## Abstract

A simple and practical apparatus enabling repetition rate (*f _{rep}*) noise, carrier-envelope frequency (

*f*) noise and n

_{ceo}^{th}optical comb mode (

*ν*) noise spectra measurements with high precision is established. The

_{n}*f*and

_{rep}*ν*noise spectra are measured by a fiber delay line interferometer, while

_{n}*f*noise spectrum is measured by an

_{ceo}*f-2f*interferometer. We utilize this apparatus to characterize the noise performance of an Er-fiber optical frequency comb (OFC) and analyze the origin of dominant noise sources. Moreover, this apparatus provides a powerful tool for diagnosing noise dynamics intrinsic in mode-locked lasers and OFCs. To this end, we uncover the anti-correlation between

*f*and

_{rep}*f*noise as well as the impact of servo loops on noise characteristics in the stabilized OFC.

_{ceo}© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Passively mode-locked laser based optical frequency combs (OFCs) have become indispensable tools for a number of metrological scientific research purposes, such as absolute optical frequency metrology, molecular spectroscopy, time-frequency remote transfer, to name a few [1–6]. The emergence of OFC has pushed the precision of these applications to unprecedented levels, benefiting from the advances in ultra-low noise OFC technologies [7–8]. Owing to the superior noise characteristics and well-established phase locking technique, OFC systems can routinely emit pulse trains with sub-100 as timing jitter, sub-10 as carrier envelop phase (CEP) jitter and < 1 Hz optical linewidth [9–12].

Repetition rate (*f _{rep}*) noise, carrier-envelope frequency (

*f*) noise and the n

_{ceo}^{th}optical comb mode (ν

_{n}) noise characterization are essential for OFC performance evaluation. Dramatic progress has been made in recent years. Optical-electric conversion based on a high-speed photodiode is commonly used for

*f*noise measurement [13,14], while the timing resolution is limited by AM-PM conversion noise induced by nonlinearities at photodetection. To solve this problem, Schibli

_{rep}*et al.,*proposed a balanced optical cross-correlation (BOC) technique in 2003 [15]. The BOC method relies on two lasers, one is laser under test, the other serves as reference. A nonlinear crystal converts their relative timing jitter to intensity fluctuations of a sum-frequency signal. This method has been widely applied in

*f*noise measurement of various fiber lasers [10,16–18] and solid-state lasers [19–21]. In 2015, Hou

_{rep}*et al.,*proposed an optical heterodyne method to measure

*f*noise [9]. Still, two lasers are required to accomplish a measurement. Compared with the BOC method, absence of second-order nonlinear progress allows a higher signal-to-noise ratio of error signal and a yoctosecond-level measurement resolution has been achieved. In order to avoid usage of the second laser as reference, Jung

_{rep}*et al.,*proposed an asymmetric fiber delay line interferometer method [22]. In this approach, a kilometer-long dispersion-compensated fiber delay line has been used as reference, providing a practical and low-cost approach for

*f*noise measurement, which even applies to noise characterization of supercontinuum sources [23].

_{rep}Carrier envelope frequency noise characterization typically relies on octave-spanning supercontinuum generation which is indispensable to obtain the *f _{ceo}* beat note via

*f-2f*self-referencing technique [24–27]. The

*f*beat note can be directly analyzed based on radio frequency signal source analyzers, allowing a noise power spectral density (PSD) measurement up to ∼ 1 MHz offset frequency, which is typically limited by photo-detector’s shot-noise [28]. Further interferometric detection of the supercontinuum enables shot noise limited

_{ceo}*f*noise PSD characterization with extended frequency range from microhertz to tens of MHz [27]. On the other side, the

_{ceo}*f*noise spectrum can be retrieved by beating the OFC with a CW laser [29]. Octave-spanning supercontinuum is not necessary. Beside the frequency domain measurements, Kim

_{ceo}*et al.,*explored a time domain method to stabilize CEP of an Er fiber mode-locked laser with the combination of BOC and interferometric cross-correlation (IXCOR) [30].

