1. Introduction

Lower timing jitter optical pulse trains from femtosecond mode-locked lasers have become more important for various high-precision scientific and engineering applications. Recently emerging examples include low-noise frequency combs [1], timing and synchronization of large-scale scientific facilities [2] (such as free-electron lasers, ultrafast electron sources and phased-array antennas), ultralow phase noise microwave generation [3], long-distance frequency comb transfer [4], photonic analog-to-digital converters (ADCs) [5], optical interconnection [6], and coherent communication, to name a few. In particular, for attosecond-precision synchronization and stabilization between different laser systems, the high-frequency (e.g., >10 kHz) jitter of laser oscillators should be minimized because the free-running jitter outside the locking bandwidth (typically limited to <100 kHz when using piezoelectric transducers) cannot be suppressed by the phase-locked loop (PLL) operation. High-frequency timing jitter also impacts the performance of high-speed, high-resolution photonic ADCs, and the reduction of jitter is important for further advancements in photonic ADCs.

Due to their compactness, ease of operation, robustness and low implementation cost, femtosecond mode-locked fiber lasers can be an ideal candidate as the master oscillator in such ultralow timing jitter applications. With the well-established mode-locked laser noise theory [7–10] and advances in ultra-sensitive timing detection method called the balanced optical cross-correlation (BOC) method [11], the accurate measurement and optimization of timing jitter in mode-locked fiber lasers have been possible in recent years. Using sub-20-attosecond-resolution BOC method, the timing jitter of nonlinear polarization evolution (NPE)-based mode-locked Er-fiber and Yb-fiber lasers was optimized to the unprecedented level of 70-attosecond [12] and 175-attosecond [13], respectively, in the 10 kHz – 40 MHz (Nyquist frequency) integration bandwidth. The timing jitter could be reduced down to the sub-100-attosecond level by setting the intra-cavity dispersion of stretched-pulse fiber lasers at close-to-zero regime. This was possible because both the quantum-noise-limited, directly coupled timing jitter and the indirectly coupled timing jitter could be minimized. Here, the directly coupled timing jitter is originated directly from the amplified spontaneous emission (ASE) noise, whereas the indirectly coupled jitter is originated from the center frequency fluctuations via intra-cavity dispersion. Because the minimal pulse duration is shorter and the average intra-cavity pulse energy is higher for stretched-pulse operation, compared to the soliton mode-locked operation, the timing jitter directly coupled from the ASE noise can be reduced. Operating at close-to-zero intra-cavity dispersion also lowers the indirectly coupled timing jitter induced from the center frequency fluctuations, which is also called the Gordon-Haus timing jitter [14].

However, long-term stable operation of the NPE-based mode-locked fiber lasers is often limited due to the birefringence drift and high polarization-state sensitivity in the fiber. For more practical applications of ultralow-jitter lasers outside laboratory environments, it is necessary to implement the fiber laser in a self-starting, robust, stable, and continuous-operating (e.g., uninterrupted operation for more than a few months) way. For this, an all-fiber laser configuration, mode-locked by a real saturable absorber device such as semiconductor saturable absorbers mirrors (SESAMs) [15], carbon nanotubes (CNTs) [16] or graphene [17], is highly desirable. Recently we have shown that an 80-MHz soliton Er-fiber laser mode-locked by a fiber tapered CNT saturable absorber can have 3-fs level timing jitter (10 kHz-40 MHz) when operated at −0.055 ps² intra-cavity dispersion [18]. It was identified that the resulting jitter is mostly limited by the Gordon-Haus jitter from large negative intra-cavity dispersion, and not by other detrimental effects caused by a slow saturable absorber. This prior result underlines the importance of dispersion engineering for minimizing the laser jitter, and further suggests that the timing jitter may be significantly reduced when operating the CNT-mode-locked laser at a smaller intra-cavity dispersion magnitude (closer to zero) while still maintaining the laser in a stable soliton mode-locking regime.

In this paper, we demonstrate that the timing jitter of a CNT-mode-locked soliton Er-fiber laser can be scaled down to the sub-femtosecond regime by intra-cavity dispersion engineering. The demonstrated minimal jitter is 490 attoseconds, integrated from 10 kHz to 39.4 MHz (Nyquist frequency) offset frequency, when operating the laser at −0.02 ps² intra-cavity dispersion. To our knowledge, this is the first time to directly measure the sub-femtosecond-level timing jitter from an all-fiber laser mode-locked by a real saturable absorber device. The measured jitter level is not as low as the lowest jitter condition (~100 as) of the NPE-based fiber lasers at zero dispersion in the stretched-pulse regime [12,13], mainly due to the required negative dispersion for stable soliton mode-locking with a slow saturable absorber [19]. One interesting observation is that the jitter level of CNT-mode-locked all-fiber lasers can be engineered to be lower than that of NPE fiber lasers at similar intra-cavity dispersion (−0.021 ps²) in the soliton regime [11]. Our result shows that sub-femtosecond-level jitter optical pulse trains are readily achievable from self-starting, robust and low-cost all-fiber lasers, which may enable more widespread applications of such ultralow-jitter photonic signal sources outside well-controlled laboratory environments.

