Ultralow-Noise Photonic Microwave Synthesis using a Soliton Microcomb-based Transfer Oscillator
Erwan Lucas, Pierre Brochard, Romain Bouchand, Stéphane Schilt, Thomas Südmeyer, Tobias J. Kippenberg
UUltralow-Noise Photonic Microwave Synthesis using a Soliton Microcomb-basedTransfer Oscillator
Erwan Lucas, ∗ Pierre Brochard, ∗ Romain Bouchand, Stéphane Schilt, Thomas Südmeyer, and Tobias J. Kippenberg † Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland Laboratoire Temps-Fréquence, Université de Neuchâtel, CH-2000 Neuchâtel, Switzerland
The synthesis of ultralow-noise microwaves isof both scientific and technological relevancefor timing, metrology, communications andradio-astronomy. Today, the lowest reportedphase noise signals are obtained via opticalfrequency-division using mode-locked laser fre-quency combs. Nonetheless, this technique ide-ally requires high repetition rates and tight combstabilisation. Here, a soliton microcomb witha 14 GHz repetition rate is generated with anultra-stable pump laser and used to derive anultralow-noise microwave reference signal, withan absolute phase noise level below −
60 dBc / Hzat 1 Hz offset frequency and −
135 dBc / Hz at10 kHz. This is achieved using a transfer os-cillator approach, where the free-running micro-comb noise (which is carefully studied and min-imised) is cancelled via a combination of elec-tronic division and mixing. Although this proof-of-principle uses an auxiliary comb for detectingthe microcomb’s offset frequency, we highlightthe prospects of this method with future self-referenced integrated microcombs and electro-optic combs, that would allow for ultralow-noisemicrowave and sub-terahertz signal generators.
Introduction
The synthesis of microwave signals via photonic sys-tems, such as dual frequency lasers , optoelectronic oscil-lators , Brillouin oscillators , or electro-optical dividers ,hold promise for their ability to synthesise low-noise orwidely tunable microwave signals with compact form fac-tor. An additional approach is based on optical fre-quency 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 . If the combline of index N is tightly phase-locked to the USL (af-ter subtraction of the carrier envelope offset (CEO) fre-quency f ceo or simultaneous stabilisation of f ceo ), thecomb repetition rate f rep is directly phase-stabilised tothe ultra-stable frequency ν usl by frequency division: f rep = ν usl /N . Importantly, owing to the carrier fre-quency division from optics to microwaves, the absolute ∗ These authors contributed equally to this work. † tobias.kippenberg@epfl.ch phase noise power spectral density is reduced by a factor N ∼ .This method has been mostly implemented using fibre-based fs-lasers with repetition rates of a few hundredmegahertz. A fast actuator (e.g., an intra-cavity electro-optic modulator ) is required to achieve a tight opti-cal lock of the comb tooth to the optical reference andperform the frequency division over a wide bandwidth.Moreover, a high harmonic of the comb repetition ratemust be used to synthesise a microwave signal beyond10 GHz. Consequently, repetition rate multipliers aretypically employed to reduce the impact of shot-noise inthe photo-detection of the pulse train, such as optical fil-tering cavities or fibre interleavers , which increasesthe system complexity. Therefore, the use of frequencycombs directly operating at ∼
10 GHz repetition rateswould be highly beneficial, but their optical lock and self-referencing are challenging.Microresonator-based Kerr frequency combs (i.e., ‘mi-crocombs’), which naturally produce multi-GHz combspectra generated via four-wave mixing in an opticalmicroresonator , are natural candidate in this con-text. Pumping a cavity resonance with a continuous-wave laser can initiate and sustain a circulating dissipa-tive Kerr soliton (DKS) pulse that is intrinsicallyphase-coherent with the input pump laser. The resultingcomb coupled out of the micro-cavity is inherently per-fectly phase-locked to the pump laser, without any ac-tuator locking bandwidth limitation. Direct soliton gen-eration from an ultra-stable pump laser holds potentialfor compact and powerful optical-to-microwave dividers.Although self-referenced optical microcombs and clockshave been demonstrated , optical frequency divisionfor ultralow-noise microwave generation using such de-vices has not been demonstrated so far, mainly due tothe complex crosstalk occurring between their two de-grees of freedom and the limited performance of theavailable actuators .Here, we demonstrate the generation of an ultralow-noise microwave signal using a microcomb-based trans-fer oscillator method to realise optical-to-microwave fre-quency division. The transfer oscillator method by-passes the need for tight optical phase-locking of the fre-quency comb to the optical reference. Instead, it relieson an adequate manipulation and combination of sig-nals to cancel the comb phase noise and to provide abroadband electronic division of the USL frequency tothe microwave domain. The frequency division by a large a r X i v : . [ phy s i c s . a pp - ph ] N ov Microresonator& soliton generationUltra-stable cw pump laserDivided laserfrequencyOctave spanning or broadenedKerr frequencycombSelf referencing& comb parametersdetection × R F S p e c t r u m R F S p e c t r u m R F S p e c t r u m Frequency a b
Figure 1 : Principle of operation of the Kerr comb-based transfer oscillator for optical-to-microwave fre-quency division. (a)
Schematic illustration of the transfer oscillator applied to a Kerr comb (or electro-optic combsequivalently). (b)
