Precise length measurement has served a fundamental role in many fields, such as metrology, laser-based light detection and ranging (lidar), precision machining, deformation and posture measurement of large equipment, three-dimensional (3D) surface profilometry of micro-nano devices, and gravitational wave detection. Various optical methods have been proposed for distance measurement, and optical interferometry has been a powerful and well-established approach for high-precision measurement. Compared with incremental distance measurement, continuous detection is no longer necessary for absolute distance measurement. Therefore, absolute distance measurement has been extensively developed, based on the methods of time of flight and multiwavelength interferometry during the past several decades. For the scheme of time of flight, the precision could be limited by the time resolution of the measurement system. Taking into account the multiwavelength interferometry, the distances can be determined by exploiting a synthetic wavelength chain, while multiple laser sources are needed, leading to a complex and expensive system.1

Laser frequency combs, since their invention, have given rise to revolutionary progress in the field of metrology. Actually, the initial motivation of the frequency comb is to measure the optical frequency (hundreds of terahertz) with subhertz precision ($<{10}^{-14}$ in relative precision),2 which is rather difficult when using the sophisticated frequency chain. When the two key parameters (repetition frequency $f_{\mathrm{rep}}$ and carrier-envelope-offset frequency $f_{\mathrm{ceo}}$) are tightly locked to an external frequency reference, the frequency comb consequently becomes a stable ruler in both time and frequency domains. More importantly, the excellent frequency stability of the time base (microwave clocks or optical clocks) can be coherently transferred to the other frequencies with high fidelity, where frequency combs serve as a bidirectional bridge. Typically, this kind of comb laser can emit a femtosecond-pulse train with an ultrastable interval and therefore inherently provides a powerful tool for the time-of-flight measurement. In addition, frequency combs are composed of numerous coherent wavelengths and naturally support multiwavelength interferometry.3 Inspired by these advantageous features, a variety of comb-based ranging methods have been proposed, such as pulse cross correlation,4^–6 dispersive interferometry,7^–10 intermode beat,11^,12 multiheterodyne interferometry,13^–15 dual-comb interferometry,16^–24 and pulse-to-pulse alignment.25^,26 At the moment, the distances can be already determined at long range with high precision and accuracy (approaching the quantum limit), large non-ambiguity range, and high speed.

Several mechanisms are capable of stimulating the frequency combs, such as passive mode locking,27 electro-optics modulation,28^,29 quantum cascaded lasers,30 and nonlinear evolution in microcavities.31^,32 By far, all these types of comb lasers can be self-referenced to the external frequency reference. Very recently, frequency combs based on microresonators (MRRs), also referred to as microcombs, have attracted extensive attention due to their small package, low power consumption, high repetition rate, large power of single tooth, broad bandwidth, and rich nonlinear phenomena.33^,34 Although the comb-based methods for distance measurement have been deliberated for a long time, microcombs could open a new window and offer new opportunities for the area of optical ranging. In 2021, Zheng et al. demonstrated the pulse cross correlation of microcombs for distance measurement.35 With the high $f_{\mathrm{rep}}$ of microcombs, the mechanical stage with a small scanning range is sufficient to realize arbitrary distance measurement, also resulting in better mechanical stability in the meantime. In 2020, Wang et al. described dispersive interferometry in kilometer-level distance measurement with a fast spectrometer.36 Because of the small pulse-to-pulse interval, the dead zones do not exist anymore. Subsequently, in 2021, Jang et al. demonstrated a hybrid method using soliton microcomb (SMC), which can support nanometric-precision distance metrology.37 The non-ambiguity range can be extended by the amplified-spontaneous-emission dispersive interferometry, and the ultimate performance can be linked to the homodyne interferometry of one comb tooth. In 2018, Trocha et al. realized ultrafast distance measurement using dual-comb multiheterodyne interferometry.38 A microwave comb can be coherently obtained whose phases can be used to measure the distances. As a consequence of the high $f_{\mathrm{rep}}$ of microcombs, a high update rate (close to 100 MHz) can be obtained, and a fast-moving target can be clearly captured after the data process based on the Fourier transform. In the same year, Suh et al. exploited dual-comb interferometry to measure distances with high precision, and Hilbert transform was used to pick up the peak position of the interferograms.39 Particularly, the configuration of the common pump and common cavity maintains the high coherence between the two combs. In 2020, Riemensberger et al. developed a frequency modulation continuous wave (FMCW) lidar using microcombs, and each comb tooth can work as one channel, giving rise to the 3D massive parallel ranging with one comb source.40 Importantly, the sweep rate of the optical frequency can reach tens or even hundreds of megahertz, indicating the potential for future ultrafast microcomb lidar. However, a photodetector (PD) array is needed at the receiving side to acquire each individual channel. After that and in 2022, Lukashchuk et al. resolved this limitation by the dual-comb method, and 64 optical channels can be easily separated, although only one PD is designed.41 In addition, microcombs can be also used as the calibrating source to monitor the optical frequency in traditional FMCW lidar42 to further improve the measurement performance. As is well known, plenty of optical states (which are potentially useful to the distance measurement) can be experienced before reaching the SMC in the MRR. In 2023, the chaotic state was used to determine the distances based on the intensity cross correlation.43^,44 As a result of the excellent property of the intensity autocorrelation, the cross talk between the different channels can hardly be observed. The distance and the velocity can be simultaneously determined with millimeter-level precision. Although the precision could not achieve the micro- or nanometer region like the soliton state, chaos ranging has provided new ideas for microcomb applications.

