Ultralow-Noise Photonic Microwave Synthesis using a Soliton Microcomb-based Transfer Oscillator

TL;DR

This paper demonstrates a method to generate ultralow-noise microwave signals using a soliton microcomb and transfer oscillator technique, achieving phase noise levels comparable to the best optical frequency division methods.

Contribution

It introduces a transfer oscillator approach with a soliton microcomb at 14 GHz, enabling ultralow-noise microwave synthesis without requiring high repetition rates or tight comb stabilization.

Findings

Achieved phase noise below -60 dBc/Hz at 1 Hz offset

Demonstrated transfer oscillator noise cancellation

Highlighted potential for integrated microcombs in future applications

Abstract

The synthesis of ultralow-noise microwaves is of both scientific and technological relevance for timing, metrology, communications and radio-astronomy. Today, the lowest reported phase noise signals are obtained via optical frequency-division using mode-locked laser frequency combs. Nonetheless, this technique ideally requires high repetition rates and tight comb stabilisation. Here, a soliton microcomb with a 14 GHz repetition rate is generated with an ultra-stable pump laser and used to derive an ultralow-noise microwave reference signal, with an absolute phase noise level below -60 dBc/Hz at 1 Hz offset frequency and -135 dBc/Hz at 10 kHz. This is achieved using a transfer oscillator approach, where the free-running microcomb noise (which is carefully studied and minimised) is cancelled via a combination of electronic division and mixing. Although this proof-of-principle uses an auxiliary comb for detecting the microcomb's offset frequency, we highlight the prospects of this method with future self-referenced integrated microcombs and electro-optic combs, that would allow for ultralow-noise microwave and sub-terahertz signal generators.

Equations

$$f_{\mathrm{signal}}=\frac{f_{\text{textsc}ceo}}{N}+f_{\mathrm{rep}}=\frac{{\nu }_{\text{textsc}usl}}{N}$$

$$f_{\mathrm{rep}}^{\mathrm{K}}=\frac{1}{2\pi }\left(D_1+\frac{D_2}{D_1}\Omega (\delta )\right)$$

$$\frac{\delta D_1(\omega )}{2\pi }=\underset{\alpha }{\underbrace{\left(\frac{D_1}{2\pi }\frac{4\eta c{n}_2}{\kappa V_{\mathrm{eff}}{n}_0^2}\right)}}\delta P_{\mathrm{in}}(\omega )$$

$$S_{D_1/2\pi }^{\varphi }(f)={\left(\frac{\alpha }{f}{P}_{\mathrm{in}}\right)}^2{S}_{\mathrm{RIN}}(f)$$

Full text

††thanks: These authors contributed equally to this work.††thanks: These authors contributed equally to this work.

Ultralow-Noise Photonic Microwave Synthesis using a Soliton Microcomb-based Transfer Oscillator

Erwan Lucas

Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

  

Pierre Brochard

Laboratoire Temps-Fréquence, Université de Neuchâtel, CH-2000 Neuchâtel, Switzerland

  

Romain Bouchand

Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

  

Stéphane Schilt

Laboratoire Temps-Fréquence, Université de Neuchâtel, CH-2000 Neuchâtel, Switzerland

  

Thomas Südmeyer

Laboratoire Temps-Fréquence, Université de Neuchâtel, CH-2000 Neuchâtel, Switzerland

  

Tobias J. Kippenberg

tobias.kippenberg@epfl.ch

Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

**The synthesis of ultralow-noise microwaves is of both scientific and technological relevance for timing, metrology, communications and radio-astronomy. Today, the lowest reported phase noise signals are obtained via optical frequency-division using mode-locked laser frequency combs. Nonetheless, this technique ideally requires high repetition rates and tight comb stabilisation. Here, a soliton microcomb with a 14 GHz repetition rate is generated with an ultra-stable pump laser and used to derive an ultralow-noise microwave reference signal, with an absolute phase noise level below $-60\text{\,}\mathrm{d}\mathrm{B}\mathrm{c}\mathrm{/}\mathrm{H}\mathrm{z}$ at 1 Hz offset frequency and $-135\text{\,}\mathrm{d}\mathrm{B}\mathrm{c}\mathrm{/}\mathrm{H}\mathrm{z}$ at 10 kHz. This is achieved using a transfer oscillator approach, where the free-running microcomb noise (which is carefully studied and minimised) is cancelled via a combination of electronic division and mixing. Although this proof-of-principle uses an auxiliary comb for detecting the microcomb’s offset frequency, we highlight the prospects of this method with future self-referenced integrated microcombs and electro-optic combs, that would allow for ultralow-noise microwave and sub-terahertz signal generators. **

