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Abstract
Long-distance ranging is a crucial tool for both industrial and scientific applications. Laser-based distance metrology offers unprecedented precision making it the ideal approach for many deployments. In particular, dual-comb ranging is favorable due to its inherently high precision and sampling rate. To make high-performance long-range dual-comb LiDAR more accessible by reducing both cost and complexity, here we demonstrate a fiber-based dual-comb LiDAR frontend combined with a free-running diode-pumped solid-state dual-comb laser that allows for sub-µm measurement precision while offering a theoretical ambiguity range of more than 200 km. Our system simultaneously measures distance with the role of each comb interchanged, thereby enabling Vernier-based determination of the number of ambiguity ranges. As a proof-of-principle experiment, we measure the distance to a moving target over more than 10 m with sub-µm precision and high update rate, corresponding to a relative precision of 10−7. For a static target at a similar distance, we achieve an instantaneous precision of 0.29 µm with an update time of 1.50 ms. With a longer averaging time of 200 ms, we reach a precision of around 33 nm, which corresponds to a relative precision of about 3·10−9 with a time-of-flight-based approach.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Laser-based distance metrology can offer long range, fast acquisition and high precision [1,2]. Those are key parameters for any ranging system, which makes laser-based light detection and ranging (LiDAR) attractive for scientific and industrial applications such as autonomous driving, drone navigation, industrial process monitoring, analysis of various environments, or satellite formation flying.
Interferometric distance measurement systems based on homodyne or heterodyne detection exhibit high precision [3]. In its traditional form, a continuous wave (cw) laser is used together with a Michelson interferometer to determine the difference in length between the reference and the target arm from the interferometric phase difference. This allows for sub-wavelength measurement resolution, but simultaneously it exhibits a sub-wavelength ambiguity range. Advanced interferometric measurements based on heterodyne detection with multiple wavelengths support an increased (but still limited) ambiguity range [4–9]. Thus, for an interferometric measurement it is challenging to achieve long-range absolute distance measurements as required for many practical applications.
To overcome this limitation, the interferometric measurements can be replaced or combined with time-of-flight-based measurements which rely on modulated or pulsed light sources. The distance is inferred from the optical time delay between a pulsed or modulated waveform reflected from a target and a reference plane [10,11]. Typically, time-of-flight-based methods exhibit lower precision than interferometric approaches. Combining the two measurement principles thus brings together long-distance and high-precision ranging which is desirable for many practical applications. This combination is naturally enabled by optical frequency combs consisting of highly-stable equidistant modes in the frequency domain, perfectly suited for interferometric measurements, and a periodic pulse train in the time domain as required for time-of-flight-based distance measurements.
Since the frequency comb revolution in the late 1990s [12–14], combs have become a key technology for metrology [10,15]. They have enabled great progress in many fields such as spectroscopy or ranging [16–18]. There exist different approaches for measuring the distance with combs such as optical cross-correlation [19], synthetic wavelength interferometry [20–22], multiple wavelengths referenced to a stabilized comb [23,24], or dual-comb ranging [25].
Dual-comb ranging relies on the combination of two optical frequency combs with slightly different repetition rates. The repetition rate difference $\Delta f_{\text {rep}}$ determines the update rate, while the repetition rate $f_{\text {rep}}$ of the comb used for remote sensing determines the ambiguity range υgroup/($2f_{\text {rep}}$) of the measurement, where υgroup is the group velocity at the carrier frequency. Given pulse repetition rates from about 100 MHz to a few GHz, the ambiguity range is typically on the order of a few meters to a few centimeters. However, many ranging applications require measurements over much longer distances, so it is necessary to determine the number of ambiguity ranges to the target. This step can be accomplished by interchanging the roles of the two combs in the measurement, which allows exploiting the Vernier effect to extend the ambiguity range to a theoretical limit of υgroup/($2\Delta f_{\text {rep}}$) [26]. In addition to this approach, there exist also other techniques for extending the ambiguity range by leveraging the Vernier effect such as adding an extra measurement path [27] adjusting the repetition rate [28–30], or by introducing cross-talk between the two combs as illustrated with a fiber-based dual-comb [31]. Consequently, dual-comb ranging allows for long ambiguity-free measurement distances while simultaneously offering high precision and fast acquisition. Moreover, it does not require any mechanically moving parts for scanning the delay, which simplifies the measurement significantly.
