The cross correlation between a pair of femtosecond lasers with slightly different repetition rates enables high precision, high update rate time-of-flight (TOF) distance measurements against multiple targets. Here, we investigate the obtainable ranging precision set by the timing jitter from femtosecond lasers. An analytical model governing dual femtosecond laser TOF distance measurement in the presence of pulse train timing jitter is built at first. A numerical study is conducted by involving typical timing jitter sources in femtosecond lasers in the following. Finally, the analytical and numerical models are verified by a TOF ranging experiment using a pair of free running femtosecond Er-fiber lasers. The timing jitter of the lasers is also characterized by an attosecond resolution balanced optical cross correlation method. The comparison between experiment and numerical model shows that the quantum-limited timing jitter of femtosecond lasers sets a fundamental limit on the performance of dual femtosecond laser TOF distance measurements.
© 2015 Optical Society of America
Time-of-flight (TOF) of pulsed lasers allows for non-contact absolute distance measurements at long ranges, enabling a number of applications, such as altimeter, terrain mapping and surface profilometry of large-scale structures [1, 2]. The precision of TOF distance measurements can be significantly improved by using passively mode-locked lasers, which emit a uniformly spaced pulse train with typical pulse duration down to tens of femtoseconds [3–5]. In 2000, K. Minoshima et al.  utilized a set of radio frequency (RF) harmonics encoded in the femtosecond laser pulse train for an absolute distance measurement in a 240 m tunnel. Alternatively, the optical cross correlation between the target-reflected pulses and reference pulses can function as a timing gate providing sub-femtosecond temporal resolution [4, 5]. In addition, a transition from incoherent, TOF measurement to coherent, fringe-resolved interferometry can be achieved by taking advantage of a phase-stabilized optical frequency comb  from a passively mode-locked femtosecond laser, resulting in wavelength resolution over arbitrary range [7–10].
The capability of femtosecond laser based distance measurements can be further advanced by an asynchronous optical sampling (ASOPS) implementation [11–15], where a slight repetition rate difference between the two femtosecond lasers allows for rapid update rate against multiple targets with an extended ambiguity range up to pulse train spacing. I. Coddington, et al. have demonstrated 5 nm precision in an 1.5 m ambiguous range when a pair of tightly phase locked optical frequency combs are employed . Note that the unprecedented performance relies on tight phase locking of the comb teeth to a common optical standard, and the nanometer precision is achieved by coherent averaging [12, 16]. For the sake of industrial applications, where a simple setup and high speed measurement are always preferred, the sophisticated phase-locking between the two optical combs is not required because the ranging precision at high update rate is only determined by the TOF measurement using pulse envelope [12, 13]. Several groups [13–15] have demonstrated micrometer precision TOF ranging at several kilohertz update rate using a pair of incoherent femtosecond lasers.
The performance of ASOPS based high speed ranging applications, say, at ~kHz update rate, is determined by short term stability of femtosecond lasers on the time scale of ~millisecond. In this study, we will show that, quantum-limited pulse train timing jitter of femtosecond lasers, that characterizes a intrinsic “flywheel” stability of pulse train , sets a fundamental limit on precision obtainable from ASOPS based TOF ranging. A number of principles emerge for advancing the design of ASOPS based TOF ranging system towards improved ranging precision.
The content of this paper is outlined as follows.
Firstly, a theoretical model governing ASOPS based TOF distance measurement in the presence of pulse train timing jitter is developed.
Secondly, a numerical simulation is conducted so as to visualize the theoretical model. Numerical simulation of ASOPS based TOF distance measurement is performed by making use of several typical probability distributions of pulse train timing jitter. The impact of timing jitter on ranging performance is evaluated from numerical simulation results.
