# Photonic microwave signals with zeptosecond-level absolute timing noise

Xiaopeng Xie, Romain Bouchand, Daniele Nicolodi, Michele Giunta, Wolfgang Hänsel, Matthias Lezius, Abhay Joshi, Shubhashish Datta, Christophe Alexandre, Michel Lours, et al.

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Xiaopeng Xie, Romain Bouchand, Daniele Nicolodi, Michele Giunta, Wolfgang Hänsel, et al. Photonic microwave signals with zeptosecond-level absolute timing noise. Nature Photonics, 2016, 11 (1), pp.44-47. 10.1038/nphoton.2016.215. hal-02265751

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# Photonic microwave signals with zeptosecond-level absolute timing noise

Xiaopeng Xie<fnref n="1" />, Romain Bouchand<fnref n="1" />,‡, Daniele Nicolodi<fnref n="1" />,†, Michele Giunta²,³, Wolfgang Hänsel², Matthias Lezius², Abhay Joshi<fnref n="4" />, Shubhashish Datta<fnref n="4" />, Christophe Alexandre⁵, Michel Lours<fnref n="1" />, Pierre-Alain Tremblin<fnref n="6" />, Giorgio Santarelli<fnref n="6" />, Ronald Holzwarth²,³ and Yann Le Coq<fnref n="1" />*

Photonic synthesis of radiofrequency (RF) revived the quest for unrivalled microwave purity because of its ability to convey the benefits of optics to the microwave world<fnref n="1" />⁻¹¹. In this work, we perform a high-fidelity transfer of frequency stability between an optical reference and a microwave signal via a low-noise fibre-based frequency comb and cutting-edge photodetection techniques. We demonstrate the generation of the purest microwave signal with a fractional frequency stability below 6.5 × 10⁻¹⁶ at 1 s and a timing noise floor below 41 zs Hz⁻<fnref n="1" />/² (phase noise below −173 dBc Hz⁻<fnref n="1" /> for a 12 GHz carrier). This outperforms existing sources and promises a new era for state-of-the-art microwave generation. The characterization is achieved through a heterodyne cross-correlation scheme with the lowermost detection noise. This unprecedented level of purity can impact domains such as radar systems¹², telecommunications¹³ and time-frequency metrology²,¹⁴. The measurement methods developed here can benefit the characterization of a broad range of signals.

Photonic microwave generation methods, such as the optoelectronic oscillator<fnref n="4" />, Brillouin oscillator⁵, sideband-injection-locked laser<fnref n="6" />, electro-optical-frequency division⁷ and Kerr-frequency-comb oscillator⁸, have drawn attention because of their interesting properties, such as high frequency, large bandwidth (BW), tunability and chip-scale packaging. In particular, ultrastable lasers and a frequency-comb-based optical-to-microwave division scheme can produce microwave signals with both extremely high stability and low noise²,³,⁹⁻¹¹. In this work, we advance this approach to an unprecedented level.

Optical frequency combs are lasers that emit a series of evenly spaced ultrashort pulses, corresponding to discrete equidistant lines in the optical spectrum¹⁵,¹⁶. The frequency of the comb lines, $v_N$, can be expressed as $v_N = Nf_r + f_0$, where N is an integer number, $f_r$ is the pulse repetition rate and $f_0$ is the carrier envelope offset (CEO) frequency. In our set-up, one of the comb lines is tightly phase locked to a narrow linewidth continuous-wave (CW) laser reference, $v_{CW}$, that exhibits a very high stability and spectral purity. In addition, the CEO frequency $f_0$ is measured with the $f-2f$ self-referencing technique, and removed in the phase-lock process to obtain a virtual 'f₀-free' comb that obeys $v_N - f_0 = Nf_r = v_{CW}$ (refs 2,3,17). The spectrum of the electrical output of a photodiode illuminated by the comb light consists of harmonics of the repetition rate up to the cutoff frequency of the detector. Any aforementioned harmonics of order *n* can then be isolated to yield a microwave signal $f_μ = n f_r$, whose absolute frequency stability is equal to that of the reference light. Accordingly, the stability is said to be transferred

from the optics to the microwaves: $\delta v_{CW}/v_{CW} = \delta f_r/f_r = \delta f_μ/f_μ$. Thanks to the carrier frequency division, the phase-noise power spectral density (PSD) is intrinsically reduced by $M^2$, where $M = v_{CW}/f_ν = N/n \sim 10^4$ is the frequency division factor.

