Abstract

A simple and practical apparatus enabling repetition rate (frep) noise, carrier-envelope frequency (fceo) noise and nth optical comb mode (νn) noise spectra measurements with high precision is established. The frep and νn noise spectra are measured by a fiber delay line interferometer, while fceo noise spectrum is measured by an 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 frep and fceo noise as well as the impact of servo loops on noise characteristics in the stabilized OFC.

© 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 [16]. The emergence of OFC has pushed the precision of these applications to unprecedented levels, benefiting from the advances in ultra-low noise OFC technologies [78]. 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 [912].

Repetition rate (frep) noise, carrier-envelope frequency (fceo) noise and the nth optical comb mode (νn) 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 frep 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 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 frep noise measurement of various fiber lasers [10,1618] and solid-state lasers [1921]. In 2015, Hou et al., proposed an optical heterodyne method to measure frep 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 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 frep noise measurement, which even applies to noise characterization of supercontinuum sources [23].

Carrier envelope frequency noise characterization typically relies on octave-spanning supercontinuum generation which is indispensable to obtain the fceo beat note via f-2f self-referencing technique [2427]. The fceo 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 fceo noise PSD characterization with extended frequency range from microhertz to tens of MHz [27]. On the other side, the fceo 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 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 [2426]. Typically, νn 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 [3335]. The multiple-fringe-side discriminator method is based on a low-finesse FP resonator operates as an array of optical frequency discriminators, enabling νn noise PSD measurement up to 1 MHz offset frequency [36].

In this article, we characterize frequency noise spectra of frep, fceo as well as νn 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 frep noise spectrum measurement and νn 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 fceo noise spectrum can be obtained by directly analyzing the fceo beat note from an f-2f interferometer output with a phase noise analyzer. As a demonstration, we characterized frep noise, fceo noise, as well as νn 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 frep and fceo 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.

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 frep or νn noise measurement while the remained 90% power is used for the fceo measurement.

 figure: Fig. 1.

Fig. 1. Experimental setup. PZT, piezo-electric transducer; EDFA, Er-doped fiber amplifier; FBG, fiber Bragg grating; FRM, faraday rotating mirror; WDM, 1540/1560 wavelength division multiplexer; VCO, voltage-controlled oscillator; AOFS, acoustic optical frequency shifter (Brimrose, AMF-50-1560-2FP); BPF, bandpass filter; PD, photodetector (Menlo Systems, FPD510); PI: proportional-integral servo (Newfocus, LB1005); HV Amp: high-voltage amplifier; PID, proportional-integral-differential servo (Vescent Photonics, D2-135); DCU, delay control unit.

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For frep and νn frequency noise analysis, the experimental scheme is based on an asymmetric fiber Michelson interferometer. Two segments of OFC optical spectrum centered at λ1=1540 nm and λ2=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 λ1 and λ2 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 λ12) temporally overlap with its lateral ∼120th pulse at the output of 2×2 coupler. Another 1540/1560 WDM splits λ1 and λ2, which allows the detection of the beat note signal at λ1 and λ2 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 fmod=50 MHz. For the reason that the AOFS modulates the pulse train twice, the output from the long fiber arm is 100-MHz-frequency (2fmod) 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.

The basic principle of frep and νn frequency noise analysis is delayed self-heterodyne method, which was firstly implemented in the frequency stabilization of CW lasers. The procedure for νn 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 p·δ[τ(nfrep+fceo+2fmod)], where p accounts for the entire number of comb lines within the filtered bandwidth, δ[τ(nfrep+fceo+2fmod)] 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 2fmod, error signal containing comb-line frequency noise p·(δ[τ(nfrep+fceo)]) 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(f)=Vpeak[(1-exp(-i×2πτ))/(i×f)] [V/Hz] in [22], where Vpeak 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) = 2πτVpeak [V/Hz] as frequency discriminator at low Fourier frequencies. Note that Vpeak 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[(nfrep+fceo)]) owning to the identical noise performance of these comb lines.

Repetition rate noise characterization needs to be conducted in two separated wavelengths, λ1 and λ2. Two beat notes carry comb-line frequency noise (p·δ[τ(mfrep+fceo)], p·δ[τ(nfrep+fceo)]). After the common frequency components are rejected by a frequency mixer, the repetition-rate frequency noise, δ(m−n)frep, can be retrieved with transfer function in a similar manner. More detail in frep noise characterization could be found in our previous publication [22,23]. In both frep and νn 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].

For fceo 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 fceo beat note can be free running. We use a phase noise analyzer (Rohde & Schwarz, R&S FSWP26) to characterize the free-running fceo frequency noise PSD. While, for the purpose of investigating the impact of the fceo stabilization to n×frep noise, the fceo needs to be stabilized. To achieve stabilization, the fceo 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 fceo to a RF reference (Stanford Research Systems, SG382) through pump power feedback.

3. Experimental results

In this section, we present frep, fceo and νn 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.

3.1 n×frep, fceo and νn 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 fceo free running. An FFT analyzer measures (n-m)×frep frequency noise PSD and then it is converted to n×frep frequency noise PSD, where n=2.50×105 is the mode number at 1540 nm and m=2.47×105 is the mode number at 1560 nm. The measured n×frep 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 (frep=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 nth comb mode (n×frep+fceo) 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 (νn=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 (fceo=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 frep is not stabilized.

 figure: Fig. 2.

Fig. 2. Frequency noise PSD of n×frep (red curve), fceo (blue curve) and νn noise (black curve).

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3.2 Origins of n×frep, fceo and νn noise

In this subsection, we explore the intrinsic noise origin of the measured n×frep, fceo and νn 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.

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 ps2 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/f2 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.

 figure: Fig. 3.

Fig. 3. Converted timing jitter PSD of the laser oscillator. Calculated timing jitter contribution from direct-coupled jitter from ASE (orange curve), Gorden-Haus jitter (Olive curve), RIN coupled jitter by Kramers-Krönig relation (purple curve) and RIN coupled jitter by self-steepening effect (blue curve). The dashed red curve and dashed gray curve in (a) shows the projected phase noise from intensity noise from 100 MHz signal and the measurement noise floor of the photodetector, electrical amplifier, bandpass filter and mixer, respectively.

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For fceo and νn 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 vfix. Different perturbations have different vfix and cause comb tooth to breath at different magnitudes. Using Eqs.(1.2), (1.5)-(1.12) from Ref. [24], we calculate fceo and νn 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 B, can be measured by pump modulation as mentioned in [42] by Washburn et al., In our system, B=(P0dfr/frdP)2=1.21×10−11. Detailed equations and parameters used could be found in appendix. For cavity length fluctuation, we assume vfix≈0 THz. For pump fluctuation, ASE induced quantum-limited noise and cavity loss induced noise, we assume vfixvc. Directly using values of Slength(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 fceo and νn noise, cavity loss induced noise, ASE induced quantum-limited noise and Schawlow-Townes limit all makes negligible contributions to the noise PSD. For νn noise, the PSD is dominated by cavity length fluctuations. Contribution of pump fluctuations is also negligible. However, for fceo noise, pump noise dominates. Contributions from cavity length fluctuations is extremely low (<10−11 Hz2/Hz).

 figure: Fig. 4.

Fig. 4. (a) νn noise and (b) fceo noise spectrum. Calculated frequency noise spectrum contribution from pump noise (red curve), ASE-induced noise (purple curve), environmental length fluctuations (orange curve), environmental loss fluctuations (dark yellow curve) and Shawlow-Townes limit (olive curve). The dashed red curve and dashed gray curve in (a) shows the projected frequency noise from intensity noise from 100 MHz signal and the measurement noise floor of the photodetector, electrical amplifier, bandpass filter and mixer, respectively.

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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 νn noise is not the simple sum of n×frep noise and fceonoise according to the comb equation (νn=n×frep+fceo). This fact has also been reflected by noise spectra measurement in Fig. 2(a). Considering the cross-correlation between frep noise and fceonoise, PSD of νn noise can be written as the following equation:

$${S_{\Delta {\nu _n}\Delta {\nu _n}}} = {S_{\Delta {f_{ceo}}\Delta {f_{ceo}}}} + {S_{N \cdot \Delta {f_{rep}}N \cdot \Delta {f_{rep}}}} + {\Gamma _\Delta }(\omega ) \times \sqrt {{S_{\Delta {f_{ceo}}\Delta {f_{ceo}}}} \cdot {S_{N\cdot \Delta {f_{rep}}N\cdot \Delta {f_{rep}}}}}$$
where ${S_{\Delta {\nu _n}\Delta {\nu _n}}}$, ${S_{N \cdot \Delta {f_{rep}}N \cdot \Delta {f_{rep}}}}$ and ${S_{\Delta {f_{ceo}}\Delta {f_{ceo}}}}$ are frequency noise spectra of νn noise, n×frep noise and fceo noise, respectively. ${S_{N \cdot \Delta {f_{rep}}N \cdot \Delta {f_{rep}}}}$ and ${S_{\Delta {f_{ceo}}\Delta {f_{ceo}}}}$. ГΔ(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×frep noise and fceo noise, consistent with the OFC noise theory [24] and noise characteristics observed in [43] using a commercial OFC (FC1500 from MenloSystems).

 figure: Fig. 5.

