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Abstract
We demonstrate a sub-ps resolution clock-offset measurement based on linear optical sampling technique via a 114 km fiber link by transferring a dual optical frequency comb. The time deviation between two distance clocks is 110 fs at 1 s and 22 fs at 100 s averaging, and the standard deviation of the measured clock offset is 237 fs. This sub-ps level of clock offset measurement should benefit many time synchronization applications via long fiber links.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High precision frequency dissemination and time synchronization have broadened applications in many scientific and industrial fields, such as high precision navigation and positioning [1,2], coherent radar array [1], detection of basic physics constants [3–5], and clock-based geodesy [5–7]. On the other hand, the progress in atomic/optical clocks has driven the frequency instability to $\textrm{1}{\textrm{0}^{\textrm{ - 15}}}\textrm{/s}\;\sim 1{\textrm{0}^{\textrm{ - 19}}}\textrm{/s}$ [8–10]. Yet the accuracy of transferring these frequency standards under conventional schemes based on free-space microwave propagation does not meet the requirement for comparison and distribution of such high-quality frequency standards.
As the fiber link is more insusceptible to the ambient perturbations in comparison with free space propagation, it has attracted extensive studies on the frequency dissemination and comparison over optical fiber networks [11–14]. Through the fiber link as long as 1840 km the optical time and frequency transfer have been demonstrated [15,16], which claimed a frequency transfer fractional instability of $\textrm{1}{\textrm{0}^{\textrm{ - 18}}}$. High-precision frequency signal to multiple stations was also demonstrated over 92 km fiber link with frequency Allan deviation of $\textrm{8 } \times \textrm{1}{\textrm{0}^{\textrm{ - 16}}}\textrm{/s}$ [17]. However, the demonstrated system only targeted to establish highly precise frequency dissemination. At the same time, it is much more challenging to synchronize two distant optical timescales. One of the conventional ways is to count the 1 pps (pulse per second) time ticker by a time interval counter (TIC) or Manchester-coded pseudo-random binary sequence with a Photo Detector (PD) [14,18]. Those electronics-based devices (PD with TIC) are limited to 10 ps ∼ 100 ps resolution and 1 ps instability [14,19], which is a barrier for further high precision synchronization.
In order to overcome the limitation of PD-TIC, the asynchronous linear optical sampling (LOS) with dual-comb has been proposed for time-offset measurement and synchronization. Similar to all other dual-comb applications, the LOS is characterized by its femtosecond resolution and large scanning range [20,21], and has been applied to clock-offset measurement [22–24] particularly in free-space two-way time transfer [25–28].
Compared with the free-space, synchronization through optical fibers is more attractive because it can transfer a long distance without suffering from the diffraction and the turbulence of atmosphere. However, there are some potential difficulties in long fiber links using LOS, the fiber loss and the dispersion, which results in the low contrast and asymmetrical LOS signals.
To implement the LOS timing determination in long fiber links, we took the following measures:
First, to alleviate the pulse broadening due to the fiber dispersion, beside the dispersion compensation fiber, we applied the DWDM as the spectral filter, to slice the pulse spectrum width to 100 GHz, as well as to isolate the counter-propagating signals in a single optical fiber.
Second, we designed an effective envelope extraction and Gaussian fitting algorithm to determine the center timing of the LOS interferogram that reduced the timing uncertainty due to the residual dispersion and noise acquired in long fiber transmission, which precisely determines the arrival timing of the optical pulses. A coarse timing measurement with conventional TIC was also applied.
With the above techniques, we demonstrated a sub-ps time synchronization by LOS over fiber link. The time deviation is 110 fs/s between two clocks. The sources for the uncertainty in the measurement were analyzed.
To the best of our knowledge, this is the first demonstration on clock-offset measurement by LOS via long fiber link, except for the part of our recent conference presentations [29–31].