To characterize the comb-line noise of a certain comb mode in OFC, several approaches have been reported. The routine approach is to heterodyne a certain frequency mode of OFC with a narrow-line CW reference laser [24–26]. Typically, ν_{n} noise PSD can be measured up to ∼MHz offset frequency from an optical heterodyne beat, which is routinely limited by signal-to-noise ratio (SNR) during photo-detection process. However, narrow-line CW lasers are only available at certain wavelengths and are usually expensive. To exploit a reference laser free and use a low-cost comb-line noise measurement method, a delayed self-heterodyne method and multiple-fringe-side discriminator method were proposed. The self-heterodyne method first appeared around 1980s aiming at the linewidth measurement of CW lasers [31,32]. This method utilizes an asymmetric fiber interferometer as an optical discriminator and allows PSD measurement of comb lines in actively mode-locked lasers and Brillouin OFCs [33–35]. The multiple-fringe-side discriminator method is based on a low-finesse FP resonator operates as an array of optical frequency discriminators, enabling ν_{n} noise PSD measurement up to 1 MHz offset frequency [36].

In this article, we characterize frequency noise spectra of *f _{rep}*,

*f*as well as ν

_{ceo}_{n}simultaneously for fiber OFCs with high precision. The scheme is based on an asymmetric fiber delay line interferometer which utilizes delayed self-heterodyne as an optical discriminator, enabling reference-free

*f*noise spectrum measurement and

_{rep}*ν*noise spectrum measurement. To the best of our knowledge, it is the first demonstration of comb-line noise measurement of a passive mode locked laser using a fiber delay line interferometer. The

_{n}*f*noise spectrum can be obtained by directly analyzing the

_{ceo}*f*beat note from an

_{ceo}*f-2f*interferometer output with a phase noise analyzer. As a demonstration, we characterized

*f*noise,

_{rep}*f*noise, as well as ν

_{ceo}_{n}noise of a nonlinear amplifying loop mirror (NALM) mode-locked Er-fiber OFC and compared the measurements with the well-established Paschotta’s theoretical model [37] and fixed-point model [24,38]. We also verified anti-correlation between

*f*and

_{rep}*f*noise in experiment. At last, PSD analysis has been utilized to experimentally investigate the impact of servo loops on noise performance. This setup can not only be used for routine OFC noise measurements but also allows the study of intrinsic noise dynamics in mode-locked lasers.

_{ceo}## 2. Principle and experimental setup

The experimental setup is shown in Fig. 1. The mode-locked laser under test is an all-polarization-maintaining NALM-based Er-doped fiber laser [39]. Different from the laser in [39], the splicing angle between fiber PBS and collimator is set to 0 degree to achieve single-pule-train output in our case. The laser emits a 78-MHz pulse train with average power of 20 mW. The output spectrum centers at 1580 nm with 30 nm FWHM. The pulse duration is 100 fs directly from laser cavity without compression. An intra-cavity piezo-actuated mirror enables high-bandwidth cavity length tuning. Through a home-build EDFA, the output pulse train is amplified up to ∼150 mW. The noise analysis system consists of two parts: an asymmetric fiber Michelson interferometer and an *f-2f* interferometer. Ten percent of laser power from EDFA is used for *f _{rep}* or

*ν*noise measurement while the remained 90% power is used for the

_{n}*f*measurement.

_{ceo}For *f _{rep}* and

*ν*frequency noise analysis, the experimental scheme is based on an asymmetric fiber Michelson interferometer. Two segments of OFC optical spectrum centered at λ

_{n}_{1}=1540 nm and λ

_{2}=1560 nm, respectively, have been filtered out by using reflective fiber Bragg gratings. Each segment of optical spectrum has 2 nm bandwidth. Then, the two filtered spectra are amplified to > 20 mW and directed to the asymmetric fiber Michelson interferometer. A 2×2 fiber coupler divides the two filtered spectral segments into the reference arm and delayed arm. In the reference arm, two spectral segments are simply reflected back by an FRM at the end of the arm. In the delayed arm, two spectral segments pass through a 140-m long fiber spool (95-m SMF-28 + 45-m DCF38), a delay control unit (DCU), a VCO-controlled AOFS and are reflected by an FRM at the end of the arm. The 140-long fiber spool, DCU and AOFS make up the fiber delay line system. The DCU consists of two 1540/1560 WDMs and an adjustable delay control unit. Through careful adjustment of the delay control unit, the optical path length difference between λ