Fig. 5. Frequency dependence of sum of the complex coherence among n×frep, fceo and νn frequency noise PSDs.

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3.4 Impact of n×frep noise stabilization to νn 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 frep stabilization to νn noise will be experimentally verified based on the noise measurement platform. P. Brochard, et al., already observed a cross talk between frep stabilization and νn noise in a 25-GHz ERGO laser [14], where the comb line frequency noise level has been lifted up to level of free running fceo noise due to anti-correlation between fceo noise and frep 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 fceo is left free running. The 1540 nm comb line frequency noise is characterized. The νn noise spectrum under frep stabilization condition is shown in Fig. 6(a). Inset shows the stabilized n×frep noise spectrum with 10 kHz locking bandwidth. Obviously, the νn noise spectrum is lifted up within the locking bandwidth in the presence of the frep locking. Particularly, from 1 kHz to 2 kHz Fourier frequency, the comb line frequency noise is already comparable with that of fceo noise measured in Fig. 2. However, νn noise PSD deviates from that of free running fceo, in particular, > 2 kHz Fourier frequency. The reason is that the functioning of feedback control on frep noise elimination is weakened at > 2 kHz Fourier frequency, as indicated from the inset of Fig. 6(a).

 figure: Fig. 6.

Fig. 6. (a)νn noise spectra of 1540 nm under repetition rate locking Inset: the stabilized n×frep noise spectrum. (b) Noise spectra of n×frep noise when the fceo is stabilized.

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3.5 Impact of fceo stabilization to n×frep noise

Finally, we experimentally verify the impact of the fceo servo loop to n×frep stability. The fceo is stabilized through pump current feedback. To this end, the fceo 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 fceo beat note with a locking bandwidth of several kHz is realized. After fceo stabilization, n×frep noise PSD is measured, as shown in Fig. 6(b). An obvious modulation bump could be observed in n×frep noise spectrum. Apparently, pump modulation from fceo stabilization has non-negligible impact on n×frep noise at ∼kHz frequency range. This observation is consistent with recent literature that utilizes pump modulation for repetition rate control of mode-locked lasers [4445]. On the contrary, when we phase-lock n×frep noise, no observable impact is found in the fceo noise PSD. A similar phenomenon has been observed in [43].

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 frep noise, fceo noise and νn 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×frep noise and fceo noise that results in a low level of νn noise has been revealed by noise spectrum analysis.

Based on the same setup, we further experimentally verify the cross-talk between servo loops. We found that high bandwidth frep stabilization introduces additional νn noise, while fceo stabilization by pump power modulation produces modulation bumps on n×frep noise spectrum.

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 frep noise PSD calculation

For Gordon-Haus jitter, we use

$$S_{\Delta t}^{GH}(f) = \frac{{4{D^2}f_{rep}^2{D_\omega }}}{{{{(2\pi f)}^2}[{{(2\pi f)}^2} + {\tau _{\omega c}}^{ - 2}]}}$$
$${D_\omega } = \frac{2}{{3{E_p}{\tau ^2}}}\Theta \frac{{2g}}{{{T_{rt}}}}h{v_0}$$
$$\frac{1}{{{\tau _{\omega c}}}} = \frac{4}{3}\frac{g}{{{T_{rt}}\Delta f_g^2{\tau ^2}}}$$
where Dw is diffusion constants of center frequency; D=0.01 ps/nm/km is half of net cavity dispersion; frep=78 MHz is laser’s repetition rate, τωc=0.25 ms is decay time for frequency perturbation; Δfg=4.8 THz is FWHM gain bandwidth; Ep=0.26 nJ is intra-cavity pulse energy; τ=100 fs is FWHM pulse duration; Θ is spontaneous emission factor, which can be varied from 1 to 10; g=0.63 is cavity gain; Trt=12.7 ns is round trip time and hv0 is single photon energy.

For ASE induced quantum-limited jitter, we use

$$S_{\Delta t}^{ASE,QL}(f) = \frac{{{D_T}}}{{{{(2\pi f)}^2}}}$$
$${D_T} = \frac{{{\pi ^2}{\tau ^2}}}{{6{E_p}}}\Theta \frac{{2g}}{{{T_{rt}}}}h{v_0}$$
where DT is diffusion constant of ASE-induced group velocity.

For RIN-coupled self-steepening induced jitter, we use

$$S_{\Delta t}^{RIN - SS} = {\left( {\frac{{{\varphi_{NL}}}}{{2{\pi^2}f{T_{rt}}{\nu_0}}}} \right)^2}{S_{RIN}}(f)$$
where φNL=2.2 π is nonlinear phase shift per round-trip.

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

$$S_{\Delta t}^{RIN - KK} = {\left( {\frac{1}{{2\pi \Delta {f_g}}}} \right)^2}{S_{RIN}}(f)$$

B. Equations for fceo and νn noise PSD calculation

To calculate fceo and vn noise spectra, we use

$$S_{vn}^X = {({v_n} - v_{fix}^X)^2}S_r^X(f), $$
where vn 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:
$$S_r^{pump}(f) = B\frac{1}{{1 + {{(f/{f_{3dB}})}^2}}}{S_{RINpump}}$$
$$S_r^{length}(f) = {\left( {\frac{{{f_{rep}}}}{{v_{group}^L}}} \right)^2}{S_{length}}(f)$$
$$S_r^{loss}(f) = B\frac{1}{{1 + {{(f/{f_{3dB}})}^2}}}{S_{loss}}(f)$$
$$S_r^{ASE,ti\min g}(f) = 2f_{rep}^2\left( {\frac{{(1 + {n_{sp}})h{v_0}G}}{{{P_{circ}}}}} \right) \times \left[ {t_{rms}^2 + \left( {\frac{{{\beta_2}}}{{4{D_g}{\omega_{rms}}}}} \right)} \right]$$
where SRINpump=2.8×10−13 (1/Hz) is the RIN of the pump diode; B=1.21×10−11 is ratio of pump variation induced repetition rate change; f3dB=6 kHz is roll-off frequency; $v_{group}^L$ is the group velocity corresponds to added cavity length; nsp is spontaneous emission factor; G=0.63 is cavity gain; Pcirc=20 mW is intra-cavity power; β2=-0.0086 ps2 is net cavity dispersion; Dg=0.065 ps is gain dispersion; trms=180 fs and ωrms=2.2 THz are estimated rms pulse width and rms spectral width, respectively. For Slength(f) and Sloss(f), we directly use the values in [24] by Newbury et al.; For Schawlow-Townes noise, following equation are used:
$$S_{vn}^{ASE,ST}(f) = 2\frac{{f_{rep}^2}}{{{{(2\pi )}^2}}}\left( {\frac{{(1 + {n_{sp}})h{v_0}G}}{{{P_{circ}}}}} \right)$$

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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4. H. Timmers, A. Kowligy, A. Lind, F. C. Cruz, N. Nader, M. Silfies, G. Ycas, T. K. Allision, P. G. Schunemann, S. B. Papp, and S. A. Diddams, “Molecular fingerprinting with bright, broadband infrared frequency combs,” Optica 5(6), 727–732 (2018). [CrossRef]  

5. F. R. Giorgetta, W. C. Swann, L. C. Sinclair, E. Baumann, I. Coddington, and N. R. Newbury, “Optical two-way time and frequency transfer over free space,” Nat. Photonics 7(6), 434–438 (2013). [CrossRef]  

6. J. D. Deschênes, L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, and N. R. Newbury, “Synchronization of distant optical clocks at the femtosecond level,” Phys. Rev. X 6(2), 021016 (2016). [CrossRef]  

7. J. Kim and Y. Song, “Ultralow-noise mode-locked fiber lasers and frequency combs: principles, status, and applications,” Adv. Opt. Photonics 8(3), 465–540 (2016). [CrossRef]  

8. M. Endo, T. D. Shoji, and T. R. Schibli, “Ultralow Noise Optical Frequency Combs,” IEEE J. Sel. Top. Quantum Electron. 24(5), 1–13 (2018). [CrossRef]  