Basically, microcombs can take charge of all the ranging methods demonstrated before. However, the full stabilization of microcombs is relatively challenging.45^,46 By far, the $f_{\mathrm{rep}}$ can be locked by feedback controlling the optical pump.47^–49 In another way, the passive injection locking via an electro-optic modulator (EOM) has been also demonstrated,50^,51 and the locking performance can track the frequency reference well. Nonetheless, the stabilization of the $f_{\mathrm{ceo}}$ of microcomb is not easy because the generation of the supercontinuum spectrum is difficult due to the low pulse peak power. Several groups have made great efforts to directly access the octave spectra by careful dispersion engineering,52^,53 giving rise to strong dispersive waves. Nevertheless, it is not easy to obtain the $f_{\mathrm{ceo}}$ signal with a high signal-to-noise ratio because of the relatively low power of the dispersive waves. Taking account of the microcomb ranging, optical distance measurement using a fully stabilized microcomb has not been demonstrated so far.

In this paper, we describe a ranging system involving two kinds of comb lasers, with submicrometer precision and hundreds of kilohertz update rate. With the help of one fully stabilized fiber comb, the pump laser can be locked to the hydrogen maser so as to lock the $f_{\mathrm{ceo}}$. Since the $f_{\mathrm{rep}}$ (48.943783800 GHz) can be easily locked by injection locking, one fully stabilized microcomb source has been developed. We subsequently developed a ranging system with hybrid coherent comb lasers, microcomb, and fiber comb. Here, the fiber comb can not only serve as the bridge to the hydrogen maser but also operate as the local oscillator (LO) in dual-comb interferometry. The direct advantages of our system are the distance measurement can be traceable to the SI unit benefiting from the fully locked microcomb, and in addition, the high coherence between the microcomb and fiber comb can be maintained for a long term. As a result of the large difference between the repetition frequencies, the update rate of the hybrid-comb ranging system can be improved by hundreds of folds, in contrast to the dual-fiber-comb system with about a 250-MHz repetition rate. Our hybrid approach can not only leverage the advantage of being easy to lock (inherited from the fiber comb) but also allow the ultrarapid measurement due to the microcomb.

Figure 1 shows the experimental setup for the fully stabilized single SMC generation. A narrow-linewidth laser at 1560.49 nm is frequency-shifted by the acousto-optic modulator (AOM) and then is split into two beams. One beam passes through an intensity EOM working as the pump laser. The $f_{\mathrm{rep}}$ can be actively stabilized to the hydrogen maser using injection locking. The other beam is frequency-shifted by AOM, which acts as the auxiliary laser. The on-chip power of the two beams (pump laser and auxiliary laser) is boosted to $\sim 450\mathrm{mW}$ using two erbium-doped fiber amplifiers (EDFAs). The circulators inhibit the strong light from transmitting to the EDFAs. The single SMCs coupled out from the drop port are measured using an optical spectrum analyzer (OSA), a fast PD, and an electrical spectrum analyzer (ESA). The microwave sources used in the experiment are synchronized to the hydrogen maser. See the details in Sec. 4.1.