Schematic representation of the signal evolution along the electronic division chain leading to thelow-noise output signal. The two comb parameters f ceo and f rep are detected. Both parameters can be free-runningand fluctuate. The carrier envelope offset (CEO) frequency is electronically divided by a large number N that corre-sponds to the tooth number of the ultra-stable pump ν usl . After this step, the frequency fluctuations of the dividedCEO f ceo /N = ν usl /N − f rep are dominated by the repetition rate fluctuations. These are removed by mixing f ceo /N with f rep to obtain the division result ν usl /N . A narrow-band filtering is used to reject spurs.factor N is performed electronically, thus removing theneed for high locking bandwidth actuators. In this work,the USL is used to pump the microresonator and inher-ently constitutes a tooth of the resulting frequency comb.We show how to extend the transfer oscillator techniqueto exploit this salient feature of microcombs (or equiva-lently of electro-optic combs ). In this proof-of-principledemonstration, we achieved a measured single-sidebandphase noise of −
110 dBc/Hz at 200 Hz offset from the14.09 GHz carrier, which is 15 dB below the lowest phasenoise microresonator-based photonic oscillator reportedso far , demonstrating the potential of this approach. Results
Transfer oscillator principle –
The high-level work-ing principle of our method is illustrated in Fig. 1. Amicroresonator pumped by a sub-Hz-linewidth USL atfrequency ν usl generates a soliton-Kerr comb with a GHz-range repetition rate f rep that is set by the resonator freespectral range (FSR). The reference laser is part of thefrequency comb (line N ) such that its frequency can bewritten as ν usl = f ceo + N f rep . The detection of the CEOfrequency (for example via f − f interferometry orwith an auxiliary self-referenced comb as in the presentwork) is followed by electronic division by means of acombination of frequency pre-scalers and direct digital synthesisers (DDS). The final step consists of mixing thedivided CEO signal with the repetition rate, which yields f signal = f ceo N + f rep = ν usl N (1)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 thelimited tuning of the resonator and narrow bandwidth ofthe microcomb used for this proof-of-concept, additionalhardware was needed to demonstrate the transfer oscilla-tor principle, as shown in Fig. 2a. The soliton-Kerr combis generated by pumping a crystalline magnesium fluoride(MgF ) microresonator with an FSR of 14.09 GHz usinga 1553 nm diode laser, which is initially quickly scannedacross a resonance for soliton generation . Next, thepump laser is phase-locked to a sub-Hz-linewidth USL(Menlo Systems ORS1500) at a frequency detuning of ∼ . P o w e r ( d B / d i v . ) -15 -10 -5 0 5 10 15 - 251.66 MHz (Hz)-101 - C F ( k H z ) R F po w e r ( d B / d i v ) Ultrastable pump Division chainRep.rate(cid:47)(cid:374)(cid:361)(cid:286)(cid:272)(cid:415)(cid:381)(cid:374)(cid:3)(cid:367)(cid:381)(cid:272)(cid:364)(cid:349)(cid:374)(cid:336) (cid:75)(cid:299)(cid:400)(cid:286)(cid:410)(cid:3)(cid:282)(cid:286)(cid:410)(cid:286)(cid:272)(cid:415)(cid:381)(cid:374)CW pump laser Ultra-stablelaserOPLL
PNA
ESAPDH (cid:94)(cid:286)(cid:367)(cid:296)(cid:882)(cid:396)(cid:286)(cid:296)(cid:286)(cid:396)(cid:286)(cid:374)(cid:272)(cid:286)(cid:282)(cid:302)(cid:271)(cid:396)(cid:286)(cid:3)(cid:272)(cid:381)(cid:373)(cid:271)EDFAPD OBPFEDFA FBG abc d (cid:47)(cid:374)(cid:361)(cid:286)(cid:272)(cid:415)(cid:381)(cid:374)(cid:3)(cid:62)(cid:381)(cid:272)(cid:364)(cid:349)(cid:374)(cid:336)(cid:75)(cid:393)(cid:415)(cid:272)(cid:258)(cid:367)(cid:3)(cid:296)(cid:396)(cid:286)(cid:395)(cid:437)(cid:286)(cid:374)(cid:272)(cid:349)(cid:286)(cid:400)(cid:60)(cid:286)(cid:396)(cid:396)(cid:3)(cid:272)(cid:381)(cid:373)(cid:271)(cid:4)(cid:437)(cid:454)(cid:349)(cid:367)(cid:367)(cid:349)(cid:258)(cid:396)(cid:455)(cid:302)(cid:271)(cid:396)(cid:286)(cid:3)(cid:272)(cid:381)(cid:373)(cid:271)
Figure 2 : Experimental setup and CEO detection with the auxiliary comb (a)
Setup for Kerr comb-basedoptical frequency division. The details of each highlighted block can be found in the supplementary information.EDFA, Er-doped fibre amplifier; AOM, Acousto-optic modulator; EOM, Electro-optic modulator; FBG, Fibre Bragggrating for pump rejection; OBPF, Optical band-pass filter; PD, Photodiode; OPLL, Optical phase lock loop; PDH,Pound-Drever-Hall lock; PNA, Phase noise analyser; ESA, Electrical spectrum analyser. (b)
Optical spectrum of thesoliton-based Kerr comb, prior to pump suppression. (c)
Radio-frequency (RF) spectrogram showing the injection-locking effect of the Kerr comb repetition rate f Krep to the 56 th harmonic of the auxiliary comb repetition rate f auxrep ,obtained by changing the frequency of f auxrep . Here, the harmonic power applied to the EOM is ∼
11 dBm yielding alocking range of ∼ . f Krep is centred at CF = 14 . (d) Principle of the Kerr comb CEO detection with the auxiliary comb. The harmonic relation betweenthe repetition rate of both combs is ensured via injection locking for M = 56. The heterodyne beat between the twocombs thus yields the difference between their carrier-envelope offset frequency (∆ f ceo ).an effective-detuning stabilisation, achieved via a side-band Pound-Drever-Hall (PDH) lock , which feedbackson an acousto-optic modulator (AOM) that modulatesthe pump power and thus thermo-optically tunes the res-onator, in addition to a slow thermal actuation of the mi-croresonator (see details in the Supplementary Note 2).The detuning setpoint was carefully optimised in order tominimise the noise of the Kerr comb repetition rate f Krep at offset frequencies beyond ∼
100 Hz (see the Methodssection and Fig. 6b). However, the residual thermal driftof the resonator degrades the performance at lower offsetfrequencies.