9. D. Hou, C.-C. Lee, Z. Yang, and T. R. Schibli, “Timing jitter characterization of mode-locked lasers with< 1 zs/√ Hz resolution using a simple optical heterodyne technique,” Opt. Lett. 40(13), 2985–2988 (2015). [CrossRef]  

10. T. K. Kim, Y. Song, K. Jung, C. Kim, H. Kim, C. H. Nam, and J. Kim, “Sub-100-as timing jitter optical pulse trains from mode-locked Er-fiber lasers,” Opt. Lett. 36(22), 4443–4445 (2011). [CrossRef]  

11. B. Borchers, S. Koke, A. Husakou, J. Herrmann, and G. Steinmeyer, “Carrier-envelope phase stabilization with sub-10 as residual timing jitter,” Opt. Lett. 36(21), 4146–4148 (2011). [CrossRef]  

12. S. Fang, H. Chen, T. Wang, Y. Jiang, Z. Bi, and L. Ma, “Optical frequency comb with an absolute linewidth of 0.6 Hz–1.2 Hz over an octave spectrum,” Appl. Phys. Lett. 102(23), 231118 (2013). [CrossRef]  

13. J. Kim, J. Chen, J. Cox, and F. X. Kärtner, “Attosecond-resolution timing jitter characterization of free-running mode-locked lasers,” Opt. Lett. 32(24), 3519–3521 (2007). [CrossRef]  

14. P. Brochard, V. J. Wittwer, S. Bilicki, B. Resan, K. J. Weiningarten, S. Schilt, and T. Südmeyer, “Frequency noise characterization of a 25-GHz diode-pumped mode-locked laser with indirect carrier-envelope offset noise assessment,” IEEE Photonics J. 10(1), 1–10 (2018). [CrossRef]  

15. T. R. Schibli, J. Kim, O. Kuzucu, J. T. Gopinath, S. N. Tandon, G. S. Petrich, L. A. Kolodziejski, J. G. Fujimoto, E. P. Ippen, and F. X. Kaertner, “Attosecond active synchronization of passively mode-locked lasers by balanced cross correlation,” Opt. Lett. 28(11), 947–949 (2003). [CrossRef]  

16. Y. Song, C. Kim, K. Jung, H. Kim, and J. Kim, “Timing jitter optimization of mode-locked Yb-fiber lasers toward the attosecond regime,” Opt. Express 19(15), 14518–14525 (2011). [CrossRef]  

17. N. Kuse, J. Jiang, C.-C. Lee, T. R. Schibli, and M. E. Fermann, “All polarization-maintaining Er fiber-based optical frequency combs with nonlinear amplifying loop mirror,” Opt. Express 24(3), 3095–3102 (2016). [CrossRef]  

18. Y. Wang, H. Tian, Y. Ma, Y. Song, and Z. Zhang, “Timing jitter of high-repetition-rate mode-locked fiber lasers,” Opt. Lett. 43(18), 4382–4385 (2018). [CrossRef]  

19. A. J. Benedick, J. G. Fujimoto, and F. X. Kärtner, “Optical flywheels with attosecond jitter,” Nat. Photonics 6(2), 97–100 (2012). [CrossRef]  

20. E. Portuondo-Campa, R. Paschotta, and S. Lecomte, “Sub-100 attosecond timing jitter from low-noise passively mode-locked solid-state laser at telecom wavelength,” Opt. Lett. 38(15), 2650–2653 (2013). [CrossRef]  

21. H. Yang, H. Kim, J. Shin, C. Kim, S. Y. Choi, G.-H. Kim, F. Rotermund, and J. Kim, “Gigahertz repetition rate, sub-femtosecond timing jitter optical pulse train directly generated from a mode-locked Yb: KYW laser,” Opt. Lett. 39(1), 56–59 (2014). [CrossRef]  

22. K. Jung and J. Kim, “All-fibre photonic signal generator for attosecond timing and ultralow-noise microwave,” Sci. Rep. 5(1), 16250 (2015). [CrossRef]  

23. D. Kwon, C.-G. Jeon, J. Shin, M.-S. Heo, S. E. Park, Y. Song, and J. Kim, “Reference-free, high-resolution measurement method of timing jitter spectra of optical frequency combs,” Sci. Rep. 7(1), 40917 (2017). [CrossRef]  

24. N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (Invited),” J. Opt. Soc. Am. B 24(8), 1756–1770 (2007). [CrossRef]  

25. L. C. Sinclair, J.-D. Deschênes, L. Sonderhouse, W. C. Swann, I. H. Khader, E. Baumann, N. R. Newbury, and I. Coddington, “Invited Article: A compact optically coherent fiber frequency comb,” Rev. Sci. Instrum. 86(8), 081301 (2015). [CrossRef]  

26. Y. Ma, B. Xu, H. Ishii, F. Meng, Y. Nakajima, I. Matsushima, T. R. Schibli, Z. Zhang, and K. Minoshima, “Low-noise 750 MHz spaced ytterbium fiber frequency combs,” Opt. Lett. 43(17), 4136–4139 (2018). [CrossRef]  

27. A. Liehl, D. Fehrenbacher, P. Sulzer, A. Leitenstorfer, and D. V. Seletskiy, “Ultrabroadband out-of-loop characterization of the carrier-envelope phase noise of an offset-free Er:fiber frequency comb,” Opt. Lett. 42(10), 2050–2053 (2017). [CrossRef]  

28. B. Borchers, A. Anderson, and G. Steinmeyer, “On the role of shot noise in carrier-envelope phase stabilization,” Laser Photonics Rev. 8(2), 303–315 (2014). [CrossRef]  

29. P. Brochard, S. Schilt, V. J. Wittwer, and T. Südmeyer, “Characterizing the carrier-envelope offset in an optical frequency comb without traditional f-to-2f interferometry,” Opt. Lett. 40(23), 5522–5525 (2015). [CrossRef]  

30. Y.-J. Kim, I. Coddington, W. C. Swann, N. R. Newbury, J. Lee, S. Kim, and S.-W. Kim, “Time-domain stabilization of carrier-envelope phase in femtosecond light pulses,” Opt. Express 22(10), 11788–11796 (2014). [CrossRef]  

31. T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980). [CrossRef]  

32. J. W. Dawson, N. Park, and K. J. Vahala, “An improved delayed self-heterodyne interferometer for linewidth measurements,” IEEE Photonics Technol. Lett 4(9), 1063–1066 (1992). [CrossRef]  

33. M. O. Sahni, S. Trebaol, L. Bramerie, M. Joindot, SPÓ Dúill, S. G. Murdoch, L. P. Barry, and P. Besnard, “Frequency noise reduction performance of a feed-forward heterodyne technique: application to an actively mode-locked laser diode,” Opt. Lett. 42(19), 4000–4003 (2017). [CrossRef]  

34. F. K. Fatemi, J. W. Lou, and T. F. Carruthers, “Frequency comb linewidth of an actively mode-locked fiber laser,” Opt. Lett. 29(9), 944–946 (2004). [CrossRef]  

35. M. I. Kayes and M. Rochette, “Optical frequency comb generation with ultra-narrow spectral lines,” Opt. Lett. 42(14), 2718–2721 (2017). [CrossRef]  

36. N. Coluccelli, M. Cassinerio, A. Gambetta, P. Laporta, and G. Galzerano, “Frequency-noise measurements of optical frequency combs by multiple fringe-side discriminator,” Sci. Rep. 5(1), 16338 (2015). [CrossRef]  

37. R. Paschotta, “Noise of mode-locked lasers (Part II): timing jitter and other fluctuations,” Appl. Phys. B 79(2), 163–173 (2004). [CrossRef]  

38. H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens, mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74(1), 1–6 (2002). [CrossRef]  

39. R. Li, H. Shi, H. Tian, Y. Li, B. Liu, Y. Song, and M. Hu, “All-polarization-maintaining dual-wavelength mode-locked fiber laser based on Sagnac loop filter,” Opt. Express 26(22), 28302–28311 (2018). [CrossRef]  

40. D. Kwon and J. Kim, “All-fiber interferometer-based repetition-rate stabilization of mode-locked lasers to 10−14-level frequency instability and 1-fs-level jitter over 1 s,” Opt. Lett. 42(24), 5186–5189 (2017). [CrossRef]  

41. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993). [CrossRef]  

42. B. R. Washburn, W. C. Swann, and N. R. Newbury, “Response dynamics of the frequency comb output from a femtosecond fiber laser,” Opt. Express 13(26), 10622–10633 (2005). [CrossRef]  

43. V. Dolgovskiy, N. Bucalovic, P. Thomann, C. Schori, G. D. Domenico, and S. Schilt, “Cross-influence between the two servo loops of a fully stabilized Er: fiber optical frequency comb,” J. Opt. Soc. Am. B 29(10), 2944–2957 (2012). [CrossRef]  