Finally, the numerical simulation is verified by a TOF ranging experiment based on a pair of free running Er-fiber lasers. As a prerequisite, the pulse train timing jitter should be precisely characterized. In femtosecond lasers, the concentration of a large number of photons in an extremely short pulse duration makes pulse position robust against perturbations. The quantum limited timing jitter is on the level of ~1 fs . To make a valid measurement, a balanced cross-correlation technique [19, 20] that provides attosecond time resolution is employed for timing jitter characterization. In the following ranging experiments, distance measurement error mapped by the tiny quantum-limited timing jitter is observed. The experiment result matches with simulation very well.
2. ASOPS based TOF ranging principle in the presence of timing jitter
The ASOPS based TOF ranging principle is illustrated in Fig. 1, see the pulses plotted in solid lines. A signal laser emits a femtosecond pulse train with a repetition rate of (for instance, 100 MHz). The pulse train is directed to an unbalanced optical-path Michelson interferometer. The Michelson interferometer is composed of a beam splitter, a fixed reference mirror, and a target mirror. The absolute distance is measured as the differential TOF between the pulse reflections from the reference mirror and the target mirror,
Alternatively, the magnification factor can be replaced with directly measurable time based variables, then,
ASOPS based TOF ranging takes advantage of the fact that the signal pulses are intrinsically repetitive, and a slightly different part of the signal can be sampled every repetition period. In real world, the output pulses emitted from femtosecond lasers are not rigorously spaced due to amplified spontaneous emission (ASE) noise in gain medium, laser cavity mirror vibrations, pump laser noise, etc. Considering that the time interval between LO and signal laser pulses is subject to a tiny variation , the sampled pulse train may read a magnified timing variation of . The behavior of ASOPS based TOF ranging in the presence of LO timing jitter is illustrated as dashed lines in Fig. 1. The TOF error induced by pulse train timing jitter is expressed as,
Equation (5) indicates that, the accumulated timing error of LO within one ranging acquisition period of will convert to a TOF error. As an exception, at a target distance of and , we have and , respectively, and the ranging error is . A coarse conclusion can be drawn that timing jitter has negligible influence on the ranging performance in the vicinity of zero distance and ambiguity range. However, significant ranging errors could be found at middle distances, between and .
3. Numerical simulations
In order to reveal the intrinsic relationship between ranging precision and pulse train timing jitter, a numerical simulation of ASOPS based TOF distance measurement in the presence of signal laser and LO timing jitter is conducted.
3.1 Numerical model
A signal (LO) pulse train that is subject to timing jitter is modeled at first. The individual laser pulse is Gaussian shaped, and the pulse duration (half-width at 1/e-intensity point) is 600 fs. The temporal position for each pulse is defined by the pulse intensity maximum. For a regularly spaced pulse train emitted by a hypothetical laser without any noise inputs, the temporal position for the th pulse is equal to , which is labeled as a reference position, where , represents repetition period. In real lasers, a random timing error with a particular noise distribution will be introduced to the output laser pulse position after each repetition period as a joint result of various noises. The timing jitter in the th pulse, which is the deviation of the temporal position of the pulse from its reference position, can be expressed as,
Once the timing position series are created, the signal laser may be sampled by the LO. In this study, only pulse envelope sampling is required since the pulse timing is determined by pulse intensity maximum. The optical carrier is not important here. For this reason, a nonlinear ASOPS approach is employed , which is based on intensity cross-correlation between signal laser and LO and a subsequent square law photo-detection. One sample voltage will be produced during each sample period , with the voltage being proportional to the temporal overlap between the LO and the delayed signal pulses. For the th sample period, the detected voltage is given by,Eqs. (1) and (4). The same ranging simulations are conducted repeatedly, and the ranging precision is calculated as the standard deviation of the sequential ranges.
3.2 Numerical simulation results
Ranging precision at different target distance for different timing jitter probability distribution is studied when the numerical model is in place. Various noise sources may contribute to pulse train timing jitter in femtosecond lasers. These noise sources are divided into two categories of technical noise and quantum noise. Technical noise sources, such as mirror vibration, temperature variation, power supply instability, etc, feature a limited bandwidth. Their influence on pulse timing could be well corrected by a servo loop of some kHz bandwidth. Quantum noise, dominated by ASE noise, however, is white noise in nature, which causes a random walk of pulse train timing jitter. An instantaneous compensation is required to fully remove this quantum timing jitter, which will set a fundamental limit for the precision of TOF acquisition in ASOPS based TOF ranging experiments.