Offset detection with an auxiliary comb –
The usedcrystalline MgF micro-comb features a relatively narrowspectrum that prevents a direct detection of its CEO fre-quency (Fig. 2b). The self-referencing of Kerr combsremains highly demanding due to the high repetitionrate, low optical power, and fairly long pulse duration(225 fs here) resulting in a low peak intensity that makes the spectral broadening for f − f interferometry chal-lenging . Therefore, we implemented an indirect de-tection scheme using an auxiliary self-referenced fibre-laser frequency comb with a repetition rate f auxrep =251 . f Krep = M f auxrep (superscripts ‘K’ and ‘aux’ re-fer to the Kerr and auxiliary combs, respectively), thenthe optical beatnote between the two combs correspondsto their relative CEO frequency ∆ f ceo = f K ceo − f aux ceo , asthe repetition rate noise contributions compensate eachother in this beat signal. The Kerr comb CEO fre-quency is then obtained by mixing out the CEO fre-quency of the auxiliary comb f aux ceo detected with an f − f interferometer (see Fig. 2a) and corresponds to f K ceo = ∆ f ceo + f aux ceo = ν pump − N f
Krep (grey box inFig. 2a). Importantly, the auxiliary comb is not stabilisedto the USL at any point and thus does not perform thedivision. Its role is limited to the offset detection in this Frequency (Hz)-150-140-130-120-110-100-90-80-70-60-50-40-30-20
SSB pha s e no i s e ( d B c / H z ) F r a c t i ona l f r eq . c hange ( % ) -1000 0 1000Frequency - offset (Hz) R e l a t i v e po w e r ( d B / d i v . ) Rep. rateDivisionresult -100 0 100Freq. - 14.0919 GHz (Hz) R e l a t i v e po w e r ( d B / d i v . ) a Shot Noise b c d
RBW: 100 mHz
CanceledNoise
Figure 3 : Characterisation of the optical-to-microwave division signal (a)
Absolute single-sideband (SSB)phase noise of the 14.09 GHz signal generated by optical-to-microwave division of the USL via the Kerr comb transferoscillator (blue) and obtained directly from the Kerr comb repetition rate (green) for comparison. The sensitivitylimit of the phase noise analyser (3000 cross correlations applied at 1 Hz) is indicated by the grey shaded area. Thered line is the limit inferred from the optical phase noise of the USL, assuming an ideal noiseless division. (b)
Precisedetermination of the optimal division factor N corresponding to the zero crossing of the linear fit (solid line) of themeasured relative frequency change of the generated RF signal for a small variation of the repetition rate (dots). (c) Comparison between the RF spectra of the Kerr comb repetition rate and the optical-to-microwave frequency divisionresult. The resolution bandwidth (RBW) is 5 Hz. (d)
RF spectrum of the frequency-divided output signal, the RBWis 100 mHz. The data was acquired with the IQ demodulation mode of the spectrum analyser.demonstration.The mutual harmonic phase-locking of the comb rep-etition rates is achieved via soliton injection-locking .The harmonic M = 56 of the repetition rate of the auxil-iary comb (at 14.093 GHz) is detected, filtered and ampli-fied to phase-modulate the pump light using an electro-optic modulator (EOM, blue box in Fig. 2a). This fre-quency is very close to the native microcomb line spacing,which gets injection-locked to this drive signal. There-fore, both repetition rates are strongly correlated over abandwidth of ∼ Transfer oscillator chain –
The Kerr comb CEO sig-nal, indirectly obtained as previously described, is de-tected at low frequency (MHz-range) and filtered tomatch the bandwidth of the injection locking of the rep-etition rate (not represented in Fig. 2a, see the Sup-plementary Note 5 and Supplementary Figure 4). Af-ter up-mixing to 15 GHz, it is frequency-divided by alarge pre-determined factor N ≈ ,
698 and is sub-tracted to the separately-detected repetition rate f Krep toobtain the frequency-divided signal of the ultra-stablepump laser: ν pump /N = f K ceo /N + f Krep (orange box inFig. 2a). The overall division of the Kerr comb CEOsignal by the factor N is realised with a frequency pre-scaler followed by two parallel DDS, which offers im-proved filtering capabilities in the electronic division . This second stage division with the DDS allows for aprecise non-integer frequency division factor and leadsto a clean single-tone output signal corresponding to thefrequency-divided USL (see Fig. 3d). The detailed de-scription of the frequency division chain is provided inthe Supplementary Note 5.The overall division factor N was accurately deter-mined experimentally, without prior knowledge of theoptical frequency of the ultra-stable pump laser, by mea-suring the frequency change of the generated microwavesignal corresponding to a small variation (140 Hz) of theKerr comb repetition rate for different programmed di-vision factors N (see Fig. 3b). This simple measurementalso provides an accurate determination of an opticalcomb line index N that can be useful for absolute op-tical frequency measurements. Microwave characterisation –
The phase noise of thegenerated ultralow-noise 14.09 GHz signal was measuredwith a cross-correlator phase noise analyser (Fig. 3a). Itreaches −
110 dBc/Hz at 200 Hz Fourier frequency, 15 dBbelow the lowest phase noise microresonator-based pho-tonics oscillator at 10 GHz. The phase noise is below −
135 dBc/Hz at 10 kHz and −
150 dBc/Hz at around1 MHz, showing that the intrinsic low-noise properties ofthe soliton Kerr comb at high Fourier frequencies are pre-served. The calculated shot-noise predicts a noise floorat −
152 dBc/Hz (thermal noise floor ∼ −
170 dBc/Hz).At 1 Hz offset, the measurement is limited by the instru-mental noise floor below −
60 dBc/Hz, even with 3000cross correlations. Nevertheless, the transfer oscillatoroffers an improvement by at least 40 dB compared to thedirect detection of the Kerr comb repetition rate (despitethe resonator being stabilised to the USL), showing itsability to cancel the residual thermal drifts of the Kerrcavity. The technical limitations of the measurement, atlow offset frequency, make it difficult to directly com-pare the method with state-of-the-art optical frequency-division using mode-locked lasers. Nevertheless, at highFourier frequencies, our results surpass some of the firstdemonstrations of optical frequency division , even if nooptimisation has been performed on the photodetectionside. Over the past 10 years, the development of mode-locked lasers, as well as the improvement of photode-tection noise , led to a reduction of the noise of thegenerated microwaves by 30 to 40 dB in some frequencybands . We believe that the transfer oscillator methodcan follow a similar path, as in particular, the high repe-tition rates of the Kerr combs should make the photode-tection optimisation less stringent. Discussion
In summary, we have reported optical-to-microwavefrequency division using a Kerr comb as transfer os-cillator. This demonstrates the potential of thismethod in microwave photonics and enlarges its pre-viously reported implementation with low repetitionrate mode-locked lasers. The approach presented herecan be further implemented with electro-optic combs,where self-referencing and feedback control were recentlyachieved . Although this proof-of-principle experi-ment required an auxiliary comb to obtain the CEOfrequency of the Kerr comb, directly self-referenced mi-crocombs are technologically feasible in silicon nitride(Si N ) photonic-chips . While octave-spanning combspectra have been achieved using dispersion control ,these implementations used THz repetition rates to coversuch a large spectral range, which made photodetectionof the repetition rate practically impossible. Nonethe-less, the residual phase noise of these combs has beenshown to be suitable for frequency division . Recentimprovements of integrated resonators have enabled soli-ton microcombs with K- and X-band (20 and 10 GHz)repetition rates in integrated resonators . However, theachieved spectral spans, although wider than in the crys-talline case, are far from covering one octave. Pulsedpumping appears as a promising approach to enableoctave spanning microcombs with detectable microwaverepetition rates. This approach uses synchronous pump-ing of the microresonator with picosecond pulses to gen-erate a soliton with a much shorter duration and a spec-trum that can cover an octave, similar to enhancementcavities . It can be seen as a hybrid between an electro-optic (EO) comb and a microcomb, with the advantagethat the spectral enlargement of the EO comb is per-formed in cavity and is therefore directly filtered . Crucially, even if the free-running phase noise of theseintegrated microcombs is typically higher than in thecrystalline platform used in this work , the addi-tional noise is cancelled over a broad frequency range viathe transfer oscillator method that constitutes a pow-erful tool for low-noise frequency division without theneed for a very low-noise comb. The free-running comboperation and the maturity of RF components, whichcan be suitably integrated, promise robust device opera-tion. Furthermore, improvements in resonator actuation,using micro-heaters , piezoelectric transducers , orthe electro-optic effect , will allow the resonator tobe tuned to the USL for direct soliton generation (asin Fig. 1a), alleviating the need for an optical phase-lock loop and greatly simplifying the detuning stabili-sation mechanism. If a lower stability level is accept-able, simpler and more compact low-noise lasers can beemployed instead of the USL. We believe that thepresented transfer oscillator method holds promising po-tential for ultralow-noise high-frequency generators witha new generation of compact photonic-based systems for radar applications , high frequency telecommunica-tions and time–frequency metrology . Methods
