# Dual comb–enhanced microwave clock synchronization over commercial fiber

Ziyang Chen  State Key Laboratory of Advanced Optical Communication Systems
and Networks, School of Electronics, and Center for Quantum Information
Technology, Peking University, Beijing 100871, China  Dongrui Yu  State Key
Laboratory of Advanced Optical Communication Systems and Networks, School of
Electronics, and Center for Quantum Information Technology, Peking University,
Beijing 100871, China  Ganbin Lu  State Key Laboratory of Information
Photonics and Optical Communications, Beijing University of Posts and
Telecommunications, Beijing 100876, China  Yufei Zhang  State Key Laboratory
of Advanced Optical Communication Systems and Networks, School of Electronics,
and Center for Quantum Information Technology, Peking University, Beijing
100871, China  Song Yu  State Key Laboratory of Information Photonics and
Optical Communications, Beijing University of Posts and Telecommunications,
Beijing 100876, China  Bin Luo<EMAIL_ADDRESS>State Key Laboratory of
Information Photonics and Optical Communications, Beijing University of Posts
and Telecommunications, Beijing 100876, China  Hong Guo<EMAIL_ADDRESS>State Key Laboratory of Advanced Optical Communication Systems and Networks,
School of Electronics, and Center for Quantum Information Technology, Peking
University, Beijing 100871, China

###### Abstract

The large-scale clock network is the key ingredient to obtain high precision
in many scenarios, from fundamental research to cutting-edge applications.
However, time synchronization among microwave clocks has become an upper limit
of the time-frequency network because of the inherent constrained precision in
optical pulse measurements. Here, we demonstrate a femtosecond-level time
synchronization of microwave clocks through a commercial link of 205.86 km via
dual comb–enhanced optical two-way time transfer, which achieves a 6.23-fs
residual time deviation between synchronized timescales at 1 s and an
instability below $6\times{10^{-18}}$ at 10,000 s. Further, the high-precision
time synchronization of microwave clocks significantly enhances the probe
ability of subtle reciprocity changes of fiber to the sub-picosecond level.
This work provides a path toward secure fiber time-frequency networks to
support future microwave clock–based precise timing and sensing systems.

##  I Introduction

High-precision clock networks offer the fundamental technical foundation for
numerous scenarios, including positioning, navigation, timing [1, 2, 3],
sensitive probes [4, 5, 6], and fundamental physics testing [7, 8, 9, 10]. The
enormous benefits are unleashed through the full interconnection between any
two nodes in a hybrid network of optical clocks and microwave clocks. In
particular, the time-frequency transfer over both fiber and free-space optical
links has seen tremendous progress [11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. The ultra-high precision frequency
comparison has been achieved using well-established optical frequency transfer
techniques by directly disseminating stable frequency signals [14, 15, 16, 25,
26], whereas the timescale comparison requires an exchange of a series of
labeled optical pulses generated by frequency standards [11, 12, 13, 17, 18,
19, 20, 21, 22, 23, 24]. Since the optical two-way time transfer (OTWTT)
method uses the reciprocity of bi-directional light travel to eliminate most
of the common-mode noises and the link drift [31, 32], the measurement of
optical pulses has become the key to improving the precision for time
transfer.

Remarkably, in recent years, time transfer has been significantly improved by
using optical measurement methods, specifically linear optical sampling (LOS)
[33], to overcome the limitation of the picosecond-level uncertainty caused by
the photoelectronic detection of pulses [13, 17, 18, 19, 20, 21, 22, 23, 24,
34, 35]. State-of-the-art time-programmable frequency comb technology has even
pushed the time synchronization to the quantum-limited region [11, 12].
Despite great improvement of the synchronization capability between optical
clocks [12, 13, 21] and between optical and microwave clocks [22, 23], the
reported time synchronization between remote microwave clocks remains limited
by the precision of photodetectors and time-interval counters (TICs) [36], so
the full potential of high-precision time-frequency networks has not been
realized.

In the time synchronization of two microwave clocks, the electrical
measurement has not been replaced by the optical approach because a phase
coherence of independent optical frequency combs (called combs for simplicity)
at both local and remote sites, which is a necessary requirement for
performing stable and low-jitter interference, is challenging to establish.
Specifically, microwave-to-microwave synchronization is difficult because both
reference and synchronized clocks have greater phase noise than optical clocks
and the pulse coherence is reduced by propagation-induced pulse impairments.