44. N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, “Frequency stabilization of mode-locked Erbium fiber lasers using pump power control,” Appl. Phys. B 78(3-4), 321–324 (2004). [CrossRef]  

45. D. Fehrenbacher, P. Sulzer, A. Liehl, T. Kälberer, C. Riek, D. V. Seletskiy, and A. Leitenstorfer, “Free-running performance and full control of a passively phase-stable Er:fiber frequency comb,” Optica 2(10), 917–923 (2015). [CrossRef]  

References

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  1. T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2008).
    [Crossref]
  2. A. Schliesser, N. Picqué, and T. W. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6(7), 440–449 (2012).
    [Crossref]
  3. I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica 3(4), 414–426 (2016).
    [Crossref]
  4. H. Timmers, A. Kowligy, A. Lind, F. C. Cruz, N. Nader, M. Silfies, G. Ycas, T. K. Allision, P. G. Schunemann, S. B. Papp, and S. A. Diddams, “Molecular fingerprinting with bright, broadband infrared frequency combs,” Optica 5(6), 727–732 (2018).
    [Crossref]
  5. F. R. Giorgetta, W. C. Swann, L. C. Sinclair, E. Baumann, I. Coddington, and N. R. Newbury, “Optical two-way time and frequency transfer over free space,” Nat. Photonics 7(6), 434–438 (2013).
    [Crossref]
  6. J. D. Deschênes, L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, and N. R. Newbury, “Synchronization of distant optical clocks at the femtosecond level,” Phys. Rev. X 6(2), 021016 (2016).
    [Crossref]
  7. J. Kim and Y. Song, “Ultralow-noise mode-locked fiber lasers and frequency combs: principles, status, and applications,” Adv. Opt. Photonics 8(3), 465–540 (2016).
    [Crossref]
  8. M. Endo, T. D. Shoji, and T. R. Schibli, “Ultralow Noise Optical Frequency Combs,” IEEE J. Sel. Top. Quantum Electron. 24(5), 1–13 (2018).
    [Crossref]
  9. D. Hou, C.-C. Lee, Z. Yang, and T. R. Schibli, “Timing jitter characterization of mode-locked lasers with< 1 zs/√ Hz resolution using a simple optical heterodyne technique,” Opt. Lett. 40(13), 2985–2988 (2015).
    [Crossref]
  10. T. K. Kim, Y. Song, K. Jung, C. Kim, H. Kim, C. H. Nam, and J. Kim, “Sub-100-as timing jitter optical pulse trains from mode-locked Er-fiber lasers,” Opt. Lett. 36(22), 4443–4445 (2011).
    [Crossref]
  11. B. Borchers, S. Koke, A. Husakou, J. Herrmann, and G. Steinmeyer, “Carrier-envelope phase stabilization with sub-10 as residual timing jitter,” Opt. Lett. 36(21), 4146–4148 (2011).
    [Crossref]
  12. S. Fang, H. Chen, T. Wang, Y. Jiang, Z. Bi, and L. Ma, “Optical frequency comb with an absolute linewidth of 0.6 Hz–1.2 Hz over an octave spectrum,” Appl. Phys. Lett. 102(23), 231118 (2013).
    [Crossref]
  13. J. Kim, J. Chen, J. Cox, and F. X. Kärtner, “Attosecond-resolution timing jitter characterization of free-running mode-locked lasers,” Opt. Lett. 32(24), 3519–3521 (2007).
    [Crossref]
  14. P. Brochard, V. J. Wittwer, S. Bilicki, B. Resan, K. J. Weiningarten, S. Schilt, and T. Südmeyer, “Frequency noise characterization of a 25-GHz diode-pumped mode-locked laser with indirect carrier-envelope offset noise assessment,” IEEE Photonics J. 10(1), 1–10 (2018).
    [Crossref]
  15. T. R. Schibli, J. Kim, O. Kuzucu, J. T. Gopinath, S. N. Tandon, G. S. Petrich, L. A. Kolodziejski, J. G. Fujimoto, E. P. Ippen, and F. X. Kaertner, “Attosecond active synchronization of passively mode-locked lasers by balanced cross correlation,” Opt. Lett. 28(11), 947–949 (2003).
    [Crossref]
  16. Y. Song, C. Kim, K. Jung, H. Kim, and J. Kim, “Timing jitter optimization of mode-locked Yb-fiber lasers toward the attosecond regime,” Opt. Express 19(15), 14518–14525 (2011).
    [Crossref]
  17. N. Kuse, J. Jiang, C.-C. Lee, T. R. Schibli, and M. E. Fermann, “All polarization-maintaining Er fiber-based optical frequency combs with nonlinear amplifying loop mirror,” Opt. Express 24(3), 3095–3102 (2016).
    [Crossref]
  18. Y. Wang, H. Tian, Y. Ma, Y. Song, and Z. Zhang, “Timing jitter of high-repetition-rate mode-locked fiber lasers,” Opt. Lett. 43(18), 4382–4385 (2018).
    [Crossref]
  19. A. J. Benedick, J. G. Fujimoto, and F. X. Kärtner, “Optical flywheels with attosecond jitter,” Nat. Photonics 6(2), 97–100 (2012).
    [Crossref]
  20. E. Portuondo-Campa, R. Paschotta, and S. Lecomte, “Sub-100 attosecond timing jitter from low-noise passively mode-locked solid-state laser at telecom wavelength,” Opt. Lett. 38(15), 2650–2653 (2013).
    [Crossref]
  21. H. Yang, H. Kim, J. Shin, C. Kim, S. Y. Choi, G.-H. Kim, F. Rotermund, and J. Kim, “Gigahertz repetition rate, sub-femtosecond timing jitter optical pulse train directly generated from a mode-locked Yb: KYW laser,” Opt. Lett. 39(1), 56–59 (2014).
    [Crossref]
  22. K. Jung and J. Kim, “All-fibre photonic signal generator for attosecond timing and ultralow-noise microwave,” Sci. Rep. 5(1), 16250 (2015).
    [Crossref]
  23. D. Kwon, C.-G. Jeon, J. Shin, M.-S. Heo, S. E. Park, Y. Song, and J. Kim, “Reference-free, high-resolution measurement method of timing jitter spectra of optical frequency combs,” Sci. Rep. 7(1), 40917 (2017).
    [Crossref]
  24. N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (Invited),” J. Opt. Soc. Am. B 24(8), 1756–1770 (2007).
    [Crossref]
  25. L. C. Sinclair, J.-D. Deschênes, L. Sonderhouse, W. C. Swann, I. H. Khader, E. Baumann, N. R. Newbury, and I. Coddington, “Invited Article: A compact optically coherent fiber frequency comb,” Rev. Sci. Instrum. 86(8), 081301 (2015).
    [Crossref]
  26. Y. Ma, B. Xu, H. Ishii, F. Meng, Y. Nakajima, I. Matsushima, T. R. Schibli, Z. Zhang, and K. Minoshima, “Low-noise 750 MHz spaced ytterbium fiber frequency combs,” Opt. Lett. 43(17), 4136–4139 (2018).
    [Crossref]
  27. A. Liehl, D. Fehrenbacher, P. Sulzer, A. Leitenstorfer, and D. V. Seletskiy, “Ultrabroadband out-of-loop characterization of the carrier-envelope phase noise of an offset-free Er:fiber frequency comb,” Opt. Lett. 42(10), 2050–2053 (2017).
    [Crossref]
  28. B. Borchers, A. Anderson, and G. Steinmeyer, “On the role of shot noise in carrier-envelope phase stabilization,” Laser Photonics Rev. 8(2), 303–315 (2014).
    [Crossref]
  29. P. Brochard, S. Schilt, V. J. Wittwer, and T. Südmeyer, “Characterizing the carrier-envelope offset in an optical frequency comb without traditional f-to-2f interferometry,” Opt. Lett. 40(23), 5522–5525 (2015).
    [Crossref]
  30. Y.-J. Kim, I. Coddington, W. C. Swann, N. R. Newbury, J. Lee, S. Kim, and S.-W. Kim, “Time-domain stabilization of carrier-envelope phase in femtosecond light pulses,” Opt. Express 22(10), 11788–11796 (2014).
    [Crossref]
  31. T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
    [Crossref]
  32. J. W. Dawson, N. Park, and K. J. Vahala, “An improved delayed self-heterodyne interferometer for linewidth measurements,” IEEE Photonics Technol. Lett 4(9), 1063–1066 (1992).
    [Crossref]
  33. M. O. Sahni, S. Trebaol, L. Bramerie, M. Joindot, SPÓ Dúill, S. G. Murdoch, L. P. Barry, and P. Besnard, “Frequency noise reduction performance of a feed-forward heterodyne technique: application to an actively mode-locked laser diode,” Opt. Lett. 42(19), 4000–4003 (2017).
    [Crossref]
  34. F. K. Fatemi, J. W. Lou, and T. F. Carruthers, “Frequency comb linewidth of an actively mode-locked fiber laser,” Opt. Lett. 29(9), 944–946 (2004).
    [Crossref]
  35. M. I. Kayes and M. Rochette, “Optical frequency comb generation with ultra-narrow spectral lines,” Opt. Lett. 42(14), 2718–2721 (2017).
    [Crossref]
  36. N. Coluccelli, M. Cassinerio, A. Gambetta, P. Laporta, and G. Galzerano, “Frequency-noise measurements of optical frequency combs by multiple fringe-side discriminator,” Sci. Rep. 5(1), 16338 (2015).
    [Crossref]
  37. R. Paschotta, “Noise of mode-locked lasers (Part II): timing jitter and other fluctuations,” Appl. Phys. B 79(2), 163–173 (2004).
    [Crossref]
  38. H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens, mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74(1), 1–6 (2002).
    [Crossref]
  39. R. Li, H. Shi, H. Tian, Y. Li, B. Liu, Y. Song, and M. Hu, “All-polarization-maintaining dual-wavelength mode-locked fiber laser based on Sagnac loop filter,” Opt. Express 26(22), 28302–28311 (2018).
    [Crossref]
  40. D. Kwon and J. Kim, “All-fiber interferometer-based repetition-rate stabilization of mode-locked lasers to 10−14-level frequency instability and 1-fs-level jitter over 1  s,” Opt. Lett. 42(24), 5186–5189 (2017).
    [Crossref]
  41. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993).
    [Crossref]
  42. B. R. Washburn, W. C. Swann, and N. R. Newbury, “Response dynamics of the frequency comb output from a femtosecond fiber laser,” Opt. Express 13(26), 10622–10633 (2005).
    [Crossref]
  43. V. Dolgovskiy, N. Bucalovic, P. Thomann, C. Schori, G. D. Domenico, and S. Schilt, “Cross-influence between the two servo loops of a fully stabilized Er: fiber optical frequency comb,” J. Opt. Soc. Am. B 29(10), 2944–2957 (2012).
    [Crossref]
  44. N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, “Frequency stabilization of mode-locked Erbium fiber lasers using pump power control,” Appl. Phys. B 78(3-4), 321–324 (2004).
    [Crossref]
  45. D. Fehrenbacher, P. Sulzer, A. Liehl, T. Kälberer, C. Riek, D. V. Seletskiy, and A. Leitenstorfer, “Free-running performance and full control of a passively phase-stable Er:fiber frequency comb,” Optica 2(10), 917–923 (2015).
    [Crossref]