2. LOS based two-way time transfer technique
2.1 Dual-comb two-way time transfer
Let us briefly review the formalization of the two-way time transfer and comparison. Consider two clocks at separate sites A and B. Suppose Site A transmits a pulse at its time of ${T_{\textrm{AA}}}$ to site B. Its measured arrival time according to Site B’s clock is ${T_{\textrm{AB}}} = {T_{\textrm{link}}} - \Delta {t_{\textrm{AB}}} + {T_{\textrm{AA}}}$, where ${T_{\textrm{link}}}$ is the time of flight (path delay) from A to B and $\Delta {t_{\textrm{AB}}}$ is the offset between the clocks (clock difference). Simultaneously, Site B transmits a pulse at its time of ${T_{\textrm{BB}}}$ in the opposite direction to Site A, where its arrival time to site A is ${T_{\textrm{BA}}} = {T_{\textrm{link}}}\textrm{ + }\Delta {t_{\textrm{AB}}} + {T_{\textrm{BB}}}$. Since site A and site B are relatively stationary in our setting, it is reasonable to assume the link is fully reciprocal. For a fully reciprocal link, the time of flight between A to B, ${T_{\textrm{link},\textrm{A} \to \textrm{B}}}$, is equal to the time of flight in opposite direction, ${T_{\textrm{link},\textrm{B} \to \textrm{A}}}$. However, if the link is nonreciprocal, it introduces a systematic bias in $\Delta {t_{\textrm{AB}}}$ [26]. Subtraction of these two arrival times yields the clock offset:
Since the photo-detection of the incoming pulses introduces picosecond-level uncertainty, we cannot implement the simple two-way comb-pulse transfer. As discussed in Introduction, we implement linear optical sampling of the optical pulse trains. Linear optical sampling requires the introduction of a third “transfer” comb that operates at an optical pulse repetition frequency offset by $\Delta {f_\textrm{r}}$ from both clocks’ pulse train repetition frequency, ${f_\textrm{r}}$. Heterodyne detection between this transfer comb and either clock comb yields an interferogram [32]. The LOS interferogram can be regarded as a “time-offset amplifier” in the relative timing measurement. This effectively stretches real femtosecond relative timing between pulse trains by a magnifying factor of ${f_\textrm{r}}/\Delta {f_\textrm{r}}$ and with the update rate of $\Delta {f_\textrm{r}}$.
2.2 Timing offset extraction from the LOS
Previous work of LOS methods infers the relative timing between pulse trains from the peak position of the interferogram [25,26,32]. However, due to the residual high order fiber dispersion, the interferogram may not be symmetrical although the DCF is applied. As shown in Fig. 1, group delay dispersion (GDD) broadens the LOS interferogram and third-order dispersion (TOD) made the LOS interferogram asymmetric. In this way, the center position of Gaussian fitting of the envelope is not necessarily the same as its peak position of the original envelope. More importantly, the peak position of the envelope is easily distorted by noise resulted in scatter data points. We propose the following center-timing Algorithm 1 to extract the center of the LOS interferogram after Gaussian fitting.
Fig. 1. LOS interferogram without residual dispersion (left) and with residual group delay and third-order dispersion (right) between two pulse trains.
In Algorithm 1, the first three steps (normalization, upper envelope extracting, interpolating) help to remove the noise. The last two steps return the center timing instead of peak-picking used in previous works. Then the center-timing is more stable and robust in the scenario where high residual dispersions exist.
The advantage of Algorithm 1 can be seen in Fig. 2, where a typical experiment LOS interferogram and fitting curves are plotted. The center timing extraction with Algorithm 1 demonstrated a notably small fitting error with an R-square of 0.96 for the final fitted curve (black curve). While the center timing determined by the simple peak-picking from the original LOS interferogram has a higher fitting error and the R-square is 0.66 (orange curve).
Fig. 2. Original LOS interferogram (red curve) has some noisy points. The upper envelope (blue curve) reduce the influence of noise. The Gaussian fitting with algorithm 1 (black curve) better matches the interferogram than directly fitting the peak of the original interferogram (orange curve).
2.3 Clock-offset measurement equation
In the arrangement as shown in Fig. 3, we can measure three LOS interferograms (1) local interferogram between comb A pulses and comb X pulses, ${I_{\textrm{AX}}}$, at site A; (2) interferogram between comb X pulses (transmitted from site A to site B) and comb B pulses, ${I_{\textrm{XB}}}$, at site B; (3) their analogous interferogram between comb B pulses (transmitted from site B to site A) and comb X pulses, ${I_{\textrm{BX}}}$, at site A; By use of Algorithm 1, we can extract the center-timing of ${I_{\textrm{AX}}}$, ${I_{\textrm{XB}}}$, ${I_{\textrm{BX}}}$ as ${t_{p\textrm{AX}}}$, ${t_{p\textrm{XB}}}$, ${t_{p\textrm{BX}}}$.