The microwave generation system is sketched in Fig. 1: a low-noise erbium-doped fibre-based optical frequency comb (FOFC) that features a 250 MHz repetition rate acts as a frequency divider; an ultrastable CW laser at 1,542 nm, with a fractional frequency stability as low as $5.5 \times 10^{-16}$ at 1 s (ref. 18), is used as a reference for stabilizing the comb; a specially designed high-linearity photodiode with flicker phase noise below $-140 f^{-1}$ dBc Hz⁻<fnref n="1" /> (with f the carrier offset frequency at which the noise is observed, in Hz) is used to convert the optical trains of pulses into a microwave signal.

During the optical-to-microwave conversion process, sundry important techniques are proposed and implemented. To increase the signal-to-noise ratio, a fibre-based optical-pulse repetition-rate multiplier is used to redistribute the photocurrent to the harmonic of interest¹⁹ (the 48th in our case). After four multiplying stages, we obtain over 2.5 mW of microwave power at 12 GHz for a d.c. photocurrent of 8 mA. A dispersion-compensation fibre unit is inserted to ensure that the optical pulses impinging on the photodiode are less than 800 fs long (Methods, Dispersion management). In this ultra-short pulse limit, the shot-noise-distribution imbalance between amplitude and phase allows us to reduce the shot-noise-induced phase noise²⁰. Careful pulse compression proved essential to achieve the final result.

Fibre-based lasers exhibit significant relative intensity noise (RIN), which can result in excess phase noise through amplitude-to-phase (AM-PM) conversion in the photodiode. We limit the RIN by meticulously choosing the laser working state (Methods, Optical frequency comb) and operate the photodiode in conditions in which nonlinear saturation effects help to zero the AM-PM conversion for the 12 GHz signal²¹. Rejection of amplitude noise by more than 33 dB is typically obtained (Methods, AM-PM conversion characterization and control).

Combining these forefront optoelectronics devices and cutting-edge techniques, microwave signals with a phase noise below −173 dBc Hz⁻<fnref n="1" /> at 10 kHz offset are made possible. As phase noise is only a reasonable figure of merit of an oscillator if its carrier frequency is provided, we choose to express it as a timing noise density, once normalized against the 12 GHz carrier. This represents an impressive timing noise below 41 zs Hz⁻<fnref n="1" />/².

The thorough characterization of such ultralow phase noise signals is a major challenge in itself, even more so for high-frequency

<footnote n="1">¹LNE-SYRTE, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, 61 avenue de l'Observatoire, 75014 Paris, France. ²Menlo Systems GmbH, Am Klopfspitz 19a, D-82152 Martinsried, Germany. ³Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany.</footnote>

<footnote n="4">⁴Discovery Semiconductors Inc., 119 Silvia Street, Ewing, New Jersey 08628, USA. ⁵CNAM, CEDRIC Laboratory, 292 rue Saint Martin, 75003 Paris, France.</footnote>

<footnote n="6">⁶LP2N, IOGS - CNRS - Université de Bordeaux 1, rue F. Mitterand, 33400 Talence, France. ⁷Present address: National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA. ⁸These authors contributed equally to this work. ⁹e-mail: yann.lecoq@obspm.fr</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1** | Experimental set-up for low-noise microwave generation and characterization. The left part of the set-up is used to generate a record low-noise 12 GHz microwave signal from an ultrastable optical reference at 1,542 nm via optical division. The right part is used to characterize the microwave-signal phase noise using a heterodyne digital cross-correlation method. The auxiliary optoelectronic microwave references A and B are also obtained by low-noise optical division of the light from two additional distinct ultrastable laser references at 1,542 nm using two separate optical frequency combs. ADC, analog-to-digital converter; CSO, cryogenic sapphire oscillator; DCF, dispersion compensation fibre; DDS, direct digital synthesizer; OCXO, oven-controlled crystal oscillator.</figcaption></figure>