In our experiments, the single SMC is generated by the auxiliary-laser-based thermal balance method. The spectra of the microcomb has a ${\mathrm{sech}}^2$ envelope with about a 100 nm optical spectral band, as shown in Fig. 2(a). To prevent the PD from saturating, the pump laser is filtered out using a fiber Bragg grating (FBG). The detected $f_{\mathrm{rep}}$ is amplified and mixed with the output of the signal generator (49 GHz) to obtain the beat signal, as shown in Fig. 2(b). When the repetition frequency is well locked, the signal-to-noise ratio can be improved to better than 70 dB, and the linewidth can be greatly narrowed, compared with the free-running case. The single-sideband (SSB) phase noise is then measured by a phase noise analyzer, as shown in Fig. 2(c). We find the SSB phase noise of the locked $f_{\mathrm{rep}}$ is $-57\mathrm{dBc}/\mathrm{Hz}$ at 1 Hz, which is reduced by about 97 dB compared to the free-running $f_{\mathrm{rep}}$. When the offset frequency is larger than 30 kHz, the SSB phase noises are almost the same for both cases (the locked $f_{\mathrm{rep}}$ and the free-running $f_{\mathrm{rep}}$). We then measure the locked $f_{\mathrm{rep}}$ for a long time to characterize the long-term stability with a frequency counter; the results are shown in Fig. 2(d). The standard deviation can reach 1.5 mHz in about 7000 s continuous measurement. The Allan deviation is $3.1\times {10}^{-14}@1\mathrm{s}$ and $2.4\times {10}^{-15}@100\mathrm{s}$, as shown in Fig. 2(e), which is consistent with the stability of the signal generator referenced to the hydrogen maser. In contrast, the stability of the free-running $f_{\mathrm{rep}}$ is $2.1\times {10}^{-9}@1\mathrm{s}$ and $4.8\times {10}^{-9}@100\mathrm{s}$. With the injection locking, the repetition frequency will follow the microwave frequency from the signal generator, which provides a new method for the fine adjustment of the repetition frequency. Figure 2(f) shows the results of the $f_{\mathrm{rep}}$ adjustment with 20 mHz step size, and the letters XIOPM (the abbreviation of Xi’an Institute of Optics and Precision Mechanics) can be depicted clearly. Figure 2(g) shows the linear modulation with $122\mathrm{kHz}/\mathrm{s}$ speed and 61 kHz range. Robust and flexible control of the repetition rate is valuable for many applications.

Next, we try to lock the $f_{\mathrm{ceo}}$. Since the pump laser is actually one of the comb teeth, we can lock the pump laser to the fully stabilized fiber comb. In this case, the single SMC can be fully stabilized. The in-loop beat frequency between the pump laser and the fiber comb is shown in Fig. 3(a), whose signal-to-noise ratio can be well above 35 dB. Meanwhile, an additional fiber comb is used to beat with the SMC, to evaluate the noise and stability in the out loop. Figure 3(b) shows the out-loop beat frequency with about a 35-dB signal-to-noise ratio. We first measure the SSB phase noises of the pump laser, as shown in Fig. 3(c). After locking, the SSB phase noise of the pump laser is $30\mathrm{dBc}/\mathrm{Hz}$ at 1 Hz offset frequency, which is actually reduced by 40 dB compared to the free-running situations. At the offset frequency from 10 Hz to 10 kHz, the SSB phase noises show similar values for both the free-running and the locked cases. There are obvious phase noise spikes around 100 and 20 kHz due to the fast- and slow-loop servo systems bandwidth as well as the performance of electrical devices in the phase-locking loop.54 The SSB phase noise of the 20th comb tooth (1568.48 nm) exhibits features similar to that of the pump laser, except that at the offset frequency larger than about 60 kHz, the noise is obviously higher than that of the pump laser. This is because the optical power of the 20th comb tooth is much lower, and two stages of optical power amplification are used to obtain the beat note with the fiber comb. In addition, the beat note is then electrically amplified so as to be identified by the phase noise analyzer. We consider that the optical and electrical power amplifications make great contributions to the phase noise at offset frequencies larger than 60 kHz.

In the stability evaluation, the long-term measurement result of the pump laser is shown in Fig. 3(d). We find the standard deviation of 24.83 Hz, and the Allan deviation can reach $1.4\times {10}^{-13}@1\mathrm{s}$ and $1.1\times {10}^{-14}@100\mathrm{s}$. We then measure the stability of the 20th comb tooth as shown in Fig. 3(e); the standard deviation is 24.81 Hz. The Allan deviation is at the same level as that of the pump laser, which is $1.4\times {10}^{-13}@1\mathrm{s}$ and $1.1\times {10}^{-14}@100\mathrm{s}$. In contrast, the stabilities of the pump laser and the 20th comb tooth are all above ${10}^{-10}$ in the free-running case, as shown in Fig. 3(f). These results show that our microcomb has been fully locked to the hydrogen maser.