Figure 4 illustrated the LOS interferogram between two combs. Comb A generates a pulse train that is coherent with the clock at a repetition rate of ${f_\textrm{r}}$. Comb X generates a pulse train with a repetition rate of ${f_\textrm{r}} + \Delta {f_\textrm{r}}$. LOS interferogram between comb A and comb X detects the arrival of the received pulses analogy with sampling oscilloscopes. The comb A pulse train optically samples (with a sample rate of ${f_\textrm{r}}$) the received comb X pulses at varying delays to generate an interferogram (measured data). Therefore, an interferogram is generated by many successive partially overlapping pulses of comb A and comb X. The repetition rate of the interferogram is $\Delta {f_\textrm{r}}$.
The equations for two-way time transfer are briefly outlined below, referencing [26]. In LOS, the center position of the interferogram (with maximum light intensity) denotes the matched interference between two pulses with the phase difference of zero modulo $2\mathrm{\pi }$, as shown in Fig. 4. They repeat at the difference in repetition rates, $\Delta {f_\textrm{r}}$. Consider the stream of interferograms represented by ${I_{\textrm{AX}}}$. Let us introduce the integer ${p_{\textrm{AX}}}$ that counts successive interferogram centers that occurs at time $t = {t_{p\textrm{AX}}}$, so that:
For the interferograms ${I_{\textrm{XB}}}$, ${I_{\textrm{BX}}}$, we introduce the analogous integers ${p_{\textrm{XB}}}$ and ${p_{\textrm{BX}}}$ that count successive interferogram centers occurring at times ${t_{p\textrm{XB}}}$(against site B’s time base) and ${t_{p\textrm{BX}}}$ (against site A’s time base). On site B, from interferogram ${I_{\textrm{XB}}}$:
On site A, from interferogram ${I_{\textrm{BX}}}$:
To formally obtain the two-way time-transfer equation, we identify two events that are related to the arrival timing of the pulse at two sites, which occur at every update time interval,${\; }\Delta {f_\textrm{r}}$. Consider effective transmission time of the comb X from site A to site B, as recorded by the ${p_{\textrm{XB}}}$th center of the ${I_{\textrm{XB}}}$ interferogram at time $t = {t_{p\textrm{XB}}}$. In conventional time transfer of Eq. (1), we would record the departure time of the pulse from site A is labeled as ${T_{\textrm{AA}}}$ (site A’s time base) and the arrival time of the pulse from comb A at comb B as ${T_{\textrm{AB}}}$ (site B’s time base). By use of Eq. (5), from interferogram ${I_{\textrm{XB}}}$, we record the comb X’s pulse arrival time ${T_{\textrm{AB}}}$ by comb B’s phase at site B’s time base $t = {t_{p\textrm{XB}}}$ and location ${z_\textrm{B}}$:
Considering link time ${T_{\textrm{link}}}$, we record the departure time ${T_{\textrm{AA}}}$ by comb A’s phase at site A’s time base $t = ({{t_{p\textrm{XB}}} - {T_{\textrm{link}}}} )$ and location ${z_\textrm{A}}$:
In analogy, consider effective transmission time of the comb B from site B to site A, we record the departure time of comb B’s pulse as ${T_{\textrm{BB}}}$ and the arrival time of the pulse at site A as ${T_{\textrm{BA}}}$:
By connecting the Eqs. (8)–(11) and the LOS measurements Eqs. (2)–(7) to eliminate the unmeasurable phase and obtain departure/arrival time with actual measurements:
After combining Eqs. (12)–(15) with two-way time transfer Eq. (1), we find LOS based time transfer equation as following:
It is worth emphasizing that, there is a periodic waveform ambiguity risk of assigning the time of arrival to the wrong pulse. In other words, we cannot obtain the value of ${p_{\textrm{AX}}}$, ${p_{\textrm{BX}}}$ and ${p_{\textrm{XB}}}$ from LOS interferogram. To counter these problems, we used PD-TIC methods [14] for coarse but absolute time comparisons between clocks. We used the PD-TIC method to determine the value of ${p_{\textrm{AX}}}$, ${p_{\textrm{BX}}}$, ${p_{\textrm{XB}}}$, then we combine the results from the PD-TIC method and from the LOS method.