2018 (6)

2017 (5)

2016 (4)

N. Kuse, J. Jiang, C.-C. Lee, T. R. Schibli, and M. E. Fermann, “All polarization-maintaining Er fiber-based optical frequency combs with nonlinear amplifying loop mirror,” Opt. Express 24(3), 3095–3102 (2016).
[Crossref]

I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica 3(4), 414–426 (2016).
[Crossref]

J. D. Deschênes, L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, and N. R. Newbury, “Synchronization of distant optical clocks at the femtosecond level,” Phys. Rev. X 6(2), 021016 (2016).
[Crossref]

J. Kim and Y. Song, “Ultralow-noise mode-locked fiber lasers and frequency combs: principles, status, and applications,” Adv. Opt. Photonics 8(3), 465–540 (2016).
[Crossref]

2015 (6)

D. Hou, C.-C. Lee, Z. Yang, and T. R. Schibli, “Timing jitter characterization of mode-locked lasers with< 1 zs/√ Hz resolution using a simple optical heterodyne technique,” Opt. Lett. 40(13), 2985–2988 (2015).
[Crossref]

K. Jung and J. Kim, “All-fibre photonic signal generator for attosecond timing and ultralow-noise microwave,” Sci. Rep. 5(1), 16250 (2015).
[Crossref]

L. C. Sinclair, J.-D. Deschênes, L. Sonderhouse, W. C. Swann, I. H. Khader, E. Baumann, N. R. Newbury, and I. Coddington, “Invited Article: A compact optically coherent fiber frequency comb,” Rev. Sci. Instrum. 86(8), 081301 (2015).
[Crossref]

P. Brochard, S. Schilt, V. J. Wittwer, and T. Südmeyer, “Characterizing the carrier-envelope offset in an optical frequency comb without traditional f-to-2f interferometry,” Opt. Lett. 40(23), 5522–5525 (2015).
[Crossref]

N. Coluccelli, M. Cassinerio, A. Gambetta, P. Laporta, and G. Galzerano, “Frequency-noise measurements of optical frequency combs by multiple fringe-side discriminator,” Sci. Rep. 5(1), 16338 (2015).
[Crossref]

D. Fehrenbacher, P. Sulzer, A. Liehl, T. Kälberer, C. Riek, D. V. Seletskiy, and A. Leitenstorfer, “Free-running performance and full control of a passively phase-stable Er:fiber frequency comb,” Optica 2(10), 917–923 (2015).
[Crossref]

2014 (3)

2013 (3)

E. Portuondo-Campa, R. Paschotta, and S. Lecomte, “Sub-100 attosecond timing jitter from low-noise passively mode-locked solid-state laser at telecom wavelength,” Opt. Lett. 38(15), 2650–2653 (2013).
[Crossref]

F. R. Giorgetta, W. C. Swann, L. C. Sinclair, E. Baumann, I. Coddington, and N. R. Newbury, “Optical two-way time and frequency transfer over free space,” Nat. Photonics 7(6), 434–438 (2013).
[Crossref]

S. Fang, H. Chen, T. Wang, Y. Jiang, Z. Bi, and L. Ma, “Optical frequency comb with an absolute linewidth of 0.6 Hz–1.2 Hz over an octave spectrum,” Appl. Phys. Lett. 102(23), 231118 (2013).
[Crossref]

2012 (3)

A. Schliesser, N. Picqué, and T. W. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6(7), 440–449 (2012).
[Crossref]

A. J. Benedick, J. G. Fujimoto, and F. X. Kärtner, “Optical flywheels with attosecond jitter,” Nat. Photonics 6(2), 97–100 (2012).
[Crossref]

V. Dolgovskiy, N. Bucalovic, P. Thomann, C. Schori, G. D. Domenico, and S. Schilt, “Cross-influence between the two servo loops of a fully stabilized Er: fiber optical frequency comb,” J. Opt. Soc. Am. B 29(10), 2944–2957 (2012).
[Crossref]

2011 (3)

2008 (1)

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2008).
[Crossref]

2007 (2)

2005 (1)

2004 (3)

N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, “Frequency stabilization of mode-locked Erbium fiber lasers using pump power control,” Appl. Phys. B 78(3-4), 321–324 (2004).
[Crossref]

R. Paschotta, “Noise of mode-locked lasers (Part II): timing jitter and other fluctuations,” Appl. Phys. B 79(2), 163–173 (2004).
[Crossref]

F. K. Fatemi, J. W. Lou, and T. F. Carruthers, “Frequency comb linewidth of an actively mode-locked fiber laser,” Opt. Lett. 29(9), 944–946 (2004).
[Crossref]

2003 (1)

2002 (1)

H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens, mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74(1), 1–6 (2002).
[Crossref]

1993 (1)

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993).
[Crossref]

1992 (1)

J. W. Dawson, N. Park, and K. J. Vahala, “An improved delayed self-heterodyne interferometer for linewidth measurements,” IEEE Photonics Technol. Lett 4(9), 1063–1066 (1992).
[Crossref]

1980 (1)

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
[Crossref]

Allision, T. K.

Anderson, A.

B. Borchers, A. Anderson, and G. Steinmeyer, “On the role of shot noise in carrier-envelope phase stabilization,” Laser Photonics Rev. 8(2), 303–315 (2014).
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Barry, L. P.

Baumann, E.