In the distance meter, the microcomb serves as the signal source, which is split into the measurement arm and the reference arm. The fiber comb operates as the LO. Two PDs are used to detect the optical interferograms. We carried out absolute distance measurement in our lab; the results are shown in Fig. 4. The environmental conditions are 25.1°C, 98.7 kPa, and 53.1% humidity, and the group refractive index of air is 1.00026026 corrected by the Ciddor formula.55Figure 4(a) shows the interferograms of the hybrid-comb interferometry, and we find that the update rate is about 245.537 kHz, which is determined by the repetition frequencies of the microcomb and the fiber comb. In our experiments, the repetition frequency of the fiber comb $f_{\mathrm{rep},\mathrm{FC}}$ is 249.711929 MHz, and that of the microcomb $f_{\mathrm{rep},\mathrm{MC}}$ is 48.943783800 GHz after injection locking, which is about 196 times $f_{\mathrm{rep},\mathrm{FC}}$. The Fourier transform of the interferograms is shown in Fig. 4(b), and a microwave comb can be obtained with 245.537 kHz frequency separation. The power of the center frequency at 12 MHz is relatively lower because the pump laser is degraded. We measure the distance for a relatively long term; the results are shown in Fig. 4(c). Figure 4(d) shows the comparison to the reference interferometer with a 0.05-m step size. At each position, the distances are fast-measured 10 times. The black solid points show the differences between the average of 10 individual measurements and the reference values, and the error bar shows the standard deviation with an averaging time of $40\mu \mathrm{s}$. The uncertainty can be within $2.5\mu \mathrm{m}$ in the 1-m range limited by the space of our lab. The Allan deviation can reach $3.572\mu \mathrm{m}$ at $4.136\mu \mathrm{s}$ (i.e., 245.537 kHz update rate), and further can be improved to 432 nm at $827.2\mu \mathrm{s}$ averaging time (i.e., 1.208 kHz update rate), showing that our system can support submicrometer precision and rapid distance measurement in Fig. 4(e). The uncertainty evaluation can be seen in Sec. 4.

To examine the ability of the rapid distance measurement, we measure a spin disk with grooves of different depths. The radius of the disk is about 15 cm, and the rotation speed is about $\mathrm{10,000}\mathrm{r}/\min$. Therefore, the line speed of the disk edge can reach about $150\mathrm{m}/\mathrm{s}$. The results are shown in Fig. 4(f), and the grooves with 3.104 and 6.508 mm depths can be clearly captured. We use the measurement results of the coordinate measuring machine (CMM, Leica PMM-Ultra) as the reference values, and the measurement uncertainty can be within $2.5\mu \mathrm{m}$.

Submicrometer measurement mainly serves precision manufacturing and micromachining technology to characterize the geometric and mechanical properties of microstructures. The effect of geometric manufacturing accuracy on the final performance is nonlinear. Any precision mismatches between parts or components in the process of manufacturing sequence will transfer to the whole machine. It is necessary to have the measurement capability of large-range, high-precision, highly dynamic, real-time monitoring of the whole process. On the other hand, the changes of material properties, geometric structure, environmental conditions, and other factors due to the stress should be monitored in real time to avoid the overall performance of putting the equipment out of control. Furthermore, the establishment of a complete submicrometer measurement traceability system is a guarantee for high-tech industries, advanced manufacturing, and scientific development. Microcombs, the chip-scale comb lasers, have been finding increasing applications, from spectroscopy to precision measurement and metrology. Together with the rapid progress of microcombs, fiber combs are becoming mature and user-friendly nowadays. We, for the first time we believe, have developed a hybrid-comb distance meter that allows for high precision and high-speed ranging. Our hybrid strategy can utilize the merit of the microcombs, leading to ultrarapid distance measurement (hundreds of kilohertz or megahertz update rate). It can also operate in a fully stabilized manner, benefiting from the locking scheme of the fiber comb. Our results indicate the high coherence between the two types of combs (even a large repetition frequency difference exists), and the precision can reach the submicrometer level with coherent averaging. Nowadays, on-chip high gain waveguides or semiconductor optical amplifiers have been well developed, which can replace the gain fiber to realize compact mode-locked laser-based OFCs.56^,57 The gain waveguides could be integrated with a high-quality microcavity for integrated dual-hybrid comb formation. Viewed more broadly, the hybrid technique is universally applicable for different kinds of combs, and our work can provide a new insight into the technique of dual-comb interferometry, potentially giving rise to the promotion of its overall performance.

In conclusion, we develop a fully stabilized microcomb source with the help of one fiber comb. The $f_{\mathrm{rep}}$ of 48.943783800 GHz is actively stabilized by injection locking with a 1.5 mHz standard deviation, and the relative stability can reach $3.1\times {10}^{-14}@1\mathrm{s}$ and $2.4\times {10}^{-15}@100\mathrm{s}$. The pump laser (actually one tooth of the microcomb) is locked to one tooth of the fiber comb, and consequently, all the comb teeth have been referenced to the hydrogen maser. We measure the 20th tooth with respect to the pump laser to evaluate the locking performance. The results indicate the 24.81 Hz standard deviation, and the Allan deviation can achieve $1.4\times {10}^{-13}@1\mathrm{s}$ and $1.1\times {10}^{-14}@100\mathrm{s}$. Since we include a fiber comb in our whole system, a hybrid-comb source with high coherence has been inherently obtained. Subsequently, we carry out optical ranging using the dual-hybrid comb interferometry. As a result of the large effective difference between the repetition frequencies of the hybrid combs, the measurement speed can be enhanced by 196 folds (i.e., $f_{\mathrm{rep},\mathrm{MC}}/f_{\mathrm{rep},\mathrm{FC}}$), in contrast to the dual-fiber-comb system with about 250 MHz repetition rate. The experimental results show the submicrometer precision in the distance measurement, and a fast spinning disk with about $150\mathrm{m}/\mathrm{s}$ can be clearly measured, representing the rapid measurement speed.

