Review of low timing jitter mode-locked fiber lasers and applications in dual-comb absolute distance measurement Open Access

Since its first demonstration in 1972,¹ passively mode-locked laser technology has been growing rapidly and has become an enabling technology for a number of advanced fields. Passively mode-locked lasers have unique time-frequency domain characteristics. In the time-domain, these lasers output uniformly spaced ultrashort optical pulse trains with duration from a sub-picosecond down to a few optical cycles. The ultrashort pulse duration is favored in ultrafast-phenomena resolving applications in chemistry, biology, and physics.^2–4 The short pulse duration also delivers ultrahigh peak power, making the mode-locked laser an excellent candidate for material processing and laser manufacturing.^5,6 Moreover, it provides an extreme physical environment in light-matter interaction study.⁷ In the frequency domain, the mode-locked laser outputs a wide optical spectrum that consists of thousands of phase-locked discrete comb lines with their spacing equal to the repetition rate of the pulse train. Phase-stabilized optical frequency combs (OFCs) based on mode-locked lasers have revolutionized the precision metrology by providing a bridge between optical and radio frequencies. The OFCs provide a precisely determined frequency ruler for optical frequency measurement^8–10 and becomes a dispensable building block for optical clocks,¹¹ astronomy,¹² and molecular spectroscopy.^13–15 

Apart from the short pulse duration and wide optical spectrum, mode-locked lasers also present excellent low timing noise performance. Compared with traditional electronic oscillators that generate clock signals with picosecond timing jitter, the pulse train generated by a passively mode-locked laser exhibits outstanding short-term stability. The high Fourier frequency timing jitter of a standard mode-locked laser working at room temperature can easily reach sub-femtosecond scale, and this jitter has been reduced to tens of attoseconds recently.¹⁶ The ultralow timing jitter characteristics facilitate numerous applications that require ultralow jitter optical oscillators, such as optical communication,¹⁷ large-scale timing synchronization/distribution,^18,19 generation of purified microwaves,^20,21 and photonic analog-digital converters.²² 

In addition to traditional mode-locked lasers based on solid-state crystals, passively mode-locked fiber lasers have attracted much attention in recent years due to their simple implementation, compact structure, excellent long-term stability, low cost, and turnkey operation. In particular, mode-locked fiber lasers can achieve an all-fiber configuration with standard telecommunication fiber-optic components in both erbium (Er-) and ytterbium (Yb)-doped fiber lasers.^23–25 The performance of the mode-locked fiber laser has caught up with that of solid-state lasers through careful management of nonlinearity and dispersion in the laser cavity. In terms of timing jitter, a nonlinear amplifying loop mirror mode-locked fiber laser has demonstrated nearly quantum-noise limited performance,²⁶ and the integrated timing jitter is as low as 40.

Consequently, a low jitter mode-locked fiber laser has become a competitive candidate for high-precision applications. Specifically, the combination of the short pulse duration and ultralow timing jitter pulse trains make the mode-locked fiber laser perfectly match the critical demands in absolute distance measurements.^27,28 The high timing resolution provides a fine discriminator for time-of-flight distance measurement and guarantees sub-micrometer ranging resolution.^29–32 Meanwhile, the uniformly spaced pulse trains in several nanosecond levels provide a meter-level non-ambiguity range. Moreover, when the spectral information is simultaneously utilized, the interferometric scheme can achieve nanometer-level ranging precision.^30,33,34

In this paper, we review the ultralow timing jitter mode-locked fiber lasers and the applications in absolute distance measurements, specifically based on a dual-comb configuration composed of a pair of OFCs with slight offset repetition rate. We present the high-precision timing jitter characterization method, which optimizes timing jitter, by tailoring the intracavity pulse evolution dynamics and dual-comb absolute distance measurement technology in sequence. The fundamental limit of the timing jitter on the high update rate dual-comb ranging precision and the route to the advanced absolute ranging performance are also discussed.

Timing jitter is the time deviation of the optical pulse position relative to its ideal equally spaced pulse position. In principle, the timing deviation can be related to the phase noise of the repetition rate of the pulse train. To this end, the simplest and most intuitive method to characterize the timing jitter of the mode-locked laser pulse train is to use a signal source analyzer to measure the phase noise of the high harmonics frequency of the repetition rate detected with a fast photodetector (i.e., with a bandwidth up to GHz).^35,36 However, as the timing jitter of the mode-locked laser is always lower than traditional microwave frequency standards at high Fourier frequencies, accurate characterization of the timing jitter of the pulse train cannot be conducted with standard electronic instruments. Moreover, the amplitude noise also couples into the measured phase noise, making it indistinguishable from the timing jitter noise. Thus, the routinely used phase noise detection method can only achieve a timing jitter measurement resolution of approximately 10 fs.