2. Laser design and measured parameters

Figure 1(a) shows the structure of the laser we investigated in this work: a soliton, all-fiber Er ring laser mode-locked by a fiber tapered CNT saturable absorber (ft-CNT-SA), which is similar to the design shown in [18] and [20]. The evanescent field interaction type CNT-SA is based on a tapered-fiber design in a similar manner reported in [20]. The ft-CNT-SA employed in the laser shows 77% loss, 6% modulation depth, 150 μJ/cm² saturation fluence, and about 500 – 800 fs recovery time. Although the fiber taper has the thinnest region of ~3 μm diameter, this section is only a few mm long and the total dispersion from the fiber taper is much smaller than 0.001 ps². The used Er gain fiber is 78.7 cm long, 110 dB/m, 4/125 fiber (from Liekki). Except for the 20 cm section of OFS-980 fiber (in the WDM section), the rest of non-gain fiber section is SMF-28 fiber. To investigate the timing jitter and relative intensity noise (RIN) performance of a soliton ft-CNT-mode-locked fiber laser at different net cavity dispersion, the intra-cavity dispersion is tuned by adding or cutting SMF-28 fiber (labeled as X section in Fig. 1(a)) inside the laser cavity. The used net cavity dispersion values in this work are −0.05 ps², −0.04 ps², −0.03 ps², −0.02 ps² and −0.016 ps² at 1562 nm center wavelength. Note that −0.016 ps² was the minimum (in magnitude) dispersion that we could obtain a stable soliton mode-locking operation with the ft-CNT-SA used. Figure 1(b) shows the measured optical spectra at various net cavity dispersion conditions. At each dispersion condition, the pump power is set to the maximum value that allows the broadest optical spectrum and highest pulse energy while maintaining a stable mode-locking condition without multi-pulsing. Table 1 summarizes the measured repetition rate, 3-dB optical bandwidth, intra-cavity pulse energy, pump power, and average output power (from a 30% output coupler) for each net cavity dispersion condition.

 

Fig. 1 (a) The structure of all-fiber Er laser mode-locked by a fiber tapered carbon nanotube saturable absorber (CNT-SA). (b) Measured broadest optical spectra at different intra-cavity dispersion condition.

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Table 1. Summary of laser parameters

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In order to assess the noise performance at each dispersion condition, we first measured the relative intensity noise (RIN). For each measurement, part of optical output power (~0.6 mW) is applied to a 48-MHz bandwidth, 900 Ω gain InGaAs transimpedance amplifier photodetector. The projected shot noise limited measurement noise floor is ~–152 dBc/Hz. Figure 2(a) shows the RIN spectral density measurement results, from 1 Hz to 10 MHz offset frequency, at each dispersion condition. Figure 2(b) shows the integrated RIN (in rms) for different integration bandwidths (1 Hz – 100 kHz, 1 kHz – 10 MHz, and 1 Hz – 10 MHz) at each dispersion condition. For dispersion conditions lower than −0.02 ps², the integrated RIN is low (<0.05% (rms) from 1 Hz to 10 MHz; <0.022% (rms) from 1 Hz to 100 kHz), which shows the excellent amplitude stability of the all-fiber CNT-mode-locked Er lasers.

 

Fig. 2 (a) Measured relative intensity noise (RIN) spectra for different intra-cavity dispersion conditions. (b) Integrated RIN (rms) with different integration bandwidth and intra-cavity dispersion condition.

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In addition, to assess the long-term stability of the constructed 78.7 MHz repetition rate CNT-SA-based ring laser at −0.02 ps² intra-cavity dispersion (in Fig. 1(a)), we measured the output power (using a 17-MHz photodetector) and optical spectrum (using an optical spectrum analyzer) over 10 days, as shown in Fig. 3. The output power measurement shows 0.61% rms and 2.8% peak-to-peak changes over 10 days. The optical spectrum shape and 3-dB bandwidth were also maintained well with 1.3% rms change over 10 days. Note that most of these changes are caused by large temperature fluctuations, up to 4.3 K, in the laboratory. When the temperature is well maintained within 0.17 K rms (1.5 K peak-to-peak) from Day 8 to 10, the fluctuations in output power and 3-dB bandwidth are only 0.17% rms (1.6% peak-to-peak) and 0.19% rms, respectively, for 3 days. These measurement results show the excellent long-term stability of the all-fiber laser. We anticipate that the mode-locked state and major laser properties (such as output power and optical spectrum shape) will be well maintained over many months, as many commercially available all-fiber lasers do.