_{1}and λ

_{2}which originates from the fiber delay line’s group velocity dispersion can be compensated. One can either change the total length of the fiber delay line or slightly tune the laser’s cavity to guarantee that the optical path length difference between the reference arm and delay arm is integral multiple of the laser’s cavity length. Consequently, pulses of λ

_{1}(λ

_{2}) temporally overlap with its lateral ∼120

^{th}pulse at the output of 2×2 coupler. Another 1540/1560 WDM splits λ

_{1}and λ

_{2}, which allows the detection of the beat note signal at λ

_{1}and λ

_{2}separately. In practice, the noise spectra can be retrieved by a simple heterodyne scheme. To this end, an AOFS is inserted to the fiber delay line and modulates the optical signals at

*f*=50 MHz. For the reason that the AOFS modulates the pulse train twice, the output from the long fiber arm is 100-MHz-frequency (2

_{mod}*f*) offset from that in the reference arm. The 100-MHz heterodyne beats of two wavelengths are detected by two PDs, respectively. Then we utilize two bandpass filters (K&L microwave, 6LB30-100/T24-0/0) with 100 MHz center frequency to filter out these two beats and amplify them to > 0 dBm.

_{mod}The basic principle of *f _{rep}* and

*ν*frequency noise analysis is delayed self-heterodyne method, which was firstly implemented in the frequency stabilization of CW lasers. The procedure for

_{n}*ν*frequency noise measurement is as follows. Assuming that the frequency noise difference of the optical modes within the narrow filter bandwidth is negligible, the heterodyne beat carries the frequency noise

_{n}*p*·δ[τ(n

*f*2

_{rep}+f_{ceo}+*f*)], where

_{mod}*p*accounts for the entire number of comb lines within the filtered bandwidth, δ[τ(n

*f*2

_{rep}+f_{ceo}+*f*)] represents the frequency noise of the specific comb line that overlaps with the filter center wavelength. Here, the delayed self-heterodyne process has magnified the frequency noise of the laser under test by a factor of τ, which is the round-trip delay time between the two arms in the interferometer. Through mixing the beat note with 2

_{mod}*f*, error signal containing comb-line frequency noise

_{mod}*p*·(δ[τ(n

*f*)]) can be obtained. Voltage noise PSD of output from the mixer is acquired by an FFT analyzer (Standard Research Systems, SR770). Using the transfer function T(

_{rep}+f_{ceo}*f*)=V

_{peak}[(1-exp(-

*i*×2πτ))/(

*i*×

*f*)] [V/Hz] in [22], where V

_{peak}is the amplitude of the low-pass filtered mixer output voltage from interference pattern, voltage noise PSD is converted to frequency noise PSD. For simplicity, one can use T(

*f*) = 2πτV

_{peak}[V/Hz] as frequency discriminator at low Fourier frequencies. Note that V

_{peak}in transfer function accounts for the contribution from all the comb lines within the filtering bandwidth. Then, the converted result represents frequency noise PSD of a single comb line (δ[(n

*f*)]) owning to the identical noise performance of these comb lines.

_{rep}+f_{ceo}Repetition rate noise characterization needs to be conducted in two separated wavelengths, λ_{1} and λ_{2}. Two beat notes carry comb-line frequency noise (*p*·δ[τ(m*f _{rep}*+

*f*)],

_{ceo}*p*·δ[τ(n

*f*+

_{rep}*f*)]). After the common frequency components are rejected by a frequency mixer, the repetition-rate frequency noise, δ(m−n)

_{ceo}*f*, can be retrieved with transfer function in a similar manner. More detail in

_{rep}*f*noise characterization could be found in our previous publication [22,23]. In both

_{rep}*f*and

_{rep}*ν*frequency noise measurement, error signals from mixers needs to be fed back to the intra-cavity piezo actuator to catch the error signal at the linear range of discrimination slope. Stable, cycle-slip-free phase-locking ensures effective power spectra measurement beyond locking bandwidth. Meanwhile, the length of fiber delay is carefully chosen to avoid null frequencies in transfer function in the Fourier frequency range under test. For frequency stabilization, longer fiber is required due to a higher frequency discrimination sensitivity [40].