Operating conditions –
The pump power after theEOM used for the PDH lock of the microresonator andthe injection locking of f rep is ∼
10 mW and is am-plified to ∼
250 mW in an EDFA. The power level af-ter the AOM that controls the pump power coupled tothe resonator (see Fig. 2a) is set to ∼
210 mW. Aftercomb generation and residual pump rejection with a fi-bre Bragg grating, the comb power of ∼ & ∼ − . th harmonic of the auxil-iary comb repetition rate f auxrep at 14.09 GHz is detected,selected using a narrow band-pass filter and amplified to ∼
19 dBm. This signal drives the phase modulator andcreates an estimated phase deviation of ∼ . spans & ∼ Resonator characteristics –
The MgF whisperinggallery mode resonator was fabricated via precision di-amond turning and hand polishing on a lathe. The in-trinsic linewidth of the pumped mode is ∼
80 kHz (in-trinsic quality factor of 2 . × ). The evanescent cou-pling to the resonance is achieved via a tapered opticalfibre. The fibre is operated in contact with the resonatorto damp its vibrations. Careful adjustment of the fibreposition is required to maximise the coupling rate andincrease the out-coupled comb power. The loaded reso- P o w e r ( d B / d i v . ) P o w e r ( d B / d i v . ) - - - - R ep . R a t e s h i ft ( k H z ) S o li t on r e c o il ( G H z ) - . - . - . R ep . R a t e s h i ft ( k H z ) S o li t on r e c o il ( G H z ) Weaker coupling Larger couplinga cdb
Figure 4 : Optimisation of f Krep phase noise (a)
Soliton spectrum for lower coupling case (Detuning 10 MHz). (b)
Evolution of the repetition rate (blue, solid) and of the soliton recoil (Ω / π ) retrieved by fitting the opticalspectrum (red), in the lower coupling case. The blue crosses and dashed line show the residual repetition rate changeafter subtraction of the recoil induced shift (using eq. (2)). (c) Soliton spectrum for larger coupling case (Detuning10 MHz). (d)
Evolution of the repetition rate (blue, solid) and of the soliton recoil (Ω / π ) retrieved by fitting theoptical spectrum (red), in the larger coupling case. The blue dashed line shows the residual repetition rate changeafter subtraction of the recoil induced shift (using eq. (2)).nance linewidth is estimated at ∼ . ∼
40 mW.The detuning setpoint was chosen to minimise the noiseof the Kerr comb repetition rate, as described in the nextsection.
Soliton noise minimisation –
The laser-resonator de-tuning δ = ν cav − ν laser is known to have a major impacton the noise and stability of Kerr frequency combs. Thisparameter not only sets the soliton pulse duration , butwas also shown to modify the repetition rate frequencythrough the Raman self-frequency shift Ω Raman ( δ ) andthe soliton recoil Ω recoil corresponding to dispersive waveemission . Indeed, these two effects lead to an overallshift of the spectral centre of the soliton (i.e., the soli-ton spectral maximum relative to the pump frequency)Ω = Ω Raman + Ω recoil , which induces in turn a changein the group velocity of the pulse and therefore of therepetition rate according to f Krep = 12 π (cid:18) D + D D Ω( δ ) (cid:19) (2)where D / π = 14 .
09 GHz is the resonator FSR and D / π = 1 .
96 kHz is the group velocity dispersion(GVD) parameter at the pump frequency . Thus,residual laser-resonator detuning noise can degrade thespectral purity of the repetition rate . A solution tothis problem was already identified by Yi et al. , whoproposed to use the balance of dispersive-wave recoiland Raman-induced soliton-self-frequency shift to en-hance the repetition-rate stability of a silica wedge-basedKerr comb. A similar concept is applied here to min-imise the repetition rate noise of the crystalline MgF microresonator-based comb. Importantly, in MgF , theRaman self frequency shift can be neglected, due to thevery narrow gain bandwidth, and the soliton shift is dom-inated by the soliton recoil Ω ≈ Ω recoil .We measured the variation in repetition rate of thesoliton comb as a function of detuning in two couplingconditions (weak and large coupling). The coupling ratewas modified by changing the position of the taperedfibre along the resonator. The detuning was scanned(forward and backward) by changing the PDH modula-tion frequency, while the phase lock loop offset frequencywas adapted accordingly to keep the total frequency off-set between the USL and the microresonator resonanceconstant. At each detuning point, the optical spectrumwas acquired and the repetition rate frequency f Krep wascounted. The results are displayed in Fig. 4. The phasemodulation at the cavity FSR used for injection-lockingwas disabled in this measurement.The weak coupling of the resonator allows for a rela-tively wide detuning range to be accessed (5 to 25 MHz,see Fig. 4b). Over this span, the repetition rate changesin total by 22 kHz, but not linearly. The non-monotonicevolution of f Krep ( δ ) is caused by the soliton recoil in-duced by dispersive waves through avoided mode cross-ings . The soliton shift Ω / π is extracted by fittingthe optical spectrum with a sech function and the as-sociated repetition rate variation can be estimated usingeq. (2). Interestingly, after subtracting this contribution,the residual shift of the repetition rate follows a lineartrend with a slope of ∼ − Frequency (Hz)-160-140-120-100-80-60-40-20
SSB pha s e no i s e ( d B c / H z ) R ep r a t e - . G H z ( H z ) -100-90-80-70-60 O p t i c a l t o R F c oe ff i c i en t ( d B ) ab Laser noise
Figure 5 : ‘Quiet’ operating point (a) Evolution ofthe repetition rate with the detuning (blue) and associ-ated optical phase modulation to RF phase modulationconversion coefficient calibrated with the 9 kHz phasemodulation tone on the laser (red). (b)
Phase noisespectra of the soliton repetition rate at the two operat-ing points highlighted in (a). The solid black line showsthe laser noise (PDH-stabilised to the microcavity). Thedashed black line shows the noise of the laser scaled by −
100 dB to match the 9 kHz phase calibration tone.modal crossings, or third order dispersion, although weobserved that the value of this slope changes with thecoupling as detailed below.Increasing the coupling rate of the resonator (seeFig. 4d) shrinks the accessible detuning range (5.5 to10 MHz) and radically changes the dependence of f Krep with δ . The overall variation is reduced to ∼ . ∼ −
160 Hz/MHz, which is very close to the value ex-pected from the nonlinear self-steepening effect . Quiet operation point –
More notably, under thislarger coupling condition, the relation f Krep ( δ ) exhibits astationary point around δ = 7 MHz, where the couplingof pump-laser frequency noise into the soliton repetitionrate is expected to be minimal since ∂f Krep /∂δ ≈
0. Toverify this prediction, the phase noise of the detectedsoliton pulse train was measured at different detuning points. The pump laser was phase-modulated by a lowfrequency tone at 9 kHz to provide a reference point. Fur-thermore, instead of phase-locking the pump laser to theUSL, the PDH feedback was applied to the pump lasercurrent in these measurements, and the resonator wasslowly stabilised to the USL via power and thermal feed-back. The larger laser noise obtained in this case helpsvisualising its impact on the repetition rate frequencyand could be calibrated via a heterodyne measurementwith the USL. The results are displayed in Fig. 5. At theoperating point 2, where the slope of f Krep ( δ ) is maximum,the noise of f Krep follows the same features as the lasernoise. Rescaling the laser noise to match the 9 kHz mod-ulation peaks indicates that the optical noise is reducedby 56 dB. Conversely the point 1, where the slope of f Krep ( δ ) is minimum, corresponds to the lowest optical-to-RF noise transduction (dip in Fig. 5a), with a conversioncoefficient below −
100 dB. As expected, this point yieldsthe lowest achieved phase noise, and it appears that thelaser phase noise is no longer the overall limiting factorof the Kerr comb repetition rate noise.