3. Experiment of two-way time transfer over the fiber
3.1 Experimental implementation
Figure 5 shows the experiment setup for synchronization over 114 km telecommunication fiber link (a combination of 100 km SMF and 14 km DCF) based on LOS. There are three combs in the system and labeled as comb A, comb B, and comb X, where comb X is located to site A. The pulse repetition rate of both comb A and comb B is 100 MHz and is locked to atomic clocks (or locked each other via the fiber link). Comb X has a repetition rate of 100.001 MHz so that the LOS can be obtained by asynchronized pulse trains from comb A with comb X and comb B with comb X.
Fig. 5. Configuration for LOS based fiber link synchronization system of two optical timescales, one located at Site A and one at Site B, via optical two-way time transfer over ∼ 114 km fiber link. The upper part is PD-TIC coarse synchronization system via #32 and #33 channels of the DWDM. The lower part is LOS based two-way time transfer system via #34 and #35 channels of the DWDM. The Three LOS interferogram signals of LOS1, LOS2, and LOS3 are measured at corresponding three BPDs. We leave two channels for data communication (not shown in the figure). DWDM: dense wavelength division multiplexing; BPD: balanced photo detector; DAQ: data acquisition; EDFA: erbium-doped fiber amplifier; SMF: single mode fiber; DCF: dispersion compensation fiber; PGM: pulse generate module; DFB: distributed feedback laser; EOM: electro-optic modulator; PD: photo detector; TIC: time interval counter.
In this experiment, two simplifications were made: 1) The so-called combs are not exactly the frequency comb. They are only repetition rate stabilized mode-locked Er:fiber lasers without locking the carrier envelope phase off-set frequency ${f_{\textrm{CEO}}}$. This is based on the fact that the measurement time is very short so that during the measurement, ${f_{\textrm{CEO}}}$ should not change very much. Also, because the LOS is not based on interference, ${f_{\textrm{CEO}}}$ plays little role in obtaining the LOS patterns. 2) To focus on the fiber link and to avoid the complicated frequency transfer between comb A and comb B, in this preliminary experiment, we used one laser to play the role of comb A and comb B, and comb B is connected with comb X via a 114 km fiber. Thus, they are virtually synchronized. The experiment is then to prove that the time difference is around zero.
To measure the time offset between remote clocks over 114 km fiber link by LOS, the following cares were also taken:
- • Spectral filter. As in our previous Ref. [14], the pulse spectrum was separated and filtered by a Dense Wavelength Division Multiplexing (DWDM) to alleviate the dispersion and to remove interactions of the reflection in the interconnectors. Each channel has a bandwidth of 100 GHz (0.8 nm @ 1550 nm) so that the pulses were broadened to about 2 ps. The counter-propagated signals travel on different channels. Dispersion compensation fiber (DCF) was also applied to recover the pulses from the fiber.
- • Center-timing algorithms for LOS interferograms. The center-timing of the LOS interferogram was determined by Algorithm 1, which reduces the effect of noise and higher-order dispersion. The extracted center-timing of the LOS interferogram by Algorithm 1 will be used for determining the departure and arrival timing of pulses.
- • Coarse synchronization. To counter the periodic waveform ambiguities in LOS measurement, a TIC-PD method was used along with the LOS measurement.
In the fiber system, channel #32 and #34 of the DWDM were used to forward the 100 MHz repetition rate mode-locked pulse train from site A to site B, and channel #33 and #35 was taken to return the pulse train to site A. A complete two-way time transfer system occupies another two channels for data communication (not shown in Fig. 5).
The “coarse” PD-TIC measurement is shown in the upper part of Fig. 5. At site A, a Rb atomic clock was used as the external trigger to synchronize the PGM that consists of a 1 pps generator. The 1 pps signal was used to drive an EOM that modulates a CW DFB laser to form an optical 1 pps signal. The optical 1 pps signal was coupled to channel #32 of the DWDM system as the frequency dissemination. Similarly, on the receiver side, the timing signal was also regenerated from the modulation of a DFB laser and channel #33 was chosen for sending back the timing signal for time interval comparison. The role of the “coarse” PD-TIC measurement is to resolve the ambiguity associated with the comb’s pulse train ${p_{\textrm{BX}}}$ and ${p_{\textrm{XB}}}$(${p_{\textrm{AX}}}$ is determined from the straightforward tracking of the number of interferogram peaks between the co-located comb A and comb X). This PD-TIC measurement generates its own set of two-way approximate values based on the departure and arrival of pulses: {${\hat{T}_{\textrm{AA}}}$, ${\hat{T}_{\textrm{AB}}}$, ${\hat{T}_{\textrm{BB}}}$, ${\hat{T}_{\textrm{BA}}}$}. Then we used these four measurements to calculate ${p_{\textrm{BX}}}$ and ${p_{\textrm{XB}}}$ via the Eqs. (12)–(15). If the uncertainty of “coarse” synchronization is below than 20 ps ($\textrm{1 fs } \times 2{f_\textrm{r}}/{f_\textrm{r}}\textrm{ = 20 ps}$), uncertainty in final $\Delta {t_{\textrm{AB}}}$ with LOS measurement caused by the “coarse” synchronization is below than 1 fs. Note that ${p_{\textrm{AX}}}$, ${p_{\textrm{BX}}}$, ${p_{\textrm{XB}}}$ are integers, so we do not need to very precise “coarse” synchronization.