J. D. Deschênes, L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, and N. R. Newbury, “Synchronization of distant optical clocks at the femtosecond level,” Phys. Rev. X 6(2), 021016 (2016).
[Crossref]

L. C. Sinclair, J.-D. Deschênes, L. Sonderhouse, W. C. Swann, I. H. Khader, E. Baumann, N. R. Newbury, and I. Coddington, “Invited Article: A compact optically coherent fiber frequency comb,” Rev. Sci. Instrum. 86(8), 081301 (2015).
[Crossref]

F. R. Giorgetta, W. C. Swann, L. C. Sinclair, E. Baumann, I. Coddington, and N. R. Newbury, “Optical two-way time and frequency transfer over free space,” Nat. Photonics 7(6), 434–438 (2013).
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A. J. Benedick, J. G. Fujimoto, and F. X. Kärtner, “Optical flywheels with attosecond jitter,” Nat. Photonics 6(2), 97–100 (2012).
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J. D. Deschênes, L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, and N. R. Newbury, “Synchronization of distant optical clocks at the femtosecond level,” Phys. Rev. X 6(2), 021016 (2016).
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Besnard, P.

Bi, Z.

S. Fang, H. Chen, T. Wang, Y. Jiang, Z. Bi, and L. Ma, “Optical frequency comb with an absolute linewidth of 0.6 Hz–1.2 Hz over an octave spectrum,” Appl. Phys. Lett. 102(23), 231118 (2013).
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P. Brochard, V. J. Wittwer, S. Bilicki, B. Resan, K. J. Weiningarten, S. Schilt, and T. Südmeyer, “Frequency noise characterization of a 25-GHz diode-pumped mode-locked laser with indirect carrier-envelope offset noise assessment,” IEEE Photonics J. 10(1), 1–10 (2018).
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Borchers, B.

B. Borchers, A. Anderson, and G. Steinmeyer, “On the role of shot noise in carrier-envelope phase stabilization,” Laser Photonics Rev. 8(2), 303–315 (2014).
[Crossref]

B. Borchers, S. Koke, A. Husakou, J. Herrmann, and G. Steinmeyer, “Carrier-envelope phase stabilization with sub-10 as residual timing jitter,” Opt. Lett. 36(21), 4146–4148 (2011).
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Brochard, P.

P. Brochard, V. J. Wittwer, S. Bilicki, B. Resan, K. J. Weiningarten, S. Schilt, and T. Südmeyer, “Frequency noise characterization of a 25-GHz diode-pumped mode-locked laser with indirect carrier-envelope offset noise assessment,” IEEE Photonics J. 10(1), 1–10 (2018).
[Crossref]

P. Brochard, S. Schilt, V. J. Wittwer, and T. Südmeyer, “Characterizing the carrier-envelope offset in an optical frequency comb without traditional f-to-2f interferometry,” Opt. Lett. 40(23), 5522–5525 (2015).
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Carruthers, T. F.

Cassinerio, M.

N. Coluccelli, M. Cassinerio, A. Gambetta, P. Laporta, and G. Galzerano, “Frequency-noise measurements of optical frequency combs by multiple fringe-side discriminator,” Sci. Rep. 5(1), 16338 (2015).
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Cermak, M.

J. D. Deschênes, L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, and N. R. Newbury, “Synchronization of distant optical clocks at the femtosecond level,” Phys. Rev. X 6(2), 021016 (2016).
[Crossref]

Chen, H.

S. Fang, H. Chen, T. Wang, Y. Jiang, Z. Bi, and L. Ma, “Optical frequency comb with an absolute linewidth of 0.6 Hz–1.2 Hz over an octave spectrum,” Appl. Phys. Lett. 102(23), 231118 (2013).
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Chen, J.

Choi, S. Y.

Coddington, I.

J. D. Deschênes, L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, and N. R. Newbury, “Synchronization of distant optical clocks at the femtosecond level,” Phys. Rev. X 6(2), 021016 (2016).
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I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica 3(4), 414–426 (2016).
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L. C. Sinclair, J.-D. Deschênes, L. Sonderhouse, W. C. Swann, I. H. Khader, E. Baumann, N. R. Newbury, and I. Coddington, “Invited Article: A compact optically coherent fiber frequency comb,” Rev. Sci. Instrum. 86(8), 081301 (2015).
[Crossref]

Y.-J. Kim, I. Coddington, W. C. Swann, N. R. Newbury, J. Lee, S. Kim, and S.-W. Kim, “Time-domain stabilization of carrier-envelope phase in femtosecond light pulses,” Opt. Express 22(10), 11788–11796 (2014).
[Crossref]

F. R. Giorgetta, W. C. Swann, L. C. Sinclair, E. Baumann, I. Coddington, and N. R. Newbury, “Optical two-way time and frequency transfer over free space,” Nat. Photonics 7(6), 434–438 (2013).
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N. Coluccelli, M. Cassinerio, A. Gambetta, P. Laporta, and G. Galzerano, “Frequency-noise measurements of optical frequency combs by multiple fringe-side discriminator,” Sci. Rep. 5(1), 16338 (2015).
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Cox, J.

Cruz, F. C.

Dawson, J. W.

J. W. Dawson, N. Park, and K. J. Vahala, “An improved delayed self-heterodyne interferometer for linewidth measurements,” IEEE Photonics Technol. Lett 4(9), 1063–1066 (1992).
[Crossref]

Deschênes, J. D.

J. D. Deschênes, L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, and N. R. Newbury, “Synchronization of distant optical clocks at the femtosecond level,” Phys. Rev. X 6(2), 021016 (2016).
[Crossref]

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L. C. Sinclair, J.-D. Deschênes, L. Sonderhouse, W. C. Swann, I. H. Khader, E. Baumann, N. R. Newbury, and I. Coddington, “Invited Article: A compact optically coherent fiber frequency comb,” Rev. Sci. Instrum. 86(8), 081301 (2015).
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Dolgovskiy, V.

Domenico, G. D.

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N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, “Frequency stabilization of mode-locked Erbium fiber lasers using pump power control,” Appl. Phys. B 78(3-4), 321–324 (2004).
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S. Fang, H. Chen, T. Wang, Y. Jiang, Z. Bi, and L. Ma, “Optical frequency comb with an absolute linewidth of 0.6 Hz–1.2 Hz over an octave spectrum,” Appl. Phys. Lett. 102(23), 231118 (2013).
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Fatemi, F. K.

Fehrenbacher, D.

Fermann, M. E.

Fujimoto, J. G.

Galzerano, G.

N. Coluccelli, M. Cassinerio, A. Gambetta, P. Laporta, and G. Galzerano, “Frequency-noise measurements of optical frequency combs by multiple fringe-side discriminator,” Sci. Rep. 5(1), 16338 (2015).
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Gambetta, A.

N. Coluccelli, M. Cassinerio, A. Gambetta, P. Laporta, and G. Galzerano, “Frequency-noise measurements of optical frequency combs by multiple fringe-side discriminator,” Sci. Rep. 5(1), 16338 (2015).
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Giorgetta, F. R.

J. D. Deschênes, L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, and N. R. Newbury, “Synchronization of distant optical clocks at the femtosecond level,” Phys. Rev. X 6(2), 021016 (2016).
[Crossref]

F. R. Giorgetta, W. C. Swann, L. C. Sinclair, E. Baumann, I. Coddington, and N. R. Newbury, “Optical two-way time and frequency transfer over free space,” Nat. Photonics 7(6), 434–438 (2013).
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Gopinath, J. T.

Hänsch, T. W.

A. Schliesser, N. Picqué, and T. W. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6(7), 440–449 (2012).
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T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2008).
[Crossref]

Haus, H. A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993).
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Haverkamp, N.

N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, “Frequency stabilization of mode-locked Erbium fiber lasers using pump power control,” Appl. Phys. B 78(3-4), 321–324 (2004).
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Heo, M.-S.

D. Kwon, C.-G. Jeon, J. Shin, M.-S. Heo, S. E. Park, Y. Song, and J. Kim, “Reference-free, high-resolution measurement method of timing jitter spectra of optical frequency combs,” Sci. Rep. 7(1), 40917 (2017).
[Crossref]

Herrmann, J.

Holzwarth, R.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2008).
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Hou, D.

Hu, M.

Hundertmark, H.

N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, “Frequency stabilization of mode-locked Erbium fiber lasers using pump power control,” Appl. Phys. B 78(3-4), 321–324 (2004).
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Husakou, A.

Ippen, E. P.

Ishii, H.

Jeon, C.-G.

D. Kwon, C.-G. Jeon, J. Shin, M.-S. Heo, S. E. Park, Y. Song, and J. Kim, “Reference-free, high-resolution measurement method of timing jitter spectra of optical frequency combs,” Sci. Rep. 7(1), 40917 (2017).
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Jiang, J.

Jiang, Y.

S. Fang, H. Chen, T. Wang, Y. Jiang, Z. Bi, and L. Ma, “Optical frequency comb with an absolute linewidth of 0.6 Hz–1.2 Hz over an octave spectrum,” Appl. Phys. Lett. 102(23), 231118 (2013).
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Joindot, M.

Jung, K.