The underlying measurement principle of dual-comb ranging was first introduced in 2009 with two stabilized fiber lasers [26]. However, the high complexity of stabilized dual-comb systems makes it challenging to use this technique in industrial applications. It was thus an important step for dual-comb ranging when the measurement principle was demonstrated with free-running fiber lasers [32], including fiber-based single-cavity dual-comb lasers [31,33,34], and microresonator-based frequency combs [35,36]. More recently, dual-comb ranging was reported with a single-cavity dual-comb solid-state laser [37], but this experiment was limited to a single 1.1-meter ambiguity range and used a free-space setup. Solid-state lasers bring the advantages of ultra-low intensity and timing noise properties [38,39], which makes recently developed dual-comb solid-state lasers appealing for ranging applications [40–43]. Since the measurement precision of time-of-flight-based dual-comb ranging is mainly determined by the high-frequency timing noise above $\Delta f_{\text{rep}}$ [44], solid-state lasers are ideal for high-performance ranging applications. Moreover, they can achieve high average power without any amplifiers and high repetition rates, which is valuable for fast data acquisition and system simplicity. For example, the advantage of high repetition rates has been shown with microresonator-based dual-comb ranging systems [35,36]. Solid-state oscillators can bridge the gap between traditional fiber lasers (operating in the 100-megahertz range) and soliton microcombs (operating at many gigahertz). However, high repetition rates imply a very limited ambiguity range.
Here, we explore ambiguity range extension for long-distance ranging with a free-running dual-comb laser. For the demonstration, we use a single-cavity dual-comb solid-state laser oscillator delivering two nearly identical pulse trains with a center wavelength of 1056 nm and a repetition rate of 160 MHz [42]. Dual-comb operation was achieved with polarization multiplexing by introducing intracavity birefringent crystals [45]. Using this laser, we demonstrate the Vernier-based extension of the ambiguity range by interchanging the roles of the two combs in the measurement via a novel fiber-based dual-comb LiDAR frontend.
As a proof-of-principle demonstration of the capability of this setup, we measure the distance to both a static and a moving target placed more than 10 m away. High-precision ranging over such distances is critical for modern precision engineering applications such as aircraft engineering or in the automotive industry [8]. For the moving target we report distance measurements with micrometer precision, which corresponds to a relative precision of more than 10−7 without averaging. For the static target, we report an instantaneous precision of 0.29 µm with an update time of 1.50 ms, while for an averaging time of 200 ms we reached a precision of around 33 nm, which corresponds to a relative precision of about 3·10−9. These results show that rapid and precise long-distance ranging is possible with a highly simple setup, by combining our single-cavity dual-comb laser with a fiber-based dual-comb LiDAR frontend. Thus, this work paves the way for a new class of versatile dual-comb LiDAR systems for both scientific and industrial applications.
2. Experimental setup
In this section, we introduce a novel measurement setup for long-distance dual-comb ranging which is illustrated in Fig. 1. Since such setups are sensitive to environmental disturbances, for this work we have developed a robust fiber-based setup suitable for high-performance dual-comb ranging.
Fig. 1. Schematic of the single-cavity dual-comb in combination with a fiber-based dual-comb LiDAR frontend (grey area). The two pulse trains emitted by the dual-comb laser are fiber-coupled with the polarization along the slow-axis of polarization-maintaining (PM) fibers. Isolators (ISO) prevent feedback from the LiDAR back into the source. Each comb is split by a 50:50 PM-fiber beam splitter (FBS) into a signal beam (S) which samples the distance d between the reference plane (R) and the target (T) while the other comb is used as a local oscillator (LO). The facet of an FC/PC-fiber tip serves as a reference plane, while a moving mirror mounted on a home-built shaker is used as a target. After the fiber beam splitters, the combs are combined with a PM-fiber polarization combiner (COM). Hence, in the free-space region, both combs reach the target with crossed polarization states. The reflected light passes back through the polarization combiner and becomes polarized along the slow axis again. This returning light (S) is combined interferometrically with the other comb (LO) via the fiber beam splitters. Hence, the measurement on PD1 (PD2) corresponds to comb 1 (comb 2) as the signal and comb 2 (comb 1) as the LO. A helium-neon (He-Ne) laser-based Michelson interferometer is integrated into the free-space path to the target via a dichroic mirror (DM) to provide a reference measurement.