Figure 1: Schematic of the microwave clock time synchronization. (a) Block
diagram of the experimental link. Two microwave clock–based optical timescales
at two sites are synchronized over two geographically folded optical fibers
linked in Liuxinji with the one-way link length of 205.86 km. Two repeater
stations are co-located at Wuhan North to recover the impairments of optical
pulses. DWDM: dense wavelength division multiplexing; DCF: dispersion
compensating fiber; EDFA: erbium doped fiber amplifier. (b) Detailed
configuration of site A. The repetition frequencies of the clock comb and
local oscillator (LO) comb, which has a small frequency offset
$\Delta{f_{\rm{r}}}=1$ kHz, are phase-locked to a microwave clock using an
asynchronous locking system. The clock comb filters an output centered at
${\lambda_{1}}$ (1,551.72 nm) followed by a Mach-Zehnder modulator (MZM). One
split modulated comb with a filtered LO comb filter centered at
${\lambda_{1}}$ is used for cross-correlation, which is followed by an analog-
to-digital converter (ADC) to generate reference interferograms. Another
output of the modulated comb is split for transmission. MZM is used to encode
time markers to remove the ambiguity. Another output centered at
${\lambda_{2}}$ (1,550.12 nm) of the LO comb is filtered with the incoming
clock comb compensated by a DCF and an in-line electronic polarization
controller (EPC) to generate the target interferograms. Post-processing is
performed to calculate the time difference ${t_{\rm{A}}}$; then, the data are
sent to site B using a modulated laser. The out-of-loop verification part is
not shown here. (c) Principle of linear optical sampling. (d) Simplified
configuration of site B. Site B is identical to site A, except that all of the
positions of wavelengths ${\lambda_{1}}$ and ${\lambda_{2}}$ are exchanged,
and a synchronization feedback system is added, which contains a real-time
time offset calculation, a proportional integral (PI) loop filter and a direct
digital synthesizer (DDS).

To overcome the limitations imposed by measurement devices on the time
synchronization of microwave clocks, here, we propose a tight microwave-clock
time synchronization scheme based on the dual comb–enhanced OTWTT and provide
a demonstration over 205.86 km commercial fiber. We achieved a 6.23-fs
residual time deviation between synchronized timescales at 1 s and an
instability below $6\times{10^{-18}}$ at 10,000 s due to a series of
breakthroughs. The intensity-encoded comb pulses were measured using a Kalman
filter–assisted LOS, where the centroids of reference and target
interferograms at each site served as the start and stop times of the time
interval, to reduce the impact of the intensity noise and pulse distortion on
the peak measurements. We used the phase time of the pulse envelope instead of
the carrier phase for time synchronization [20] to remove the limit of the
carrier envelope offset (CEO) frequency, since the stable carrier phase is
vulnerable to fluctuations in independent sources and fiber transmission.
Dispersion compensation, fully automatic polarization control, and real-time
feedback are essential to guarantee the mutual optical coherence.

An interesting extension of the fiber-based time synchronization is the
perception of subtle reciprocity changes of the fiber, which enhances the
ability against potential delay attacks [37, 38, 39]. We achieved sub-
picosecond (ps)-level sensing of asymmetric delay attacks within 200
kilometers, which significantly surpasses the capabilities of the existing TIC
measurement instruments of a sub-nanosecond level [37].

The improvement in microwave clock synchronization capabilities has greatly
expanded the application range of the entire high-precision clock networks.

##  II Results

Experimental layout. The experiment was performed in Wuhan commercial link. As
shown in Fig. 1 (a), two sites A and B are adjacent to each other and located
at Wuhan laboratory with independent rubidium clocks to easily quantify the
residual transfer noise. The master site (site A) and the synchronized site
(site B) were connected through two geographically folded optical fibers
linked in Liuxinji (round-trip node), and the total link length was 205.86 km
(the total link loss was 57.18 dB). The long-haul fiber transmission
necessitated the installation of two optical repeater stations at Wuhan North,
each of which contained an erbium-doped fiber amplifier (EDFA) and a
dispersion compensating fiber (DCF) along the terrestrial link to recover the
signal impairments caused by loss and dispersion. OTWTT was achieved by using
the dense wavelength division multiplexing (DWDM) technology over the shared
fiber link.