To precisely determine the timing jitter of mode-locked lasers, several optical characterization methods have been proposed. First, to solve the problem that timing jitter is lower than the noise floor of the RF instrument, another mode-locked laser with identical or lower timing jitter to the laser under test is used as the local oscillator.¹⁶ As shown in Fig. 1(a), the high harmonics of the repetition rates of both the laser under test and the local oscillator are detected with photodetectors. The sampled signals are then bandpass filtered and mixed to detect the phase error between the laser under test and the reference oscillator. To realize the relative phase noise between the two laser pulse trains, the error signal is used to lock the repetition of the reference laser to the laser under test by tuning the intracavity PZT with a low feedback bandwidth. Then, the residual phase error is used to characterize the phase noise of the free running mode-locked laser outside the locking bandwidth. To increase the sensitivity as much as possible, higher harmonics of the repetition rate is desired. As a result, the bandwidth and amplitude-to-phase-noise process of the photodetector limit the measurement resolution and dynamic range.

To achieve better timing resolution, balanced optical cross-correlation (BOC) method is proposed to realize timing jitter characterization to the Nyquist frequency.³⁷ The BOC-based timing jitter measurement scheme is shown in Fig. 1(b). The setup is similar to the aforementioned method except that the photo detection and frequency mixer are substituted by the balanced optical cross-correlator. The optical pulse itself is used as the high sensitivity timing discriminator that guarantees the sub-femtosecond or even attosecond timing resolution. As the two optical pulses from the laser under test and the local oscillator overlap in the time-domain, a sum frequency signal is generated by the nonlinear crystal and the intensity of the sum frequency signal is proportional to the relative timing between the two optical pulses. As a result, when the repetition rates of the two lasers are synchronized with a low-bandwidth servo loop, the residual timing jitter is encoded into the optical cross-correlation signal. The power spectral density (PSD) of the optical cross-correlation signal is used to indicate the timing stability of the mode-locked laser. Meanwhile, another optical cross-correlator with a certain delay can be added, and a balanced detector is used to eliminate the impact of laser intensity noise on the measured timing jitter result. To this end, ultrahigh timing jitter sensitivity and wide dynamic range can be achieved.^38–41 

Although the BOC method has successfully characterized the timing jitter of mode-locked laser down to a range of several attoseconds, several disadvantages hinder its widespread implementation. For example, the timing resolution is determined by the ultrashort optical pulse duration, and the measurement dynamic range can be significantly limited when the laser under test has long pulse durations (hundreds of femtoseconds to few picoseconds). In addition, two tightly phase-locked lasers should be used, which greatly complicate the experimental setup. To solve these problems, a number of novel optical heterodyne methods have been proposed.^42,43 The ultrahigh timing discrimination capability is provided by comb-line interference. Remarkably, reference-source-free characterization of timing jitter spectra of passively mode-locked lasers is allowed by using only a single-mode fiber delay line and optical carrier interference.⁴³ 

The existing jitter characterization methods mostly measure PSD. In the optical communication field, eye diagram analysis is routinely used to characterize the timing jitter of the transmitted signals directly in the time-domain with nanosecond- to picosecond-level resolution. As a complementary approach to spectral analysis, the time-domain timing jitter characterization is able to directly “visualize” the timing jitters, and the measurement is in real time. In mode-locked lasers, however, the eye diagram-based method is not feasible because of the limited timing resolution of the photodetector for femtosecond laser pulse. Asynchronous optical sampling (ASOPS) method has been used to solve this problem.^4,44,45 Fig. 2(a) shows the principle of the ASOPS process. In this case, a low-noise mode-locked laser with a certain repetition rate offset against the laser under test (LUT) is used as the local oscillator (LO). In an ideal condition (where no timing jitter occurs) as indicated by the solid line, the repetition rate offset leads to a process where LO naturally samples a different part of the LUT pulse in each repetition period by utilizing optical cross-correlation. During the time span equal to the inverse repetition rate offset, the LO walks across the LUT repetition rate period, thereby stretching the time scale of the optical pulse train with a factor of N = f_r/Δf_r, where f_r is LUT's repetition rate and Δf_r is the repetition rate offset between LUT and LO. This condition effectively maps the fast-varying optical phenomenon into the slow-varying RF frequency domain. The down converted signals are readily detected and sampled with slow electronics and the optical pulse information can be retrieved with sub-femtosecond or even attosecond timing resolution as long as the magnification factor N is determined in advance.

The high-precision time-domain timing jitter characterization based on ASOPS is described as follows. The effect of LUT timing jitter can be considered while the LO is still assumed to be noise free. The measurement principle is based on the fact that the ASOPS process not only stretches the optical pulse but also significantly magnifies the accumulated timing error, as indicated by the dashed lines in Fig. 2(a). To implement this principle, the LUT pulse train is directed to a Michelson-interferometer-based delay line that divides each pulse into a pulse pair with temporal delay of t_p. The pulse pair is stretched in time by the ASOPS process and then the separation becomes T_p. As the stretched-pulse pairs are repetitive, they can be recorded by a photodetector and displayed on an oscilloscope operating in the persistent mode. Proper triggering aligns all the pulse pairs with the front pulses. The statistics of the timing of the rear pulses come up with a visual period jitter standard deviation (STD) in the stretched timescale, as shown in Fig. 2(b). This resembles eye diagram analysis in telecommunications. The period jitter STD of mode-locked lasers can be mapped back into the original femtosecond or even attosecond timescale based on $σ 0 = σ visual / ( N ⋅ M )$⁠,⁴⁵ where M ≈ [T_p/T_r] and T_r = 1/f_r.