carriers. In high-precision measurements, phase-noise characteriz-
ation is usually a process that involves comparing a signal from
the device under test (DUT) with a reference source. When the
signal under test has a lower phase noise than the available refer-
ence, two separate but identical systems can be built and compared⁹.
The data are then analysed assuming that the two identical systems
contribute equally to the phase noise. However, to realize two
equally good systems is not straightforward and a minute excess
phase noise of the phase-comparison system can strongly impact
the final result obtained by such methods.

To overcome these limitations, we demonstrate a heterodyne
digital cross-correlation scheme based on three similar but indepen-
dent optoelectronic microwave-generation systems, as sketched in
Fig. 1. By frequency mixing the DUT 12 GHz microwave signal
with signals from the two auxiliary systems, two beat notes around
5 MHz are obtained. These beat-note signals, both of which carry
phase information from the DUT, are sent to a field-programmable
gate array (FPGA)-based heterodyne cross-correlator where they
are sampled by fast analog-to-digital converters, digitally down-
converted and processed to generate two independent phase-
comparison data sets. Cross-correlation of these two phase-noise
data sets converges to the absolute phase-noise PSD of the 12 GHz
signal that we want to characterize. The microwave signals generated
by the two auxiliary systems act as phase references but do not need to
be as good as the signal being characterized. Their uncorrelated noise
determines the uncertainty of the estimates of the phase-noise PSD.
This uncertainty scales as 1/√m, with m the number of averages, and
can thus be made arbitrarily small in the assumption of a stationary
DUT phase noise²². The measurement noise floor is below
-180 dBc Hz⁻¹ for Fourier frequencies beyond 1 kHz offset (Fig. 2,
black dashed curve).

To assess the additional phase noise introduced by the optical fre-
quency division process, including comb and photodetectors, we
measure the microwave phase noise obtained when locking the two
auxiliary microwave references and the DUT to the same ultrastable
optical reference. In this configuration, the laser noise is common to
the three systems and then suppressed. The obtained phase-noise
PSD is shown in Fig. 2 (red curve). At Fourier frequencies up to
3 kHz, although there are numerous peaks caused by acoustic
noise, the photodiode flicker noise can be inferred to be below
-140 f⁻¹ dBc Hz⁻¹, which, to our knowledge, is the lowest level
ever reported<sup>23</sup>. At lower Fourier frequencies, the phase noise
departs from a 1/f behaviour, which we explain by residual long-
term drift that results from the noisy laboratory environment and
the significant amount of non-noise-cancelled fibred optics
(several tens of metres, mostly in the pulse-rate multipliers). From
3 to 100 kHz, we measure a shot-noise-limited white-phase-noise
floor at -173 dBc Hz⁻¹ that is more than 10 dB below previously
reported results obtained with comparable schemes<sup>9,24</sup>. This shot-
noise floor corresponds to a timing noise of 41 zs Hz⁻¹/² and
depends on the optical pulse duration (Methods, Shot-noise limit).
Thermal noise is intrinsically rejected in the cross-correlation
process<sup>25</sup>, but from the measured microwave power it can be
deduced as -181 dBc Hz⁻¹. Between 100 kHz and 1 MHz, we are
limited by a servo bump because of the residual in-loop errors
from the comb phase-lock loop.