Dual comb–based LOS measurement. At each site, we used a dual comb–based LOS
measurement to replace the TIC in the measurement of pulses generated by
microwave clocks. As shown in Fig. 1 (b), at each site, we first created a
local timescale by phase-locking the repetition frequency of the
${f_{\rm{r}}}=$ 100-MHz comb to the local microwave clock, which served as the
clock comb. Then, a 100.001-MHz comb, which was also locked to the local
clock, output optical pulses with a repetition frequency offset by
$\Delta{f_{\rm{r}}}=1$ kHz relative to the clock comb, which served as the
local oscillator (LO) comb to sample clock combs for performing LOS, as shown
in Fig. 1 (c). Phase locking was achieved with a locking-induced noise of 45
fs (time deviation at 1 s) through an asynchronous locking system (see
Methods).

Both combs had center frequencies of 1,560 nm and a 3-dB spectrum bandwidth of
approximately 15 nm. The spectral effectively covered the commonly used C-band
range in fiber optic communication, which is necessary for two-way
transmission. Here, we directly applied the international-telecommunication-
union-standard (ITU) DWDM module to the output of each comb to generate band-
limited pulses with different spectra. We filtered the LO comb to generate two
outputs with a bandwidth of 0.8 nm (100 GHz), which were centered at 1,551.72
nm (ITU-C32) and 1,550.12 nm (ITU-C34). We filtered the clock comb to generate
an output centered at ITU-C32 (for Site A) or ITU-C34 (for Site B). Since
directly using the LOS for synchronization will introduce the ambiguity of
${1\mathord{\left/{\vphantom{1{{f_{\rm{r}}}}}}\right.\kern-1.2pt}{{f_{\rm{r}}}}}=10$
ns, which causes cycle slips, we provided an unambiguous synchronization
method by encoding time markers on the optical pulse using Mach-Zehnder
modulator (MZM) (Methods). For each site, the output after the intensity
modulator was separated into two parts. One part was sent to the link for
transmission with the output power of 1 mW; the other part was to generate the
local reference interferogram.

The use of DWDM offered two advantages. First, the spectral bandwidth of the
filtered comb signals were limited to stay below Nyquist condition:
${{f_{\rm{r}}^{2}}\mathord{\left/{\vphantom{{f_{\rm{r}}^{2}}{\left({2\Delta{f_{\rm{r}}}}\right)}}}\right.\kern-1.2pt}{\left({2\Delta{f_{\rm{r}}}}\right)}}$
[24]. Second, combs with a constrained optical bandwidth could better
propagate through fibers to decrease the dispersion and nonlinear effects
[40]. However, this behavior increases the uncertainty of timing
discrimination compared to the free-space scenario with narrower pulses (where
the typical pulse width is 0.1-1 ps) [21].

Figure 2: Noise floor of the LOS measurement. (a) Centroid-finding process of
different interferograms by locally changing the length of the DCF. FWHM: full
width at half maximum. (b) Time deviation of the noise floor over an averaging
time of 1 s, calculated by the centroids, under different pulse FWHMs. (c)
Power spectral densities (PSDs) before and after Kalman filtering. (d) Time
deviation of the noise floor. The shaded region indicates the short-term noise
suppression by the Kalman filter. (e) Modified Allan deviation (MDEV) of the
noise floor. In both plots, the orange squares or solid lines indicate the
noise floor of the LOS; the blue circles or solid lines indicate the noise
floor of the Kalman filter–assisted LOS. The dashed lines highlight the main
contributed noise types with typical slopes. The error bars indicate the
68$\%$ confidence intervals.