The dual-comb system used is based on the polarization multiplexing technique [45]. This technique is a simple and compact solution to dual-comb generation since only a single passively-stable oscillator is required. To allow for a continuously adjustable repetition rate difference between the combs which is sufficiently small to avoid aliasing, we rely on a pair of intracavity birefringent α-BBO crystals which are cut at an angle of 45° with respect to the c-axis. We set the two slightly different repetition rates of the dual-comb laser to $f_{\text {rep,1 }}$ = 160.339608 MHz and $f_{\text {rep}, 2}=f_{\text {rep}, 1}+\Delta f_{\text {rep}}$ with $\Delta f_{\text{rep}}$ = 581 Hz, corresponding to an intrinsic ambiguity range υgroup/($2f_{\text {rep}}$) = 0.93 m. The Yb:CaF2 based laser produces 115-fs pulses with a center wavelength of 1056 nm and average power of 130 mW per comb [42].
To perform LiDAR measurements, as depicted in Fig. 1, each comb is coupled into the input port of a polarization-maintaining (PM) fiber isolator (IO-G-1064-APC, Thorlabs) to avoid feedback into the laser oscillator. The polarization of the input beams is aligned with the slow-axis of the PM fiber to guarantee optimal functionality of the commercial components. Each isolator is then followed by a 50:50 fiber splitter (PN1064R5A2, Thorlabs) of which one output port is connected to one input port of a polarization combiner (PFC1064A, Thorlabs) while the other two output ports of the two 50:50 splitters are connected to each other as illustrated in Fig. 1. The polarization combiner is designed such that it combines two slow-axis-aligned input beams into an output beam of two superimposed, orthogonal, linearly polarized components. The polarization combiner can operate in reverse, i.e. if light composed of two orthogonal linear polarizations enters from the opposite direction the light is split into its polarized components. In most practical applications, the target is not in fiber but in free-space. Thus, the output port of the polarization combiner is connected to a PM-fiber with an FC/PC-fiber tip followed by a collimation unit to couple the light into free-space. The facet of the uncoated FC/PC-fiber tip serves as a partially reflective reference plane (R) as it reflects a few percent of the incoming light. The transmitted beam is expanded with a telescope to make sure it does not diverge significantly when propagating over long distances towards a target (T).
To explain how the setup solves the Vernier measurement problem of dual-comb ranging, we begin by discussing the path of comb 1. The beam component propagating to the polarization combiner is referred to as signal beam (Scomb1) since it will sample the distance to a target. The light reflected by the target travels back along the same path in free-space so that it is automatically coupled into the same fiber again. The back-reflected light then leaves the polarization-combiner along the same port it entered because the polarization of the light did not rotate by propagating between the reference plane and the target. As the signal beam (Scomb1) was partially reflected on the reference plane (R) and on the target (T), it created two replicas of the pulse with different delays. The distance d between the reference plane and the target is encoded in the optical delay τTR,o between the two replicas:
where υgroup is the group velocity at the carrier frequency. The subscript ‘o’ indicates that τTR,o is the delay between optical pulses, and ‘TR’ indicates the delay between the target and the reference. Since this delay is difficult to measure precisely enough in the optical domain, dual-comb ranging uses the second comb as a broadband local oscillator (LO) for the signal beam to map the signal from the optical domain to the radio-frequency domain in an approach equivalent to linear optical sampling [46]. For that purpose, the local oscillator (LOcomb2) is combined with the signal beam (Scomb1) in the 50:50 fiber splitter, and the combined beam is measured on photodiode PD1.A single measurement according to Eq. (1) is subject to a limited ambiguity range, as described in more detail in Section 3. The dual-comb Vernier principle addresses this by performing an identical measurement with the role of the two combs interchanged. In our setup, this occurs automatically because of the symmetry between the paths taken by the two combs. Specifically, the ‘signal beam’ component of comb 2 reaches the polarization combiner and thereafter takes the same physical path as Scomb1 before free-space propagation to the target. This signal, denoted Scomb2, is reflected in the same way, and the interference of the reflected Scomb2 with LOcomb1 is measured at PD2. Thus, by recording the signals from both PD1 and PD2 we have simultaneous measurements of Scomb1 and Scomb2. The photodiodes used are standard reverse-biased InGaAs detectors (DET01CFC, Thorlabs).
In the preceding description of the measurement principle, we assumed that the polarization of the signal beams is not rotated along the measurement path. This assumption is valid if the target is for example a silver mirror which does not alter the polarization state of the reflected light. However, for many fabricated and natural materials this is not necessarily the case [47]. We expect that a change in polarization along the measurement path would affect the strength of the interference signal since the signal portion that propagates back along the correct path after the polarization combiner is reduced. Meanwhile, the remaining signal portion that propagates along the incorrect path after the polarization combiner does not distort the interference signal recorded on the corresponding photodetector, since the signal and local oscillator pulses, which in this case have the same repetition rate, do not move relative to each other so that this signal portion is not sampled by the local oscillator. Thus, the signal portion in the incorrect path would just result in background noise at the repetition frequency of the signal comb, which can be filtered out either digitally or electronically. This makes the proposed LiDAR system suitable for a wide range of targets. We found that for example a retroreflector leads to a similar strength in the target interferogram as the silver mirror used, but with the additional benefit that it is less sensitive to the angle of incidence.