Fig. 3(a) illustrates stretched-pulse pairs by the ASOPS process in the simulation, where T_p is equal to 30 μs. Here the period jitter of the pulse train is preset as 1 fs RMS. Fig. 3(b) provides the statistical results of the eye diagrams at different T_p. As the timing jitter of mode-locked lasers in the high-frequency range is mostly dominated by quantum-noise, which leads to a random walk of the pulse position, the visual timing jitter STD increases monolithically with T_p, as shown in the upper part of Fig. 3(b). However, all of the retrieved optical period timing jitters are close to the preset simulation number, as shown in the bottom of Fig. 3(b), thereby indicating the validity of the high-precision timing jitter characterization method.

The main performance metric of the time-domain timing jitter measurement method is based on the fact that the timing resolution is determined by the ASOPS magnification factor, which can be easily improved by reducing the repetition rate offset between the LUT and LO. In this manner, the measurement dynamic range is independent of pulse duration. On the other hand, despite the use of two lasers, they do not need to be phase-locked. This condition greatly simplifies the experimental configuration. Most importantly, the measured visual timing jitter can be expressed with an eye diagram, realizing direct and intuitive estimation of sub-femtosecond timing jitter in real time. This setup is expected to become a routine approach to attosecond precision jitter measurement, which is affordable to most optics laboratories.

Based on various fiber dispersion and nonlinearity parameters, optical pulses in the laser cavity experience different formation and shaping mechanisms that lead to distinct mode-locking operation regimes.⁴⁶ In an Er-doped fiber laser working at 1550 nm band, where the laser cavity consists of fibers with anomalous dispersion, soliton mode-locking is easy to achieve. Solitons can be generated with the balance of the net cavity dispersion; the nonlinearity of the fiber and soliton pulses propagate inside the fiber cavity with a fixed pulse shape and optical spectrum. As the effect of the fiber dispersion counteracts the nonlinearity, the output optical solitons are free of chirp with high pulse quality. However, owing to the limitation of the nonlinear phase shift, the pulse energy of soliton pulses generated by a single-mode mode-locked fiber laser is always limited to several picojoule levels. When the dispersion signs of the fibers in the cavity are opposite to manage the net cavity dispersion, the laser operates in stretch-pulse mode-locking regime, where the optical pulse is periodically stretched and compressed in different segments of the laser cavity.⁴⁷ In this case, the average pulse width in a round trip is much broader and the fiber nonlinearity effect is reduced accordingly. When working in a stretched-pulse regime with close-to-zero net cavity dispersion, the output pulse energy is raised while the pulse duration is reduced. Furthermore, novel mode-locking approaches draw on the use of the laser cavity operating in net positive cavity dispersion. When the net cavity dispersion is further tuned toward the normal dispersion area, it tends to scale down the peak power inside the fiber core by stretching the pulse during propagation so that nonlinearity is under control. By utilizing an intracavity spectral filter to realize the pulse-shaping mechanism, dissipative solitons are supported in an all-normal-dispersion (ANDi) laser.⁴⁸ The dissipative soliton originates from a double balance of the dispersion and nonlinearity and the gain and loss. As the net dispersion of the cavity is positive, the output optical pulse is greatly chirped but it can be compressed outside the cavity. The ANDi laser supports pulse energy up to tens of nanojoules or even microjoules. In addition to the solitons and quasi-solitons that can be achieved in the mode-locked fiber laser, similariton pulses with a parabolic shape also exist.^49,50

Timing jitter performance of the mode-locked fiber laser is closely related to the pulse evolution dynamics in the laser cavity. In theory, the amplified spontaneous emission (ASE) from the cavity sets a quantum limit for the timing jitter of mode-locked lasers.⁵¹ In general, there are two different types of coupling mechanisms of the ASE to pulse train timing jitter. First, the ASE can directly act on the timing jitter by imposing a perturbation on the pulse envelope amplitude, which disturbs the pulse weight center and causes a timing fluctuation.⁵² The timing jitter from the direct coupling effect of ASE noise is dependent on the pulse width, pulse energy, and cavity loss. In principle, the shorter pulse width and larger pulse energy lead to a better timing jitter performance, while the increase of output coupling ratio deteriorates the timing jitter character. On the other hand, the net cavity dispersion also contributes to the timing jitter through the coupling from ASE to the spectral center fluctuation.⁵³ The ASE-induced carrier frequency jitter is indirectly translated into a pulse position fluctuation mediated by the group velocity dispersion of the fiber cavity. At the same time, the finite gain bandwidth provides a certain restoring force to stabilize the pulse jitter to a certain extent. When considering both the direct and indirect ASE coupling mechanisms, we can express the timing jitter PSD of the mode-locked fiber laser as follows:^51–53 

where f_r is the repetition rate of the mode-locked fiber laser, E_p is the intracavity pulse energy, D is the net cavity dispersion, and τ_f corresponds to the limited gain bandwidth. As shown in Eqs. (1) and (2), the final timing jitter is solely determined by the laser parameters. In particular, the timing jitter performance is dependent on the net cavity dispersion.

In the experiment, the timing jitter of mode-locked fiber lasers working in different operation regimes is characterized by the BOC technique.^39,54 The LUT is a nonlinear polarization-rotation based mode-locked Yb-fiber laser with a repetition rate of 80 MHz. A grating pair is inserted into the cavity to adjust the net cavity dispersion from negative to positive, switching the operation regimes among soliton mode-locking, stretched-pulse mode-locking, and self-similar mode-locking.