Noise limitations in microcombs –
In a nonlinearresonator, the free spectral range D / π depends on thecirculating optical power. Therefore, the relative inten-sity noise (RIN) of the pump laser (power P in ) eventuallyinduces timing jitter of the repetition rate, according toeq.(2). Assuming a laser on resonance, the self phasemodulation induced shift follows : δD ( ω )2 π = (cid:18) D π η c n κV eff n (cid:19)| {z } α δP in ( ω ) (3)where κ/ π ≈ .
35 MHz is the cavity energy decay rate, η = κ ex /κ ≈ .
94 is the coupling impedance of the res-onator ( κ ex is the coupling rate), V eff ≈ . × − m is the mode volume, n = 9 × − m / W is the(Kerr) nonlinear index and n = 1 .
37 is the refrac-tive index. These values yield a conversion coefficient α ≈ . / W. We measured the relative intensitynoise S RIN ( f ) of the pump laser (Fig. 6a) and the asso-ciated induced phase noise was estimated using: S φD / π ( f ) = (cid:18) αf P in (cid:19) S RIN ( f ) (4)for the measured input pump power of P in ≈
212 mW.The results are displayed in Fig. 6b. The estimated levelmatches remarkably the repetition rate phase noise at off-sets between 500 Hz and 100 kHz (blue and green curvesin Fig. 6b), suggesting that the pump laser RIN is lim-iting the performances in this range. The phase noisereaches ∼ −
143 dBc/Hz at 10 kHz, which outperformsany other microresonator-based approach .At lower offset frequencies (50 – 500 Hz), the thermalfluctuations and drift of the resonator, which are beyondthe power stabilisation bandwidth, are the limiting fac-tor . Frequency (Hz)-160-140-120-100-80 R I N ( d B c / H z ) Pump laser optical RIN Comb microwave RIN10 Frequency (Hz)-170-160-150-140-130-120-110-100-90-80-70
SSB pha s e no i s e ( d B c / H z ) ab PDH mod.& CW shot noiseCW shot noise
Figure 6 : Pump laser RIN and estimated limita-tion on the phase noise (a)
Optical RIN of the pumplaser (green) and microwave amplitude noise of the soli-ton repetition rate (purple). (b)
Phase noise spectrumof the repetition rate in the ‘quiet’ point (blue) and esti-mated limitation from the pump laser RIN (green). Thegrey curve corresponds to the estimated AM-to-PM con-version in the photodiode (microwave amplitude noisescaled by −
25 dB).At higher offset frequencies, two noise bumps appearrelated to the characteristic double resonant response ( S and C ) of the resonator in the soliton regime . In theseresonant features, the transduction of the pump lasernoise is enhanced . Beyond 100 kHz offset, the con-tributions of various factors are more difficult to identify.We observed nonetheless a correlation between the mi-crowave RIN (Fig. 6a) and the phase noise, which sug-gests that amplitude-to-phase noise conversion is occur-ring in the photodiode , with a conversion of ∼ −
25 dB(grey curve in Fig. 6b), which is in agreement with re-ported values for similar photodiodes . We report herethe microwave amplitude noise, as our measurement de-vice offered a better sensitivity in this configuration,but our observations showed that this amplitude noisematches well the optical RIN (measured at DC with adiplexer).Finally, the continuous-wave shot-noise floor is ex-pected to be at −
159 dBc/Hz (photocurrent of 6.85 mA, microwave power of − . . Data availability statement
The data and code used to produce the results of thismanuscript are available on Zenodo: http://doi.org/10.5281/zenodo.3515211 . Authors contributions
E.L. and P.B. designed the experimental setup and per-formed the experiments with assistance of R.B. and S.S.E.L. analysed the data and wrote the manuscript, withinput from other authors. T.S. and T.J.K. supervisedthe project.
Competing interests
The authors declare no competing interests.