For the data acquisition, we used a digital oscilloscope (WavePro 254HD, Teledyne LeCroy). To avoid distortions of the signal, we used a low-pass filter at the Nyquist frequency determined by the sampling rate of the local oscillator signal.
3. Vernier principle for dual-comb ranging
In this section, we give an overview of how we analyze the interferograms from the photodiodes to extend the ambiguity range via the Vernier effect [26]. These interferograms encode the relevant information about the optical domain. In particular, the delay between two subsequent interferograms corresponds to the delay between the replicas in the optical domain via a scale factor of $\Delta f_{\mathrm{rep}} / f_{\mathrm{rep}}$ = 3.63·10−6:
Since the delay between subsequent reference interferograms τRR,el corresponds to the inverse pulse repetition rate difference, $\Delta f_{\mathrm{rep}}$ is reassessed at every measurement instance meaning that slow drifts in $\Delta f_{\mathrm{rep}}$ are removed, so only noise components with frequencies of order $\Delta f_{\mathrm{rep}}$ or higher are relevant for the uncertainty in the distance measurement.
The intrinsic ambiguity range of the ranging system using only one of the two photodiode signals is determined by the repetition rate according to
Both signal combs sample the same absolute distance d, and therefore with Eq. (4) the difference in the measured values is given by
The general principle of this process could be compared to a distance measurement with two rulers with different lengths RA,1 and RA,2 which are both shorter than the absolute distance to be measured. Both rulers are used for measuring the absolute distance by stacking them until the complete distance is covered. The number of times the ruler fits completely within the total distance determines the integer m as illustrated schematically in Fig. 2 for m = 2. To retrieve this information after the measurement, it is sufficient to know the difference between the two distances measured with each ruler in their (m+1)-th stacking position as this corresponds to the distance Δdmod. Together with Eq. (7), this information allows then to retrieve the integer number m and consequently the absolute distance.
Fig. 2. Illustration of the ambiguity range extension by leveraging the Vernier effect at the example of two rulers with different lengths RA,1 (blue) and RA,2 (green). Both rulers fit twice within the absolute distance between reference (R) and target (T), meaning that for this example m = 2. According to Eq. (7), the difference between their length ΔRA is thus exactly one half of Δdmod.
The Vernier-based extension of the ambiguity range fails once the value of m is different for the two measurement channels, as in this case, Eq. (5) becomes invalid. The highest number of intrinsic ambiguity ranges RA,i of comb i that can be stacked to cover the distance d before this happens is labeled mmax,i. It is defined by the requirement that the accumulated difference in ambiguity range mmax,i · ΔRA matches the intrinsic ambiguity range of comb j. This condition can be framed mathematically as
with i ≠ j. Accordingly, the new effective ambiguity range is defined as the distance that can be covered by stacking mmax,i times the intrinsic ambiguity range of comb i:Notice that $R_\textrm{A}^{\textrm{vernier}}$ is on the order of 258 km for our measurement system which far exceeds the requirements of many practical applications. This gives room for increasing the repetition rate difference between the two combs which leads to higher update rates while still maintaining an ambiguity range suitable for most applications.
4. LiDAR measurements
In this section we present our experimental results. Since our goal is to perform proof-of-principle demonstrations of the new dual-comb LiDAR approach, we do not account for the group index of air or the lenses. Hence, all distances are inferred via the speed of light in vacuum. Inferring the physical path length over long distances, e.g. by installing environmental monitoring along the path [48,49] or by a two-color approach [50] to correct for the air dispersion, will be the subject of future work. The absolute delays that are used to infer the distance are themselves inferred from our knowledge of the laser repetition rates. Since these repetition rates are measured with a spectrum analyzer (8592L, Hewlett-Packard), the clock on that device currently defines the scaling of the delay axis.