 

Fig. 3 Long-term output power and optical bandwidth measurement results of the constructed CNT-SA-based all-fiber Er laser over 10 days.

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3. Timing jitter prediction, measurement, and analysis

With the measured optical spectrum bandwidth, intra-cavity pulse energy, RIN spectrum and other known and calculated laser parameters, we can predict the achievable timing jitter at each dispersion condition by using analytic laser noise models shown in [7,9,10]. The three major contributions to the total timing jitter in a soliton fiber laser are following: (i) quantum-limited timing jitter directly originated from the ASE noise (which will be denoted as “direct jitter” in the following sections), (ii) quantum-limited timing jitter indirectly originated from center frequency fluctuation via intra-cavity dispersion (which will be denoted as “Gordon-Haus jitter” in the following sections), and (iii) RIN-originated timing jitter, including the Kramers-Kronig relationship and the coupling via a slow saturable absorber (which will be denoted as “RIN-originated jitter” in the following sections). Each of these contributions can be calculated using soliton perturbation theory-based timing jitter models (in [7,9,10]). For the jitter calculation, we used the measured laser parameters (such as intra-cavity pulse energy, optical spectrum bandwidth, repetition rate) and RIN spectra shown in Section 2. In addition to the measured values, we further assumed that the excess noise factor is 2 and the gain bandwidth is 40 nm [21].

Figure 4 shows the predicted integrated rms timing jitter (integrated from 10 kHz to 25 MHz offset frequency) for each dispersion condition. Plots (a), (b), and (c) show the amount of direct jitter (Δt_direct), Gordon-Haus jitter (Δt_G-H), and RIN-originated jitter (Δt_RIN), respectively. Plot (d) corresponds to the total integrated rms jitter, which is obtained by $\Delta t_{total}=\sqrt{\Delta t_{direct}^2+\Delta t_{G-H}^2+\Delta t_{RIN}^2}$. As can be seen in Fig. 4, the dominant contribution to the total jitter is from the Gordon-Haus jitter (plot (b)), which is proportional to the dispersion magnitude ($\Delta t_{G-H}~\left|D\right|/\left(\Delta {\nu }_p\sqrt{E_p}\right)$, where $D$ is the intra-cavity 2nd order dispersion, $\Delta {\nu }_p$ is the spectral bandwidth, and $E_p$ is the intra-cavity pulse energy). As a result, the Gordon-Haus jitter and the total timing jitter decreases as the dispersion becomes closer to zero. The only exception is at −0.016 ps², where the Gordon-Haus jitter increases compared to the −0.02 ps² condition. This is caused by the combined effects of lower intra-cavity pulse energy and narrower spectral bandwidth. Although the direct jitter and RIN-originated jitter increase as the dispersion magnitude decreases (as shown in plots (a) and (c) in Fig. 4), their magnitudes are much lower than that of the Gordon-Haus jitter (plot (b) in Fig. 4) and they do not affect the total jitter much. From this analysis, we can expect that around −0.02 ps², where the Gordon-Haus jitter is minimized, the total timing jitter can be minimized.

 

Fig. 4 Predicted integrated rms timing jitter as a function of intra-cavity dispersion (integration bandwidth: 10 kHz – 25 MHz offset frequency) based on the measured and known laser parameters. (a) Quantum-limited timing jitter directly originated from the ASE noise (direct jitter). (b) Quantum-limited timing jitter indirectly originated from center frequency fluctuation via intra-cavity dispersion (Gordon-Haus jitter). (c) RIN-originated timing jitter, including the Kramers-Kronig relationship and the coupling via a slow saturable absorber (RIN-originated jitter). (d) Total timing jitter, including all effects of direct jitter, Gordon-Haus jitter, and RIN-originated jitter. The result shows that Gordon-Haus jitter contribution is the dominant effect to the total jitter.