A simulation for distance measurement based on femtosecond lasers with quantum-limited timing jitter is conducted at the beginning. The quantum noise source is functioned by a computer pseudorandom number generator, which produces a sequence of random timing errors with a specific variance of following standard normal distribution. The pulse train timing jitter can thus be calculated according to Eq. (6). The resulted timing jitter can be quantified by its single-sided power spectral density (PSD), which is obtained following the numerical method in . Note that, the quantum noise limited timing jitter PSD of free running femtosecond lasers can be routinely characterized with high precision by utilizing balanced cross-correlation methods. In our simulation work, a specific value of noise variance is selected so that the simulated timing jitter PSD is comparable with that from practical femtosecond laser sources.
Considering two identically made free running signal laser and LO, their simulated timing jitter PSD is shown as black curve in Fig. 2(a). The PSD falls off at a −20 dB/decade, which indicates a random walk nature of quantum-limited timing jitter. The jitter level is same with that from a standard stretched pulse Er-fiber laser characterized in . At a repetition rate difference of 2 kHz between the two femtosecond lasers, a ranging simulation is conducted with an update time of 0.5 ms. The black curve in Fig. 2(b) shows the calculated ranging precision within the 0.5 ms acquisition period at target mirror positions up to the ambiguity range of ~1.5 m. Interestingly, there is an arc-shaped correlation between ranging precision and target mirror position. The ranging performance at a distance of and is much better than that close to half of the ambiguity range. This observation visualizes the ranging behavior drawn from Eq. (5). Similar phenomenon was also observed from ranging experiments by using two free running femtosecond lasers . Here, we know that this behavior is closely related with pulse train timing jitter.
Technical noise is added to quantum noise in practical femtosecond lasers, especially at low Fourier frequencies. Without loss of generality, a technical noise is added to the laser so that the timing jitter PSD characterizes a −30 dB/decade slope below a Fourier frequency of 5 kHz, and the high frequency is still quantum-noise dominated, as shown in the blue curve of Fig. 2(a). The simulated ranging result is shown as the blue curve in Fig. 2(b). Obviously, the ranging error is significantly increased in the middle of ambiguity range due to technical noise. However, the ranging performance in the vicinity of zero distance and ambiguity range is not affected by technical noise at all.
The timing jitter, both from technical noise sources and quantum noise sources, can be corrected to some extend by locking the repetition rates of femtosecond lasers to an external RF or optical frequency standard. For simplicity, only quantum noise is considered. Under a locking bandwidth of 2 kHz equal to ranging update rate, the resulted timing jitter PSD is shown in the magenta curve in Fig. 2(a), with a residual timing jitter of 11 fs RMS integrated from DC to Nyquist frequency (ie, 50 MHz). Interestingly, there is only minor improvement for ranging performance in comparison with results from free running femtosecond lasers, as shown in magenta curve of Fig. 2(b). The reason is that ranging performance is highly related with the short term stability of pulse timing. The timing compensator could marginally follow, and thus correct the temporal error accumulated within the 0.5 ms ranging acquisition period. When locking bandwidth is increased to 5 kHz, the resulted timing jitter PSD is shown in the red curve in Fig. 2(a), with a residual timing jitter of 7 fs RMS integrated from DC to Nyquist frequency (ie, 50 MHz). The corresponding ranging performance is shown as red curve in Fig. 2(b), which reveals a one-fold reduction of measurement deviation around half ambiguity range. However, the residual timing jitter at high Fourier frequencies still sets a fundamental limit for ranging precision of ~1.3 μm.