4.1 Measurement of the Vernier splitting
For a first proof-of-principle demonstration, we chose a target at a distance of 10.77 m away from the FC/PC-fiber tip reference plane in free-space. This measurement distance is more than eleven times longer than the intrinsic ambiguity range of the laser $\upsilon_{\text {group }} /\left(2 \cdot f_{\text{rep}}\right)$ = 0.93 m. The target in the signal arm of the LiDAR is a mirror mounted on a home-built shaker oscillating with a frequency of 9.9 Hz. To provide a reference measurement that can be used to continuously track the motion of the moving mirror and thereby validate the results from the dual-comb LiDAR measurement, we integrate a helium-neon (He-Ne, λ = 632.8 nm) laser-based Michelson interferometer into our setup. The beam in the measurement arm of the Michelson interferometer is spatially overlapped with the signal beam of the LiDAR by using a dichroic mirror (DM) as illustrated in Fig. 1.
To avoid saturating the photodetectors used for this experiment (PD1 and PD2 in Fig. 1), the optical power coupled into the dual-comb LiDAR frontend is attenuated so that only around 1 mW per comb is sent towards the target. In case of less ideal or more distant targets, the optical power in the target arm could be increased significantly to enhance the signal strength. This would allow to resolve low-power back reflections by interfering them with a high-power local oscillator. If necessary, coherent averaging of subsequent interferograms could even give access to signals buried within the noise floor of the measurement.
The interferograms, obtained by linear optical sampling of the returning signal comb by the local oscillator comb, are shown in Figs. 3(a-b). In Fig. 3(a), comb 1 serves as local oscillator while comb 2 is used as the signal beam. Since $f_{\text {rep},1}<f_{\text {rep},2}$, in this configuration the local oscillator pulses advance in time compared to the signal pulses. Consequently, the order in which the pulse replicas from reflections at the reference plane and the target appear in the electronic signal is equivalent to the order of those pulses in the optical signal. Thus, the time τTR,1 is measured from a reference peak to the following target peak. In Fig. 3(b) the role of signal beam and local oscillator is interchanged compared to Fig. 3(a). In this case, the local oscillator pulses are delayed compared to the signal pulses so that the order of reference and target peaks is reversed in the electronic domain with respect to the order of these pulses in the optical domain. Thus, the delay τTR,2 is measured from a reference peak to the preceding target peak as indicated in Fig. 3(b). The small signals of tertiary height just above the noise floor are caused by stray reflections of the pulses in the LiDAR setup. They can be easily identified by their amplitude and do not affect the measurement.
Fig. 3. LiDAR measurement: (a) Interferograms in the electronic time domain (see Eq. (2)) generated by linear optical sampling of the signal beam (comb 2) which is partially reflected at the reference plane (R) and the target (T). The delay between the reflections from the reference plane and the target τTR,1 in the electronic domain encodes their separation, while the delay between subsequent reference pulses indicates the update rate of the distance measurement which is given by the repetition rate difference of the laser τRR,1 = τRR,2 = 1/Δ$f_{\mathrm{rep}}=1.72 \; \text{ms}$. Note that the pulse replica generated from the reflection of the signal beam at the reference plane is slightly smaller than the back-reflection from the target. The linear optical sampling of the signal beam by the local oscillator thus results in smaller reference interferograms compared to the target interferograms. (b) Similar to (a) but with the roles of signal beam and local oscillator interchanged, i.e. comb 2 now serves as local oscillator while comb 1 is used as the signal beam. (c) Zoom on one interferogram (T) corresponding to the pulse replica generated from the reflection at the target together with the corresponding envelope.
To determine the delay between subsequent target and reference pulses indicated by τTR,1 in Fig. 3(a) and τTR,2 in Fig. 3(b), we need to find the peak of the respective interferograms. For that purpose, we first determine their envelope by computing the magnitude of the analytic signal in the time domain as obtained from the Hilbert-transform algorithm (Fig. 3(c)) to make the signal insensitive to fluctuations in the carrier-envelope offset phase of the two pulse trains. The peak of the resulting signal can then be extracted with a first-order moment integral of the magnitude squared of the envelope which is robust against intensity noise on the signal. This kind of data processing yields time-of-flight information.
Since the target and reference plane are separated by more than the intrinsic ambiguity range of the system, the delay between the reflections from the reference plane and the target differs for the two simultaneous distance measurements, i.e. τTR,1 ≠ τTR,2. Figure 4(a) shows the resulting Vernier splitting Δdmod as defined in Eq. (5).
Fig. 4. (a) Simultaneous measurement of the distance dmod,j, j ∈ 1,2 (absolute distance modulo ambiguity range RA,j, j ∈ 1,2) with the roles of signal beam and local oscillator interchanged. The distance Δdmod that separates those distance measurements is referred to as the Vernier splitting. Each measurement dot is obtained within a time period of $1 / \Delta f_{\text {rep }}=1.72 \mathrm{~ms}$. (b) Measurement of the integer number of ambiguity ranges between the reference plane and the target using Eq. (6) and Eq. (7) with υgroup = c (speed of light in vacuum). By averaging those values, we find m = 11.008 ± 0.025 which is close to an integer. (c) Deviation of the estimate of m from the true value (m = 11) as a function of the averaging time (blue dots, left axis) and probability to estimate the incorrect value for m (red triangles, right axis) if we assume normally distributed noise for the individual measurements using Eq. (10).