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For the timing jitter measurement experiment, we used a timing detector method in [18], also summarized in Fig. 5. Here, the repetition rates of the laser under test (LUT, CNT-mode-locked all-fiber Er-laser in this work) and a reference laser (with lower intrinsic jitter than the LUT) are synchronized with low (~kHz) bandwidth, and the timing jitter spectral density is measured outside the locking bandwidth using a PPKTP-based BOC. In this way, we can measure the jitter spectral density of free-running lasers, which follows the sum of jitter spectral densities of the LUT and the reference laser, with sub-100-attosecond resolution over the full Nyquist frequency. For the reference laser, we used an NPE-based mode-locked Er-fiber laser operating at close-to-zero intra-cavity dispersion, which is similar to the design in [12], with an estimated timing jitter of ~100-attosecond level. As the repetition rate of the LUT changes by adding or cutting SMF-28 (for net cavity dispersion tuning), we also added or removed OFS-980 fiber in the reference laser to match the repetition rate between them. Since OFS-980 fiber has a much smaller dispersion magnitude than SMF-28 fiber (OFS-980: + 2 ps²/km, SMF-28: −24 ps²/km at the NPE-laser center wavelength 1580 nm), we could change the repetition rate of the reference laser without much altering the net intra-cavity dispersion, which maintains the jitter performance of the reference laser. For the jitter spectral density measurements, we tested at −0.03 ps² (67.1 MHz repetition rate) and −0.02 ps² (78.7 MHz repetition rate) net cavity dispersion conditions, where the integrated jitter is expected to be the lowest as shown in Fig. 4.

 

Fig. 5 Experimental set-up for timing jitter characterization of the laser under test (CNT-mode-locked all-fiber Er laser). BPF, bandpass filter; EDFA, Er-doped fiber amplifier; HWP, half-wave plate; PBS, polarization beam splitter; PD, photodetector.

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Figure 6 shows the timing jitter spectral density measurement results. Figure 6(a) shows the jitter spectral densities when the net cavity dispersion of LUT is set to −0.02 ps² (red curve) and −0.03 ps² (black curve). The output power and the optical spectrum bandwidth of LUT are set to the maximally allowed values as shown in Table 1: (3.61 mW, 13.8 nm) for −0.02 ps² and (4.20 mW, 14.0 nm) for −0.03 ps². For both cases, the measured jitter spectrum is more than 10 dB higher than the estimated jitter spectrum of the NPE-based reference laser. Thus, we can assume that the measured jitter spectrum follows that of the LUT. The predicted timing jitter spectral density, based on the analytic model [10] and the known and measured laser parameters, also matches the measured data fairly well at −0.02 ps² condition. For comparison, we also plot the previous measurement result at −0.055 ps² (which was reported in [18]). As expected, the jitter spectra at −0.02 ps² and −0.03 ps² are significantly lower than that at −0.055 ps², especially at lower offset frequency (e.g., <1 MHz) where the Gordon-Haus jitter dominates. This measurement shows that reduced Gordon-Haus jitter by lower dispersion magnitude indeed leads to lower timing jitter in a soliton fiber laser. Note that the optical bandwidth (8.4 nm) and the intra-cavity pulse energy (0.14 nJ) of the −0.055 ps² laser in [18] are much lower than those of the laser in this work (as shown in Table 1), and this is why the spectrum of the −0.055 ps² case in Fig. 6(a) is much higher than the prediction at −0.05 ps² in Fig. 4. Another interesting observation is that, unlike the previous NPE-based fiber laser measurement results [11–13], we could observe both broadband and narrowband modulations at high offset frequency range (i.e., from ~10 MHz to the Nyquist frequency) for the CNT-mode-locked lasers. We believe that this might be caused by slight instability in the soliton mode-locked lasers as we push the intra-cavity dispersion closer to zero. However, in terms of magnitude, this additional noise contributes very little (<100 as) to the total integrated jitter.

 

Fig. 6 Timing jitter spectral density measurement results of CNT-mode-locked soliton Er-fiber laser (LUT). (a) Timing jitter spectral density of LUT at −0.02 ps2 and −0.03 ps2 net cavity dispersion. For comparison, jitter data at −0.055 ps2 (from [18]) is included. (b) Timing jitter spectral density of LUT at different optical bandwidth and intra-cavity pulse energy at −0.02 ps2 net cavity dispersion.

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We further tested the impact of optical spectral bandwidth and intra-cavity pulse energy when the net cavity dispersion is fixed. Figure 6(b) shows the measurement result at −0.02 ps² net cavity dispersion. By adjusting the pump power, we set the 3-dB optical bandwidth at 8 nm (pumped by 74 mW; intra-cavity pulse energy 0.07 nJ) and 13.8 nm (pumped by 82 mW; intra-cavity pulse energy 0.15 nJ). As expected, broader optical spectrum and higher intra-cavity pulse energy lead to lower timing jitter. The integrated rms timing jitters of 8 nm and 13.8 nm bandwidth conditions are 1.06 fs and 490 as, respectively, when integrated from 10 kHz to 39.4 MHz (Nyquist frequency) offset frequency.