Introduction

The synthesis of microwave signals via photonic systems, such as dual frequency lasers Pillet et al. (2008), optoelectronic oscillators Maleki (2011), Brillouin oscillators Li et al. (2013), or electro-optical dividers Li et al. (2014), hold promise for their ability to synthesise low-noise or widely tunable microwave signals with compact form factor. An additional approach is based on optical frequency division, which makes use of a self-referenced fs-laser comb optically-locked to an ultra-stable laser (USL) with a typical linewidth at the Hz-level Millo et al. (2009); Fortier et al. (2011); Portuondo-campa et al. (2015); Xie et al. (2017). If the comb line of index $N$ is tightly phase-locked to the USL (after subtraction of the carrier envelope offset (CEO) frequency $f_{\textsc{ceo}}$ or simultaneous stabilisation of $f_{\textsc{ceo}}$), the comb repetition rate $f_{\rm rep}$ is directly phase-stabilised to the ultra-stable frequency $\nu_{\textsc{usl}}$ by frequency division: $f_{\rm rep}=\nu_{\textsc{usl}}/N$. Importantly, owing to the carrier frequency division from optics to microwaves, the absolute phase noise power spectral density is reduced by a factor $N^{2}\sim 10^{8}$.

This method has been mostly implemented using fibre-based fs-lasers with repetition rates of a few hundred megahertz. A fast actuator (e.g., an intra-cavity electro-optic modulator Hudson et al. (2005)) is required to achieve a tight optical lock of the comb tooth to the optical reference and perform the frequency division over a wide bandwidth. Moreover, a high harmonic of the comb repetition rate must be used to synthesise a microwave signal beyond 10 GHz. Consequently, repetition rate multipliers are typically employed to reduce the impact of shot-noise in the photo-detection of the pulse train, such as optical filtering cavities Diddams et al. (2009) or fibre interleavers Haboucha et al. (2011), which increases the system complexity. Therefore, the use of frequency combs directly operating at $\sim 10$ GHz repetition rates would be highly beneficial, but their optical lock and self-referencing are challenging.

Microresonator-based Kerr frequency combs (i.e., ‘microcombs’), which naturally produce multi-GHz comb spectra generated via four-wave mixing in an optical microresonator Gaeta et al. (2019); Kippenberg et al. (2018), are natural candidate in this context. Pumping a cavity resonance with a continuous-wave laser can initiate and sustain a circulating dissipative Kerr soliton (DKS) pulse Herr et al. (2013); Yi et al. (2015); Brasch et al. (2015); Joshi et al. (2016) that is intrinsically phase-coherent with the input pump laser. The resulting comb coupled out of the micro-cavity is inherently perfectly phase-locked to the pump laser, without any actuator locking bandwidth limitation. Direct soliton generation from an ultra-stable pump laser holds potential for compact and powerful optical-to-microwave dividers. Although self-referenced optical microcombs and clocks have been demonstrated Jost et al. (2015); Del’Haye et al. (2016); Newman et al. (2019), optical frequency division for ultralow-noise microwave generation using such devices has not been demonstrated so far, mainly due to the complex crosstalk occurring between their two degrees of freedom Del’Haye et al. (2016); Stone et al. (2018) and the limited performance of the available actuators Papp et al. (2013); Joshi et al. (2016).