Kaertner, F. X.

Kälberer, T.

Kärtner, F. X.

A. J. Benedick, J. G. Fujimoto, and F. X. Kärtner, “Optical flywheels with attosecond jitter,” Nat. Photonics 6(2), 97–100 (2012).
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J. Kim, J. Chen, J. Cox, and F. X. Kärtner, “Attosecond-resolution timing jitter characterization of free-running mode-locked lasers,” Opt. Lett. 32(24), 3519–3521 (2007).
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Kayes, M. I.

Khader, I. H.

L. C. Sinclair, J.-D. Deschênes, L. Sonderhouse, W. C. Swann, I. H. Khader, E. Baumann, N. R. Newbury, and I. Coddington, “Invited Article: A compact optically coherent fiber frequency comb,” Rev. Sci. Instrum. 86(8), 081301 (2015).
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Kikuchi, K.

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
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Kim, C.

Kim, G.-H.

Kim, H.

Kim, J.

D. Kwon, C.-G. Jeon, J. Shin, M.-S. Heo, S. E. Park, Y. Song, and J. Kim, “Reference-free, high-resolution measurement method of timing jitter spectra of optical frequency combs,” Sci. Rep. 7(1), 40917 (2017).
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D. Kwon and J. Kim, “All-fiber interferometer-based repetition-rate stabilization of mode-locked lasers to 10−14-level frequency instability and 1-fs-level jitter over 1  s,” Opt. Lett. 42(24), 5186–5189 (2017).
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J. Kim and Y. Song, “Ultralow-noise mode-locked fiber lasers and frequency combs: principles, status, and applications,” Adv. Opt. Photonics 8(3), 465–540 (2016).
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K. Jung and J. Kim, “All-fibre photonic signal generator for attosecond timing and ultralow-noise microwave,” Sci. Rep. 5(1), 16250 (2015).
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H. Yang, H. Kim, J. Shin, C. Kim, S. Y. Choi, G.-H. Kim, F. Rotermund, and J. Kim, “Gigahertz repetition rate, sub-femtosecond timing jitter optical pulse train directly generated from a mode-locked Yb: KYW laser,” Opt. Lett. 39(1), 56–59 (2014).
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Y. Song, C. Kim, K. Jung, H. Kim, and J. Kim, “Timing jitter optimization of mode-locked Yb-fiber lasers toward the attosecond regime,” Opt. Express 19(15), 14518–14525 (2011).
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T. K. Kim, Y. Song, K. Jung, C. Kim, H. Kim, C. H. Nam, and J. Kim, “Sub-100-as timing jitter optical pulse trains from mode-locked Er-fiber lasers,” Opt. Lett. 36(22), 4443–4445 (2011).
[Crossref]

J. Kim, J. Chen, J. Cox, and F. X. Kärtner, “Attosecond-resolution timing jitter characterization of free-running mode-locked lasers,” Opt. Lett. 32(24), 3519–3521 (2007).
[Crossref]

T. R. Schibli, J. Kim, O. Kuzucu, J. T. Gopinath, S. N. Tandon, G. S. Petrich, L. A. Kolodziejski, J. G. Fujimoto, E. P. Ippen, and F. X. Kaertner, “Attosecond active synchronization of passively mode-locked lasers by balanced cross correlation,” Opt. Lett. 28(11), 947–949 (2003).
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Kim, S.

Kim, S.-W.

Kim, T. K.

Kim, Y.-J.

Koke, S.

Kolodziejski, L. A.

Kowligy, A.

Kuse, N.

Kuzucu, O.

Kwon, D.

D. Kwon, C.-G. Jeon, J. Shin, M.-S. Heo, S. E. Park, Y. Song, and J. Kim, “Reference-free, high-resolution measurement method of timing jitter spectra of optical frequency combs,” Sci. Rep. 7(1), 40917 (2017).
[Crossref]

D. Kwon and J. Kim, “All-fiber interferometer-based repetition-rate stabilization of mode-locked lasers to 10−14-level frequency instability and 1-fs-level jitter over 1  s,” Opt. Lett. 42(24), 5186–5189 (2017).
[Crossref]

Laporta, P.

N. Coluccelli, M. Cassinerio, A. Gambetta, P. Laporta, and G. Galzerano, “Frequency-noise measurements of optical frequency combs by multiple fringe-side discriminator,” Sci. Rep. 5(1), 16338 (2015).
[Crossref]

Lecomte, S.

Lee, C.-C.

Lee, J.

Leitenstorfer, A.

Li, R.

Li, Y.

Liehl, A.

Lind, A.

Lipphardt, B.

H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens, mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74(1), 1–6 (2002).
[Crossref]

Liu, B.

Lou, J. W.

Ma, L.

S. Fang, H. Chen, T. Wang, Y. Jiang, Z. Bi, and L. Ma, “Optical frequency comb with an absolute linewidth of 0.6 Hz–1.2 Hz over an octave spectrum,” Appl. Phys. Lett. 102(23), 231118 (2013).
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Ma, Y.

Matsushima, I.

Mecozzi, A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993).
[Crossref]

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Minoshima, K.

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Nakayama, A.

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
[Crossref]

Nam, C. H.

Newbury, N.

Newbury, N. R.

J. D. Deschênes, L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, and N. R. Newbury, “Synchronization of distant optical clocks at the femtosecond level,” Phys. Rev. X 6(2), 021016 (2016).
[Crossref]

L. C. Sinclair, J.-D. Deschênes, L. Sonderhouse, W. C. Swann, I. H. Khader, E. Baumann, N. R. Newbury, and I. Coddington, “Invited Article: A compact optically coherent fiber frequency comb,” Rev. Sci. Instrum. 86(8), 081301 (2015).
[Crossref]

Y.-J. Kim, I. Coddington, W. C. Swann, N. R. Newbury, J. Lee, S. Kim, and S.-W. Kim, “Time-domain stabilization of carrier-envelope phase in femtosecond light pulses,” Opt. Express 22(10), 11788–11796 (2014).
[Crossref]

F. R. Giorgetta, W. C. Swann, L. C. Sinclair, E. Baumann, I. Coddington, and N. R. Newbury, “Optical two-way time and frequency transfer over free space,” Nat. Photonics 7(6), 434–438 (2013).
[Crossref]

N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (Invited),” J. Opt. Soc. Am. B 24(8), 1756–1770 (2007).
[Crossref]

B. R. Washburn, W. C. Swann, and N. R. Newbury, “Response dynamics of the frequency comb output from a femtosecond fiber laser,” Opt. Express 13(26), 10622–10633 (2005).
[Crossref]

Okoshi, T.

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
[Crossref]

Papp, S. B.

Park, N.

J. W. Dawson, N. Park, and K. J. Vahala, “An improved delayed self-heterodyne interferometer for linewidth measurements,” IEEE Photonics Technol. Lett 4(9), 1063–1066 (1992).
[Crossref]

Park, S. E.

D. Kwon, C.-G. Jeon, J. Shin, M.-S. Heo, S. E. Park, Y. Song, and J. Kim, “Reference-free, high-resolution measurement method of timing jitter spectra of optical frequency combs,” Sci. Rep. 7(1), 40917 (2017).
[Crossref]

Paschotta, R.

Petrich, G. S.

Picqué, N.

A. Schliesser, N. Picqué, and T. W. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6(7), 440–449 (2012).
[Crossref]

Portuondo-Campa, E.

Resan, B.

P. Brochard, V. J. Wittwer, S. Bilicki, B. Resan, K. J. Weiningarten, S. Schilt, and T. Südmeyer, “Frequency noise characterization of a 25-GHz diode-pumped mode-locked laser with indirect carrier-envelope offset noise assessment,” IEEE Photonics J. 10(1), 1–10 (2018).
[Crossref]

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Rochette, M.

Rotermund, F.

Sahni, M. O.

Schibli, T. R.

Schilt, S.

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A. Schliesser, N. Picqué, and T. W. Hänsch, “Mid-infrared frequency combs,” Nat. Photonics 6(7), 440–449 (2012).
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Schori, C.

Schunemann, P. G.

Seletskiy, D. V.

Shi, H.

Shin, J.

D. Kwon, C.-G. Jeon, J. Shin, M.-S. Heo, S. E. Park, Y. Song, and J. Kim, “Reference-free, high-resolution measurement method of timing jitter spectra of optical frequency combs,” Sci. Rep. 7(1), 40917 (2017).
[Crossref]

H. Yang, H. Kim, J. Shin, C. Kim, S. Y. Choi, G.-H. Kim, F. Rotermund, and J. Kim, “Gigahertz repetition rate, sub-femtosecond timing jitter optical pulse train directly generated from a mode-locked Yb: KYW laser,” Opt. Lett. 39(1), 56–59 (2014).
[Crossref]

Shoji, T. D.