The measured absolute single-sideband (SSB) phase-noise PSD for the 12 GHz microwave signal generated by our optical-frequency-comb frequency division scheme, obtained by locking the two auxiliary microwave references and the DUT to three independent ultrastable optical references, is displayed in Fig. 3 (red curve). To our knowledge, this is the lowest absolute phase noise
<figure><img src="image_3.png" /></figure>

**Figure 2** | Additive phase-noise contribution of the frequency division scheme. The green and blue lines are the phase noise of the 5 MHz RF signals obtained from the beat between the signal under test (at 12 GHz) and the auxiliary sources (at 11.995 and 12.005 GHz). The phase data that correspond to these two curves are cross-correlated to yield the phase noise of the DUT (red line). In this measurement, all the lasers are stabilized using the same laser reference so that it represents the residual phase noise introduced in the optical division process and, thus, the quality of the frequency-stability transfer. It reaches a white timing noise floor of 41 zs Hz⁻¹/² (thick blue line). The outstanding photodiode flicker can be inferred to be below -140 f⁻¹ dBc Hz⁻¹ (thick grey line). The orange line is the projected RIN-induced phase noise after 33 dB of AM-PM conversion rejection. The dashed black line is the measurement noise floor.

<figure><img src="image_4.png" /></figure>

**Figure 3** | Absolute transfer of spectral purity from optics to microwave. The absolute phase noise of the 12 GHz microwave signal we generated (red line) is limited by the optical phase noise of the laser reference at 1,542 nm (blue line) and by the residual in-loop error of the phase-lock loop used to synchronize the repetition rate of the comb with the optical reference (bump above 700 kHz offset frequency). The phase noise of the 12 GHz signal sticks almost entirely to the optical phase noise, which indicates a close-to-complete absolute transfer of spectral purity from optics to microwave. Between 3 and 100 kHz Fourier frequency, the transfer remains partial but offers an original opportunity to probe shot-noise levels. The spurs that range from 10 Hz to 1 kHz result from acoustic noise picked up in the various fibre links and from electromagnetic interference. The optical phase noise has been characterized independently by the cross-correlation of RF optical beats from ultrastable cavities.
ever reported both close and far from the carrier⁹. At low offset Fourier frequencies, from 1 to 400 Hz, the phase noise is almost fully determined by the CW laser reference, which indicates that we performed a close-to-complete transfer of frequency stability from optics to microwave. The -106 dBc Hz⁻¹ at a 1 Hz offset phase-noise level is only 3 dB higher than that inferred from the measured optical reference phase noise. The spurs in the Fourier frequency range from 10 Hz to 3 kHz originate from the 50 Hz power line harmonics as a result of imperfect electromagnetic shielding and acoustic noise coupling to the fibre-optics set-up despite the laser being acoustically isolated and the fibre link from the CW laser to the comb system being noise-cancelled. Between 3 kHz and 1 MHz, the transfer is only limited by the residual phase noise of the optical frequency division scheme and the in-loop errors from the reference laser Pound-Drever-Hall (PDH) lock.

Improvements to the close-to-carrier phase noise could be obtained with better ultrastable laser frequency stability. Longer or cryogenic reference cavities²⁶,²⁷ with crystalline coatings²⁸ or spectral hole-burning stabilization²⁹ could lower the phase-noise limit by one order of magnitude. To be limited solely by the optical reference, shot noise could be reduced by using a photodiode with higher power-handling capabilities³⁰. However, the most stringent requirement is photodiode flicker below the state-of-the-art -140 $f^{-1}$ dBc Hz⁻¹ demonstrated here. Our extremely low-noise set-up is the ideal testbed for that development, as it offers the opportunity to study fundamental photodetection limits and is suitable for any phase-noise characterization beyond comb-based systems.

Terabit communication systems with high-speed data transmission, high-stability fountain atomic clocks, very long baseline radio astronomy, high-accuracy navigation and radar systems are direct applications that could benefit from the new level of microwave purity demonstrated in this Letter. In particular, low phase noise in defence pulse-Doppler radar will enhance the detection of moving targets to an unprecedented level of resolution. Furthermore, this result paves the way to compact, robust and mobile microwave sources with ultralow phase noise based on reliable technologies that have become readily accessible.

## Methods

Methods and any associated references are available in the online version of the paper.

