise (PM-to-PM) transduction coefficient (β2) is estimated at −73dB, as shown in black in Fig. 2(a). This is evidenced by the ECDL noise features imprinted on the microwave beat note at an offset below 2 kHz as well as a slight noise bump at the 20 kHz offset. This is further verified by modulating the phase of the pump laser by a 5 kHz sinusoidal signal to act as a calibration tone in the soliton microwave beat power spectral density [31]. Compared to the best existing Si3N4-based low repetition rate free-running DKS [22], our PM-to-PM transduction is 18 dB lower.

The laser PM-to-PM is only 7 dB higher than the limit imposed by phase noise reduction due to the optical-to-microwave division (20log10fpumpfrep=80dB). In addition, we investigate the noise contribution preventing us from achieving even lower noise. The intensity fluctuation of the laser pump and EDFA is measured as the relative intensity noise (RIN), as shown in Fig. 2(c). The intensity noise of the pump transduction to the phase noise in the microresonator does not seem to be a limiting factor at any frequency offset, as shown in orange in Fig. 2(a), where the fit represents the maximum possible noise transduction. The dominant noise at offsets higher than 2 kHz corresponds to the dispersive wave noise due to the AMX among the soliton mode, and fundamental transverse electric mode (TE0), and the fundamental transverse magnetic mode (TM0). The calculation of the AMX-induced spatiotemporal thermal noise fits well with the measured phase noise of the soliton at offsets higher than 2 kHz with γ2=58dB in our microcomb subsystem (see Appendix B).

The fractional frequency Allan deviation of our free-running microcomb is shown in Fig. 2(e). For short integration times, the Allan deviation is calculated by integrating the phase noise from Fig. 2(a); the result obtained from a frequency counter with a gate time of 10 ms [shown in Fig. 2(d)], without removing the linear drift, is used to compute the Allan deviation for longer integration times. The obtained Allan deviations for both cases show good overlap between the 10 ms and 1 s integration time, with a free-running Allan deviation at 8.1×10−9 in 1 s. Furthermore, the long-term stability of the free-running microcomb is dominated by the pump laser frequency fluctuations shown in the same figure.

We examine the phase noise of the microwave beat signal along the large soliton existence range to probe the possibility of degraded phase noise for different detuning. Figure 3(a) shows the 19.69 GHz SSB phase at 10 kHz offset for different detunings and the corresponding repetition rate shift. First, the monotonic linear behavior of the repetition rate frequency shift with respect to detuning demonstrates the absence of quiet point [31,41,55]. This also confirms that our low-noise operation does not manifest itself as a quiet-point operation. This is also experimentally verified by the absence of a significant soliton spectrum frequency shift with respect to our pump wavelength in Fig. 3(c). However, we still observe an increase of more than 20 dB in additional noise as we change the detuning. Such excess noise potentially originates from the formation of breather solitons due to AMX [56]. The amplitude modulation of the soliton increases the amplitude to phase noise conversion inside the microresonator, as shown in Fig. 3(b) where the pump frequency noise features at 2 kHz and 20 kHz offset (highlighted in the blue shaded regions) are now hidden (under the excess noise). Figures 3(d)–3(f) show the different RF spectra for different detunings before and after the aforementioned soliton breather formation, which starts at a detuning of ≈4.2GHz, where the formation of sidebands on the microwave beat note of the soliton with a fundamental breathing frequency of ≈291.6MHz starts to appear.