The integrated add-drop high-Q MRR is fabricated on a CMOS-compatible, high-index doped silica glass platform and packaged in a butterfly package with a thermo-electric cooler. The cross sections of the ring and bus waveguides are both $2\mu \mathrm{m}\times 3\mu \mathrm{m}$. The MRR exhibits anomalous dispersion in the communication band. In our experiments, the auxiliary optical field and pump laser locate in the same TM mode and are counter-coupled into the MRR to realize thermal balance for the SMC generation. By sequentially decreasing and increasing the operation temperature, the soliton switching and annihilation can be clearly recognized from the optical spectra. The $Q$ factors of TM00 mode are $1.69\times {10}^6$. The MRR is pigtailed with a standard fiber array with coupling loss of about 2.5 dB per facet and insert loss from other passive optical devices of about 2.0 dB. The operation temperature of the MRR can be precisely tuned through an external TEC controller.

In our experiments, the microcomb is pumped by a continuous-wave (CW) laser (NKT Koheras ADJUSTIK E15, 1560.49 nm wavelength, 100 Hz linewidth, 40 mW output power). The output of the CW laser is shifted by 100 MHz by the first AOM and works as the pump laser. In the phase-locking loop, the pump laser is combined with the fiber comb (Menlo Systems FC1500, 1560 nm center wavelength, 50 nm bandwidth, about 250 MHz repetition frequency) at a polarization beam splitter, and then diffracted at a grating. After a pinhole to improve the signal-to-noise ratio, a PD is used to detect the beat notes between the pump laser and the fiber comb. Consequently, the amplified beat is fed into a servo system (Moku Lab), to produce an error signal. Finally, the AOM and piezoelectric ceramic transducer inside the CW laser are employed as the actuating element to lock the pump laser frequency.

The injection-locking technique provides an easy way to stabilize the repetition frequency of the microcomb. The second-order sidebands of the EOM are utilized for repetition frequency injection locking. Therefore, the driving frequency of the EOM is one half of $f_{\mathrm{rep},\mathrm{MC}}$, i.e., 24.471891900 GHz in our experiments. A high-speed PD (Finisar XPDV2120RA, 50 GHz) is used to detect the repetition frequency $f_{\mathrm{rep}}$, which is mixed with the output of the signal generator (R&S SMF100B), and the intermediate frequency signal of the mixer is filtered out using a low-pass filter and measured by the phase noise analyzer (R&S FSWP26). The repetition frequency stability is evaluated by a frequency counter (Keysight 53230A).

Subsequently, we use an additional fiber comb to evaluate the frequency stability of the microcomb lines. Taking the 20th comb tooth as an example, an optical tunable filter (SATEC OTF-950) is used to pick up the individual mode, which is then amplified and beats with the additional fiber comb. Finally, the beat note is used to measure the long-term stability and the SSB phase noise.

We have developed a fully stabilized microcomb source, and one self-referenced fiber comb is involved so as to lock the pump laser. Here, we actually have gotten one set of dual-comb sources with high coherence. In general, dual-comb interferometry, which has been widely used in metrology and spectroscopy, relies on two frequency combs with slightly different repetition frequencies. For example, the repetition frequency difference is several kilohertz for the fiber combs with $f_{\mathrm{rep}}$ of hundreds megahertz.18 This value can be increased to several megahertz when the $f_{\mathrm{rep}}$ reaches tens of gigahertz in the cases of electro-optic combs14^,15 or microcombs.38 In our work, one set of hybrid dual-comb sources can be naturally aroused, which comprises a microcomb ($f_{\mathrm{rep},\mathrm{MC}}$, 48.943783800 GHz) and a fiber comb ($f_{\mathrm{rep},\mathrm{FC}}$, 249.711929 MHz). The distances can be precisely determined using this hybrid dual-comb source, despite the large repetition frequency difference. We note that dual-comb interferometry with quasi-integer-ratio repetition rates has been previously described.58^,59 Nonetheless, we report here the coherent dual-hybrid-comb interferometry for distance measurement for the first time. The comparison among the different schemes of dual-comb interferometry is shown in Fig. 5. Table 1 shows the comparison with other dual-comb ranging methods.