Here, we demonstrate the generation of an ultralow-noise microwave signal using a microcomb-based transfer oscillator method to realise optical-to-microwave frequency division. The transfer oscillator method Telle et al. (2002); Brochard et al. (2018) bypasses the need for tight optical phase-locking of the frequency comb to the optical reference. Instead, it relies on an adequate manipulation and combination of signals to cancel the comb phase noise and to provide a broadband electronic division of the USL frequency to the microwave domain. The frequency division by a large factor $N$ is performed electronically, thus removing the need for high locking bandwidth actuators. In this work, the USL is used to pump the microresonator and inherently constitutes a tooth of the resulting frequency comb. We show how to extend the transfer oscillator technique to exploit this salient feature of microcombs (or equivalently of electro-optic combs Beha et al. (2017)). In this proof-of-principle demonstration, we achieved a measured single-sideband phase noise of $-110$ dBc/Hz at 200 Hz offset from the 14.09 GHz carrier, which is 15 dB below the lowest phase noise microresonator-based photonic oscillator reported so far Liang et al. (2015a), demonstrating the potential of this approach.

Results

Transfer oscillator principle

The high-level working principle of our method is illustrated in Fig. 1. A microresonator pumped by a sub-Hz-linewidth USL at frequency $\nu_{\textsc{usl}}$ generates a soliton-Kerr comb with a GHz-range repetition rate $f_{\rm rep}$ that is set by the resonator free spectral range (FSR). The reference laser is part of the frequency comb (line $N$) such that its frequency can be written as $\nu_{\textsc{usl}}=f_{\textsc{ceo}}+N\,f_{\rm rep}$. The detection of the CEO frequency (for example via $f-2f$ interferometry Telle et al. (1999); Jones et al. (2000) or with an auxiliary self-referenced comb as in the present work) is followed by electronic division by means of a combination of frequency pre-scalers and direct digital synthesisers (DDS). The final step consists of mixing the divided CEO signal with the repetition rate, which yields

[TABLE]

Importantly, this process can be carried out with a free-running Kerr comb and circumvents the need for a high-bandwidth repetition rate lock.

Microcomb generation with the USL

Due to the limited tuning of the resonator and narrow bandwidth of the microcomb used for this proof-of-concept, additional hardware was needed to demonstrate the transfer oscillator principle, as shown in Fig. 2. The soliton-Kerr comb is generated by pumping a crystalline magnesium fluoride (\ceMgF2) microresonator with an FSR of 14.09 GHz using a 1553 nm diode laser, which is initially quickly scanned across a resonance for soliton generation Herr et al. (2013). Next, the pump laser is phase-locked to a sub-Hz-linewidth USL (Menlo Systems ORS1500) at a frequency detuning of $\sim 1.7$ GHz, thereby acquiring a comparable level of purity and stability (green box in Fig. 2, a detailed description is provided in the Supplementary Note LABEL:SI:PLL_soliton).

To ensure the long-term stable operation of the Kerr comb and prevent the decay of the soliton, the microresonator resonance is slowly locked to the pump laser with an effective-detuning stabilisation, achieved via a sideband Pound-Drever-Hall (PDH) lock Thorpe et al. (2008), which feedbacks on an acousto-optic modulator (AOM) that modulates the pump power and thus thermo-optically tunes the resonator, in addition to a slow thermal actuation of the microresonator (see details in the Supplementary Note LABEL:SI:res_stab). The detuning setpoint was carefully optimised in order to minimise the noise of the Kerr comb repetition rate $f_{\rm rep}^{\rm K}$ at offset frequencies beyond $\sim 100$ Hz (see the Methods section and Fig. 6). However, the residual thermal drift of the resonator degrades the performance at lower offset frequencies.