First, the effect of pulse-shaping mechanisms on the timing jitter is shown in Fig. 4 when the LUT works in different mode-locked regimes. The measured timing jitter PSD presents distinct characteristics.³⁹ As the results show, the stretch-pulse mode-locking shows a relatively lower timing jitter value compared with the other two operation regimes. Furthermore, the PSD curve has a shape with 1/f² slope, indicating that the timing jitter is dominated directly by the ASE noise. The lowest timing jitter is attributed to three key factors, where the relative close-to-zero dispersion greatly reduces the indirectly coupled timing jitter from the ASE noise while the shortest pulse width and higher pulse energy ensure that the directly coupled timing jitter from the ASE noise is minimized. By contrast, the soliton and self-similar regimes suffer from Gordon-Haus jitter originating from the large residual net cavity dispersion, which greatly increased the timing jitter of the output optical pulse train in the low-frequency range.

Second, when the laser works in the stretched-pulse regime with a close-to-zero net cavity dispersion, the residual dispersion value still plays a role in the pulse train timing jitter.⁵⁴ When tuning the grating pair distance and thus changing the net cavity dispersion finely toward 0, the timing jitter of the stretched-pulse trains is evaluated, as shown in Fig. 5(a). When the intracavity net dispersion is set to a slightly negative value near 0, the timing jitter of the optical pulse trains can reach a minimum value. However, when the net cavity dispersion surpasses 0 and reaches the positive dispersion regime, the timing jitter rapidly increases as the pulse chirp severely deteriorates the timing jitter performance.

To summarize, the timing jitter performance of the mode-locked laser can be engineered with careful tailoring of the laser parameters, especially by adjusting the cavity dispersion into the close-to-zero regime. Ultralow timing jitter performance can be achieved in this manner but it is not a method for practical application. The reason is that when the mode-locked laser operates near the close-to-zero dispersion regime, various mode-locking states occur with different output parameters and timing jitter characteristics. One needs to carefully adjust the mode-locking status and optimize the timing jitter properties; these tasks are complex and time consuming.

To solve this problem, the dissipative soliton mode-locking regime was proposed, where an intracavity bandpass filter is used to introduce the dissipative effect to the laser.^40,55 The double balance between the gain and loss as well as the nonlinearity and dispersion secures a singular stable mode-locking state that promises excellent long-term stability.⁵⁶ At the same time, the timing jitter performance can be improved with proper bandpass filter. As shown in Eq. (2), the indirect ASE coupled timing jitter originated from the central wavelength fluctuation is relevant to the spectral bandwidth. The finite gain bandwidth effectively reduces the indirectly coupled timing jitter. As the intracavity bandpass filter eventually causes an equivalent limitation of the optical spectrum and reduces the Gordon-Haus jitter, it can suppress the ASE indirectly coupled timing jitter to a large extent.

The effect of the bandpass filtering on the timing jitter performance of an ANDi mode-locked fiber laser is shown in Fig. 6. As shown by the measurements, the timing jitter of the mode-locked laser reduces roughly 10 dB by inserting a 7 nm bandwidth filter in the cavity.⁴⁰ When no filter is used, the timing jitter of the laser increases with the net intracavity dispersion. However, the timing jitter PSD barely changes at different cavity dispersion conditions when the bandpass filter is in place and it indicates that the bandpass filter can effectively eliminate the indirectly coupled ASE noise induced Gordon-Haus jitter. If the bandwidth is highly limited, then the optical pulse width would increase. This condition in turn leads to an increased directly coupled ASE timing jitter. To make a balance between the directly and indirectly coupled timing jitters, the bandwidth of the bandpass filter actually has an optimal value that can lead to the best timing jitter performance where the pulse width is short enough to resist the direct perturbation of the ASE noise and the pulse spectral bandwidth is also narrow enough to counteract the ASE-induced central frequency fluctuation.⁴¹ 

When the laser parameters are carefully designed, mode-locked fiber lasers have the potential to achieve ultralow quantum-noise limited timing jitter performance. Now, the record of 40 as timing jitter in the range of 10 kHz–10 MHz has been realized in an Er-doped mode-locked fiber laser²⁶ and it is very close to the short-term stability of the solid-state titanium-sapphire mode-locked laser. The low pulse train noise performance in the time-domain also facilitates the noise properties of the comb-line frequency noise. The lower ASE-dominated timing jitter corresponds to an intrinsic lower comb linewidth if the low-frequency technical noise is well suppressed. As a result, the low-noise mode-locked fiber lasers are highly attractive for metrological applications.

The distinct time-frequency features of passively mode-locked laser-based OFCs have been explored for high-precision length metrology applications.²⁸ In the time-domain, the cross-correlation between the retro-reflected laser pulses and LO laser pulses work as a timing gate with femtosecond resolution,^29,32,57,58 which enables a direct time-of-flight ranging. In the frequency domain, millions of phase-coherent discrete comb lines allow multi-wavelength interferometry with extended ranging ambiguity.^34,59–62 Remarkable achievements of nanometer precision-absolute ranging over arbitrary distance has been demonstrated, which is impossible from a conventional continuous wave (CW) laser-based optical range sensor.