_{n}For *f _{ceo}* noise detection and stabilization, 90% part of the pulse train from the EDFA is injected into a home-built collinear

*f-2f*interferometer. For noise characterization, the

*f*beat note can be free running. We use a phase noise analyzer (Rohde & Schwarz, R&S FSWP26) to characterize the free-running

_{ceo}*f*frequency noise PSD. While, for the purpose of investigating the impact of the

_{ceo}*f*stabilization to n×

_{ceo}*f*noise, the

_{rep}*f*needs to be stabilized. To achieve stabilization, the

_{ceo}*f*beat note is firstly filtered out with a low pass filter. An integrated digital phase discriminator and PID servo unit allow us to effectively phase lock the

_{ceo}*f*to a RF reference (Stanford Research Systems, SG382) through pump power feedback.

_{ceo}## 3. Experimental results

In this section, we present *f _{rep}*,

*f*and

_{ceo}*ν*noise spectra of the Er-optical frequency comb obtained from the noise measurement platform. The noise origins have been analyzed. The correlation between noise spectra and the influence of servo loops on noise performance have also been investigated based on this setup.

_{n}#### 3.1 n×f_{rep}, f_{ceo} and ν_{n} noise

The OFC under test and fiber delay line are all enclosed in an 11-mm thickness aluminum box with glued foam on the inner side for acoustic noise damping. The whole system is built on a floating optical table. At first, we stabilize the laser’s repetition rate to the fiber delay line with ∼1 kHz bandwidth, leaving the *f _{ceo}* free running. An FFT analyzer measures (n-m)×

*f*frequency noise PSD and then it is converted to n×

_{rep}*f*frequency noise PSD, where n=2.50×10

_{rep}^{5}is the mode number at 1540 nm and m=2.47×10

^{5}is the mode number at 1560 nm. The measured n×

*f*frequency noise PSD ranging from 100 Hz to 100 kHz is shown in Fig. 2 in red curve. Converting to phase noise, the integrated repetition rate (

_{rep}*f*=78 MHz) phase error from 1 kHz to 100 kHz is 2 μrad, corresponding to timing jitter of 4.35 fs. After that, we break the PLL of repetition rate locking and stabilize the n

_{rep}^{th}comb mode (n×

*f*+

_{rep}*f*) to the fiber delay line through cavity length feedback (< 200 Hz locking bandwidth). Using the same approach, the frequency noise of the comb mode of 1540 nm could be retrieved. The comb-line (

_{ceo}*ν*=194.8 THz) frequency noise PSD result is shown in Fig. 2 in black, corresponding to a 0.4-rad integrated phase error (integrated from 1 kHz to 100 kHz). Finally, we utilize a phase noise analyzer to characterize the free-running carrier-envelope frequency (

_{n}*f*=8 MHz) noise PSD, which is shown in Fig. 2 in blue, corresponding to a 6.5-rad integrated phase error (integrated from 1 kHz to 100 kHz). In this case, the

_{ceo}*f*is not stabilized.

_{rep}#### 3.2 Origins of n×f_{rep}, f_{ceo} and ν_{n} noise

In this subsection, we explore the intrinsic noise origin of the measured n×*f _{rep}*,

*f*and

_{ceo}*ν*noise based on the above measurement. The numerical analysis allows us to verify the validity of our measured noise spectra and to get insight into the noise dynamics across the entire optical frequency comb spectrum.