Offset detection with an auxiliary comb

The used crystalline \ceMgF2 micro-comb features a relatively narrow spectrum that prevents a direct detection of its CEO frequency (Fig. 2). The self-referencing of Kerr combs remains highly demanding due to the high repetition rate, low optical power, and fairly long pulse duration (225 fs here) resulting in a low peak intensity that makes the spectral broadening for $f-2f$ interferometry challenging Del’Haye et al. (2016); Lamb et al. (2018). Therefore, we implemented an indirect detection scheme using an auxiliary self-referenced fibre-laser frequency comb Tian et al. (2018a) with a repetition rate $f_{\rm rep}^{\rm aux}=251.7$ MHz, as schematised in Fig. 2. Provided the repetition rates of both combs are harmonically phase-locked, i.e., $f_{\rm rep}^{\rm K}=M\,f_{\rm rep}^{\rm aux}$ (superscripts ‘K’ and ‘aux’ refer to the Kerr and auxiliary combs, respectively), then the optical beatnote between the two combs corresponds to their relative CEO frequency $\Delta f_{\textsc{ceo}}=f_{\textsc{ceo}}^{\rm K}-f_{\textsc{ceo}}^{\rm aux}$, as the repetition rate noise contributions compensate each other in this beat signal. The Kerr comb CEO frequency is then obtained by mixing out the CEO frequency of the auxiliary comb $f_{\textsc{ceo}}^{\rm aux}$ detected with an $f-2f$ interferometer (see Fig. 2) and corresponds to $f_{\textsc{ceo}}^{\rm K}=\Delta f_{\textsc{ceo}}+f_{\textsc{ceo}}^{\rm aux}=\nu_{\text{pump}}-Nf_{\rm rep}^{\rm K}$ (grey box in Fig. 2). Importantly, the auxiliary comb is not stabilised to the USL at any point and thus does not perform the division. Its role is limited to the offset detection in this demonstration.

The mutual harmonic phase-locking of the comb repetition rates is achieved via soliton injection-locking Weng et al. (2019). The harmonic $M=56$ of the repetition rate of the auxiliary comb (at 14.093 GHz) is detected, filtered and amplified to phase-modulate the pump light using an electro-optic modulator (EOM, blue box in Fig. 2). This frequency is very close to the native microcomb line spacing, which gets injection-locked to this drive signal. Therefore, both repetition rates are strongly correlated over a bandwidth of $\sim 2$ kHz (see the Supplementary Note LABEL:SI:InjLocking and Supplementary Figure LABEL:fig:INJLOCK).

Transfer oscillator chain

The Kerr comb CEO signal, indirectly obtained as previously described, is detected at low frequency (MHz-range) and filtered to match the bandwidth of the injection locking of the repetition rate (not represented in Fig. 2, see the Supplementary Note LABEL:SI:TO_chain and Supplementary Figure LABEL:fig:DivisionChain). After up-mixing to 15 GHz, it is frequency-divided by a large pre-determined factor $N\approx 13,698$ and is subtracted to the separately-detected repetition rate $f_{\rm rep}^{\rm K}$ to obtain the frequency-divided signal of the ultra-stable pump laser: $\nu_{\text{pump}}/N=f_{\textsc{ceo}}^{\rm K}/N+f_{\rm rep}^{\rm K}$ (orange box in Fig. 2). The overall division of the Kerr comb CEO signal by the factor $N$ is realised with a frequency pre-scaler followed by two parallel DDS, which offers improved filtering capabilities in the electronic division Brochard et al. (2018). This second stage division with the DDS allows for a precise non-integer frequency division factor and leads to a clean single-tone output signal corresponding to the frequency-divided USL (see Fig. 3). The detailed description of the frequency division chain is provided in the Supplementary Note LABEL:SI:TO_chain.

The overall division factor $N$ was accurately determined experimentally, without prior knowledge of the optical frequency of the ultra-stable pump laser, by measuring the frequency change of the generated microwave signal corresponding to a small variation (140 Hz) of the Kerr comb repetition rate for different programmed division factors $N$ (see Fig. 3). This simple measurement also provides an accurate determination of an optical comb line index $N$ that can be useful for absolute optical frequency measurements.