The capability of optical frequency comb-based arbitrary distance ranging can be further advanced by a dual-comb implementation,^{30,31,63–67} which is equivalent to the linear optical sampling used in an optical telecommunication system. A slight repetition rate difference between the two combs allows a rapid update rate against multiple targets with an extended ambiguity range up to pulse train spacing. Coddington et al.³⁰ demonstrated 5 nm precision in a 1.5 m ambiguous range by taking advantage of dual-comb multi-wavelength interferometry. A simplified time-of-flight dual-comb cross-correlation configuration without delicate inter-laser carrier envelope phase lock has also been developed for industrial applications, where a sub-micrometer measurement precision is maintained.⁶³ The pulse timing is determined by the peak center of the cross-correlation trace. The envelope of the cross-correlation trace can be extracted by removing the detrimental unstable carrier interference via Hilbert transform algorithm and a subsequent Gaussian fit. Alternatively, an intensity cross-correlation is inherently immune to interference impact,³⁰ where only the Gaussian fit of the cross-correlation trace is required. In theory, the dual-comb ranging method can be adapted to kilometer range. On the other hand, balanced optical cross-correlator can be utilized as the time-of-flight detector, where the zero crossing of the BOC curve is treated as the timing of the signal and greatly simplifies the pulse timing retrieval algorithm.⁶⁵ 

High-precision dual-comb ranging has been demonstrated in a number of studies.^{30,31,63–67} Here, we developed a compact dual-comb absolute distance measurement system and demonstrated micrometer measurement accuracy over 70 m range through comparison with the commercial interferometer-based length standard. A Kalman filter-based post-processing algorithm^68,69 is used to improve the precision of the dual-comb ranging system instead of multiple averaging without scarifying the update rate. The working principle of the dual-comb absolute distance ranging system is based on the ASOPS method.^4,44,45 In principle, this is a time-of-flight ranging system. A signal laser (SL) is used to encode the distance between the target and the reference into time interval information, and then another LO is used to sample against the ultrafast optical pulses. The basic measurement principle of dual-comb distance measurement is shown in Fig. 7. An SL emits a femtosecond pulse train with a repetition rate of f_r. The pulse train is reflected from the target mirror and the reference mirror with distance d, introducing a delayed interval τ = 2d ⋅ n_g/c, where c is the speed of light and n_g is the air group refractive index. The reflected SL pulses are combined and subsequently sampled by a local pulse train with a slightly different repetition rate of f_r ‐ Δf_r through asynchronous optical sampling. The ASOPS process is achieved by optical intensity cross-correlation (XCOR) employing sum frequency generation in a type II periodically poled KTiOPO₄ (PPKTP) crystal to eliminate the influence of the carrier frequency interference. The natural scanning of LO pulses through the target and reference pulses generates a sampled pulse train. A full scan of the signal pulses is accomplished every T_update = 1/Δf_r and the effective sampling step is ΔT_r ≈ Δf_r/f_r². As a result, the ultrafast optical pulses are stretched to slower electrical pulses by a factor of N = f_r/Δf_r and the time interval of the sampled target and reference becomes T_d = N ⋅ τ. The distance between the target and reference mirror becomes

where m is an integer, T_r is the repetition period of the SL, and L_a accounts for the non-ambiguity range. With a typical 75 MHz repetition rate SL, the ambiguity range is approximately 2 m, which is easy to remove with a number of simple ranging methods.

The structure of the dual-comb absolute distance ranging system is depicted in Fig. 8. Four modules are shown, namely, pump module, SL module, LO module, and an optical XCOR module.

The pump module comprises two 976 nm pump diodes with PM fiber pigtails. Both diodes are divided by a 40:60 ratio coupler to pump the oscillator and amplifier of the SL and LO at the same time. The SL and LO are two all-polarization maintained Er-doped fiber laser systems. The oscillator is a typical ring oscillator with 0.8 m PM Er-doped fiber (Nufern PM-ESF-7/125) and carbon nanotube (CNT) mode locker. A hybrid component serving as isolator, wavelength division multiplexer, and 20% tap is used to minimize the length of the cavity. The repetition rates of the oscillators are approximately 76 MHz. A free space alignment is used in the SL cavity to finely tune the repetition rate difference to approximately 2 kHz. Both the SL and LO are free running, without repetition rate locking devices, thereby greatly reducing the system complexity. The output power of the two oscillators is approximately 1.5 mW at 60 mW pump power. The power is upscaled by an Er-doped fiber amplifier (EDFA) to 15 mW. The optical spectrum of the amplified pulses is shown in Fig. 9(a), with approximately 5 nm full-width-half-maximum spectral width, and 10% of the amplified light is coupled to monitor the repetition rate.

While in the LO module the amplified pulse is directly sent into the optical XCOR module, an optical circulator is used in the SL module to guide the measurement pulse to a beam expander. The output beam is then launched to the reference and target mirror. The reflected light is recollected and guided through the circulator into the optical XCOR module. The optical XCOR module consists of a polarized beam splitter (PBS), a PPKTP, an avalanche photo diode (APD) and a low-pass filter. The SL and LO are combined on the PBS and then focuses on the PPKTP to accomplish type-II cross-correlation. The signals received by the APD is filtered and then sent into the data sampling system, which consists of a 14 bit 100 MHz digitizer (National Instrument, PXIe-5122) and a frequency counter (Agilent, 53220A), all referenced to an Rb clock (Stanford Research System, FS725).