_{n}We analyze the origin of the measured timing jitter following the well-established analytic model based on Haus master equation [41]. The laser parameters used in the calculation are from the Er-fiber laser in experiment: 78-MHz repetition rate, -0.0086 ps^{2} net intra-cavity dispersion, and 100-fs FWHM pulse width. The saturated gain is 0.693 calculated from a round-trip cavity loss of 50%. Intra-cavity pulse energy is 0.26 nJ and center wavelength is 1580 nm. More detailed equations and parameters could be found in the appendix. Based on these parameters, we reproduce the pulse dynamics of the fiber laser by numerically solving nonlinear Schrödinger equation with split step Fourier algorithm. Numerical simulation allows us to precisely obtain the nonlinear phase shift, which equals to ∼2.2 π radians per cavity round-trip. We also measure the relative intensity noise (RIN) spectrum of the Er-fiber laser, so as to estimate RIN coupled jitter. Below 1 kHz offset frequency, the RIN spectrum is flat at a level of -120 dBc/ Hz, which starts to roll off at higher offset frequencies following a 1/*f*^{2} slope. The measured timing jitter spectrum and projected timing jitter spectra originated from different effects: quantum-limited jitter directly from ASE noise, Gordon-Haus jitter, RIN-coupled jitter by Kramers-Krönig relationship and RIN-coupled jitter by self-steepening effect are all plotted in Fig. 3(a). The calculation indicates that for Fourier frequency > 10 kHz, Gordon-Haus jitter, i.e. the quantum limited timing jitter originated from the center frequency fluctuation coupled with the intra-cavity dispersion is dominant in this fiber laser. However, for 1∼10 kHz Fourier frequency, RIN-coupled jitter by self-steepening effect plays the most important role.

For *f _{ceo}* and

*ν*noise, the well-established ‘elastic tape’ model depicts the ‘breathing motion’ of the optical comb modes. A certain perturbation on laser parameter, i.e. cavity length fluctuation or pump fluctuation, will cause the comb to expand or contract at fixed-point frequency

_{n}*v*. Different perturbations have different

_{fix}*v*and cause comb tooth to breath at different magnitudes. Using Eqs.(1.2), (1.5)-(1.12) from Ref. [24], we calculate

_{fix}*f*and

_{ceo}*ν*noise PSD induced by various noise sources, as shown in Figs. 4(a) and 4(b). Besides the aforementioned parameters, we measure the RIN of pump diode, which is at -128 dBc/Hz level. The ratio of pump variation induced repetition rate change, noted as

_{n}*B*, can be measured by pump modulation as mentioned in [42] by Washburn

*et al.,*In our system,

*B*=(

*P*d

_{0}*f*/

_{r}*f*d

_{r}*P*)

^{2}=1.21×10

^{−11}. Detailed equations and parameters used could be found in appendix. For cavity length fluctuation, we assume

*v*≈0 THz. For pump fluctuation, ASE induced quantum-limited noise and cavity loss induced noise, we assume

_{fix}*v*≈

_{fix}*v*. Directly using values of

_{c}*S*(

_{length}*f*) from Ref. [24] to calculate length fluctuations induced noise spectra, the calculated result is one order of magnitude higher than our experimental result at low frequency. This may be attributed to different environmental conditions. From Figs. 4(a) and 4(b), it can be seen that, for both

*f*and

_{ceo}*ν*noise, cavity loss induced noise, ASE induced quantum-limited noise and Schawlow-Townes limit all makes negligible contributions to the noise PSD. For

_{n}*ν*noise, the PSD is dominated by cavity length fluctuations. Contribution of pump fluctuations is also negligible. However, for

_{n}*f*noise, pump noise dominates. Contributions from cavity length fluctuations is extremely low (<10

_{ceo}^{−11}Hz

^{2}/Hz).