Microwave characterisation

The phase noise of the generated ultralow-noise 14.09 GHz signal was measured with a cross-correlator phase noise analyser (Fig. 3). It reaches $-110$ dBc/Hz at 200 Hz Fourier frequency, 15 dB below the lowest phase noise microresonator-based photonics oscillator Liang et al. (2015a) at 10 GHz. The phase noise is below $-135$ dBc/Hz at 10 kHz and $-150$ dBc/Hz at around 1 MHz, showing that the intrinsic low-noise properties of the soliton Kerr comb at high Fourier frequencies are preserved. The calculated shot-noise predicts a noise floor at $-152$ dBc/Hz (thermal noise floor $\sim-170$ dBc/Hz). At 1 Hz offset, the measurement is limited by the instrumental noise floor below $-60$ dBc/Hz, even with 3000 cross correlations. Nevertheless, the transfer oscillator offers an improvement by at least 40 dB compared to the direct detection of the Kerr comb repetition rate (despite the resonator being stabilised to the USL), showing its ability to cancel the residual thermal drifts of the Kerr cavity. The technical limitations of the measurement, at low offset frequency, make it difficult to directly compare the method with state-of-the-art optical frequency-division using mode-locked lasers. Nevertheless, at high Fourier frequencies, our results surpass some of the first demonstrations of optical frequency division Millo et al. (2009), even if no optimisation has been performed on the photodetection side. Over the past 10 years, the development of mode-locked lasers, as well as the improvement of photodetection noise Ivanov et al. (2005); Bouchand et al. (2017), led to a reduction of the noise of the generated microwaves by 30 to 40 dB in some frequency bands Xie et al. (2017). We believe that the transfer oscillator method can follow a similar path, as in particular, the high repetition rates of the Kerr combs should make the photodetection optimisation less stringent.

Discussion

In summary, we have reported optical-to-microwave frequency division using a Kerr comb as transfer oscillator. This demonstrates the potential of this method in microwave photonics and enlarges its previously reported implementation with low repetition rate mode-locked lasers. The approach presented here can be further implemented with electro-optic combs, where self-referencing and feedback control were recently achieved Carlson et al. (2018); Beha et al. (2017). Although this proof-of-principle experiment required an auxiliary comb to obtain the CEO frequency of the Kerr comb, directly self-referenced microcombs are technologically feasible in silicon nitride (\ceSi3N4) photonic-chips Brasch et al. (2017). While octave-spanning comb spectra have been achieved using dispersion control Pfeiffer et al. (2017); Newman et al. (2019), these implementations used THz repetition rates to cover such a large spectral range, which made photodetection of the repetition rate practically impossible. Nonetheless, the residual phase noise of these combs has been shown to be suitable for frequency division Drake et al. (2019a). Recent improvements of integrated resonators have enabled soliton microcombs with K- and X-band (20 and 10 GHz) repetition rates in integrated resonators Liu et al. (2019). However, the achieved spectral spans, although wider than in the crystalline case, are far from covering one octave. Pulsed pumping Obrzud et al. (2017) appears as a promising approach to enable octave spanning microcombs with detectable microwave repetition rates. This approach uses synchronous pumping of the microresonator with picosecond pulses to generate a soliton with a much shorter duration and a spectrum that can cover an octave, similar to enhancement cavities Lilienfein et al. (2019). It can be seen as a hybrid between an electro-optic (EO) comb and a microcomb, with the advantage that the spectral enlargement of the EO comb is performed in cavity and is therefore directly filtered Anderson et al. (2019); Brasch et al. (2019). Crucially, even if the free-running phase noise of these integrated microcombs is typically higher than in the crystalline platform used in this work Liu et al. (2019); Huang et al. (2019); Drake et al. (2019b), the additional noise is cancelled over a broad frequency range via the transfer oscillator method that constitutes a powerful tool for low-noise frequency division without the need for a very low-noise comb. The free-running comb operation and the maturity of RF components, which can be suitably integrated, promise robust device operation. Furthermore, improvements in resonator actuation, using micro-heaters Joshi et al. (2016), piezoelectric transducers Liang et al. (2017); Tian et al. (2018b); Dong et al. (2018), or the electro-optic effect Alexander et al. (2018); He et al. (2019), will allow the resonator to be tuned to the USL for direct soliton generation (as in Fig. 1), alleviating the need for an optical phase-lock loop and greatly simplifying the detuning stabilisation mechanism. If a lower stability level is acceptable, simpler and more compact low-noise lasers can be employed Kéfélian et al. (2009); Liang et al. (2015b); Gundavarapu et al. (2019) instead of the USL. We believe that the presented transfer oscillator method holds promising potential for ultralow-noise high-frequency generators with a new generation of compact photonic-based systems Marpaung et al. (2019) for radar applications Ghelfi et al. (2014), high frequency telecommunications Koenig et al. (2013) and time–frequency metrology Millo et al. (2009).