We test whether the thermorefractive noise poses a fundamental limit when actively stabilizing the soliton repetition rate to an external reference, as shown in Fig. 4(a). We synthesize an ultralow phase noise 20 GHz microwave signal using a commercial self-referenced fiber frequency comb (FFC) system with a repetition rate of 250 MHz. One of the comb lines of the FFC is locked to an ultrastable laser with 1 Hz linewidth after subtraction of the carrier envelope offset frequency fceo to make the locking independent of fceo [57]. The low repetition rate of the FFC results in low power in the 80th harmonic of the repetition rate (80frepFFC=20GHz). Cascaded Mach–Zehnder fiber interferometers are used as interleavers for repetition rate multiplication to overcome this problem. This setup increases the signal level of the photodetected pulse train and reduces the amplitude-to-phase noise conversion in the photodetection [58]. An RF bandpass filter is used to select the 20 GHz microwave signal, and then amplified by a low-phase noise amplifier to 2 dBm. The synthesized 20 GHz phase noise is more than 10 dB lower than the TRN of our microresonator at 1 kHz offset. The 20 GHz microwave signal for the FFC is then mixed with the soliton microwave beat to generate a ≈310MHz signal that is further mixed down to DC with a low-noise microwave generator (Rohde & Schwarz SMA100B) to generate the error signal. This error signal is fed to the PID controller that actuates the pump power using an intensity modulator (IM) before the microresonator. In our setup, the modulation of the pump power, rather than pump detuning, via ECDL current modulation provides a better lock-in sensitivity (see Appendix C). Figure 4(b) shows the phase noise of the stabilized soliton microwave beat, with SSB phase noise of −102dBc/Hz at 100 Hz offset (instrument limited), −127dBc/Hz at 1 kHz offset, and −136dBc/Hz at 10 kHz offset. At 1 kHz the SSB phase noise is 14 dB lower than the optical-to-microwave divided TRN limit. This implies that one can overcome the TRN limit of SiN microresonators via active stabilization. Figure 4(c) shows the RF spectrum with 1 Hz RBW to demonstrate the long-term drift difference between the free-running and stabilized microwave beat notes. The stabilized soliton microwave beat note can track the frequency of the microwave signal 80frepFFC over a 25 kHz range before losing the servo lock, as shown in Fig. 4(d).

With the verified stabilization below the TRN bounds, we next stabilize the soliton microwave beat note using all-optical stabilization through the technique of two-point optical frequency division [59,60]. Figure 5(a) shows the experimental setup where the soliton spectrum is combined with the fiber comb spectrum, and then two optical lines of the soliton microcomb are selected using dense wavelength-division multiplexing (DWDM) filters. Each channel is sent into a photodiode to detect the beat between the soliton microcomb line and the fiber comb line. The two generated RF beat notes are mixed to generate a frequency that contains the relative noise between the two soliton microcomb lines. The resultant beat is mixed down to DC using an RF signal generator and sent into the servo controller. Figure 5(b) shows the three different optical channel spacings used in this measurement: 100 GHz, 400 GHz, and 1 THz. The corresponding stabilized phase noise for each of the three channel spacings is shown in Fig. 5(c), where the experimental improvement in each case follows the 20log10fspacing. We note that the close-in phase noise, such as at 100 Hz, shows more than a 40 dB improvement, even below the microresonator TRN. Further phase noise reduction can be achieved by increasing the optical spacing of the two references. This demonstration verifies that the microwave-rate DKS microcombs can aid in two-point optical frequency division efforts, inheriting the stability of the two references to generate stable carrier signals below thermorefractive noise bounds.

In this work, we demonstrated a low repetition rate soliton based on a Si3N4 microresonator with a record low phase noise over a large pump detuning range without requiring active pump cavity detuning stabilization. The low-phase noise Si3N4 microresonator provides a compact and scalable solution for clean microwave signal generation. The measured free-running phase noise (scaled to 10 GHz) of −123dBc/Hz at 10 kHz offset frequency is lower than that of existing chip-based dissipative Kerr soliton sources [22,32,33,35,36,39,40]. The low-noise performance was achieved with the assistance of an auxiliary mode and the minimal soliton spectrum shift yielding a −73dB PM-to-PM noise conversion from laser noise to the soliton repetition rate. We have also observed AMX-induced spatiotemporal noise limiting the performance of our Si3N4 microresonator. Furthermore, we demonstrated that the fundamental thermorefractive noise of the soliton microwave signal can be suppressed by actively stabilizing our soliton to an ultrastable fiber frequency comb oscillator. The achieved phase noise of the stabilized soliton is 10 dB lower than the thermorefractive noise of the soliton. The use of an auxiliary mode with orthogonal polarization and a similar intensity profile increases the thermal correlation between the soliton mode and the dispersive wave mode, hence reducing the AMX-induced thermorefractive noise due to the difference in the intensity mode profile. Further delocalization of the mode with larger mode volumes and low confinement can lower the thermal noise of the microresonator frequency combs.

Acknowledgment. We acknowledge discussions with Tara Drake and Michele Giunta. We also acknowledge the loan of the Mach–Zender interleaver from Menlo Systems. A.A. acknowledges support from the Kuwait University scholarship program.

Author Contributions.A.A. conducted the experiments with assistance from H.C. and D.I.L. A.A. analyzed the data and performed the simulations. W.W. performed the device characterization. T.M. and A.C. contributed to the experiments. M.Y., G.-Q.L., and D.L.K. performed the device nanofabrication. A.A. and C.W.W. initiated the project. A.A. and C.W.W. wrote the manuscript.