2.1.

Fully Stabilized Single SMC Source

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.

Fig. 1

Experimental setup. ECDL, external cavity diode laser; AOM, acousto-optic modulator; EOM, electro-optic modulator; EDFA, Er-doped fiber amplifier; MRR, microresonator; TEC, temperature controller; PD, photodetector; LPF, low-pass filter; PBS, polarization beam splitter; BS, beam splitter; FBG, fiber Bragg grating; OSA, optical spectrum analyzer; ESA, electrical spectrum analyzer; PNA, phase noise analyzer; OSC, oscilloscope; SG, signal generator; Servo, moku lab; OTF, optical bandpass filter; $\lambda /2$, half-wave plate; Col, collimator; $\lambda /4$, quarter-wave plate; LI, laser interferometer; Sg, sliding guide.

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.

Fig. 2

Experimental results. (a) Optical spectrum of the single soliton. The spectrum of a ${\mathrm{sech}}^2$ shape with about 100 nm optical band can be found. (b) Electrical spectrum of the free-running repetition frequency [blue, measured with 1 kHz resolution bandwidth (RBW)] and the locked repetition frequency (red, measured with 30 Hz RBW). (c) SSB phase noise of the free-running repetition frequency (blue) and the locked repetition frequency (red). The noise floor is shown as the black line. (d) The long-term measurement of the locked repetition frequency. The gate time of the frequency counter is 1 s. (e) Relative Allan deviation of the free-running repetition frequency (green), the locked repetition frequency (red), and the signal generator at 49 GHz (black). (f) The letters XIOPM obtained by programming the locked repetition frequency. (g) Linear modulation of the locked repetition frequency.

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.

Fig. 3

Experimental results. (a) Electrical spectrum of the in-loop beat frequency between the pump laser and the fiber comb. The signal-to-noise ratio is better than 35 dB with 1 kHz RBW. (b) Electrical spectrum of the out-loop beat frequency between the pump laser and the additional fiber comb. The signal-to-noise ratio is 35 dB with 1 kHz RBW. (c) SSB phase noises of the free-running pump laser (red), the free-running 20th comb tooth (blue), the locked pump laser (green), and the locked 20th comb tooth (yellow). The noise floor is shown as the black line. (d) The long-term measurement of the locked pump laser with 1 s gate time. (e) The long-term measurement of the locked 20th comb tooth with 1 s gate time. (f) Relative Allan deviation of the free-running pump laser (black), the free-running 20th comb tooth (red), the locked pump laser (green), the locked 20th comb tooth (pink), and the hydrogen maser (blue).

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.

2.2.

Precise and Rapid Distance Measurement with Hybrid Comb Lasers

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.⁵⁵ Figure 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.

Fig. 4

Experimental results. (a) Interferograms of the hybrid-comb interferometry. (b) Fourier transform of the interferograms. (c) Ranging results at a standoff distance of 0.5 m. The red points indicate the results when the measuring time for the single measurement is $4\mu \mathrm{s}$, and the blue line represents the results with $40\mu \mathrm{s}$ averaging time. (d) Results of the distance measurement. (e) Allan deviation measured at the distance of 0.5 m. (f) Measured surface profile of the disk. The black points show the measurement results and the red lines are the results measured by CMM.

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 $2.5\mu \mathrm{m}$.

4.1.

Development of the Fully Stabilized Single SMC

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.

4.2.

Distance Measurement Using Hybrid Comb Lasers

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 combs^14,15 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.

Fig. 5

The interferometry characteristics of different types of dual-comb sources. (a) Time-domain description of dual-fiber-comb interferometry. The difference between the repetition frequencies of the signal source (SS) and LO is often at the kilohertz level, implying that the update of the distance measurement is at the kilohertz level. The advantage of this system is that it is relatively easy to develop two fully stabilized fiber combs. (b) Time-domain description of hybrid-comb interferometry. The $f_{\mathrm{rep}}$ of the microcomb is usually high at the gigahertz level, or even at the terahertz level, and in contrast, the fiber comb often holds tens or hundreds of megahertz repetition frequency. In spite of this highly different repetition frequency, hybrid-comb interferometry can occur. With the aid of the self-referenced fiber comb, the microcomb can be fully stabilized. In addition, the update rate can reach hundreds of kilohertz, or even megahertz, which is capable of rapid measurement. (c) Time-domain description of dual-microcomb interferometry. Since the high $f_{\mathrm{rep}}$, the detuning between the repetition frequencies of SS and LO can easily reach tens of megahertz. Consequently, the update rate can also achieve tens of megahertz, showing the capability of ultrafast measurement. However, it is rather challenging to fully stabilize two microcombs without the help of the self-referenced fiber comb. (d) Frequency-domain description of dual-fiber-comb interferometry. Because of the (relatively) small repetition frequency and the broad optical band, the optical band pass filter is generally required to circumvent the spectrum aliasing. The beat notes should be smaller than $f_{\mathrm{rep}}/2$, with a frequency separation of $\mathrm{\Delta }f$. $\mathrm{\Delta }f$ is the difference between the repetition frequencies. (e) Frequency-domain description of hybrid-comb interferometry. Interestingly, the beat components are composed of a series of groups with different center frequencies. An electrical filter can be used to select the different groups of the beat notes, whose phases can be used to calculate the distances. (f) Frequency-domain description of dual-microcomb interferometry. Similar to dual-fiber-comb interferometry, the beat notes are located with the separation of $\mathrm{\Delta }f$. The optical bandpass filter is not strictly required since the repetition frequency of microcomb is large.

Table 1

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

Similarly, the field $E_{\mathrm{FC}}$ of the fiber comb can be written as

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

4.3.

Uncertainty Budget

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:

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 ${[(2.6\mu \mathrm{m}{)}^2+{(5\times {10}^{-8}·\mathrm{L})}^2]}^{0.5}$.