Methods

Operating conditions

The pump power after the EOM used for the PDH lock of the microresonator and the injection locking of $f_{\rm rep}$ is $\sim 10$ mW and is amplified to $\sim 250$ mW in an EDFA. The power level after the AOM that controls the pump power coupled to the resonator (see Fig. 2) is set to $\sim 210$ mW. After comb generation and residual pump rejection with a fibre Bragg grating, the comb power of $\sim 1$ mW is amplified to $\gtrsim 5$ mW. The largest part of this power (90%) is sent onto a high power handling photodiode (Discovery Semiconductors DSC40S, generating a photocurrent of 5.12 mA and a microwave power of $\sim-7.4$ dBm), while the remaining fraction is used for the intercomb beatnote detection. The shot noise level is estimated for a CW laser detection, based on the generated photocurrent and microwave power. The 56th harmonic of the auxiliary comb repetition rate $f_{\rm rep}^{\rm aux}$ at 14.09 GHz is detected, selected using a narrow band-pass filter and amplified to $\sim 19$ dBm. This signal drives the phase modulator and creates an estimated phase deviation of $\sim 1.4$ rad. The injection-locking range of the Kerr comb repetition rate Weng et al. (2019) spans $\gtrsim 2$ kHz and the locking bandwidth is $\sim 2$ kHz.

Resonator characteristics

The \ceMgF2 whispering gallery mode resonator was fabricated via precision diamond turning and hand polishing on a lathe. The intrinsic linewidth of the pumped mode is $\sim 80$ kHz (intrinsic quality factor of $2.4\times 10^{9}$). The evanescent coupling to the resonance is achieved via a tapered optical fibre. The fibre is operated in contact with the resonator to damp its vibrations. Careful adjustment of the fibre position is required to maximise the coupling rate and increase the out-coupled comb power. The loaded resonance linewidth is estimated at $\sim 2.4$ MHz. The threshold power for comb formation is estimated at $\sim 40$ mW. The detuning setpoint was chosen to minimise the noise of the Kerr comb repetition rate, as described in the next section.

Soliton noise minimisation

The laser-resonator detuning $\delta=\nu_{\rm cav}-\nu_{\rm laser}$ is known to have a major impact on the noise and stability of Kerr frequency combs. This parameter not only sets the soliton pulse duration Lucas et al. (2017), but was also shown to modify the repetition rate frequency through the Raman self-frequency shift Yi et al. (2016) $\Omega_{\rm Raman}(\delta)$ and the soliton recoil $\Omega_{\rm recoil}$ corresponding to dispersive wave emission Yi et al. (2017); Lucas et al. (2017). Indeed, these two effects lead to an overall shift of the spectral centre of the soliton (i.e., the soliton spectral maximum relative to the pump frequency) $\Omega=\Omega_{\rm Raman}+\Omega_{\rm recoil}$, which induces in turn a change in the group velocity of the pulse and therefore of the repetition rate according to Matsko and Maleki (2013)

[TABLE]

where $D_{1}/2\pi=14.09$ GHz is the resonator FSR and $D_{2}/2\pi=1.96$ kHz is the group velocity dispersion (GVD) parameter at the pump frequency Herr et al. (2012). Thus, residual laser-resonator detuning noise can degrade the spectral purity of the repetition rate Stone et al. (2018). A solution to this problem was already identified by Yi et al. Yi et al. (2017), who proposed to use the balance of dispersive-wave recoil and Raman-induced soliton-self-frequency shift to enhance the repetition-rate stability of a silica wedge-based Kerr comb. A similar concept is applied here to minimise the repetition rate noise of the crystalline \ceMgF2 microresonator-based comb. Importantly, in \ceMgF2, the Raman self frequency shift can be neglected, due to the very narrow gain bandwidth, and the soliton shift is dominated by the soliton recoil $\Omega\approx\Omega_{\rm recoil}$.