A frame of the sampled ASPOS signal is shown in Fig. 9(b). The sampled pulses are fitted with Gaussian functions and the pulse center is used as the pulse timing. The time interval between the first two adjacent pulses and the interval of two separated pulses are viewed as T_d and T_update, respectively. The distance is calculated in real time according to Eq. (3), while the repetition rate is updated with 100ms gate time by the frequency counter.

We test the ranging performance in an 80 m underground optical tunnel that belongs to the National Institute of Metrology of China (NIM). A retroreflector is used as the target to ensure precise back reflection. It is placed on a floating platform to maintain the stability of the laser measurement system. The platform is placed on an 80-m long granite rail system for continuous measurements. The true distance is provided by a commercial interferometer system calibrated by NIM. The output of the dual-comb system and the partial reflect mirror that serves as the reference mirror are placed on a fixed breadboard. The light path of the dual-comb ranging system is carefully adjusted to be parallel to the reference interferometer to minimize the cosine error. The alignment between the reference interferometer and dual-comb ranging laser is <10 mm along the 70 m range, corresponding to a cosine error <1 × 10⁻⁸ in relative. The environment parameter is updated in real time to compensate the air refraction index using Ciddor equation.⁷⁰ The average temperature is 15.63 °C measured by 40 sensors near the optical path, air pressure is 101.642 kPa, and humidity is 15.47%. The calculated group refractive index for center wavelength 1,561.4 nm is 1.00027467.

First, the ranging accuracy is tested by comparing the dual-comb laser ranger and the standard interferometer measurement values at a close range of 0–3 m with a step of 0.3 m. The measured results are shown in Fig. 10(a). The discrepancy of the measured distance by the dual-comb laser ranger and the standard interferometers is within ±0.5 μm along the 3 m range. The measurement is updated at 200 Hz based on Kalman filtering. Then, the long-range distance measurement performance is tested by moving the floating platform with a step of 4 m along the 70 m range. The experiment is conducted twice; each time, 500 consecutive measured data are used for statistics at discrete positions. The measured result is shown in Fig. 10(b). The measured distances exhibit good linearity (R² = 1) with the true distance. The difference between the dual-comb ranging system and the standard interferometer system is within the −6–3 μm range in both experiment processes, corresponding to an accuracy of 8 × 10⁻⁸ in relative.

The inherent uncertainty of the empirical equation is 2 × 10⁻⁸ and the uncertainty of the measured environmental parameters contribute to the total uncertainty. The air pressure measurement uncertainty for air pressure, humidity, and temperature is 7 Pa, ±1%, and 10 mK, respectively, corresponding to air group index uncertainty of 1.9 × 10⁻⁸, 1.33 × 10⁻⁸, and 9.6 × 10⁻⁹. The combined uncertainty of the estimated air group index is 5 × 10⁻⁸ (k = 2) by considering the environment parameter measurement uncertainty.

The performance of ASOPS-based high precision and high update rate time-of-flight measurement intrinsically relies on the short-term timing stability of femtosecond lasers. Specifically, the quantum-limited pulse train timing jitter on the time scale of milliseconds that characterizes an intrinsic “flywheel” stability of pulse train⁷¹ sets a fundamental limit on the obtainable dual-comb absolute ranging precision. Therefore, ultralow timing jitter mode-locked laser design is crucial for high-precision and high-speed dual-comb ranging applications.

The inherent relationship between timing jitter of mode-locked lasers and the dual-comb ranging precision has been described.⁷² As introduced in Section 2, the time magnification effect of the ASOPS process not only temporally stretches the optical pulses but also magnifies the pulse timing jitter at the same time. As a result, a timing fluctuation of the down-sampled pulses acquired by slow electronics inevitably leads to uncertainty in time-of-flight determination. This uncertainty is further converted to ranging uncertainty by

where σ₀ is the quantum noise-limited relative period timing jitter between the two femtosecond lasers used in the ranging system. As illustrated by Eq. (4), the timing jitter relevant ranging uncertainty varies against the distance to be measured. The ranging system reaches the highest precision in the vicinity of the integer non-ambiguity range (L_a) while it deteriorates in the middle of the L_a. The impact of f_r and Δf_r on ranging uncertainty is also not negligible in practice. Generally, pump laser noise can couple to f_r and Δf_r fluctuations via complex nonlinear intracavity pulse dynamics, thereby giving rise to dual-comb ranging uncertainty. This effect can be minimized by using low-noise telecommunication-class laser diodes and controllers. In addition, the two femtosecond lasers can be pumped by one laser diode whose power is split by a 50:50 fiber coupler. In this way, Δf_r fluctuation induced by the differential mode pump noise can be eliminated to a large extent.