M. Endo, T. D. Shoji, and T. R. Schibli, “Ultralow Noise Optical Frequency Combs,” IEEE J. Sel. Top. Quantum Electron. 24(5), 1–13 (2018).
[Crossref]

Silfies, M.

Sinclair, L. C.

J. D. Deschênes, L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, and N. R. Newbury, “Synchronization of distant optical clocks at the femtosecond level,” Phys. Rev. X 6(2), 021016 (2016).
[Crossref]

L. C. Sinclair, J.-D. Deschênes, L. Sonderhouse, W. C. Swann, I. H. Khader, E. Baumann, N. R. Newbury, and I. Coddington, “Invited Article: A compact optically coherent fiber frequency comb,” Rev. Sci. Instrum. 86(8), 081301 (2015).
[Crossref]

F. R. Giorgetta, W. C. Swann, L. C. Sinclair, E. Baumann, I. Coddington, and N. R. Newbury, “Optical two-way time and frequency transfer over free space,” Nat. Photonics 7(6), 434–438 (2013).
[Crossref]

Sonderhouse, L.

L. C. Sinclair, J.-D. Deschênes, L. Sonderhouse, W. C. Swann, I. H. Khader, E. Baumann, N. R. Newbury, and I. Coddington, “Invited Article: A compact optically coherent fiber frequency comb,” Rev. Sci. Instrum. 86(8), 081301 (2015).
[Crossref]

Song, Y.

Steinmeyer, G.

B. Borchers, A. Anderson, and G. Steinmeyer, “On the role of shot noise in carrier-envelope phase stabilization,” Laser Photonics Rev. 8(2), 303–315 (2014).
[Crossref]

B. Borchers, S. Koke, A. Husakou, J. Herrmann, and G. Steinmeyer, “Carrier-envelope phase stabilization with sub-10 as residual timing jitter,” Opt. Lett. 36(21), 4146–4148 (2011).
[Crossref]

Stenger, J.

H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens, mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74(1), 1–6 (2002).
[Crossref]

Südmeyer, T.

P. Brochard, V. J. Wittwer, S. Bilicki, B. Resan, K. J. Weiningarten, S. Schilt, and T. Südmeyer, “Frequency noise characterization of a 25-GHz diode-pumped mode-locked laser with indirect carrier-envelope offset noise assessment,” IEEE Photonics J. 10(1), 1–10 (2018).
[Crossref]

P. Brochard, S. Schilt, V. J. Wittwer, and T. Südmeyer, “Characterizing the carrier-envelope offset in an optical frequency comb without traditional f-to-2f interferometry,” Opt. Lett. 40(23), 5522–5525 (2015).
[Crossref]

Sulzer, P.

Swann, W.

Swann, W. C.

J. D. Deschênes, L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, H. Bergeron, M. Cermak, I. Coddington, and N. R. Newbury, “Synchronization of distant optical clocks at the femtosecond level,” Phys. Rev. X 6(2), 021016 (2016).
[Crossref]

L. C. Sinclair, J.-D. Deschênes, L. Sonderhouse, W. C. Swann, I. H. Khader, E. Baumann, N. R. Newbury, and I. Coddington, “Invited Article: A compact optically coherent fiber frequency comb,” Rev. Sci. Instrum. 86(8), 081301 (2015).
[Crossref]

Y.-J. Kim, I. Coddington, W. C. Swann, N. R. Newbury, J. Lee, S. Kim, and S.-W. Kim, “Time-domain stabilization of carrier-envelope phase in femtosecond light pulses,” Opt. Express 22(10), 11788–11796 (2014).
[Crossref]

F. R. Giorgetta, W. C. Swann, L. C. Sinclair, E. Baumann, I. Coddington, and N. R. Newbury, “Optical two-way time and frequency transfer over free space,” Nat. Photonics 7(6), 434–438 (2013).
[Crossref]

N. R. Newbury and W. C. Swann, “Low-noise fiber-laser frequency combs (Invited),” J. Opt. Soc. Am. B 24(8), 1756–1770 (2007).
[Crossref]

B. R. Washburn, W. C. Swann, and N. R. Newbury, “Response dynamics of the frequency comb output from a femtosecond fiber laser,” Opt. Express 13(26), 10622–10633 (2005).
[Crossref]

Tandon, S. N.

Telle, H. R.

N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, “Frequency stabilization of mode-locked Erbium fiber lasers using pump power control,” Appl. Phys. B 78(3-4), 321–324 (2004).
[Crossref]

H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens, mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74(1), 1–6 (2002).
[Crossref]

Thomann, P.

Tian, H.

Timmers, H.

Trebaol, S.

Udem, T.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2008).
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Figures (6)

Fig. 1.
Fig. 1. Experimental setup. PZT, piezo-electric transducer; EDFA, Er-doped fiber amplifier; FBG, fiber Bragg grating; FRM, faraday rotating mirror; WDM, 1540/1560 wavelength division multiplexer; VCO, voltage-controlled oscillator; AOFS, acoustic optical frequency shifter (Brimrose, AMF-50-1560-2FP); BPF, bandpass filter; PD, photodetector (Menlo Systems, FPD510); PI: proportional-integral servo (Newfocus, LB1005); HV Amp: high-voltage amplifier; PID, proportional-integral-differential servo (Vescent Photonics, D2-135); DCU, delay control unit.
Fig. 2.
Fig. 2. Frequency noise PSD of n×frep (red curve), fceo (blue curve) and νn noise (black curve).
Fig. 3.
Fig. 3. Converted timing jitter PSD of the laser oscillator. Calculated timing jitter contribution from direct-coupled jitter from ASE (orange curve), Gorden-Haus jitter (Olive curve), RIN coupled jitter by Kramers-Krönig relation (purple curve) and RIN coupled jitter by self-steepening effect (blue curve). The dashed red curve and dashed gray curve in (a) shows the projected phase noise from intensity noise from 100 MHz signal and the measurement noise floor of the photodetector, electrical amplifier, bandpass filter and mixer, respectively.
Fig. 4.
Fig. 4. (a) νn noise and (b) fceo noise spectrum. Calculated frequency noise spectrum contribution from pump noise (red curve), ASE-induced noise (purple curve), environmental length fluctuations (orange curve), environmental loss fluctuations (dark yellow curve) and Shawlow-Townes limit (olive curve). The dashed red curve and dashed gray curve in (a) shows the projected frequency noise from intensity noise from 100 MHz signal and the measurement noise floor of the photodetector, electrical amplifier, bandpass filter and mixer, respectively.
Fig. 5.
Fig. 5. Frequency dependence of sum of the complex coherence among n×frep, fceo and νn frequency noise PSDs.
Fig. 6.
Fig. 6. (a)νn noise spectra of 1540 nm under repetition rate locking Inset: the stabilized n×frep noise spectrum. (b) Noise spectra of n×frep noise when the fceo is stabilized.

Equations (14)

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S Δ ν n Δ ν n = S Δ f c e o Δ f c e o + S N Δ f r e p N Δ f r e p + Γ Δ ( ω ) × S Δ f c e o Δ f c e o S N Δ f r e p N Δ f r e p
S Δ t G H ( f ) = 4 D 2 f r e p 2 D ω ( 2 π f ) 2 [ ( 2 π f ) 2 + τ ω c 2 ]
D ω = 2 3 E p τ 2 Θ 2 g T r t h v 0
1 τ ω c = 4 3 g T r t Δ f g 2 τ 2
S Δ t A S E , Q L ( f ) = D T ( 2 π f ) 2
D T = π 2 τ 2 6 E p Θ 2 g T r t h v 0
S Δ t R I N S S = ( φ N L 2 π 2 f T r t ν 0 ) 2 S R I N ( f )
S Δ t R I N K K = ( 1 2 π Δ f g ) 2 S R I N ( f )
S v n X = ( v n v f i x X ) 2 S r X ( f ) ,
S r p u m p ( f ) = B 1 1 + ( f / f 3 d B ) 2 S R I N p u m p
S r l e n g t h ( f ) = ( f r e p v g r o u p L ) 2 S l e n g t h ( f )
S r l o s s ( f ) = B 1 1 + ( f / f 3 d B ) 2 S l o s s ( f )
S r A S E , t i min g ( f ) = 2 f r e p 2 ( ( 1 + n s p ) h v 0 G P c i r c ) × [ t r m s 2 + ( β 2 4 D g ω r m s ) ]
S v n A S E , S T ( f ) = 2 f r e p 2 ( 2 π ) 2 ( ( 1 + n s p ) h v 0 G P c i r c )

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