There are three sets of LOS signals in this system, as shown in Fig. 5. LOS1: the local interferogram at site A between comb A pulses and comb X pulses. LOS2: the interferogram including fiber link delay at site A between comb X pulses and comb B pulses over the fiber link. LOS3: the counterpart of LOS2, the interferogram at site B between comb B pulses and comb X pulses over the fiber link. The last two interferograms (LOS2 and LOS3) are similar to a typical two-way transmission system, and the first interferogram (LOS1) is the local reference signal. The three interferograms were detected by the BPD and sampled by DAQ at each site. Then the center-timing extraction Algorithm 1 was performed to calculate the value of ${t_{\textrm{AX}}}$, ${t_{\textrm{BX}}}$ and ${t_{\textrm{XB}}}$ with the recorded LOS1, LO2, LOS3 signals. When the values of ${p_{\textrm{AX}}}$, ${p_{\textrm{BX}}}$, ${p_{\textrm{XB}}}$, ${t_{\textrm{AX}}}$, ${t_{\textrm{BX}}}$, ${t_{\textrm{XB}}}$ were all set, the clock-offset $\Delta {t_{\textrm{AB}}}$ can be calculated by Eq. (16).
3.2 Results
The measured LOS1 and LOS2 interferograms are shown in Fig. 6, which repeat in a period of $\textrm{1}/\Delta {f_\textrm{r}}$. However, due to loss, environmental noise, and residual dispersion in the long fiber, there are many disordered structures in LOS2. It is also seen that the interferogram is slightly asymmetrical. Although our Algorithm 1 for finding center-timing of LOS is applied, the uncertainty of the center-timing is still the main uncertainty of clock-offset $\Delta {t_{\textrm{AB}}}$. The derived clock-offset $\Delta {t_{\textrm{AB}}}$ measured by solving Eq. (16) is shown in Fig. 7 for several samplings.
Fig. 6. LOS1 and LOS2 interferogram in the experiment. The scale is 1µs/div. (We do not show LOS3, because LOS3 is similar to LOS2. LOS3 is a counterpart of LOS2 with transmitting pulses via 114 km fiber link in the opposite direction.)
It is seen from Fig. 7 that the derived clock offset $\Delta {t_{\textrm{AB}}}$ varies within the range of ± 700 fs for 400 sampling points at 1 Hz sampling rate and the standard deviation of the measured clock offset is 237 fs. The resolution of LOS based clock offset measurement is 10 fs level. As shown in Fig. 8, the time deviation is 110 fs at 1 s and 22 fs at 100 s averaging. Over durations from 0.1 s to 100 s, the time deviation of clock offset measurement is below 210 fs.