We measured the variation in repetition rate of the soliton comb as a function of detuning in two coupling conditions (weak and large coupling). The coupling rate was modified by changing the position of the tapered fibre along the resonator. The detuning was scanned (forward and backward) by changing the PDH modulation frequency, while the phase lock loop offset frequency was adapted accordingly to keep the total frequency offset between the USL and the microresonator resonance constant. At each detuning point, the optical spectrum was acquired and the repetition rate frequency $f_{\rm rep}^{\rm K}$ was counted. The results are displayed in Fig. 4. The phase modulation at the cavity FSR used for injection-locking was disabled in this measurement.

The weak coupling of the resonator allows for a relatively wide detuning range to be accessed (5 to 25 MHz, see Fig. 4). Over this span, the repetition rate changes in total by 22 kHz, but not linearly. The non-monotonic evolution of $f_{\rm rep}^{\rm K}(\delta)$ is caused by the soliton recoil induced by dispersive waves through avoided mode crossings Yang et al. (2016); Yi et al. (2017); Lucas et al. (2017). The soliton shift $\Omega/2\pi$ is extracted by fitting the optical spectrum with a $\operatorname{sech}^{2}$ function and the associated repetition rate variation can be estimated using eq. (2). Interestingly, after subtracting this contribution, the residual shift of the repetition rate follows a linear trend with a slope of $\sim-1$ kHz/MHz. This significant variation is independent from any recoil-associated effect and could originate from more complex forms of avoided modal crossings, or third order dispersion, although we observed that the value of this slope changes with the coupling as detailed below.

Increasing the coupling rate of the resonator (see Fig. 4) shrinks the accessible detuning range (5.5 to 10 MHz) and radically changes the dependence of $f_{\rm rep}^{\rm K}$ with $\delta$. The overall variation is reduced to $\sim 1.4$ kHz and is dominated by solitonic recoil. Once this contribution is subtracted, the residual slope is on the order of $\sim-160$ Hz/MHz, which is very close to the value expected from the nonlinear self-steepening effect Bao et al. (2017).

Quiet operation point

More notably, under this larger coupling condition, the relation $f_{\rm rep}^{\rm K}(\delta)$ exhibits a stationary point around $\delta=7$ MHz, where the coupling of pump-laser frequency noise into the soliton repetition rate is expected to be minimal since $\partial f_{\rm rep}^{\rm K}/\partial\delta\approx 0$. To verify this prediction, the phase noise of the detected soliton pulse train was measured at different detuning points. The pump laser was phase-modulated by a low frequency tone at 9 kHz to provide a reference point. Furthermore, instead of phase-locking the pump laser to the USL, the PDH feedback was applied to the pump laser current in these measurements, and the resonator was slowly stabilised to the USL via power and thermal feedback. The larger laser noise obtained in this case helps visualising its impact on the repetition rate frequency and could be calibrated via a heterodyne measurement with the USL. The results are displayed in Fig. 5. At the operating point 2, where the slope of $f_{\rm rep}^{\rm K}(\delta)$ is maximum, the noise of $f_{\rm rep}^{\rm K}$ follows the same features as the laser noise. Rescaling the laser noise to match the 9 kHz modulation peaks indicates that the optical noise is reduced by $56$ dB. Conversely the point 1, where the slope of $f_{\rm rep}^{\rm K}(\delta)$ is minimum, corresponds to the lowest optical-to-RF noise transduction (dip in Fig. 5), with a conversion coefficient below $-100$ dB. As expected, this point yields the lowest achieved phase noise, and it appears that the laser phase noise is no longer the overall limiting factor of the Kerr comb repetition rate noise.