#### 3.3 Noise correlation analysis

This platform can be utilized to study the intrinsic noise correlations of an OFC. It has been well known that *ν _{n}* noise is not the simple sum of n×

*f*noise and

_{rep}*f*noise according to the comb equation (

_{ceo}*ν*=n×

_{n}*f*+

_{rep}*f*). This fact has also been reflected by noise spectra measurement in Fig. 2(a). Considering the cross-correlation between

_{ceo}*f*noise and

_{rep}*f*noise, PSD of

_{ceo}*ν*noise can be written as the following equation:

_{n}*ν*noise, n×

_{n}*f*noise and

_{rep}*f*noise, respectively. ${S_{N \cdot \Delta {f_{rep}}N \cdot \Delta {f_{rep}}}}$ and ${S_{\Delta {f_{ceo}}\Delta {f_{ceo}}}}$. Г

_{ceo}_{Δ}(ω) is the sum of complex coherence for this laser remains nearly constant at a value of ∼ -2 from 1 kHz to 100 kHz offset frequency, as shown in Fig. 5. This indicates a 100% anti-correlation between n×

*f*noise and

_{rep}*f*noise, consistent with the OFC noise theory [24] and noise characteristics observed in [43] using a commercial OFC (FC1500 from MenloSystems).

_{ceo}#### 3.4 Impact of n×f_{rep} noise stabilization to ν_{n} noise

For full-stabilization of an OFC, the cross-talk between servo loops needs to be taken into account because excess noise e.g. servo bump from one loop could be coupled into another one. Therefore, inappropriate servo loop design would deteriorate the OFC performance. In this subsection, the impact of *f _{rep}* stabilization to

*ν*noise will be experimentally verified based on the noise measurement platform. P. Brochard,

_{n}*et al*., already observed a cross talk between

*f*stabilization and

_{rep}*ν*noise in a 25-GHz ERGO laser [14], where the comb line frequency noise level has been lifted up to level of free running

_{n}*f*noise due to anti-correlation between

_{ceo}*f*noise and

_{ceo}*f*noise. Similar phenomenon in the mode-locked Er-fiber laser has been observed based on our noise measurement platform. Here, laser the repetition rate is stabilized while the

_{rep}*f*is left free running. The 1540 nm comb line frequency noise is characterized. The

_{ceo}*ν*noise spectrum under

_{n}*f*stabilization condition is shown in Fig. 6(a). Inset shows the stabilized n×

_{rep}*f*noise spectrum with 10 kHz locking bandwidth. Obviously, the

_{rep}*ν*noise spectrum is lifted up within the locking bandwidth in the presence of the

_{n}*f*locking. Particularly, from 1 kHz to 2 kHz Fourier frequency, the comb line frequency noise is already comparable with that of

_{rep}*f*noise measured in Fig. 2. However,

_{ceo}*ν*noise PSD deviates from that of free running

_{n}*f*, in particular, > 2 kHz Fourier frequency. The reason is that the functioning of feedback control on

_{ceo}*f*noise elimination is weakened at > 2 kHz Fourier frequency, as indicated from the inset of Fig. 6(a).

_{rep}#### 3.5 Impact of f_{ceo} stabilization to n×f_{rep} noise

Finally, we experimentally verify the impact of the *f _{ceo}* servo loop to n×

*f*stability. The

_{rep}*f*is stabilized through pump current feedback. To this end, the

_{ceo}*f*beat note and an RF reference signal are sent into a digital phase discriminator, generating phase error signal that drives a PID servo to close to loop. The phase locking of the

_{ceo}*f*beat note with a locking bandwidth of several kHz is realized. After

_{ceo}*f*stabilization, n×

_{ceo}*f*noise PSD is measured, as shown in Fig. 6(b). An obvious modulation bump could be observed in n×

_{rep}*f*noise spectrum. Apparently, pump modulation from

_{rep}*f*stabilization has non-negligible impact on n×

_{ceo}*f*noise at ∼kHz frequency range. This observation is consistent with recent literature that utilizes pump modulation for repetition rate control of mode-locked lasers [44–45]. On the contrary, when we phase-lock n×

_{rep}*f*noise, no observable impact is found in the

_{rep}*f*noise PSD. A similar phenomenon has been observed in [43].