3.3 Uncertainty estimation
According to Eq. (16), in order to calculate clock-offset $\Delta {t_{\textrm{AB}}}$, we need to measure ${t_{p\textrm{AX}}},{t_{p\textrm{BX}}}$, ${t_{p\textrm{XB}}},{p_{\textrm{AX}}},{p_{\textrm{BX}}},$ ${p_{\textrm{XB}}},{f_\textrm{r}},\textrm{ and }\Delta {f_\textrm{r}}$. The contribution of the uncertainty of each independent variable to the total clock-offset depends on the corresponding partial differential. Using this relation, we estimated the impact of independent measurement uncertainty on the final clock-offset uncertainty. It is assumed that the uncertainty of the same physical variables at site A and site B are subject to the same distribution. By defining the uncertainty of ${f_\textrm{r}}$ as ${e_f}$, the uncertainty of ${t_{p\textrm{AX}}},{t_{p\textrm{BX}}},{t_{p\textrm{XB}}}$ as ${e_t}$, and the uncertainty of ${p_{\textrm{AX}}},{p_{\textrm{BX}}},{p_{\textrm{XB}}}$ as ${e_p}$, those are:
The final uncertainty of the clock-offset is then:
whereWe calculate these parameters using Eq. (16) and approximate simplify as:
In our experiment, ${f_\textrm{r}} = 100$MHz, $\Delta {f_\textrm{r}} = 1$kHz, and combs are all locked to the atomic clock. With locked to an atomic clock, assuming that the uncertainty of ${f_\textrm{r}}$ (including ${f_{\textrm{r, A}}}$, ${f_{\textrm{r, B}}}$, ${f_{\textrm{r, X}}}$) is ${e_f}$ = 10 mHz, and the maximum initial clock-offset $\textrm{max}({\Delta {t_{\textrm{AB, init}}}} )= 1\textrm{ ns}$. The uncertainty of the clock-offset caused by the uncertainty of ${f_\textrm{r}}$ is very small and negligible, as:
Incorrect p will bring large uncertainty to clock offset measurement. The ambiguity is proportional to $1/{f_\textrm{r}}$, which is called periodic waveform ambiguities. The clock offset measurement with LOS alone is unable to determine the value of ${p_{\textrm{AX}}}$, ${p_{\textrm{BX}}}$, ${p_{\textrm{XB}}}$. That is why the coarse time synchronization PD-TIC method was introduced. As mentioned in Section 3.1, if the uncertainty of supplementary “coarse” synchronization (such as PD-TIC) is below 20 ps, the value of ${p_{\textrm{AX}}}$, ${p_{\textrm{BX}}}$, ${p_{\textrm{XB}}}$ can be correctly determined. Thus, periodic waveform ambiguities can be resolved by introducing “coarse” synchronization methods.
As mentioned in Section 2.2, the main source of clock offset uncertainty is caused by the uncertainty of center-timing determination of LOS interferograms. In our experiment, the sampling rate of the data acquisition card is 300 MHz. Note that, LOS optical sampling rate is $1/{f_\textrm{r}} = \textrm{10 ns}$. We first consider short-term uncertainty, i.e. consider uncertainty in one-time measurement. Although sampling rate of the data acquisition card much higher than the LOS optical sampling rate, we cannot determine ${t_{p\textrm{AX}}},{t_{p\textrm{BX}}},{t_{p\textrm{XB}}}$ more precisely because of the high noise of the long fiber link. In our experiment, the short-term uncertainty of t (including ${t_{p\textrm{AX}}},{t_{p\textrm{BX}}},{t_{p\textrm{XB}}}$) is about ${e_t} \approx \textrm{10 ns}$, then short-term uncertainty of the clock-offset caused by the uncertainty of LOS center-timing determination is:
The result is consistent with our experimental result of short-term time deviation of clock offset. With long-period averaging, the uncertainty of t (including ${t_{p\textrm{AX}}},{t_{p\textrm{BX}}},{t_{p\textrm{XB}}}$) is about ${e_t} \approx \textrm{1 ns}$, then long-term uncertainty of the clock-offset caused by the uncertainty of LOS center-timing determination is:
The result is consistent with our experiment of time deviation in long-term averaging.
More importantly, we still need to improve the LOS interferogram by more precision dispersion compensation and noise reduction, which can narrow and smooth the interferogram patterns. Meanwhile, searching for noise-immune algorithms for determining the center of LOS interferogram are always necessary.
4. Conclusions
We have demonstrated clock-offset measurement of two combs separated by 114 km fiber link using the LOS technique with sub-picosecond time resolution. The time deviation of clock-offset measurement is 110 fs at 1 s and 22 fs at 100 s averaging. The main uncertainty source of the clock-offset measurement is the uncertainty of the center-timing extraction of LOS interferograms. More accurate center-timing extraction algorithms are in effort.
Funding
National Key Scientific Instrument and Equipment Development Projects of China (2012YQ140005); National Natural Science Foundation of China (61761136002).
Acknowledgement
The authors are in debt to Laura C. Sinclair and Nathan R. Newbury of NIST for invaluable and stimulating discussions.
Disclosures
The authors declare no conflicts of interest.
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