At each site, two cross-correlations were generated. The reference
interferogram was generated by the cross-correlation between the LO comb and
the local clock comb. The target interferogram was generated by the cross-
correlation between the LO comb and the incoming clock comb from the remote
site, which correspond to the “time-stretched” start and stop signals of the
measuring time intervals, respectively. The cross-correlations were generated
with the equivalent sampling rate
${f_{\rm{s}}}={{f_{\rm{r}}^{2}}\mathord{\left/{\vphantom{{f_{\rm{r}}^{2}}{\Delta{f_{\rm{r}}}}}}\right.\kern-1.2pt}{\Delta{f_{\rm{r}}}}}=10$
THz in the optical frequency domain and the updating rate of
${T_{{\rm{update}}}}={1\mathord{\left/{\vphantom{1{\Delta{f_{\rm{r}}}}}}\right.\kern-1.2pt}{\Delta{f_{\rm{r}}}}}=1$
ms in the laboratory timescale. The equivalent timescale stretching parameter
was obtained as
$\alpha={{{f_{\rm{r}}}}\mathord{\left/{\vphantom{{{f_{\rm{r}}}}{\Delta{f_{\rm{r}}}}}}\right.\kern-1.2pt}{\Delta{f_{\rm{r}}}}}{\rm{=1}}{{\rm{0}}^{5}}$.
Then, we implemented an analog-to-digital converter (ADC) with a 400-MHz
sampling rate and a 10-bit resolution to digitize the train of interferograms.
The data were subsequently transmitted to a field programmable gate array
(FPGA) controller and synchronized with the local clock for data processing.

Figure 3: Results of the dual comb–enhanced microwave clock synchronization.
(a) Time deviations of the LOS noise floor (blue circles), one-way free-
running link (purple diamonds) and synchronized link (red triangles). The free
running performance was obtained by comparing the position of the centroids
between local reference and target interferograms. The performance of the
synchronized link was calculated from the out-of-loop measurement
$\Delta{T_{{\rm{AB}}}}$ by comparing the position of the centroids between
out-of-loop reference interferograms from two sites. (b) Residual MDEV of the
corresponding results. The insert shows the PSDs of OTWTT. The synchronized
result was well below the instabilities of state-of-the-art microwave clocks
(shaded region [44, 45, 46, 47]). The dashed lines highlight the main
contributed noise types with typical slopes. The error bars indicate the
68$\%$ confidence intervals.

Timing discrimination. The fundamental task of OTWTT is to recover timing
information from exchanged optical pulses. Because signal distortion and noise
accumulation were caused by the long-haul fiber transmission, high-precision
timing discrimination was challenging.

Instead of the commonly used heterodyne-voltage-to-time scaling [12], peak
location [13] or carrier phase [20] to define the arrival time in free-space
scenarios, we used the centroid of the interferogram to discriminate the
temporal position of the optical pulses, which is defined by

${t_{i}}=\frac{{\int{dt}{A_{i}}\left(t\right)\times
t}}{{\int{dt}{A_{i}}\left(t\right)}},$ (1)

for the $i$-th labeled interferogram. Here, ${{A_{i}}\left(t\right)}$ is the
envelope extracted by the Hilbert transformation, since it is not sensitive to
the pulse shape and can smooth out the intensity noise superimposed on optical
pulses. Then, the time difference was obtained by comparing the centroid
position of two interferograms.

To quantify the noise floor of the microwave clock–based LOS measurement, we
replaced the incoming clock comb by the local generated clock comb at each
site and calculated the time difference between reference and target
interferograms. Due to the DWDM filtering, the initial width of the filtered
pulse was approximately 13.6 ps in the effective time (which corresponds to
1.36 ${\rm{\mu s}}$ in lab time). Then, we used local DCFs with different
length to locally simulate the effect of the fiber transmission on the pulse
shape, as shown in Fig. 2 (a). When the dispersion-induced pulse width
increased, the peak became more difficult to find because the pulse flattened.
By contrast, as shown in Fig. 2 (b), in the pulse width of 13.62-217.97 ps,
the short-term time deviation remained stable (65 $\pm$ 2 fs for 1 s of
averaging time) by the centroid estimation, which demonstrates the
insensitivity to waveforms.

Kalman filter–assisted LOS measurement. In contrast to the signal fading from
turbulence in the free-space scenario [12, 19], the decrease in visibility of
the target interferogram mainly attributes to the polarization disturbance in
fiber, which is eliminated by the in-line electronic polarization controller.
Therefore, the hold-over behavior of the Kalman filter was not considered in
our experiments. In this paper, we focused on the Kalman filter to limit the
amount of measurement noise [41, 42]. We assumed two zero-mean Gaussian random
processes in the prior noise model of Kalman filter to represent the random-
walk phase noise and random-walk frequency noise for real-time data processing
[22].