_{ceo}## 4. Summary

Noise measurement in OFC is a prerequisite for deep insight into intra-cavity noise dynamic analysis and low noise OFC design. In this report, we characterize *f _{rep}* noise,

*f*noise and ν

_{ceo}_{n}noise, separately. The whole setup is based on an asymmetric fiber delay line interferometer and a home-build

*f-2f*interferometer. The noise measurements match well with the well-established Paschotta’s theoretical model and fixed-point model. An anti-correlation between n×

*f*noise and

_{rep}*f*noise that results in a low level of

_{ceo}*ν*noise has been revealed by noise spectrum analysis.

_{n}Based on the same setup, we further experimentally verify the cross-talk between servo loops. We found that high bandwidth *f _{rep}* stabilization introduces additional ν

_{n}noise, while

*f*stabilization by pump power modulation produces modulation bumps on n×

_{ceo}*f*noise spectrum.

_{rep}We expect that this setup can be used for routine characterization of OFC performances and for the study of intrinsic noise dynamics, not only in mode-locked lasers, but also in novel optical frequency combs, e.g. electro-optical modulated comb, micro-resonator comb, etc. Besides providing a powerful diagnostical tool, this setup also shows great potential on OFC stabilizations. By translating the premium short-term stability of fiber delay line to the OFC, significantly reduced comb linewidth is expected, as long as large bandwidth feedback control on laser cavity is allowed. The resulting narrow linewidth OFC could be suitable for various high precision applications, such as molecular fingerprint spectroscopy, time-of-flight based ranging, optical frequency transfer, to name a few.

## Appendix

## A. Equations for f_{rep} noise PSD calculation

For Gordon-Haus jitter, we use

*D*is diffusion constants of center frequency;

_{w}*D=*0.01 ps/nm/km is half of net cavity dispersion;

*f*=78 MHz is laser’s repetition rate, τ

_{rep}_{ωc}=0.25 ms is decay time for frequency perturbation; Δ

*f*=4.8 THz is FWHM gain bandwidth;

_{g}*E*=0.26 nJ is intra-cavity pulse energy; τ=100 fs is FWHM pulse duration;

_{p}*Θ*is spontaneous emission factor, which can be varied from 1 to 10;

*g=*0.63 is cavity gain;

*T*=12.7 ns is round trip time and

_{rt}*hv*is single photon energy.

_{0}For ASE induced quantum-limited jitter, we use

where*D*is diffusion constant of ASE-induced group velocity.

_{T}For RIN-coupled self-steepening induced jitter, we use

*φ*

_{NL}=2.2 π is nonlinear phase shift per round-trip.

For RIN-coupled jitter through Kramers-Krönig relationship, we use

## B. Equations for f_{ceo} and ν_{n} noise PSD calculation

To calculate *f _{ceo}* and

*v*noise spectra, we use

_{n}*v*is the frequency of a certain mode, $v_{fix}^X$ the fixed-point frequency of a certain noise factor. $S_r^X(f)$ is fractional frequency noise PSD by a certain noise factor. For $S_r^X(f)$ from pump, length, loss and ASE-induced noise, we use the following equations:

_{n}*S*=2.8×10

_{RINpump}^{−13}(1/Hz) is the RIN of the pump diode;

*B*=1.21×10

^{−11}is ratio of pump variation induced repetition rate change;

*f*=6 kHz is roll-off frequency; $v_{group}^L$ is the group velocity corresponds to added cavity length;

_{3dB}*n*is spontaneous emission factor; G=0.63 is cavity gain;

_{sp}*P*=20 mW is intra-cavity power;

_{circ}*β*=-0.0086 ps

_{2}^{2}is net cavity dispersion;

*D*=0.065 ps is gain dispersion;

_{g}*t*=180 fs and

_{rms}*ω*=2.2 THz are estimated rms pulse width and rms spectral width, respectively. For

_{rms}*S*and

_{length}(f)*S*, we directly use the values in [24] by Newbury

_{loss}(f)*et al.*; For Schawlow-Townes noise, following equation are used:

## Funding

National Natural Science Foundation of China (61675150, 61827821,61535009); National Research Foundation of Korea (2018R1A2B3001793); Science and Technology Planning Project of Guangdong Province (2018B090944001).

## Disclosures

The authors declare no conflicts of interest.

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