Noise limitations in microcombs

In a nonlinear resonator, the free spectral range $D_{1}/2\pi$ depends on the circulating optical power. Therefore, the relative intensity noise (RIN) of the pump laser (power $P_{\rm in}$) eventually induces timing jitter of the repetition rate, according to eq.(2). Assuming a laser on resonance, the self phase modulation induced shift follows Wilson et al. (2018):

[TABLE]

where $\kappa/2\pi\approx 1.35$ MHz is the cavity energy decay rate, $\eta=\kappa_{\rm ex}/\kappa\approx 0.94$ is the coupling impedance of the resonator ($\kappa_{\rm ex}$ is the coupling rate), $V_{\rm eff}\approx 2.32\times 10^{-12}\leavevmode\nobreak\ \mathrm{m}^{3}$ is the mode volume, $n_{2}=9\times 10^{-21}\leavevmode\nobreak\ \mathrm{m}^{2}\mathrm{/}\mathrm{W}$ is the (Kerr) nonlinear index and $n_{0}=1.37$ is the refractive index. These values yield a conversion coefficient $\alpha\approx 3.8\leavevmode\nobreak\ \mathrm{k}\mathrm{Hz}\mathrm{/}\mathrm{W}$. We measured the relative intensity noise $S_{\rm RIN}(f)$ of the pump laser (Fig. 6) and the associated induced phase noise was estimated using:

[TABLE]

for the measured input pump power of $P_{\rm in}\approx 212$ mW. The results are displayed in Fig. 6. The estimated level matches remarkably the repetition rate phase noise at offsets between 500 Hz and 100 kHz (blue and green curves in Fig. 6), suggesting that the pump laser RIN is limiting the performances in this range. The phase noise reaches $\sim-143$ dBc/Hz at 10 kHz, which outperforms any other microresonator-based approach Liang et al. (2015a); Li et al. (2014, 2013); Yi et al. (2017); Liang et al. (2017).

At lower offset frequencies (50 – 500 Hz), the thermal fluctuations and drift of the resonator, which are beyond the power stabilisation bandwidth, are the limiting factor Gorodetsky and Grudinin (2004); Liang et al. (2015a).

At higher offset frequencies, two noise bumps appear related to the characteristic double resonant response ($\mathcal{S}$ and $\mathcal{C}$) of the resonator in the soliton regime Guo et al. (2017). In these resonant features, the transduction of the pump laser noise is enhanced Stone et al. (2018). Beyond 100 kHz offset, the contributions of various factors are more difficult to identify. We observed nonetheless a correlation between the microwave RIN (Fig. 6) and the phase noise, which suggests that amplitude-to-phase noise conversion is occurring in the photodiode Zhang et al. (2011), with a conversion of $\sim-25$ dB (grey curve in Fig. 6), which is in agreement with reported values for similar photodiodes Bouchand et al. (2017). We report here the microwave amplitude noise, as our measurement device offered a better sensitivity in this configuration, but our observations showed that this amplitude noise matches well the optical RIN (measured at DC with a diplexer).

Finally, the continuous-wave shot-noise floor is expected to be at $-159$ dBc/Hz (photocurrent of 6.85 mA, microwave power of $-3.8$ dBm). However, we noticed that the phase noise floor of our measurement stands 4.1 dB below this value (at frequency offsets above 20 MHz), while the amplitude noise floor is 6.3 dB above. This imbalance between amplitude and phase needs further investigation and could be related to shot noise correlations in the detection of optical pulses Niebauer et al. (1991); Quinlan et al. (2013a, b).

Data availability statement

The data and code used to produce the results of this manuscript are available on Zenodo:http://doi.org/10.5281/zenodo.3515211.

Authors contributions

E.L. and P.B. designed the experimental setup and performed the experiments with assistance of R.B. and S.S. E.L. analysed the data and wrote the manuscript, with input from other authors. T.S. and T.J.K. supervised the project.

Competing interests

The authors declare no competing interests.

Acknowledgements.

The authors acknowledge Menlo Systems and the Observatoire de Paris (SYRTE) for providing access to the raw data of the measurement of the USL phase noise (the methods can be found in ref Giunta et al. (2018)). The authors thank M. H. Anderson for the proofreading the manuscript. This publication was supported by the Swiss National Science Foundation (SNF) under grant agreement 176563, as well as Contract W31P4Q-14-C-0050 (PULSE) from the Defense Advanced Research Projects Agency (DARPA), Defense Sciences Office (DSO). This material is based upon work supported by the Air Force Office of Scientific Research, Air Force Material Command, USAF under Award No. FA9550-15-1-0099. E.L. acknowledges support from the European Space Technology Centre, with ESA Contract No. 4000118777/16/NL/GM.