Fig. 2 (c) shows noise power spectral densities (PSDs) of the measurement
process. The effective bandwidth of the Kalman filter was 0.01 Hz, and the PSD
was reduced by approximately three orders of magnitude reduction at 0.5 Hz to
1.2 ${\rm{f}}{{\rm{s}}^{\rm{2}}}{\rm{H}}{{\rm{z}}^{{\rm{-1}}}}$, which is
close to the noise floor of the optical clock–based LOS measurement [12, 13,
22].

In addition, Figs. 2 (d) and (e) show that the Kalman filter–assisted LOS
optimizes the measurement instability for a short averaging time. The shaded
region shows the noise suppression below 100 s. In Fig. 2 (d), the time
deviation of noise floor was optimized to 3.7 $\pm$ 0.1 fs for 1 s of
averaging time, which can support the measurement of femtosecond-level fiber
changes. In Fig. 2 (e), the modified Allan deviation (MDEV) changed the slope
of the noise floor from
${\tau^{-{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}}}$ to
${\tau^{-1}}$ after the Kalman filtering ($\tau$ is the averaging time in
seconds). Thus, the dominate white phase measurement noise was suppressed to
the lower flicker phase noise region at short timescales (below 100 s), which
determined the noise limitation of the time offset measurement.

In-loop time offset calculation. After the precision of the measurement part
had been quantified, the real-time timing discrimination of both sites and
calculation of the in-loop time offset $\Delta{t_{{}_{{\rm{AB}}}}}$ were
performed in the two FPGA controllers. First, at both sites, the respective
time differences under laboratory timescale $t_{\rm{A}}^{{\rm{lab}}}$ and
$t_{\rm{B}}^{{\rm{lab}}}$ were obtained by locally comparing the centroids of
the labeled reference and target interferograms using the time difference
calculation algorithm. The effective (actual) time differences at both sites
are
${t_{\rm{A}}}={{t_{\rm{A}}^{{\rm{lab}}}}\mathord{\left/{\vphantom{{t_{\rm{A}}^{{\rm{lab}}}}\alpha}}\right.\kern-1.2pt}\alpha}$
and
${t_{\rm{B}}}={{t_{\rm{B}}^{{\rm{lab}}}}\mathord{\left/{\vphantom{{t_{\rm{B}}^{{\rm{lab}}}}\alpha}}\right.\kern-1.2pt}\alpha}$.
Then, the local generated time difference ${t_{\rm{A}}}$ at site A was
transmitted to site B using a modulated laser centered at 1,550.92 nm
(ITU-C33) over the DWDM communication link. Combining with the local
calculated time information ${t_{\rm{B}}}$, the in-loop time offset
$\Delta{t_{{}_{{\rm{AB}}}}}$ was calculated in the FPGA controller of Site B.

Based on OTWTT, the equivalent time offset between clocks was obtained by
subtracting the two time difference results as follows:

$\Delta{t_{{\rm{AB}}}}=\frac{1}{2}\left({{t_{\rm{B}}}-{t_{\rm{A}}}}\right)+{t_{{\rm{NR}}}},$
(2)

where
${t_{{\rm{NR}}}}={{\left({\Delta{\tau_{{\rm{BA}}}}-\Delta{\tau_{{\rm{AB}}}}}\right)}\mathord{\left/{\vphantom{{\left({\Delta{\tau_{{\rm{BA}}}}-\Delta{\tau_{{\rm{AB}}}}}\right)}2}}\right.\kern-1.2pt}2}$
is the non-reciprocal propagation time delay between forward transmission time
${\Delta{\tau_{{\rm{AB}}}}}$ and backward transmission time
${\Delta{\tau_{{\rm{BA}}}}}$. In the experiment, ${\Delta{\tau_{{\rm{AB}}}}}$
and ${\Delta{\tau_{{\rm{BA}}}}}$ were estimated in advance based on the link
length, and the results were pre-stored in FPGA. Based on the reciprocity
assumption, ${t_{{\rm{NR}}}}$ is commonly set to zero. However, many scenarios
such as asymmetric delay attacks in the fiber [37] will break this assumption,
which will be discussed later.

