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Abstract
A multi-pulse sampling dual-comb ranging (MS-DCR) method is proposed in this paper. Four sampling pulses and two signal pulses separated in the time domain are generated in a repetition period by fiber delay. Through multi-pulse linear optical sampling, eight cross-correlation interferograms (IGMs) are generated in an updating period. The proposed method realizes the multiplication of IGMs so that additional ranging results can be obtained. The experimental results demonstrate that we suppress any random noise by averaging the ranging results and improve the precision of the time-of-flight (TOF) method and carrier-wave interferometric (CWI) method simultaneously. The precision of TOF is improved from 3.85 µm to 1.39 µm without time averaging and that of CWI is improved from 25 nm to 11 nm. The TOF result can link to the interferometric phase with 15 ms averaging, and a precision of 0.48 nm is reached with 0.5 s averaging. The proposed technique overcomes the limitations of linear optical sampling in conventional dual-comb interferometers and achieves faster and higher precision distance measurements without decreasing the unambiguity range.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the time domain, an optical frequency comb (OFC) emits an ultra-short pulse train with a certain repetition period. In the frequency domain, the Fourier spectrum of OFC is composed of evenly spaced, narrow spectral lines over a wide spectrum range, and the frequency of each spectral line f(n) obeys the basic relationship f(n) = nfr + fceo [1,2]. Here, fr is the repetition frequency determined by the laser cavity length, and fceo represents the carrier–envelope offset frequency given by the phase shift between the carrier-wave and pulse envelope from pulse to pulse. In recent years, several OFC-based methods have demonstrated OFC’s superiority in precise optical metrology, especially using the dual-comb method [3,4]. This method employs two OFCs with slightly varied repetition frequency (Δfr). One serves as a signal comb with repetition frequency (fr1), and the other serves as a sampling comb with repetition frequency fr2 = fr1 +Δfr. Through linear optical sampling over a certain measurement period, one interferogram (IGM) can be generated [3]. Based on the multi-heterodyne process between the spectral lines in two OFCs, a sub-comb in the radio frequency (RF) domain with an interval of Δfr is generated. A stable dual-comb system provides a new set of powerful tools for dynamic and high precision amplitude and phase spectra measurements, and thus is being applied in an increasing number of fields [5–13].
Absolute distance measurement is one of the typical applications of the dual-comb system in optical metrology, and it plays an important role in LIDAR, satellite ranging, and other applications [10–13]. Normally, based on the time-of-flight (TOF) technique, we can obtain the distance by calculating the time delay (Δt) from the reference IGM (IR) to the measurement IGM (IM) [10,13]. This method is similar to the dispersive interferometric method, and can achieve micrometer-level precision with an unambiguity range equal to Lpp/2 = vg/(2fr1), where vg is the optical pulse velocity [12]. To achieve nanometer-level precision, stable phase information is required. Based on the carrier-wave interferometric (CWI) technique, the precise distance result can be calculated by the carrier phase difference (Δφc) between IR and IM within only the half-carrier wavelength (λc/2) unambiguity range [4,11]. However, the carrier phase noise and the timing jitter of envelopes in the conventional fiber dual-comb can reach nearly 104 rad and 100 ns, respectively [14]. To suppress these noise sources, we often use ultra-stable CW lasers as precise references to stabilize fr and fceo [15,16]. Because our main concern is the relative frequency noise in a dual-comb system, a synchronous locking approach and post-compensation method are also widely used to compensate for the relative frequency noise and thus improve the mutual coherence of the two combs [17–21]. It should be noted that the ranging result is still affected by intensity noise and other residual noise after noise suppression. Here, the carrier phase difference is not sensitive to intensity noise. However, affected by other residual noise, the precision is only ∼0.2 rad, which is not sufficiently high [12,14]. In addition, the intensity noise exerts a substantial effect on Δt, which limits the precision of the TOF method to micrometer levels [22–24]. Therefore, to achieve nanometer-level precision, additional time is needed to average the results of the TOF method until the precision can link to the interferometric phase [4,11].
To shorten the averaging time, one solution is to use synthetic-wavelength interferometry as a bridge to connect the TOF and CWI methods [25]. Another solution is to improve the precision of the TOF method. Normally, we can choose two OFCs with higher repetition frequency. In this case, we can increase Δfr to improve both the precision and measurement rate [26–28]. Such high-repetition-frequency dual-comb ranging (DCR) systems can be realized by using micro-resonator combs, broadband electro-optic frequency combs, or mode-filtering techniques [29–31], and usually provide megahertz update speeds and sub-micron TOF precision. However, use of a high-repetition-frequency DCR system yields not large unambiguity range, and thus limits its application regarding long distance measurements.