Received 19 August 2016; accepted 4 October 2016; published online 21 November 2016

## References

1. Capmany, J. & Novak, D. Microwave photonics combines two worlds. *Nat. Photon.* **1**, 319–330 (2007).

2. Millo, J. *et al*. Ultralow noise microwave generation with fiber-based optical frequency comb and application to atomic fountain clock. *Appl. Phys. Lett.* **94**, 141105 (2009).

3. Zhang, W. *et al*. Sub-100 attoseconds stability optics-to-microwave synchronization. *Appl. Phys. Lett.* **96**, 211105 (2010).

4. Maleki, L. Sources: the optoelectronic oscillator. *Nat. Photon.* **5**, 728–730 (2011).

5. Li, J., Lee, H. & Vahala, K. J. Microwave synthesizer using an on-chip Brillouin oscillator. *Nat. Commun.* **4**, 2097 (2013).

6. Schneider, G. J., Murakowski, J. A., Schuetz, C. A., Shi, S. & Prather, D. W. Radiofrequency signal-generation system with over seven octaves of continuous tuning. *Nat. Photon.* **7**, 118–122 (2013).

7. Li, J., Yi, X., Lee, H., Diddams, S. A. & Vahala, K. J. Electro-optical frequency division and stable microwave synthesis. *Science* **345**, 309–313 (2014).

8. Liang, W. *et al*. High spectral purity Kerr frequency comb radio frequency photonic oscillator. *Nat. Commun.* **6**, 7957 (2015).

9. Fortier, T. M. *et al*. Generation of ultrastable microwaves via optical frequency division. *Nat. Photon.* **5**, 425–429 (2011).

10. Didier, A. *et al*. Ultra-low phase noise all-optical microwave generation setup based on commercial devices. *Appl. Opt.* **54**, 3682–3686 (2015).

11. Portuondo-Campa, E., Buchs, G., Kundermann, S., Balet, L. & Lecomte, S. Ultra-low phase-noise microwave generation using a diode-pumped solid-state laser based frequency comb and a polarization-maintaining pulse interleaver. *Opt. Express* **23**, 32441–32451 (2015).

12. Ghelfi, P. *et al*. A fully photonics-based coherent radar system. *Nature* **507**, 341–345 (2014).

13. Koenig, S. *et al*. Wireless sub-THz communication system with high data rate. *Nat. Photon.* **7**, 977–981 (2013).

14. Santarelli, G. *et al*. Quantum projection noise in an atomic fountain: a high stability cesium frequency standard. *Phys. Rev. Lett.* **82**, 4619–4622 (1999).

15. Hänsch, T. W. Nobel lecture: passion for precision. *Rev. Mod. Phys.* **78**, 1297–1309 (2006).

16. Hall, J. L. Nobel lecture: defining and measuring optical frequencies. *Rev. Mod. Phys.* **78**, 1279–1295 (2006).

17. Nicolodi, D. *et al*. Spectral purity transfer between optical wavelengths at the 10⁻¹⁸ level. *Nat. Photon.* **8**, 219–223 (2014).

18. Argence, B. *et al*. Prototype of an ultra-stable optical cavity for space applications. *Opt. Express* **20**, 25409–25420 (2012).

19. Haboucha, A. *et al*. Optical-fiber pulse rate multiplier for ultralow phase-noise signal generation. *Opt. Lett.* **36**, 3654–3656 (2011).

20. Quinlan, F. *et al*. Exploiting shot noise correlations in the photodetection of ultrashort optical pulse trains. *Nat. Photon.* **7**, 290–293 (2013).

21. Zhang, W. *et al*. Amplitude to phase conversion of InGaAs pin photo-diodes for femtosecond lasers microwave signal generation. *Appl. Phys. B* **106**, 301–308 (2012).

22. Rubiola, E. & Giordano, V. Correlation-based phase noise measurements. *Rev. Sci. Instrum.* **71**, 3085–3091 (2000).

23. Baynes, R. N. *et al*. Attosecond timing in optical-to-electrical conversion. *Optica* **2**, 141–146 (2015).

24. Quinlan, F. *et al*. Optical amplification and pulse interleaving for low-noise photonic generation. *Opt. Lett.* **39**, 1581–1584 (2014).