In the distance measurement, the amplified microcomb is split into two parts. One part is used as a measurement beam after passing through the circulator, and the other part is the reference arm. The fiber comb actually serves as the LO. Two PDs are exploited to detect the interferograms, which are measured by an oscilloscope (LeCroy HDO6104A).

Assuming the field $E_{\mathrm{MC}}$ of the microcomb can be expressed as $$E_{\mathrm{MC}}=\sum _p{A}_{p,\mathrm{MC}}\exp [2\pi i(p\times f_{\mathrm{rep},\mathrm{MC}}+f_{\text{pump},\mathrm{MC}})t+i{\phi }_{0,\mathrm{MC}}],$$(1)where $p$ is the relative mode number with respect to the pump resonance mode, $A_{p,\mathrm{MC}}$ is the field amplitude of the $p$th comb tooth, $f_{\mathrm{rep},\mathrm{MC}}$ is the repetition frequency of the microcomb, $f_{\text{pump},\mathrm{MC}}$ is the pump laser frequency of the microcomb, and ${\phi }_{0,\mathrm{MC}}$ is the initial phase.

Similarly, the field $E_{\mathrm{FC}}$ of the fiber comb can be written as $$E_{\mathrm{FC}}=\sum _q{A}_{q,\mathrm{FC}}\exp [2\pi i(q\times f_{\mathrm{rep},\mathrm{FC}}+f_{\text{center},\mathrm{FC}})t+i{\phi }_{0,\mathrm{FC}}],$$(2)where $A_{q,\mathrm{FC}}$ is the field amplitude of the $q$th comb tooth, $f_{\mathrm{rep},\mathrm{FC}}$ is the repetition frequency of the fiber comb, $f_{\text{center},\mathrm{FC}}$ is the center frequency of the fiber comb, ${\phi }_{0,\mathrm{FC}}$ is the initial phase, and $q$ is the relative mode number with respect to the center frequency of the fiber comb. For the convenience of description, we assume that $f_{\text{pump},\mathrm{MC}}$ is close to $f_{\text{center},\mathrm{FC}}$, with the beat of $f_b$. Consequently, the heterodyne frequencies between the microcomb and the fiber comb can be expressed as $$\begin{gathered} E_{\text{beat}}\propto \sum _p\sum _q{A}_{p,\mathrm{MC}}{A}_{q,\mathrm{FC}}\exp [2\pi i(p\times f_{\mathrm{rep},\mathrm{MC}}-q\times f_{\mathrm{rep},\mathrm{FC}}+f_{\text{pump},\mathrm{FC}}-f_{\text{center},\mathrm{FC}})t+i{\phi }_{0,\mathrm{MC}}-i{\phi }_{0,\mathrm{FC}}] \\ \propto \sum _p\sum _q\exp [2\pi i(p\times f_{\mathrm{rep},\mathrm{MC}}-q\times f_{\mathrm{rep},\mathrm{FC}}+f_b)t+i{\phi }_{0,\mathrm{MC}}-i{\phi }_{0,\mathrm{FC}}]. \end{gathered}$$(3)

A number of beat notes can be obtained based on Eq. (3). In our experiments, $f_{\mathrm{rep},\mathrm{MC}}$ is 48.943783800 GHz, and $f_{\mathrm{rep},\mathrm{FC}}$ equals 249.711929 MHz, with the ratio of 196 (i.e., $\text{round}(f_{\mathrm{rep},\mathrm{MC}}/f_{\mathrm{rep},\mathrm{FC}})$, and round means the nearest integer). First, we consider the mode $f_{\text{pump},\mathrm{MC}}+f_{\mathrm{rep},\mathrm{MC}}$ (the first mode), and correspondingly, the nearest mode in the fiber comb goes to $f_{\text{center},\mathrm{FC}}+196\times f_{\mathrm{rep},\mathrm{FC}}$. The beat frequency can be calculated by $f_b+f_{\mathrm{rep},\mathrm{MC}}-196\times f_{\mathrm{rep},\mathrm{FC}}$. Let us go further and consider the second mode of the microcomb (i.e., $f_{\text{pump},\mathrm{MC}}+2\times f_{\mathrm{rep},\mathrm{MC}}$). The nearest mode in the fiber comb is then $f_{\text{center},\mathrm{FC}}+2\times 196\times f_{\mathrm{rep},\mathrm{FC}}$, and the beat frequency is then $f_b+2\times (f_{\mathrm{rep},\mathrm{MC}}-196\times f_{\mathrm{rep},\mathrm{FC}})$. Consequently, we can reach a group of beat notes, which can be expressed as $f_b\pm p\times (f_{\mathrm{rep},\mathrm{MC}}-196\times f_{\mathrm{rep},\mathrm{FC}})$ ($p\sim \pm 100$). We find that all the microcomb modes can be downconverted to the radio frequency (RF) region by multiheterodyne interferometry. $f_b$ can be easily tuned by the phase-locking loop, which is 12 MHz in our experiments. The value of $f_{\mathrm{rep},\mathrm{MC}}-196\times f_{\mathrm{rep},\mathrm{FC}}$ is 245.537 kHz, which is actually the frequency separation of this RF comb. To avoid spectral leakage, $f_b$ is recommended to be larger than $p\times (f_{\mathrm{rep},\mathrm{MC}}-196\times f_{\mathrm{rep},\mathrm{FC}})$.