The ranging precision at different update rates is also studied. The simulation result is shown in Fig. 3. With other ranging parameters fixed, ranging precision will improve with the increase of update rate. This behavior is highly related with the random walk nature of timing jitter. See Eq. (5), the timing jitter of the sampled target (reference) pulse reflects the accumulated timing errors from successive laser repetition periods within one ranging acquisition period. This accumulated timing error can be reduced by decreasing ranging acquisition time, or by increasing ranging update rate equivalently. As a result, ranging precision can be improved.
Finally, a ranging simulation for distances longer than one ambiguity range is conducted. The ranging performance over 10 ambiguity ranges under a 2 kHz update rate is shown in Fig. 4, where the timing jitter distributions in Fig. 2(a) are used. The ranging performance does not degrade over longer distances, regardless of noise type. An intuitive explanation is proposed as follows. For a target distance beyond one ambiguity range, say times of the ambiguity range, the interval between subsequent target and reference pulses will thus be subject to an additional timing jitter of . Then, the timing jitter between successive target and reference pulses in Eq. (5) can be generalized as,
Considering a target distance of m, we have. However, since , we have . Apparently, , the contribution from multiple ambiguity ranges is negligible. In case that ranging at tens of kilometers is required, a low bandwidth repetition rate locking will be necessary to remove the timing error accumulated between subsequent target and reference pulses.
A number of important facts can be drawn from the above simulations.
Most importantly, timing jitter of femtosecond lasers sets a fundamental limit on the precision of ASOPS based TOF distance measurement. Note that this fundamental ranging precision refers in particular to the standard deviation of the distance acquired at the ranging update rate. Considering a quantum-limited signal laser (LO) timing jitter, this ranging precision can always be averaged down by increase of integration time. However, ranging cannot be acquired in real time at a cost.
It will be interesting to discuss how to beat the fundamental limit. A natural way is to lock the laser repetition rates to a common external frequency standard. We have already shown that the locking bandwidth should be much larger than ranging update rate so as to observe an obviously improved ranging performance. This actually sets a critical requirement for the short term stability of frequency standards. The short term (< 1 ms) stability of standard RF frequency standards is much worse than that of femtosecond lasers. Therefore, RF frequency standards are not suitable for improving ranging precision because they do not allow for phase correction for >1 kHz locking bandwidth. Then, only optical standard can be used . However, this demanding approach only works in a few metrology labs around the world.
Other than repetition rate locking, a practical method is to choose free running femtosecond lasers with lower quantum-limited timing jitter. Timing jitter of femtosecond lasers from different mode-locking regimes has been intensively studied [24, 25]. A stretched pulse laser outperforms other lasers regarding timing jitter. As a result, stretched pulse lasers are expected to demonstrate a better ranging performance.
Besides, one can also increase ranging update rate so as to improve ranging precision. However, when ranging update rate is doubled, the effective samples reduce by half. For the sake of adequate sampling, lasers with longer pulse duration are preferred. However, longer pulse duration will, in turn, limit time resolution in TOF based ranging experiments. One should carefully find a balance between the pulse duration and update rate in ASOPS based ranging experiment design.
The theory and simulation are verified by experiments. An absolute distance measurement experiment is launched by using a pair of home-made femtosecond Er-fiber lasers. Their timing jitter is characterized at first. Then, a ranging experiment is conducted over an entire ambiguity range. The ranging performance by using a stretched pulse fiber laser and a soliton fiber laser as LO is also compared. A numerical simulation is conducted using practical parameters from the ASOPS based TOF ranging experiment. Quantum noise source is modeled so that the jitter PSD of the computer generated pulse train matches that from timing jitter characterization experiment. The simulated ranging performance is compared with experiments.