The time at which dmod is measured – which corresponds to the time of the target peaks – is in general not the same for the two measurement channels. The target peaks in Fig. 3(a-b), for example, exhibit a delay of around 0.8 ms. For a stationary target, it is still possible to just take the difference between the two measurements to infer the Vernier splitting, but for a moving target, it is important to account for movements of the target during the time between two measurements since this could otherwise lead to systematic errors in Δdmod. To account for this, we linearly interpolate between subsequent distance measurements for one of the traces to estimate dmod at the time when dmod is measured for the other trace. This can then be used to infer the Vernier splitting. This approach requires no prior knowledge about the motion of the moving target.
The resulting prediction of the integer number of ambiguity ranges between reference plane and target is shown in Fig. 4(b), using Eq. (6), Eq. (7) and the data shown in Fig. 4(a). The estimated values exhibit a standard deviation of 0.26 which is a direct consequence of the uncertainty in the distance measurement. By averaging those estimates for a time τavg, we can improve the precision of the final prediction of m (Fig. 4(c)). We validated that the m value is 11 with an independent measurement using a ruler. The mean of the predicted values after averaging is µm(τavg), and σm(τavg) describes their standard deviation. If we assume normally distributed noise for the individual measurements, we can calculate the probability to estimate m correctly after an averaging time τavg with an integral of the corresponding normal distribution over the values that would be rounded to the correct integer as
By averaging 6 measurements, which corresponds to an averaging time of only 10 ms, we can reduce the risk to find the incorrect value for m to less than 10−6 (Fig. 4(c)). Furthermore, for an averaging time of 200 ms, we obtain m = 11.008 ± 0.025 which is close to the correct integer demonstrating that our measurement exhibits a negligible systematic error.
4.2 Absolute distance measurement of moving target
As a proof-of-principle measurement on a moving target, we capture the motion of a mirror at a distance of 10.77 m which oscillates sinusoidally at a frequency of 9.9 Hz (Fig. 5(a)). To provide a reference measurement, the relative changes in the target mirror position are simultaneously recorded with a He-Ne-laser-based Michelson interferometer. The relative distance changes recorded in the reference measurement are shifted to the absolute target mirror position as retrieved from the dual-comb LiDAR to allow for an assessment of the measurement precision. The absolute distances measured with the dual-comb LiDAR are in close agreement with the distance changes retrieved from the He-Ne-laser-based Michelson interferometer, as indicated by the residuals shown in Fig. 5(b) which exhibit a root mean square (rms) deviation of only 0.87 µm. It should be noted that the dual-comb and He-Ne lasers do not share a complete common path from reference to target. Hence, various air path length changes along the >10-m path can contribute to the residuals as well as noise of the lasers themselves.
Fig. 5. (a) Distance between the reference plane and the moving target (mirror mounted on home-build shaker) measured with the dual-comb LiDAR and a He-Ne-laser-based Michelson interferometer. Each LiDAR measurement dot is obtained within a time period of $1 / \Delta f_{\text {rep }}=1.72 \mathrm{~ms}$. The distance measurement with the He-Ne laser is used as a reference to validate the LiDAR ranging results. (b) Residuals between the distance inferred from the LiDAR measurement and the reference measurement with the He-Ne-laser-based Michelson interferometer. Without any averaging of the distance measurements, they already amount to an rms deviation of only 0.87 µm.
4.3 Absolute distance measurement of static target
For static or slowly moving targets, the high update rate inherent to the dual-comb LiDAR technology is still beneficial as it allows to average the retrieved distances thereby improving the measurement precision.
To demonstrate the capability of the proposed LiDAR system for high-precision distance metrology, we replaced the shaker in Fig. 1 with a fixed mirror serving as a static target. Furthermore, we increased the repetition rate difference to $\Delta f_{\text {rep}}=666 \mathrm{~Hz}$. The distance measurements over an observation time of 1 s are shown in Fig. 6(a). The corresponding Allan deviation is shown in Fig. 6(b). Without averaging, i.e. with an update time of $1 / \Delta f_{\text {rep}}=1.50 \mathrm{~ms}$, we achieve a measurement precision of 0.29 µm.