In this paper, we propose a multi-pulse linear sampling technique, which provides for the multiplication of IGMs in a measurement period so that additional ranging results can be obtained. By averaging the ranging results, we can suppress the random noise effectively. The experimental results demonstrate that the precision of both the TOF and CWI methods can be improved simultaneously and faster and higher precision distance measurements are achieved without decreasing the unambiguity range.
2. Experimental setup and principle
Figure 1(a) shows a schematic of the proposed multi-pulse sampling dual-comb ranging (MS-DCR) system. A signal pulse emitted from Comb 1 (fr1 = 56.090 MHz) is divided into m signal pulses by a 1 × m optical splitter, and is coupled again by an m × 1 optical coupler after the m signal pulses pass through the m optical fibers of different lengths. The m signal pulses separated in the time domain are generated in a repetition period of 1/fr1. These pulses are incident on the reference arm and measurement arm respectively, and m reference pulses and m measurement pulses are produced. Here, the time delay (Δτ) between each pair of reference and measurement pulses is equal, and is determined by the path length difference between the two arms. Similarly, n sampling pulses from Comb 2 are generated in a repetition period of 1/fr2 after passing through a 1 × n optical splitter and an n × 1 optical coupler. As illustrated in Fig. 1(b), through linear optical sampling, the n sampling pulse trains from Comb 2 interfere with the m reference pulse trains and m measurement pulse trains from Comb 1, generating m × n IR and m × n IM with a certain measurement period Tupdate = 1/Δfr. This realizes the multiplication of IGMs, which is different from the conventional dual-comb system. Here, the maximum number of signal pulses m and sampling pulses n in a repetition period mainly depends on how many IGMs can fill the gap between two adjacent IGMs in a measuring period without pulse multiplication. In the proposed method, the width of each IGM in the time domain is ∼0.5 µs, and the corresponding “dead zone” of ranging is wd = 1.3 mm calculated from the TOF method. The “dead zone” means that IR and IM overlap, resulting in a zone that cannot be measured. Consider for example the unambiguity range of ∼2.67 m when fr1 is 56.090 MHz; then ${m \times n}$ should be less than 1026 (calculated by 2670/2/1.3). For simplicity, m is set to 2 and n is set to 4 to verify the principle of the present method. The delay between the pulses depends on the fiber length difference. The design rules are as follow: L4 > L3 + 2wd/n1 > L2 + 4wd/n1 > L1 + 6wd/n1, and L′2 – L′1 > L4 – L1 + 2wd/n1, where the n1 is the refractive index of the optical fiber. Such design is to ensure that all IGMs do not overlap in the time domain and are arranged in order. In the experiments, we built the MS-DCR system according to these rules, but it is not necessary to precisely control the delay between each pulse to a fixed value. It should be noted that although the IGMs overlapping positions will increase with the increase of m and n, we can set the time delay between multiple pulses to avoid overlapping simultaneously. In this case, we still can obtain the ranging results from other pairs of non-overlapping IGMs.
Fig. 1. Principle of the MS-DCR method. Two nonlinear polarization rotation (NPR) passively mode-locked Erbium-doped fiber combs with 1 kHz repetition frequency difference are used. (a) The experimental setup: P: Polarizer; BPF: an optical filter, the bandwidth is ∼5 nm to avoid spectral aliasing; PD: photodetector; BS1–2: beam splitter; LPF: low-pass filter. The fceo of both OFCs are fully stabilized by the f−2f interferometers (fceo1 = fceo2 = 10.56 MHz). (b) Two reference and measurement signals appear in a period of 1/fr1, and four sampling pulses appear in a period of 1/fr2. These pulse trains interfere with each other and generate eight reference IGMs and eight measurement IGMs in a period of 1/Δfr.
To realize a phase-stable MS-DCR system, a free-running continuous wave (CW) laser is used as an optical intermediary to obtain the beat signal between two combs fb = |f1 – f2|. We can extract the envelope timing jitter (Tjitter) and carrier phase noise (δφc) of the IGMs based on the jitter of fb. Through a digital post-correction, all of the raw IGMs can be corrected in the time domain. The corrected method can be divided into two steps:
where I0 represents the raw IGMs and I1 represents the phase-corrected IGMs, andFig. 2. (a) Reference IGMs and Measurement IGMs of MS-DCR system during 1.4 ms. (b) 1000 IR1 are moved to the first period and shown together. The width of each IGM in the time domain is ∼0.5 µs. (c) The normalized amplitude spectrum of IR1 with an interval of Δfr = 1 kHz. (d) The parts of the amplitude spectrum in the range of 12 kHz. The resolution bandwidth (RBW) of the spectrum is 1 Hz.
Fig. 3. Principle behind the TOF and CWI methods used in MS-DCR. The TOF result (Dtof) determines the multiple integer Nc of λc/2. The unambiguity ranges of the TOF and CWI methods in MS-DCR are at the meter-scale level and submicron-scale level, respectively.