We experimentally test the timing jitter effect on the ranging precision. The experiment setup is shown in Fig. 11. The SL is an NPE-based mode-locked Er-doped fiber laser and the used LO is a carbon nanotube-based all-fiber mode-locked laser with repetition rates approximately 100 MHz. The timing jitter of the LO is changed when the mode-locking regimes are switched from soliton to stretched-pulse regime by alternating the dispersion sign of the gain fiber and the dual-comb ranging precisions in both conditions are compared. The timing jitter of the LO at different working states is characterized by the BOC method in advance.

Fig. 12(a) presents the measured timing jitter PSD of the LO femtosecond laser working in distinct states. Consistent with the findings in Section 3, the timing jitter of the LO working in the soliton domain is approximately 10 dB higher than the stretched-pulse state and shows a −20 dB/decade slope for quantum noise limited in frequency range [20 kHz–1 MHz], while the technical noise dominates in the frequency range below 20 kHz. In the dual-comb ranging experiment, the repetition rate offset is set to roughly 2 kHz and the ranging precisions at various distances in one non-ambiguity range (0–1.5 m) is tested. Fig. 12(b) depicts the ranging precision. Overall, the stretched-pulse laser with lower timing jitter shows much better ranging precision. This experiment demonstrates the critical role of low jitter femtosecond laser on high-precision dual-comb range finders. We further investigate the ranging precision over the full non-ambiguity range. As shown in the inset of Fig. 12(b), the ranging precision characterizes the predicted trend of Eq. (4) that the ranging precision deteriorates in the middle of the non-ambiguity range and greatly improves in the vicinity of 0 and L_a, showing an arc-shaped dependence. In particular, when the distance is close to 0, the ranging precision can be fitted well by Eq. (4) with a relative period jitterσ₀estimated by the timing jitter PSD, which indicates that the ranging precision is solely determined by quantum-noise limited timing jitter. However, when the distance approaches half of L_a, the experiment results deviate from the theoretical prediction, indicating a technical noise effect domination.

For an in-depth investigation of the timing jitter on the dual-comb ranging precision, a numerical simulation of the dual-comb distance measurement considering various noise processes and repetition rate phase-locking conditions is conducted. In simulation, the SL contains no timing jitter, and we add random numbers that satisfy certain PSD distributions to the pulse positions of the LO pulse trains to simulate the effect of timing jitter. The ASOPS process is achieved with an intensity cross-correlation and the sampled pulse is Gaussian fitted to extract the LO pulse position. The distance is calculated according to Eq. (3). In Fig. 13(a), the timing jitter PSDs of the simulated LO pulse trains are provided including the quantum-noise with a –20 dB/decade slope feature (curve iv) and quantum-noise with extra technical noise below 5 kHz (curve v). To reproduce the absolute ranging scenarios in the presence of phase-locked loops, the timing jitter PSDs with repetition rate locking bandwidth of 5, 2, and 1 kHz are simulated, as shown respectively in curves i, ii, and iii in Fig. 13(a). Fig. 13(b) demonstrates the simulated ranging precision in the presence of the aforementioned noise processes and phase-locking conditions. The ranging precision over the full non-ambiguity range based on quantum-limited mode-locked lasers (case iv) effectively matches the theoretical predictions of Eq. (4). The additional low-frequency technical noise (case v) obviously degrades the ranging precision, especially at half of L_a. On the contrary, as the feedback loop for the repetition rate stabilization suppresses the low-frequency timing jitter, the ranging precision is improved accordingly. The feedback control improves ranging precision only when the phase-locking bandwidth is larger than the repetition rate offset between the LO and SL and the larger locking bandwidth warrants improved ranging performance.

As we have found from the theory prediction, experiment, and numerical simulation, to further improve the ranging precision of a dual-comb absolute distance measurement system, on the one hand, the mode-locked laser should be carefully designed to optimize the quantum-noise limited timing jitter. On the other hand, a large bandwidth phase-locking loop can be adopted to suppress the low-frequency technical noise and quantum-noise so that the ranging precision in the middle of the non-ambiguity range can be improved. In addition, Eq. (4) shows that the larger repetition rate offset also facilitates the ranging precision at arbitrary distance.

Fig. 13 shows that active phase-locked loops with large locking bandwidth promises an improved dual-comb ranging precision. To date, the repetition rate locking of femtosecond fiber lasers can achieve as high as 2 MHz feedback bandwidth, enabled by phase-locking cavity-stabilized CW fiber lasers to an intracavity electro-optic modulator.⁷³ Ultralow jitter RF signal extraction from passively mode-locked lasers has been demonstrated based on this approach. However, the tight phase-locking technologies have not been widely used in dual-comb absolute ranging applications due to their complicated setups and sophisticated phase-locked loop designs. Recently, a growing trend is to conduct dual-comb applications based on a single free running dual-comb mode-locked laser, which offers a much simplified configuration. A pair of pulse trains with an offset repetition rate oscillate in the same laser cavity and are expected to suffer common mode fluctuations due to the shared cavity. As a result, the significantly reduced relative timing jitter promises ultrahigh precision dual-comb ranging without any active phase-locked loops. To implement this technical approach, a dual-comb mode-locked laser-based on dual-wavelength output has been used as the laser source.^74,75 The laser system is illustrated in Fig. 14(a). The dual-comb operation relies on the equivalent Lyot filter effect induced by a section of polarization maintaining fiber (PMF) and in a line polarizer. Mode-locking is realized by a carbon nanotube absorber. The birefringence provided by the PMF is translated into a wavelength-dependent transmission filter when the light is polarized by the polarizer.⁷⁶ Balanced by the gain dispersion, the laser can achieve mode-locking at two wavelengths of 1531 nm and 1555 nm simultaneously in a single laser cavity, as shown in Fig. 14(b). Moreover, the group velocity dispersion introduces a repetition rate offset, which is approximately 2.13 kHz, between the pulse trains of different wavelengths, as resolved by two peaks on the RF spectrum shown in Fig. 14(c), representing the distinct repetition rates of approximately 50.64 MHz.