In fact, the mode $f_{\text{pump},\mathrm{MC}}$ can beat with all the modes of the fiber comb within the PD band. Considering the situation that $f_{\text{pump},\mathrm{MC}}$ beats with the mode of $f_{\text{center},\mathrm{FC}}+f_{\mathrm{rep},\mathrm{FC}}$, the beat frequency $f_b$ is then updated to $f_b+f_{\mathrm{rep},\mathrm{FC}}$. Similarly, a new group of beat notes can make an appearance, which can be written as $f_b+f_{\mathrm{rep},\mathrm{FC}}\pm p\times (f_{\mathrm{rep},\mathrm{MC}}-196\times f_{\mathrm{rep},\mathrm{FC}})$. In addition, if $f_{\text{pump},\mathrm{MC}}$ beats with the mode of $f_{\text{center},\mathrm{FC}}-f_{\mathrm{rep},\mathrm{FC}}$, the beat frequency will be $f_{\mathrm{rep},\mathrm{FC}}-f_b$, and in consequence, the corresponding group of beat notes can be expressed as $f_{\mathrm{rep},\mathrm{FC}}-f_b\pm p\times (f_{\mathrm{rep},\mathrm{MC}}-196\times f_{\mathrm{rep},\mathrm{FC}})$. Finally, a series of groups of beat frequency, at different center frequencies but with the same frequency spacing, can be produced as shown in Fig. 5(e). The center frequency can be defined as $f_b+m\times f_{\mathrm{rep},\mathrm{FC}}$ and $m\times f_{\mathrm{rep},\mathrm{FC}}-f_b$ and is smaller than $f_{\mathrm{rep},\mathrm{MC}}/2$. $m$ is an integer. We note that, despite the groups of the beats located at the different frequencies along the frequency axis, each group can actually be individually picked up and utilized to determine the distances. In our experiments, we use a low-pass filter (DC-50MHz) to acquire the group at the center frequency of $f_b$.

In the time domain, the fiber comb scans the microcomb, like the optical sampling, to generate the interferograms. The difference is that $f_{\mathrm{rep},\mathrm{MC}}$ is not slightly detuned with $f_{\mathrm{rep},\mathrm{FC}}$, but is much larger than $f_{\mathrm{rep},\mathrm{FC}}$ (196 times). As a result, 196 interferograms can be generated, when the fiber comb (LO) scans $1/f_{\mathrm{rep},\mathrm{FC}}$ in the time line. In contrast to the dual-fiber-comb interferometry, one interferogram appears with each scan of $1/f_{\mathrm{rep},\mathrm{FC}}$. Therefore, the measuring speed (the update rate) can be enhanced by 196 folds, corresponding to the frequency separation of $f_{\mathrm{rep},\mathrm{MC}}-196\times f_{\mathrm{rep},\mathrm{FC}}$ in the frequency domain.

In our ranging system, the distance can be calculated as $$L=\frac{1}{2}·\frac{c}{n_g}·(\frac{N}{f_{\mathrm{rep},\mathrm{MC}}}+\tau ·\frac{f_{\mathrm{rep},\mathrm{MC}}-196\times f_{\mathrm{rep},\mathrm{FC}}}{f_{\mathrm{rep},\mathrm{MC}}})=\frac{1}{2}·\frac{c}{n_g·f_{\mathrm{rep},\mathrm{MC}}}·(N+\tau \times \mathrm{\Delta }{f}_{\mathrm{eff}}),$$(4)where we define the parameter $f_{\mathrm{rep},\mathrm{MC}}-196\times f_{\mathrm{rep},\mathrm{FC}}$ as the effective repetition frequency difference, which is referred to as $\mathrm{\Delta }{f}_{\mathrm{eff}}$. $N$ is the integer part of the pulse-to-pulse length, $c$ is the light speed in vacuum, and $n_g$ is the group refractive index of air. $\tau$ is the real-time delay, which can be measured by Hilbert transform or Fourier transform.