In MS-DCR, each pair of IR and IM in the time domain can be regarded as the IGMs obtained by the conventional phase-stable DCR system. Therefore, we can calculate the ranging result for each pair of IGMs (IR1 and IM1, IR2 and IM2, …, IR8 and IM8) in a measurement period of 1/Δfr. Theoretically, by averaging all ranging results in a measurement period, we can suppress any random noise and improve the ranging precision. The principle of the combined ranging method is shown in Fig. 3. The TOF measurement distance of MS-DCR method is thus given by
After Δt averaging, the TOF ranging result of MS-DCR can achieve a precision below λc/4 more quickly than conventional DCR, and can be used to determine the unique integer Nc of λc/2 from
3. Experiments
The precision of the TOF and CWI methods mainly depends on the stability of the time delay (Δt) and the carrier phase difference (Δφc) between IR and IM, respectively. Therefore, the stabilities of Δt and Δφc between IR and IM are first analyzed. We calculate Δt and Δφc using conventional DCR and $\overline {\Delta t} $ and $\overline {\Delta \varphi_{\textrm{c}}} $ using MS-DCR during 1 s for comparison. Here, we use the Δt and Δφc values of the first pair of IGMs (IR1 and IM1) to evaluate Δt and Δφc in conventional DCR. The measured distance is fixed at ∼1.1 m, which is close to Lpp/4, and is where the noise-level is higher than at other positions [23]. As Figs. 4(a) and 4(c) shows, compared with the jitter of Δt ranging from −4.94 ns – 4.67 ns for conventional DCR, the jitter of $\overline {\Delta t} $ is distributed only from −1.52 ns – 1.89 ns after averaging the Δt of 8 pairs of IGMs using MS-DCR. For the carrier phase difference, as Figs. 4(b) and 4(d) shows, the jitter is decreased from −0.73 rad – 0.71 rad to −0.33 rad – 0.32 rad by averaging the 8 pairs of Δφc.
Fig. 4. (a) Jitter of Δt for 1,000 updating periods. (b) Jitter of Δφc for 1,000 updating periods. The carrier wavelength is λc ≈ 1567.193 nm. (c) Jitter of $\overline {\Delta t} $ for 1,000 updating periods. (d) Jitter of $\overline {\Delta \varphi_{\textrm{c}}} $ for 1,000 updating periods. (e) The δΔt of 8 pairs of IGMs. (f) The δΔφc of 8 pairs of IGMs.
In the present method, adding such multiple delays would naturally lead to additional noise for each of the divided pulses. To analyze the effects of the additional noise for different pairs of IGMs, the time delay jitter (δΔt) and carrier phase difference noise (δΔφc) of 8 pairs of IGMs are given. Here, we use the standard deviation of Δt and Δφc over 1 s, to evaluate the δΔt and δΔφc, respectively. As depicted in Figs. 4(e) and 4(f), the δΔt of 8 pairs of IGMs are nearly same, and the δΔφc of 8 pairs of IGMs do not show obvious difference. As a result, although the different fiber lengths bring the additional noise for signal pulses and sampling pulses, each pair of reference IGM and measurement IGM share the common path in the delayed path. Theoretically, the additional noise can be eliminated when calculating the Δt and Δφc.
To further demonstrate the capability of the proposed method, the precision of both the TOF and CWI methods in DCR by the use of Δt and Δφc and in MS-DCR by the use of $\overline {\Delta t} $ and $\overline {\Delta \varphi_{\textrm{c}}} $ at a fixed distance of ∼1.1 m are analyzed. As Fig. 5 shows, the TOF precision (Allan deviation) in DCR is δTOF ≈ 3.85 µm (Tupdate /T)1/2, in which T is the averaging time. The ranging error of TOF method is mainly caused by intensity noise, which cannot be compensated by the noise-suppression methods in a dual-comb system. To achieve a precision better than λc/4, a 120 ms averaging period is required. However, through the MS-DCR method, the precision of TOF can be improved to 1.39 µm (Tupdate /T)1/2 directly and can reach as low as 363 nm with only a 15 ms averaging time. At this time, we can measure the unique multiple integer Nc of λc/2, and combine the TOF method with the CWI method. It is clear that the random noise in the carrier phase difference is also suppressed by carrier phase difference averaging, and the corresponding precision of CWI is improved from 25 nm (Tupdate /T)1/2 in DCR to 11 nm (Tupdate /T)1/2 in MS-DCR. The precision can be improved to better than 3.1 nm in 15 ms, and reach 0.48 nm at 0.5 s. In the experiments, the jitter of both $\overline {\Delta t} $ and $\overline {\Delta \varphi \textrm{c}} $ in MS-DCR is decreased by ${\sim} \sqrt{8} $ times, and the corresponding precision of the TOF and CWI methods in MS-DCR is increased by a factor ${\sim} \sqrt{8} $. This factor depends on the number of m and n of signal pulses and sampling pulses, and is basically equal to $\sqrt {{m \times n}} $, which meets the random noise suppression rules. Therefore, to improve the precision further, we can simply increase m and n. It should be noted that to avoid data overload, we can average each pair of IGMs directly by moving coherent averaging method in the time domain. Then only the IGMs after coherent averaging need to be processed, and finally the ranging results are averaged.