Acknowledgments
The authors acknowledge Menlo Systems and the Ob-servatoire de Paris (SYRTE) for providing access to theraw data of the measurement of the USL phase noise (themethods can be found in ref ). The authors thank M.H. Anderson for the proofreading the manuscript. Thispublication was supported by the Swiss National ScienceFoundation (SNF) under grant agreement 176563, aswell as Contract W31P4Q-14-C-0050 (PULSE) from theDefense Advanced Research Projects Agency (DARPA),Defense Sciences Office (DSO). This material is basedupon work supported by the Air Force Office of ScientificResearch, Air Force Material Command, USAF underAward No. FA9550-15-1-0099. E.L. acknowledges sup-port from the European Space Technology Centre, withESA Contract No. 4000118777/16/NL/GM. 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Conference on Lasers and Electro-Optics , OSA Technical Digest (online) (Optical Society ofAmerica, San Jose, California, 2018) p. SM2L.5. upplementary Material for: Ultralow-Noise Photonic Microwave Synthesis using aSoliton Microcomb-based Transfer Oscillator
Erwan Lucas, ∗ Pierre Brochard, ∗ Romain Bouchand, Stéphane Schilt, Thomas Südmeyer, and Tobias J. Kippenberg Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland Laboratoire Temps-Fréquence, Université de Neuchâtel, CH-2000 Neuchâtel, Switzerland
Supplementary Note 1: Soliton generation proce-dure and optical phase lock loop
In a microresonator with appropriate dispersion, soli-tons emerge spontaneously (soft excitation) when thecontinuous wave pump laser is scanned from blue (higherlaser frequency) to red detuning (lower laser frequency)across a high-Q resonance . This requires a rapid tun-ability of either the pump laser or the resonator. Anultra-stable laser (USL) locked to a high-finesse refer-ence cavity is typically not tunable and the crystallineresonator used here cannot be tuned fast enough to over-come thermal effects . We circumvented this problemby implementing an optical phase lock loop (OPLL) tostabilise the frequency offset between the pump external-cavity diode laser (ECDL, Toptica CTL1550) and the ref-erence USL (Menlo Systems ORS1500). The soliton stateis generated by rapid tuning of the ECDL via current tun-ing. After soliton generation, the OPLL is activated witha frequency offset set so as to preserve the soliton. Res-onators with improved tuning capabilities could removethe need for an OPLL altogether.Solitons are sustained in the cavity within a narrowrange of ‘red’ laser–cavity detuning (i.e., the laser fre-quency is lower than the resonance frequency), which de-pends on the pump power level and the coupling rate ofthe resonator. The detuning control is crucial in soliton-based Kerr comb as it determines many of the comb prop-erties. First, in order to preserve the soliton in the cavity,the detuning must remain within the soliton existencerange. Secondly, as exposed in the Methods section, thesoliton properties, such as the pulse duration and espe-cially the repetition rate, are highly sensitive to the de-tuning operating point . However, even if an USL isused to pump the resonator, the thermal drift of the res-onator induces detuning drifts that can lead to degradednoise performance and even loss of the soliton. Therefore,following soliton generation and OPLL activation, an off-set Pound-Drever-Hall (PDH) lock actively stabilises thedetuning to a precise, defined radio-frequency (RF).The schematic of the OPLL is presented in Supple-mentary Fig. 1a. The beatnote between the two lasersis photo-detected at a frequency of 1 . ∗ These authors contributed equally to this work. Frequency (Hz)-140-120-100-80-60-40-20
SSB pha s e no i s e ( d B c / H z ) USLOPLL Synth.OPLL residual
TunableSynth.10 MHzclock DBM DBM PDServo x2 Counter
CWpump laser Ultrastablelaser ab Supplementary Figure 1 : Optical phase lock loop(a)
Detailed experimental setup. PD, photodiode; DBMdouble balanced mixer; ×
2, frequency doubler. (b)
Com-parison between the USL phase noise and the residualphase noise contribution of the OPLL components. Thered line shows the phase noise of the 1.7 GHz synthesiserand the yellow line the phase noise of the 20 MHz IFsignal at the output of the DBM, indicating a feedbackbandwidth in the range of ∼
500 kHz. The overall addednoise is negligible compared to the USL.counter, in order to preserve the pump laser within thesoliton existence range upon lock activation. The IFsignal is band-pass filtered and compared to a 20 MHzRF signal derived from a 10 MHz common clock, us-ing a double-balanced mixer (DBM). The resulting errorsignal passes through a proportional–integral–derivativecontroller (PID, Toptica MFALC) that implements a slowfeedback to the laser piezoelectric transducer and a fastfeedback to the diode laser current allowing a ∼
500 kHzactuation bandwidth (see Supplementary Fig. 1b). Theresidual noise of the OPLL and the phase noise of the a r X i v : . [ phy s i c s . a pp - ph ] N ov Frequencies (Hz) -120-110-100-90-80-70 PS D ( d B m / H z ) -30 -20 -10 0 10 20 30Frequency detuning (MHz) A m p li t ude ( a r b . un i t ) AOM FBG
EOM C o m b P u m p P u m p (cid:660) TunableSynth.EDFA
Servo
DBM frequency̴10 MHzdetuning(PDH locking)̴1.7 GHz(OPLL) cavityresonanceUSL pumpPDH sidebands a bdc
Locking bandwidth LowpassLock-pointPDH errorsignal Comb power
Supplementary Figure 2 : Pound-Drever-Hall detuning stabilisation (a)
Detailed experimental setup. EOM,electro-optic modulator; EDFA, erbium-doped fibre amplifier; AOM, acousto-optic modulator; FBG, fibre Bragggrating. (b)
Scheme of principle of the stabilisation. The pump laser is phase-locked to the USL. The cavityresonance detuning is then locked to the pump laser via PDH stabilisation using a feedback to the pump laser power(and radiative heating of the resonator). (c)
PDH error signal (blue) and generated comb power (black), as a functionof detuning (the PDH frequency is 13.4 MHz). The detuning lock-point can be set arbitrary in the soliton step bychanging the synthesiser frequency. (d)
Power spectral density (PSD) of the residual PDH error signal, when thedetuning lock is active.synthesiser are negligible compared to the USL noise, in-dicating a good transfer of the USL purity to the pumplaser (see Supplementary Fig. 1b).
Supplementary Note 2: Resonator stabilisation
The detuning stabilisation is implemented via a PDHstabilisation (see Supplementary Fig. 2). The PDH er-ror signal is obtained by phase-modulating the pumplaser (at a frequency in the range of 5 – 25 MHz) beforecoupling to the cavity, using an electro-optic modula-tor (EOM, iXblue MPX-LN-0.1). The modulated signalis detected after the resonator (on the filtered residualpump) and demodulated to DC using the same phase-shifted RF signal (to account for the unbalanced delay be-tween the modulation and demodulation paths). In prac-tice, a dual-channel arbitrary waveform generator is usedand the relative phase between the channels is adapted asa function of the modulation frequency. After demodula-tion, the baseband signal is low-pass-filtered and sent toa PID servo-controller (Toptica FALC). When the comboperates in the soliton regime, the pump laser is red-detuned. Thus, the PDH feedback setpoint correspondsto the higher frequency phase modulation sideband be-ing in resonance (Supplementary Fig. 2b). The PDHservo acts thermally on the resonator, to maintain a fixedpump-resonator detuning. The feedback is implementedto the pump laser power (using a 0 th order acousto-optic modulator (AOM)) and a slower actuation on a LED(Thorlabs MCWHL5 with a typical power of 800 mW)shining on the resonator through a microscope also usedfor imaging. This allows keeping the pump power to adetermined setpoint. The residual noise of the PDH er-ror signal indicates an actuation bandwidth of ∼
100 Hz,limited by the thermal response of the resonator (Sup-plementary Fig. 2d).Owing to this overall scheme, the resonator is stabilisedto the USL, which improves the stability of the systemand helps preserving a given operation point.