In Eq. (4), the integer $N$ can be precisely determined with uncertainty below 0.5 by changing the repetition frequency of the microcomb,60 and thus will not contribute to the measurement uncertainty. Consequently, the measurement uncertainty of the distance measurement is related to the parameters of the group refractive index $n_g$, the repetition frequency of the microcomb $f_{\mathrm{rep},\mathrm{MC}}$, the real-time delay $\tau$, and the effective repetition frequency difference $\mathrm{\Delta }{f}_{\mathrm{eff}}$, as calculated in the following equation:$$\begin{gathered} u_L^2={\left(\frac{\partial L}{\partial n_g}{u}_{n_g}\right)}^2+{\left(\frac{\partial L}{\partial f_{\mathrm{rep},\mathrm{MC}}}{u}_{{\mathrm{f}}_{\mathrm{rep},\mathrm{MC}}}\right)}^2+(\frac{\partial L}{\partial \tau }{u}_{\tau }{)}^2+(\frac{\partial L}{\partial \mathrm{\Delta }{f}_{\mathrm{eff}}}{u}_{\mathrm{\Delta }{f}_{\mathrm{eff}}}{)}^2 \\ ={\left(\frac{L}{n_g}{u}_{n_g}\right)}^2+{\left(\frac{L}{f_{\mathrm{rep},\mathrm{MC}}}{u}_{f_{\mathrm{rep},\mathrm{MC}}}\right)}^2+(\frac{1}{2}·\frac{c}{n_g}·\frac{\mathrm{\Delta }{f}_{\mathrm{eff}}}{f_{\mathrm{rep},\mathrm{MC}}}{u}_{\tau }{)}^2+{(\frac{1}{2}·\frac{c}{n_g}·\frac{\tau }{f_{\mathrm{rep},\mathrm{MC}}}{u}_{\mathrm{\Delta }{f}_{\mathrm{eff}}})}^2. \end{gathered}$$(5)

The first term on the right side is due to the group refractive index of air. In general, the measurement of the air refractive index is based on the empirical equation (Ciddor formula in this work) using environmental sensors. The uncertainty of the Ciddor formula itself is about $2\times {10}^{-8}$. The measurement uncertainties of the temperature, the pressure, and the relative humidity are determined by the uncertainties of the sensors, the environmental stability, and the homogeneity, which turns out to be 27 mK, 13 Pa, and 1.7%, respectively. Correspondingly, the uncertainties due to the measurement of the environmental parameters are $2.5\times {10}^{-8}$, $3.3\times {10}^{-8}$, and $2\times {10}^{-8}$, respectively. Therefore, the uncertainty of $n_g$ can be combined to be $5\times {10}^{-8}$. The second term is related to the uncertainty of the repetition frequency, which is well-locked to the hydrogen maser. Since the stability of the hydrogen maser is better than ${10}^{-13}$ at 1 s averaging time, this part can be estimated as ${10}^{-13}·\mathrm{L}$, which can be neglected in fact. The third term is due to the measurement of the real-time delay, which can be evaluated by the standard deviation. In our experiments, the standard deviation can reach $1.7\mu \mathrm{m}$ with $40\mu \mathrm{s}$ averaging time. We note that the part can also include the effect due to the fiber fluctuation and the stage vibration, despite the fact that we place the devices into a well-controlled enclosure. The last term is due to the uncertainty of the parameter $\mathrm{\Delta }{f}_{\mathrm{eff}}$. Here, we can give an upper limit of the fourth term, where $\tau$ equals $1/\mathrm{\Delta }{f}_{\mathrm{eff}}$ (i.e., the maximum value). $\mathrm{\Delta }{f}_{\mathrm{eff}}$ is equal to $f_{\mathrm{rep},\mathrm{MC}}-196\times f_{\mathrm{rep},\mathrm{FC}}$, and both $f_{\mathrm{rep},\mathrm{MC}}$ and $f_{\mathrm{rep},\mathrm{FC}}$ are locked to the hydrogen maser. Therefore, the uncertainty of $\mathrm{\Delta }{f}_{\mathrm{eff}}$ can be also better than ${10}^{-13}$. $f_{\mathrm{rep},\mathrm{MC}}$ is about 50 GHz. In this case, the fourth term is at ${10}^{-16}\mathrm{m}$ level, which can be neglected. Since the microcomb laser and the reference distance meter do not share the same target mirror, an Abbe error could exist. The Abbe error is always below $2\mu \mathrm{m}$ during the whole stroke, corresponding to a $20\mu \mathrm{rad}$ yaw error of the rail. Finally, the combined uncertainty can be calculated as ${[(2.6\mu \mathrm{m}{)}^2+{(5\times {10}^{-8}·\mathrm{L})}^2]}^{0.5}$.

Biographies of the authors are not available.