Figure 7 compares measured integrated rms timing jitter (blue stars) versus predicted integrated rms timing jitter (black circles). The best jitter performances at both −0.03 ps² (point (a)) and −0.02 ps² (point (b)) are very close to the predicted jitter performances of the maximum bandwidth conditions (from Fig. 4). We can also see that the integrated jitter can vary significantly at the same dispersion, depending on the spectral bandwidth and intra-cavity pulse energy conditions: at −0.02 ps², it varies from 490 as to 1.06 fs; at −0.03 ps², it varies from 630 as to 1.5 fs. For the minimal bandwidth conditions (8 nm) at −0.03 ps² and −0.02 ps², the prediction and the actual measurements (points (c) and (d)) also agree fairly well. Note that the measured jitter at 8 nm bandwidth, −0.02 ps² dispersion condition (point (d)) is slightly lower than the prediction, which is caused by the unwanted suppression of jitter spectrum in the 10 kHz - 20 kHz range by the PLL operation.

 

Fig. 7 Predicted integrated timing jitter (black circles) versus measured integrated timing jitter (blue stars). The plot of “predicted timing jitter @ maximum bandwidth” used the conditions shown in Table 1. The plot of “predicted timing jitter @ minimum bandwidth” is based on the 8-nm bandwidth condition. The measured timing jitter (blue stars) of (a) 14 nm bandwidth and 0.21 nJ pulse energy at −0.03 ps², (b) 13.8 nm bandwidth and 0.15 nJ pulse energy at −0.02 ps², (c) 8 nm bandwidth and 0.09 nJ pulse energy at −0.03 ps², and (d) 8 nm bandwidth and 0.07 nJ pulse energy at −0.02 ps².

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One interesting finding is that the targeted timing jitter performance is easily reproducible by setting optical bandwidth, intra-cavity pulse energy and net cavity dispersion to the targeted values. When we switched the laser operation conditions (between 8-nm and 13.8-nm optical bandwidth conditions) back and forth and measured the jitter spectrum repeatedly, there was good reproducibility in jitter performance. This is quite different from the NPE-based fiber laser case, where many different mode-locked conditions (in terms of optical spectral shape and pulse energy) can be obtained by adjusting the input polarization states using intra-cavity wave-plates. Due to random and uncontrollable changes in fiber birefringence, even setting the wave-plates back to the original condition does not guarantee to find the same mode-locked condition. Even when the resulting optical spectra look similar for different mode-locking conditions, the jitter performance can be significantly different, as reported in [13]. In contrast, for the soliton fiber laser mode-locked by a slow saturable absorber device (such as CNT in this work), the jitter performance follows the predicted performance fairly well and is reproducible, which will be a great advantage for producing commercial low-jitter lasers in the future.

4. Summary and outlooks

We demonstrate that the timing jitter of an ft-CNT-mode-locked soliton all-fiber ring laser can be scaled down to the sub-femtosecond regime by finding a net cavity dispersion condition where the Gordon-Haus jitter is minimized while stable soliton mode-locking is maintained at the same time. The demonstrated minimal timing jitter is 490 attoseconds, integrated from 10 kHz to 39.4 MHz (Nyquist frequency) offset frequency, when operating the laser at −0.02 ps² intra-cavity dispersion. To our knowledge, this is the first direct measurement of sub-femtosecond-level timing jitter from an all-fiber laser mode-locked by a real saturable absorber device. Our result shows that sub-femtosecond-level jitter optical pulse trains are readily achievable from self-starting, robust and low-cost all-fiber lasers, which may enable more widespread applications of such ultralow-jitter photonic signal sources outside well-controlled laboratory environments, such as clocking X-ray free-electron lasers (XFELs). Although ft-CNT-SA was used as a saturable absorber in this work, we believe that this general jitter reduction strategy in a soliton fiber laser can be applied to other types of saturable absorbers, such as SESAMs and graphenes, as well. In the future, we further have an interest in engineering new fiber- or waveguide-coupled saturable absorbers with wider bandwidth (leading to shorter pulse duration), higher damage threshold (leading to higher pulse energy), and faster recovery time (leading to lower necessary dispersion magnitude, which in turn will lower the Gordon-Haus jitter), which will enable even lower noise and more robust all-fiber lasers suitable for many field applications.