The common laser cavity provides a high common mode rejection ratio. In the experiment, the common noise suppression effect is validated by the long-term stability of the repetition rate offset. In half an hour, the repetition rate difference merely drifts about 2 Hz while the repetition rates drift >50 Hz. In addition, the STD of the offset rate in a minute is 30 mHz, reflecting a well suppressed environmental noise distribution. To test whether the relative timing jitter is eliminated by the shared cavity effectively, the relative timing jitter between the pulse trains at different wavelengths is characterized using the time-domain timing jitter measurement method based on ASOPS. The measured visual timing jitter versus observation time window T_p is depicted in Fig. 15; the results satisfy the theoretical prediction of σ_visual ∝ T^1/2, as indicated by the red line. The result indicates a quantum noise limited timing jitter and the pulse trains undergo a relative random walk process. The estimated relative timing jitter between the two pulse trains is approximately 0.82 fs ± 25 as. This means that the shared cavity effectively suppresses the technical noise, but the noise canceling effect does not apply to quantum-noise. As a result, a quantum-noise limited ranging precision is expected in dual-comb ranging experiments based on a single dual-comb dual-wavelength fiber laser.

We experimentally tested the ranging performance utilizing the dual-comb mode-locked laser as laser source,⁷⁴ and the experiment setup is shown in Fig. 16. The dual-wavelength pulse trains are separated by a coarse wavelength division multiplexer and separately amplified with an EDFA. The 1555 nm part is used as the SL and the 1531 nm is adopted as the LO. The rest of the setup resembles the concrete dual-comb ranging setup. The dual-comb ranging is conducted along a non-ambiguity range of approximately 3 m to verify the ranging precision.

Fig. 17 shows the experimental results of the dual-comb ranging when the dual-wavelength mode-locked fiber laser is used as light source.⁷⁴ At a fixed distance of 1.56 m, the measured results at 2.13 kHz update rates shows a deviation of approximately 11 μm while it can be improved to 1 μm within 100 ms averaging time, as shown by the Allan deviation in Fig. 17(a). In addition, the ranging precision at the highest update rate is evaluated at 48 positions along the entire 2.96 m non-ambiguity range, as indicated in Fig. 17(b). The experiment results show consistency with the theoretical prediction that only suffer from quantum-noise limited timing jitter. This condition verifies that the effect of technical noise has been entirely eliminated by the shared cavity and the ranging uncertainty is solely determined by the quantum-noise limited timing jitter. Recently, we have observed that the two-color pulse trains oscillating in the dual-wavelength mode-locked laser occasionally collide with one another⁷⁷ and the pulse interaction has an impact on dual-comb spectroscopic applications.⁷⁸ Nevertheless, the absolute ranging precision degradation due to pulse interactions is negligible, as proven in Fig. 17. Compared with the conventional dual-comb ranging setups that rely on two phase-locked lasers, a similar dual-comb ranging performance is achieved here with a single free running dual-wavelength Er-fiber laser, which is highly promising for practical applications of dual-comb absolute ranging technology.

We reviewed the recent progress in ultralow timing jitter mode-locked fiber laser technology and the applications in dual-comb absolute ranging. As a new interdisciplinary research field, femtosecond-laser-based absolute ranging has attracted great attention in recent years. Despite the explosive growth in the innovative measurement principles and experimental demonstrations, the effect of femtosecond laser sources, particularly in terms of noise characteristics, on the absolute ranging performance has not been fully considered. In this review, we concentrate on the timing jitter characteristics of mode-locked laser sources and their impact on absolute ranging. We show that the advances in accurate and sub-femtosecond precision timing jitter characterization method offers great opportunities for the ultralow timing jitter mode-locked fiber laser design. By manipulating the intracavity pulse dynamics, the quantum-limited timing jitter of mode-locked fiber lasers has been reduced to tens of attoseconds such that the performance already matches that of the bulk solid-state lasers. The ultralow timing jitter mode-locked fiber laser technology is promising for its high-precision metrological applications, particularly in terms of absolute time-of-flight measurement. We demonstrate that the timing jitter sets a fundamental limitation on the ranging precision of the dual-comb absolute distance measurement systems. As light source for any femtosecond laser absolute ranging system, advancements in ultralow-noise femtosecond laser sources are expected to continue promoting its absolute ranging performance in the future.

This work was supported by National Natural Science Foundation of China (Grant Nos. 61475162, 61675150, and 61535009), Tianjin Natural Science Foundation (Grant No. 18JCYBJC16900), and Tianjin Research Program of Application Foundation and Advanced Technology (Grant No. 17JCJQJC43500).