4.1 Experimental setup
The design for the femtosecond lasers is shown in Fig. 5(a). Their repetition rates are ~100 MHz. The signal laser is a nonlinear polarization evolution (NPE) mode-locked Er-fiber laser based on a sigma cavity design. The net cavity dispersion is close to zero so that the laser works in a stretched pulse regime. An 8 nm bandpass filter centered at 1550 nm is intentionally inserted before the laser output in order to limit the output laser bandwidth, so that aliasing in sampling could be prevented in ranging acquisition stage. The average output power is 40 mW @ 580 mW pump. The LO laser features an all fiber configuration using carbon nanotube (CNT) as saturable absorber. The average output power is 2 mW @100 mW pump. The output power is scaled to 20 mW by an EDFA. In particular, the mode-locking regime of LO laser can be switched by using different types of Er-fibers. When a segment of 52 cm Er-gain fiber (Liekki Er80-8/125) with an anomalous dispersion of ~-20 fs2/mm is used as gain medium, the laser features an all anomalous intra-cavity dispersion and works in a soliton regime. Alternatively, a segment of 52 cm Er-gain fiber (Liekki Er110-4/125) with a normal dispersion of ~ + 12 fs2/mm can be used, which properly balances the negative dispersion from other fiber components in the laser. In this way, the laser will operate in a stretched pulse regime.
The timing jitter of the two lasers is characterized when the lasers are in place. The experimental setup for timing jitter characterization is shown in Fig. 5(b). A periodically poled KTiOPO4 (PPKTP) based single crystal balanced optical correlation technique  is used. The output of BOC is shown in the inset of Fig. 5(b). The linear range of the BOC signal provides a timing error discriminator with sub-femtosecond resolution. To confine the jitter measurement in this range, the repetition rates of two lasers are locked by a low-bandwidth phase-locked loop (PLL) that uses a piezoelectric transducer (PZT) mounted end mirror in the signal laser as actuator. The BOC signal is received by a radio-frequency (RF) spectral analyzer, which characterizes the sum of timing jitter PSD of the signal laser and LO.
Following the timing jitter characterization, an ASOPS based TOF absolute distance measurement experiment is conducted using the same lasers. The system configuration is shown in Fig. 6. The setup is based on an un-balanced Michelson interferometer structure with one reference arm and one target arm. The reference mirror is fixed while the target mirror is installed on a high precision linear translation platform. A telescope is used to expand the laser beam and to guide the light to the target mirror. The reflected reference pulses and target pulses are combined with the LO pulses by a polarization beam splitter (PBS). Subsequently, the combined beams are directed into a PPKTP based optical intensity cross-correlator, where the reference pulses and the target pulses are sampled against the LO pulses in a manner that is governed by the principle proposed in Section 3.1. The repetition frequency difference of the two free running lasers is tuned to ~2 kHz. The effective time step is ~200 fs and each cross-correlation trace contains roughly 20 points. The cross-correlation traces are detected by a high speed low noise avalanche photo-detector (Thorlabs, APD120A). The output voltage signals are digitized and stored by a 14 bit 100 MHz digitizer (National Instrument, PXIe-5122) by using the LO repetition rate as external sampling clock. The pulse time-of-flight is calculated by Eq. (4), where the refractive index of the air is calculated from Edlen equation, the timing of the reference and target retro-reflected pulses is determined by the peaks of the Gaussian fitted cross-correlation signals, and the repetition rate of the signal laser is measured by a 12 bit frequency counter referenced to a Rb atom clock using 1 s gate time. During this gate time, the measured repetition rate uncertainty is ~10−9 for both lasers. As a result, the distance measurement uncertainty introduced by repetition rate instability is negligible.
4.2 Experimental results
The results of timing jitter characterization and the subsequent ASOPS based ranging experiment are shown as follows.
In the beginning, experiments are conducted when the CNT mode-locked LO is working at stretched pulse regime, and the results are shown in Fig. 7. Figure 7(a) shows the sum timing jitter PSD of the signal laser and the LO characterized by BOC technique. The influence of low bandwidth PLL on the pulse train timing jitter extends up to 20 kHz. Outside this bandwidth, there will be trivial correlation between the two lasers imposed by the PLL, and the measurement reflects the intrinsic short term stability of pulse train from free running lasers. The measured timing jitter PSD features a −20 dB/decade roll off beyond 20 kHz offset frequency, meaning that the pulse timing undergoes a random walk originated from quantum noise. The ranging results based on the present lasers are shown in Fig. 7(b). A full scan over one ambiguity range is conducted. The measurement standard deviations show an arc-shaped correlation with target distances, in accordance with the numerical simulation results in Section 3.2. Considering that the time domain signal to noise ratio for the sampled cross-correlation trace (defined as the cross-correlation trace peak height relative to the baseline scatter) is above 40 dB, the timing noise introduced by the sampling process is negligible. As a result, the ranging precision is identified as pulse train timing jitter limited.