25. Hati, A., Nelson, C. W. & Howe, D. A. Cross-spectrum measurement of thermal-noise limited oscillators. *Rev. Sci. Instrum.* **87**, 034708 (2016).

26. Häfner, S. *et al*. 8 × 10⁻¹⁷ fractional laser frequency instability with a long room-temperature cavity. *Opt. Lett.* **40**, 2112–2115 (2015).

27. Matei, D. G. *et al*. A second generation of low thermal noise cryogenic silicon resonator. *J. Phys. Conf. Ser.* **723**, 012031 (2016).

28. Cole, G. D. *et al*. High-performance near- and mid-infrared crystalline coatings. *Optica* **3**, 647–656 (2016).

29. Thorpe, M. J., Rippe, L., Fortier, T. M., Kirchner, M. S. & Rosenband, T. Frequency stabilization to 6 × 10⁻¹⁶ via spectral-hole burning. *Nat. Photon.* **5**, 688–693 (2011).

30. Beling, A., Xie, X. & Campbell, J. C. High-power, high-linearity photodiodes. *Optica* **3**, 328–338 (2016).

## Acknowledgements

We thank J. Pinto for help with the electronics and R. Le Targat for the reference-laser distribution. This work is funded by the Defense Advanced Research Projects Agency (DARPA) as a part of the Program in Ultrafast Laser Science and Engineering (PureComb project) under contract no. W31P4Q-14-C-0050, by the Formation, Innovation, Recherche, Services et Transfert en Temps-Fréquence (FIRST-TF) Labex, by the Eurostar Eureka program (Stable Microwave Generation and Dissemination over Optical Fiber project) and by the EU FP7 Initial Training Network FACT (Future Atomic Clock Technology).

## Author contributions

X.X., R.B. and D.N. set up the experiment and carried out the measurements. M.G., W.H., M.L. and R.H. conceived and built the low-noise optical frequency combs. A.J. and S.D. provided the photodiodes. C.A. programmed the cross-correlator hardware. M.L. made the RF chains in the cross-correlator. P.-A.T. and G.S. fabricated the pulse rate multipliers. X.X. and R.B. obtained the final data, prepared the manuscript and gathered the contributions from all the other co-authors. Y.L.C. designed the experiment and lead the project.

## Additional information

Reprints and permissions information is available online at www.nature.com/reprints.
Correspondence and requests for materials should be addressed to Y.L.C.

## Competing financial interests

The authors declare no competing financial interests.
## Methods

**Optical frequency comb.** The FOFC consists of three parts: a femtosecond laser, an f-2f interferometer module and an erbium-doped fibre amplifier (EDFA), all made with polarization-maintaining fibre. Based on the nonlinear amplifying loop mirror mode-locking principle, the laser reaches a steady state in a few seconds and can remain in the mode-locked state for several months. The repetition rate of the FOFC is coarsely tunable over 5 MHz around 250 MHz to accommodate the mode spacing and thus tune the frequency of the optically generated microwave signal. An intracavity electro-optical modulator (EOM) provides feedback on the comb repetition rate with a BW of roughly 1 MHz (ref. 31).

The laser output is pre-amplified up to 30 mW by a first EDFA and split into two parts that are used to seed the f-2f interferometer for the CEO-frequency detection and by a second EDFA for microwave generation. The second high-power EDFA boosts the optical power up to 350 mW and passively reduces the RIN through seed saturation. It is pumped by three low-noise single-mode high-power laser diodes, all set to specific currents to avoid parasitic mode hopping. For each diode, the lowest RIN is observed when the current set point is at an equal distance from two successive mode hops. The output shows a pulse width below 50 fs, with an optical spectrum broader than 60 nm FWHM.