Fig. 5. Ranging precision (Allan deviation) in DCR and MS-DCR for different averaging times, evaluated over a 1-s data. The values of TOF and CWI as well as TOF_MS and CWI_MS represent the precision of the TOF and CWI methods in the DCR and the MS-DCR systems, respectively. The TOF result of MS-DCR is sufficiently stable to link to the interferometric phase after only 15 ms averaging time compared with the 120 ms averaging time required for the DCR method.
To verify the distance measurement accuracy of the MS-DCR method, we conducted an experiment to compare the results of the proposed method to the “standard results” provided by a commercial interferometer. The reference-to-target distance was moved a total of 1.5 cm using 15 steps starting at ∼1.1 m. During the experiments, we recorded the environmental parameters to correct for the effect of the refractive index of air. Figure 6 shows that the ranging results of the MS-DCR method agree well with those of the commercial interferometer. The fitting slope and the correlation coefficient (R2) are 0.9999999998 and 0.99999999994, respectively. The comparison residuals are kept within ± 55 nm, with a standard deviation (STD, σ) of 36.6 nm. In free space measurements, the proposed method and the comparison interferometer are disturbed by different environmental perturbations, and therefore the comparison residuals are larger than the precision of MS-DCR mentioned previously.
Fig. 6. Ranging results of MS-DCR with the combined TOF and CWI methods versus the ranging results from a heterodyne interferometer. The data length is 500 ms with 4000 pairs of IGMs, and their ranging average was used for the comparison.
4. Discussion and conclusion
There is a tradeoff between measurement precision and range when selecting m and n. In this work, our main purpose is to shorten the averaging time. Therefore, the most suitable value of ${m \times n}$ is determined by the precision of the TOF method and is basically equal to (4δTOF/λc)2. For example, as depicted in Fig. 5, the precision of the TOF method in DCR is δTOF ≈ 3.85 µm without time averaging. From the previous analysis, the TOF precision of the MS-DCR can reach a precision of 385 nm when the value of ${m \times n}$ is 100. In this case, The TOF result can link to the interferometric phase, and thus realize nanometer-level precision at a Δfr update rate without time averaging. Although, the increase of ${m \times n}$ leads to more overlapping positions in the distance range, the overlapping positions only affect a few pairs of IGMs because the delay between pulses is different. We still can obtain the ranging results from other pairs of non-overlapping IGMs. In addition, we also can decrease the overlapping positions through the polarization system. As noted in [10], we can obtain the reference IGMs and the measurement IGMs separately by two PDs, which can reduce the overlapping positions by half.
It should be noted that the present method is more suitable for low-repetition-frequency dual-comb systems (tens to hundreds of Megahertz). For some high-repetition-frequency systems (Giga Hertz level or higher), the Δfr can be large, and the gap between adjacent pulses is small. Therefore, it is difficult and not necessary to add additional pulses in the time domain.
In conclusion, an MS-DCR method was proposed in this paper to realize the multiplication of IGMs using multi-pulse linear optical sampling. The ranging result can be obtained from each pair of IGMs in a measurement period of 1/Δfr. By averaging the results of Δt and Δφc over a measurement period, we can theoretically suppress any random noise and improve the precision of the TOF and CWI methods simultaneously. Stability experiments based on conventional DCR and MS-DCR systems were investigated, demonstrating that the TOF precision of MS-DCR can reach λc/4 with only 15 ms averaging, while the average time required by conventional DCR is 120 ms. Using MS-DCR, the ranging precision can be increased to 3.1 nm directly and can reach 0.48 nm at 0.5 s. We also compared the present method with a commercial heterodyne interferometer over a range of 1.5 cm starting at ∼1.1 m. The STD of the residuals is lower than 36.6 nm. The proposed technique overcomes the limitations of linear optical sampling in conventional dual-comb interferometers, and realizes faster and higher precision ranging without decreasing the unambiguity range in a relatively uncomplicated way. Furthermore, it reduces the dependence on the use of a low-noise dual-comb light source and a low-noise detection system.
Funding
National Natural Science Foundation of China (51835007, 61575105); State Key Lab of Digital Manufacturing Equipment and Technology (DMETKF2018016).
Disclosures
The authors declare no conflicts of interest.
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