The ranging experiment is compared with numerical simulation. The signal laser and LO in the numerical model share the identical timing jitter PSD, the sum of which is shown as the gray curve in Fig. 7(a). A quantum-limited timing jitter is considered in the simulation, so that the simulated timing jitter PSD only fits with the experimentally characterized timing jitter PSD at high Fourier frequencies above 20 kHz. The simulated quantum-limited distance measurement standard deviations at different target-reference separations are shown as gray curves in Fig. 7(b). The simulated ranging precision at short target distances (< 8 cm) matches very well with the experiment. A deviation from quantum-limited ranging precision starts to grow at larger distances, and reaches a maximum at half of the ambiguity range, as shown in the inset of Fig. 7(b). This behavior indicates that, the timing jitter PSD in practical lasers deviates from quantum limited domination at some point below 20 kHz Fourier frequency due to the impact from lower bandwidth technical noises.
The same experiment is conducted by switching the LO to a soliton mode-locking regime. The experimental results and the comparison with numerical model are shown in Fig. 8, which characterizes a similar behavior as Fig. 7. In particular, the experimental results from Fig. 7 and Fig. 8 are re-plotted in Fig. 9 for comparison. The characterized quantum-limited timing jitter PSD of the stretched pulse mode-locked LO is ~10 dB lower than that of the soliton mode-locked LO. This is because stretched pulse laser is immune of Gordon-Haus jitter , which couples ASE noise to pulse timing through cavity dispersion. The following ranging experiments show that, better ranging performance over the entire ambiguity range can be achieved when a stretched pulse laser that features lower quantum-limited timing jitter is used as LO. This observation is also in accordance with the prediction induced by numerical simulation in Section 3.
To summarize, we systematically study the effect of pulse train timing jitter on real-time ASOPS based TOF measurements. An analytical model governing ASOPS based TOF measurement in the presence of timing jitter of femtosecond lasers is built. To visualize the analytical model, a numerical simulation is conducted by involving various timing jitter noise distributions. The analytical and numerical model are verified by a carefully designed ASOPS based TOF ranging experiment, where the quantum-limited timing jitter of the lasers sources for ranging is characterized with sub-femtosecond precision in advance. A number of important facts can be drawn from this study.
Most importantly, the intrinsic 'flywheel' stability of pulse train emitted by femtosecond lasers sets a fundamental limit on the precision of real-time ASOPS based distance measurement.
Secondly, there is an arc-shaped correlation between ranging precision and target mirror position over one ambiguity range. It implies that the ranging performance will be subject to the least influence from timing jitter when the target is placed at the multiples of ambiguity range.
Thirdly, a practical method for improving TOF precision is to choose femtosecond lasers with lower quantum-limited timing jitter, such as stretched pulse lasers.
It is also noteworthy that the principles drawn from this study of timing jitter limited distance measurement using two femtosecond lasers are transportable to coherent dual-comb interferometry. The reason is that timing jitter brings comb line spacing fluctuations, thus broadens individual comb linewidth. This will set a fundamental limit on spectral resolution of real-time dual-comb Fourier transform spectroscopy.
This research is supported by the National High Technology Research and Development Program of China (Grant 2013AA122602), the National Basic Research Program of China (Grant 2011CB808101 and 2010CB327604), the National Natural Science Foundation of China (Grant 61205131, 11274239, 61227010 and 61322502), Tianjin Research Program of Application Foundation and Advanced Technology (Grant 13JCQNJC01400), and Program for Changjiang Scholars and Innovative Research Team in University (Grant IRT13033).
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