**Ultrastable laser and its phase noise characterization.** The reference CW laser is a 1,542 nm semiconductor laser locked to an ultrastable Fabry–Perot cavity ($Q = 380,000$) via PDH techniques (500 kHz BW), which provides a fractional frequency stability of $5.5 \times 10^{-16}$ at 1 s. The reference laser light is transferred to the FOFC using a 50 m fibre link with an acousto-optic modulator (AOM)-based fibre-noise canceller³². The laser phase noise is characterized by the same cross-correlation scheme that is used for the microwave system, but operating on optical beat notes rather than microwave-mixed signals. Three 1,542 nm ultrastable lasers locked to three distinct cavities via the PDH scheme were used. For each optical phase-noise measurement two lasers act as references to characterize the third one. The frequency difference between these three lasers is below 600 MHz.

**Optical frequency comb phase locking.** The optical phase-locking technique of the FOFC to the ultrastable reference laser is similar to that reported previously²,³,¹⁷. Briefly, the booster-amplifier output is followed by a fibbed three-port thin-film optical add-drop multiplexer. The first output port delivers a signal filtered around 1,542 nm with a BW of 0.8 nm. At the second output, the unfiltered part of the initial spectrum is used for the microwave generation. The filtered signal is beat with the ultrastable reference laser $v_{CW}$, which leads to a beat-note signal $f_0 = v_{CW} - Nf_0 - f_0$. This beat-note signal is mixed with the CEO frequency $f_0$ to obtain an $f_0$-independent signal at 880 MHz. This frequency is digitally divided by eight and compared with a fixed frequency from a direct digital synthesizer to generate an error signal. The error signal is then processed through a fast analogue loop filter, and fed to the intracavity EOM and piezoelectric actuators so as to stabilize the repetition rate of the comb.

**High-linearity low-noise photodiode.** A top-illuminated dual-depletion region InGaAs/InP photodiode with an optimized illumination profile was used for this work. The illumination profile is improved through beam shaping via gradient-index lens coupling. The power-handling capability and linearity of the photodiode, which are limited by the collapse of the applied electric field because of the photogenerated space-charge effects, increase monotonically with its reverse bias. At a 15 V reverse bias, the photodiode shows a linear response up to a 4 V<sub>PP</sub> output amplitude and delivers a compressed output up to 6 V<sub>PP</sub>. To avoid any thermally induced device failure, the photodiode chip was integrated with a thermoelectric cooler and temperature sensor and assembled in a compact eight-pin microwave package to extract the heat from the chip and ensure reliable operation. In the final measurements presented here, the photodiode is typically illuminated with 15 mW of optical power to yield 2.5 mW of 12 GHz RF power for a d.c. photocurrent of 8 mA. From the data shown in Fig. 2 we can infer a flicker phase noise below –140 f<sup>-1</sup> dBc Hz<sup>-1</sup>, comparable to the best modified unitravelling-carrier photodiodes²³.

**AM-PM conversion characterization and control.** To quantify and control the AM-PM noise conversion that occurs through the photodetection process, we implemented an FPGA-based system to modulate the intensity of the FOFC output and coherently demodulate the relative phase between the photodetected signal and a reference microwave oscillator. The intensity of the comb output light is modulated with an AOM placed right before the photodiode. The photodiode output signal is split between low frequencies (d.c., 30 MHz) and high frequencies (30 MHz to 65 GHz) via a high-quality diplexer. The d.c. part is digitally processed to extract the

effective amplitude modulation. The a.c. part is filtered at 12 GHz with a narrow-band cavity filter and directed to the RF port of a termination-insensitive triple-balanced mixer ensuring broadband impedance matching at each port. The local oscillator (LO) port of this mixer is fed with the 12.005 GHz signal from another stabilized FOFC that acts as a low-noise microwave-phase reference. The down-converted heterodyne signal (5 MHz) obtained at the intermediate frequency (IF) port of the mixer is digitally phase-demodulated at the very frequency used for the amplitude modulation. The system allows a slow active feedback to keep the AM-PM conversion near a zero point.