Fig. 6. (a) Absolute distance measured between a reference plane and a static target over a time scale of 1 s. Each measurement dot is obtained within a time period of $1 / \Delta f_{\text {rep}}=1.50 \mathrm{~ms}$. (b) Allan deviation with error bars indicating the 1σ standard deviation as estimated from five subsequent distance measurement data sets, each with a duration of 1 s. This illustrates the measurement precision for varying averaging times as detailed in Appendix A1. The instantaneous precision is 0.29 µm, while for an averaging time of 200 ms we achieve a precision of around 33 nm. For averaging times up to around 50 ms, the precision improves proportionally to the square root of the measurement time as expected for white noise in the inferred distance. The dashed red line has a slope of -0.5.
The precision can be increased by averaging out most of the zero-mean high-frequency noise such as detector shot noise or thermal electric noise with frequencies above 1/τavg, where τavg is the averaging time. This behavior can be observed in Fig. 6(b) where the Allan deviation is initially decreasing. The associated increase in precision is proportional to the square root of the averaging time, as expected for white noise (such as shot noise or electronic noise), yielding a precision of around 33 nm in 200 ms averaging time. Compared to the absolute distance of more than 10 meters, this corresponds to a relative precision of about 3·10−9. For longer averaging times, the Allan deviation starts to increase again which we attribute to the growing impact of low-frequency noise sources such as thermally induced path length changes [51] and slow-frequency drifts of the repetition rate $f_{\mathrm{rep}}$. Since we estimate the repetition rate to be stable within around 40 mHz (for a frequency noise power spectral density integration range of [10 Hz, 10 kHz]) for both lasers, which corresponds to a relative stability of 2.5·10−10 [42], we expect the latter to be negligible [44]. Due to the increasingly dominating impact of external low-frequency noise sources for longer averaging times, we limited the measurement duration to 1 second.
4.4 Extended measurement distance and resolution limitations
Based on an analytical model describing time-of-flight based dual-comb ranging with femtosecond lasers in the presence of timing jitter, it has been predicted that the ranging precision and consequently the uncertainty in the estimate of m does not significantly degrade over longer measurement distances, i.e. for larger values of m [44]. Only when m becomes comparable to $f_{\mathrm{rep}} / \Delta f_{\mathrm{rep}}$, which corresponds for our laser configuration to ranging at tens of kilometers, the timing error accumulated during the propagation time of the signal pulse along the measurement path (which is approximately $m / f_{\text {rep}}$) starts to become relevant. An increased timing error affects the measurement precision and thus the uncertainty in m. In this case, a low bandwidth repetition rate locking could be applied to mitigate the accumulated timing error.
The precision with which m can be estimated depends on the ranging precision and on the Vernier splitting after a single ambiguity range (see Eq. (7)), which corresponds to the propagation distance of light during one delay step between the two pulse trains, i.e. during $\Delta f_{\mathrm{rep}} /\left(f_{\mathrm{rep},1} f_{\mathrm{rep},2}\right)$. By increasing $\Delta f_{\mathrm{rep}}$, this delay step is increased so that for a given ranging precision the noise in estimating m is reduced. Moreover, a higher $\Delta f_{\mathrm{rep}}$ also leads to less accumulated timing jitter between corresponding reference and target pulses (since a narrower range of noise frequencies contributes to the jitter when $\Delta f_{\mathrm{rep}}$ is increased [42]), which benefits the ranging precision. Hence, with a higher repetition rate difference, the measurement precision of m is increased [44].
Finally, provided the speed of the target is significantly less than $R_{\mathrm{A}} \cdot \Delta f_{\text {rep}}$ which is typically on the order of a few hundred meters per second, the uncertainty in m can always be averaged down by increasing the integration time. This approach also benefits from an increased repetition rate difference as the associated higher update rate allows to average more data points within the same integration time which further increases the confidence in the estimated value of m. Notice, however, that the repetition rate difference should always remain low enough to satisfy
to prevent spectral aliasing, where Δνopt is the full optical bandwidth shared by the two frequency combs as detailed in Appendix A2. An increased repetition rate difference with a constant repetition rate thus comes at the cost of a narrower spectrum and thus longer pulse durations leading to reduced time resolution in time-of-flight-based ranging applications. This restriction imposed by the aliasing condition can be softened by substituting the interferometric detection with two-photon detection [52].5. Conclusion and outlook
We have demonstrated a dual-comb LiDAR system using a free-running dual-comb oscillator combined with a novel fiber-based dual-comb LiDAR frontend to extend the range. Measuring the distance simultaneously with the roles of signal beam and local oscillator interchanged allows to extend the ambiguity range to several hundred kilometers by leveraging the Vernier effect. The dual-comb LiDAR frontend is implemented as a fiber-based setup that is insensitive to misalignments and environmental disturbances. Furthermore, it relies solely on standard off-the-shelf components which makes it easy and cheap to implement. Since the reference comes from a Fresnel reflection at the output-fiber tip, the measurements are immune to refractive index changes in the fibers.