Supplementary Note 3: Auxiliary comb and offsetdetection
An auxiliary optical frequency comb (Menlo SystemsFC1500) is used here to detect the CEO frequency of theKerr-comb f K ceo , as the used crystalline micro-comb fea-tures a relatively narrow spectrum that prevents a directdetection of its CEO frequency. An optical beat-note be-tween the two combs is first detected with a photodiode(NewFocus model 1811) at a frequency of a few tens ofMHz. This low-frequency beat signal corresponds to thefrequency difference between one mode of each comb, i.e., f beat = N ( f Krep − f auxrep ) + ( f K ceo − f aux ceo ) (1)where the superscripts ‘K’ and ‘aux’ refer to the Kerrand auxiliary comb, respectively, and the 56 th harmonicof the 251.6 MHz repetition rate of the auxiliary comb isin close vicinity to the fundamental repetition rate of theKerr comb. Supplementary Note 4: Injection locking
To suppress the relative phase noise between the rep-etition rate of the two combs in their beat signal, weimprint the f rep noise of the auxiliary comb to the Kerrcomb by injection locking. This is realised by detectingand band-pass filtering the 56 th harmonic of f auxrep (auxil-iary comb) at 14.09 GHz and using this signal, after am-plification to ∼
19 dBm, to drive an EOM (iXblue MPZ-LN-10) to create a set of sidebands around the ultra-stable pump laser of the micro-resonator, which injection-lock the adjacent optical modes of the resonator. Thisstrongly correlates the noise of the repetition rate of thetwo combs, but only within the bandwidth of the injec-tion locking that is in the kHz range, as shown in Supple-mentary Fig. 3. In that case, the beat signal frequencyin eq. (1) can be re-expressed as f beat = f K ceo − f aux ceo = ∆ f ceo (2) Frequency (Hz)-150-140-130-120-110-100-90-80-70-60-50-40-30
SSB pha s e no i s e ( d B c / H z ) Rep Rate nativeRep Rate injectedAux. Comb 56 th harmonics Supplementary Figure 3 : Injection locking of therepetition rate
Comparison between the phase noise ofthe Kerr comb repetition rate when it is native (blue)and injection locked (red) by the 56 th harmonic of theauxiliary comb repetition rate (purple). Supplementary Note 5: Division chain
The transfer oscillator approach is implemented in thiswork with a 2-DDS scheme as introduced by Brochardand co-authors to perform electronic division with afinely adjustable ratio. This implementation also makespossible the generation of a low-noise single-tone RF out-put signal by efficiently filtering out other spurious peaksthat would occur with a single DDS. The practical real-isation of this scheme with the Kerr comb is depicted inSupplementary Fig. 4. The beat signal ∆ f ceo between the two combs is mixedin a double balanced mixer (DBM) with the CEO sig-nal of the auxiliary comb f aux ceo (detected using a stan-dard f − f interferometer) in order to remove this con-tribution and retrieve f K ceo . Prior to this mixing, theCEO signal of the auxiliary comb, which is stabilised at20 MHz, is frequency-up-shifted using a low-noise synthe-siser. This provides a straightforward means to arbitrar-ily tune f aux ceo without changing it optically (which wouldalso change the optical beat-note frequency by ∆ f ceo )and without frequency noise degradation, as the mea-sured phase noise of the intermediate synthesiser is com-paratively negligible. Thereby, the effective sign of f aux ceo can be quickly changed, without changing any RF com-ponent, in order to properly remove its contribution whenmixing with ∆ f ceo in the DBM.Furthermore, this additional flexibility enables us tofinely adjust the signal frequency at the DBM out-put, which contains the effective Kerr comb CEO, tomake it coincide with the frequency of a tracking oscil-lator around 40 MHz. This tracking oscillator consistsin a narrow-band low-noise voltage-controlled oscillator(VCO) that is phase-locked to f K ceo . The fine adjustmentof the VCO locking bandwidth (by tuning the PI param-eters of the feedback) enables us to filter the noise of thedetected f K ceo in order to match the ∼ f Krep = 56 f auxrep is ensured. Thereby, the contribution of the residualuncorrelated fluctuations between the two combs in thegenerated ultralow-noise RF signal is minimised. Lower( ∼
200 Hz) or higher ( ∼
20 kHz) feedback bandwidthslead to an increased noise in the final signal at low or highFourier frequencies respectively (Supplementary Fig. 5)as a result of the imperfect compensation of the auxil-iary comb noise. Furthermore, as an RF signal with asufficient signal-to-noise ratio (SNR) of more than 30 dBis needed for a proper and stable operation of the sub-sequent frequency divider, this tracking oscillator helpsimproving the signal quality, so that even a fairly lowSNR of the signal at the output of the DBM makes pos-sible to implement the transfer oscillator method.The signal after the tracking oscillator is up-convertedto 15 GHz using a synthesiser with an absolute phasenoise lower than the USL (see Supplementary Fig. 6), sothat its noise has a negligible contribution in the final sig-nal. This frequency shift is necessary to perform the sub-sequent frequency division by a large number of around13,698. Eventually, this synthesiser could be replaced bythe repetition rate of the Kerr-comb to alleviate any ofthe associated limitation. This was not implemented heredue to the lack of appropriate filters.The frequency division from 15 GHz to 1.095 MHz isrealised first by a frequency pre-scaler ( ÷
6, RF Bay FPS-6-15) followed by two DDS (AD9915 evaluation board)in parallel that respectively output signals at 100 MHzand 101.095 MHz from their 2.5 GHz input clock signal.These two signals are subsequently mixed with the repe-tition rate of the Kerr comb separately detected using a
PNAESA upmixtracking oscillatorshift
DDS1 DDS2÷ 6PC VCO
Servo
Kerr combaux.comb
DBMOBPF SSB– SSB+ SSB–DBM
Supplementary Figure 4 : Frequency chain for the transfer oscillator division
Implementation of the optical-to-microwave frequency division using the 2-DDS transfer oscillator scheme. The f rep component of the Kerr combdetected with a fast photodiode (lower path) is mixed with the CEO signal frequency-divided by a factor N (upperpath). The division from 15 GHz to 1.095 MHz is realised by a frequency pre-scaler ( ÷
6) followed by two DDS inparallel that output signals at 100 MHz and 101.095 MHz, respectively, from the 2.5 GHz input clock signal. TheCEO frequency of the Kerr comb is indirectly obtained from the subtraction of the frequency-shifted auxiliary self-referenced comb CEO f aux ceo with the optical beat-note from the two combs (∆ f ceo ), due to the fact that the phase noiseof the repetition rate of the two combs is correlated by an injection locking scheme. DBM, double-balanced mixer;VCO, voltage-controlled oscillator; PC, digital phase comparator; SSB(+/-), single sideband mixer (sum/differencefrequency); DDS, direct digital synthesiser; PNA, phase noise analyser; ESA, electrical spectrum analyser. Frequency (Hz)-140-130-120-110-100-90-80-70-60
SSB pha s e no i s e ( d B c / H z ) Device floor200 Hz2 kHz20 kHz
VCO feedback bandwidth
Supplementary Figure 5 : Tracking oscillator ef-fect.