The mixer input powers are +15 dBm on the LO port and –5 dBm on the RF port. The modulation frequency is conveniently chosen to be 1.23 kHz to avoid any disturbance from the parasitic power-supply harmonics and still preserve an efficient modulation. Typically, the measured AM-PM conversion coefficient corresponds to a rejection of RIN-induced phase noise by about 33 dB.

**Dispersion management.** When propagating through the 20 m of optical fibres that compose the set-up, the femtosecond pulses spread and acquire considerable chirp, which results in an increased pulse duration. To reduce the pulse duration below the picosecond level, dispersion is compensated by inserting negative-dispersion optical fibres (DCF38) at the output of the pulse-rate multipliers. After a theoretical estimation of the required fibre length, the optimization is done iteratively by steps of 2.5 cm with the pulse duration monitored by a commercial interferometric autocorrelator.

**Shot-noise limit.** It has been reported previously that when photodetecting a train of optical pulses, the shot-noise limit is related to the optical pulse width²⁰. Although we do observe and confirm such a dependency, the measured white-phase-noise floor of the microwave signal obtained when illuminating the photodiode with 800 fs pulses deviates by more than 20 dB from the theoretical prediction (–196 dBc Hz<sup>-1</sup> as calculated using the equation proposed in Quinlan *et al.*²⁰). This significant discrepancy has been thought to originate from photocarrier scattering and absorption in the photodetector that results in an increased timing jitter of the electrical pulse train³³.

**Phase-noise characterization.** The 12 GHz harmonics from the FOFC under test is selected by a narrow filter (10 MHz BW) and split into two parts. Each part is sent to the RF port of a termination-insensitive triple-balanced mixer ensuring broadband impedance matching at each port and is then mixed with two distinct 12 GHz signals from two independently stabilized FOFCs acting as references. The two reference signals (12.005 GHz and 11.995 GHz) are amplified to +13 dBm by low phase-noise narrow-band microwave amplifiers. Isolators (80 dB) are inserted at every mixer port to avoid any parasitic feedback that has been observed to undermine the phase-noise result and the AM-PM conversion coefficient measurement. After the mixers, two heterodyne beat notes at 5 MHz are obtained and sent to the digital heterodyne cross-correlator³⁴. Each heterodyne signal is sampled at 250 mega samples per second (MS s<sup>-1</sup>) in an analog-to-digital converter clocked by a specific low phase-noise oven-controlled crystal oscillator slowly phase locked to a cryogenic sapphire oscillator for long-term stability (30 Hz BW). The signal is digitally down-converted to d.c. and processed in an FPGA to yield in-phase and quadrature components down-sampled to 2 MS s<sup>-1</sup>. The samples are transferred via Gbit-Ethernet to a computer where they are real-time analysed by a Python-based software to obtain the phase-noise information. The typical phase-noise floor of the instrument for a 10 MHz carrier is –130 dBc Hz<sup>-1</sup> at 1 Hz offset and below –180 dBc Hz<sup>-1</sup> for Fourier frequencies beyond 1 kHz. Moreover, the heterodyne nature of the cross-correlator renders it only a little sensitive to amplitude noise with more than 20 dB of AM-PM conversion rejection. The usual averaging time required to obtain the results displayed in Fig. 2 is around 18 h.

## References

31. Zhang, W. et al. Characterizing a fiber-based frequency comb with electro-optic modulator. *IEEE Trans. Ultrason. Ferroelectr. Freq. Control* **59**, 432–438 (2012).

32. Ma, L. S., Jungner, P., Ye, J. & Hall, J. L. Delivering the same optical frequency at two places: accurate cancellation of phase noise introduced by an optical fiber or other time-varying path. *Opt. Lett.* **19**, 1777–1779 (1994).

33. Sun, W. et al. Broadband noise limit in the photodetection of ultralow jitter optical pulses. *Phys. Rev. Lett.* **113**, 203901 (2014).

34. Nelson, C. W. & Howe, D. A. A sub-sampling digital PM/AM noise measurement system. In *Proc. IEEE Frequency Control Symp.* 1–4 (IEEE, 2012).