To demonstrate the suitability of the setup for rapid and precise long-distance ranging, we first measured the absolute distance to a moving target over a distance of more than 10 m. The moving target is implemented with a mirror mounted on a shaker which performs sinusoidal oscillations at a frequency of 9.9 Hz. A comparison of the measured distance to a reference measurement with a He-Ne-laser-based Michelson interferometer reveals micrometer precision, meaning that with our system we can reliably track the motion of a moving target. Compared to the absolute distance of about 10.77 m between reference plane and target, this corresponds to a relative precision of more than 10−7 without any averaging required.
With averaging, even higher precision is possible. To demonstrate this, we performed a measurement on a static target (fixed mirror mounted at a distance of more than 10 m away from the reference plane). The instantaneous precision is 0.29 µm with an update time of 1.50 ms, and with averaging over a time scale of around 200 ms even sub-40 nm precision is possible. This corresponds to a precision significantly below λ/4, meaning that the time-of-flight-based distance measurement could be combined with an interferometric approach [26] to obtain higher precision. However, because we reach 3·10−9 relative precision further improvements are limited by the atmospheric fluctuations and not by the measurement sensitivity.
To combine the demonstrated time-of-flight-based distance measurement with an interferometric approach, while simultaneously extending the measurement distance to a longer range, we plan to account for refractive index variations along the measurement path. In combination with the inherent capability of the dual-comb LiDAR technology to detect low-power back reflections from a distant or non-cooperative target by interfering them with a high-power local oscillator [34], we expect to push the boundary of precision long-distance ranging with free-running dual-combs even further.
In addition, the solid-state dual-comb laser technology we use is well suited for scaling up the laser repetition rate to the gigahertz range, which would increase both measurement speed and accuracy [53], and still allow for km-scale ambiguity ranges after accounting for the Vernier splitting using our robust fiber-based dual-comb LiDAR frontend. This in turn would enable higher measurement rates in the 100 kHz regime. Our results thus represent an important step for practical use of long-range and high-precision dual-comb-based ranging.
6. Appendix
A1. Computation of Allan deviation and error bars
For non-stationary quantities such as the distance measured with our dual-comb LiDAR system which is subject to slow-frequency drifts, the standard deviation is not well-defined. To still describe the stability of such quantities, a broadly accepted tool is the Allan deviation [54].
For the stability analysis in this work we calculate the overlapping Allan deviation which is an adapted form of the traditional Allan deviation but with the advantage that it makes maximum use of the available data by forming all possible overlapping samples at each averaging time τ [55]. Specifically, for assessing the stability of the distance measurement we define the overlapped Allan deviation as
To determine the uncertainty in σΔd(τ) for a certain averaging time $\tau=m / \Delta f_{\mathrm{rep}}$, we repeatedly calculate the Allan deviation for five subsequent distance measurement data sets, each with a duration of 1 s. The obtained values are used to compute the 1σ standard deviation in $\sigma_{\Delta d}\left(\tau=m / \Delta f_{\text {rep}}\right)$, which is expressed in terms of error bars in Fig. 6(b).
A2. Condition to prevent spectral aliasing
Within the full bandwidth Δνopt shared by the two optical frequency combs, there are
pairs of comb lines. These N pairs get down-converted to N radio-frequency (RF) comb lines via photodetection of the heterodyne beat signal between the frequency combs. The bandwidth of the resulting RF spectrum isBecause all pairs of optical comb lines can mix, there are many replicas of the same RF comb. To unambiguously distinguish the individual replicas, the RF combs should not overlap. Therefore, the bandwidth BRF must satisfy the aliasing condition
By combining Eq. (14) and Eq. (15), we arrive at a constraint on $\Delta f_{\text {rep}}$:Funding
H2020 European Research Council (966718); Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (40B2-0_180933, 40B1-0_203709); Innosuisse - Schweizerische Agentur für Innovationsförderung (40B2-0_180933, 40B1-0_203709).
Acknowledgments
We thank our collaborator Prof. Patrice Camy, CIMAP of Caen, France for manufacturing the Yb:CaF2 gain crystals.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in [56].
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