Influence of the feedback bandwidth of the track-ing oscillator, used to filter the Kerr comb f ceo , onto thefinal generated RF signal. The lowest phase noise of thefinal RF signal is obtained for a locking bandwidth of theVCO of ∼ ∼
200 Hz,blue curve) or higher ( ∼
20 kHz, purple curve) bandwidthresults in a higher noise of the generated RF signal.fast photodiode (Discovery Semiconductors DSC40) andfiltered to select the proper component that correspondsto f RF = f Krep + f K ceo /N = ν usl /N . The sequential mix-ing followed by filtering after each DDS allows for theefficient rejection of the spurious peaks occurring at har- Supplementary Figure 6 : Upmixing and pre-scaler division.
Evolution of the signal phase noiseduring the upmixing and pre-scaler division of the fil-tered Kerr comb CEO. The carrier frequencies are indi-cated on the plot. The noise of the upmixing synthesiser(purple curve) is lower than the USL noise (black), suchthat it does not limit the final division result. Upmixingwith the comb repetition rate would avoid this potentiallimitation.monics of the DDS signals, thanks to their relatively highfrequency spacing (100 MHz range).The generated ultralow noise RF signal is characterisedusing a phase noise analyser (PNA, model FSWP26from Rohde-Schwarz) and an electrical spectrum analyser(ESA, model FSW43 from Rohde-Schwarz). The effect ofthe frequency division performed with the 2-DDS schemeis illustrated in Supplementary Fig. 7. The frequency dif-ference of the two signals at 100 MHz and 101.095 MHz,respectively, gives a signal at 1.095 MHz which corre-spond to f K ceo /N and is strongly correlated with f Krep at14.09 GHz. Mixing these two signal results in the gener-ation of a very low noise RF signal that demonstrates ahigh rejection of the Kerr comb phase noise. Frequency (Hz)-150-140-130-120-110-100-90-80-70-60-50-40-30-20-100
SSB pha s e no i s e ( d B c / H z ) Device floorDDS1 outDDS2 outDivided CEO f
CEOK /NDivision resultRep. rate f repK
Supplementary Figure 7 : Noise division demon-stration
Demonstration of the noise division achievedby the 2-DDS scheme. The orange and dashed violetcurves show the phase noise PSD separately measured atthe output of each DDS at 101.095 MHz (DDS1) and100 MHz (DDS2), respectively. The green curve dis-plays the noise of the frequency-divided CEO signal ofthe Kerr comb, which overlaps the noise of the repetitionrate (blue curve). Therefore, these two noise contribu-tions compensate each other to a large extend in the fi-nal RF signal (red curve), which demonstrates the noiseimprovement brought by the transfer oscillator schemelimited here by the Kerr comb injection locking band-width.
Supplementary Note 6: Sign effect in the transferoscillator
The high rejection of the Kerr comb phase noise offeredby the transfer oscillator scheme requires mixing signalswith the proper sign combination, so that the frequencyfluctuations of f Krep and f K ceo are indeed compensated inthe final RF signal. The sign of f K ceo is determined by theheterodyne beat between the Kerr comb and the auxiliarycomb, which can be controlled by the repetition rate ofthe auxiliary comb, and by the subtracted CEO signal ofthe auxiliary comb, whose sign can be changed using thefrequency-shifting synthesiser as previously mentioned.The sign of the f Krep contribution to be removed can beadjusted by inverting the output frequencies of the twoDDS, without changing any RF component. This is il- lustrated in Supplementary Fig. 8, which shows how thenoise is correctly compensated with the proper sign andincreases by a factor of 4 (in terms of PSD, or +6 dB)compared to the noise of f rep with the incorrect sign (asthe resulting signal corresponds to ν usl /N + 2 f rep ). Frequency (Hz)-150-140-130-120-110-100-90-80-70-60-50-40-30
SSB pha s e no i s e ( d B c / H z ) Device floorRep rateDivision output (wrong sign)Division output (right sign)
Supplementary Figure 8 : Sign effect
Demonstrationof the adjustment of the sign of the correction of the Kerrcomb f Krep noise in the low-noise RF output signal gen-erated by the transfer oscillator scheme. The blue curvedisplays the phase noise of the Kerr comb repetition rate.The purple curve shows the phase noise of the generatedRF signal obtained with the correct sign where the f Krep noise is removed, whereas the orange curve correspondsto the other sign (obtained by inverting the frequency ofthe two DDS), which leads to a 6 dB noise increase (theoutput signal contains twice the frequency fluctuationsof f Krep ). Supplementary Note 7: Source of imperfect noisecompensation
The noise compensation in the transfer oscillatormethod relies on the subtraction of various noise con-tributions of the micro-resonator comb. Perfect noisecompensation occurs when no relative delay is introducedbetween f ceo /N and f rep at the time of their final mix-ing. If a significant delay occurs between the signals,the noise compensation may be incomplete as the signalsbecome imperfectly correlated and the residual (uncom-pensated) noise scales according to S signal ϕ ( f ) = 1 N S usl ϕ ( f ) + 4 sin ( πτ f ) S rep ϕ ( f ) , (3)where τ is the relative delay, f is the Fourier frequencyand S ϕ denotes the phase noise power spectral density.If τ = 0 the phase noise of the repetition rate is prop-erly cancelled. Some care is thus needed to minimisethe delays, in order to maximise the noise cancellationbandwidth, but this factor is not critical. For example,a coarse length mismatch of 10 m would correspond toa delay of ∼
42 ns (assuming a velocity factor of 80 %),resulting in a fully uncompensated noise (0 dB rejection)reached at a Fourier frequency of ∼ Supplementary References T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kon-dratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Tem-poral solitons in optical microresonators,” Nature Photonics , 145–152 (2013). Tal Carmon, Lan Yang, and Kerry J. Vahala, “Dynamicalthermal behavior and thermal self-stability of microcavi-ties,” Optics Express , 4742 (2004). F Leo, S Coen, P Kockaert, S-P Gorza, P Emplit,and M Haelterman, “Temporal cavity solitons in one- dimensional kerr media as bits in an all-optical buffer,” Na-ture Photonics , 471–476 (2010). Xu Yi, Qi-Fan Yang, Xueyue Zhang, Ki Youl Yang, XinbaiLi, and Kerry Vahala, “Single-mode dispersive waves andsoliton microcomb dynamics,” Nature Communications ,14869 (2017). Jordan R. Stone, Travis C. Briles, Tara E. Drake, Daryl T.Spencer, David R. Carlson, Scott A. Diddams, and Scott B.Papp, “Thermal and nonlinear dissipative-soliton dynamicsin kerr-microresonator frequency combs,” Physical ReviewLetters , 063902 (2018). Pierre Brochard, Stéphane Schilt, and Thomas Südmeyer,“Ultra-low noise microwave generation with a free-runningoptical frequency comb transfer oscillator,” Optics Letters , 4651–4654 (2018). Enrico Rubiola, Ertan Salik, Shouhua Huang, Nan Yu,and Lute Maleki, “Photonic-delay technique for phase-noisemeasurement of microwave oscillators,” Journal of the Op-tical Society of America B22