Free-running link. First, we used our system to investigate the one-way free-
running link. Specifically, we set both sites that shared a common master
clock to test the open-loop link noise from A to B. After passing through the
link, the incoming clock comb accumulated link effects, which resulted in the
pulse impairments. One impairment is dispersion, which caused pulse broadening
and distortion. The results show both second-order dispersion, which caused
pulse broadening, and third-order dispersion, which resulted in pulse
asymmetry and distortion. Additionally, the link noise and amplified
spontaneous emission (ASE) noise introduced by EDFAs inevitably introduced
intensity fluctuations in the optical pulses. Both dispersion-induced noise
and ASE-induced intensity fluctuations decreased the pulse coherence, which
ultimately affected the measurement jitter of the optical cross-correlation.

For the dispersion compensation, even if we use DCFs at the repeater station,
a complete dispersion compensation for ultrashort pulse remains challenging.
Therefore, we used a small amount of DCF at the receiver site to compensate
for the residual dispersion in a fine-tuning manner. The cross-correlation
after compensation was provided by the target interferogram in the insert of
Fig. 3 (a) with a pulse width of 75.36 ps, which would not cause significant
measurement uncertainties due to pulse distortion. For more precise dispersion
compensation, techniques such as liquid crystal modulation may be used [43].
The effect of the intensity fluctuations was suppressed by the centroid
estimation to smooth the intensity noise. When we calculated the centroid, as
shown in Fig. 3 (a), the time deviation introduced by the link was 5 fs at 1
s, which was optimized by an order of magnitude compared to the direct peak-
location finding method in our experiment.

Figure 4: Result of the asymmetric delay attack. (a) Asymmetric delay
attacking module at the round-trip node. Two DWDMs were used to separate the
forward and backward optical pluses, and an optical delay line (ODL) in the
forward path was used to adjust the reciprocal time delays between forward and
backward transmissions. (b) The in-loop time offset changed when the sub-
picosecond-level asymmetric delay attack was used. The peak-to-peak time
wander was approximately 100 fs without attack.

Synchronized link. After fully characterizing the 205.86-km fiber link, by
performing an OTWTT to cancel the link’s influence and closing the
synchronized feedback loop, we obtained the performance of the synchronized
link, which was continuously acquired for over 24 hours. As shown in Fig. 1
(d), the time offset calculation, which acted as the error signal, was fed
into the digital feedback system that contained a proportional integral loop
filter and a direct digital synthesizer (DDS). It also had a 30-Hz
synchronization bandwidth to force the minimized residual timing jitter of the
clock output to synchronize two microwave clocks.

Fig. 3 (a) shows a time deviation. The starting point was the floor at
approximately 6.23 fs and reached another floor of 10 fs at 400 s, which
contributes to the short-term noise suppression by Kalman filter–assisted LOS.
The trend is consistent with the measurement noise floor for an averaging time
of up to 1,000 s, which indicates that the residual synchronized noise
strongly depends on the type of noise in the measurement part and the degree
of noise. The synchronized link exhibits slightly poorer stability for very
short averaging time than the free-running link. The main reason is the non-
reciprocity in the bidirectional amplifiers at the repeater station, which
makes it difficult to cancel the non-common-mode ASE noise. The slope
dramatically changed beyond 1,000 s, which shows that the time deviation is
limited by the thermal effect for longer averaging times. This issue will be
improved by an active thermal stabilization of experimental components.

The residual MDEV between remote parties show the performance of the time-
frequency transfer in Fig. 3 (b). The residual MDEV began at
$1\times{10^{-14}}$ at 1 s and reached below $6\times{10^{-18}}$ at 10,000 s.
The residual MDEV was below the state-of-the-art microwave clocks [44, 45, 46,
47] after 1 s of averaging time and more than two orders of magnitude better
after the long averaging time. The slope was ${\tau^{-1}}$ below 1,000 s,
which indicates that the dominant noise at short timescales is flicker phase
noise, and the residual MDEV follows ${\tau^{-1/2}}$ power laws after 1,000 s
averaging time. All of the results are consistent with the time deviation.

We also compared the stability result of $1\times{10^{-14}}{\tau^{-1}}$ in our
experiment with the free-space scenarios that synchronized the optical clocks.
The stability was approximately one order of magnitude less than that of Ref
[13], given by a few ${10^{-15}}{\tau^{-1}}$. This lower starting point was
mainly due to the higher noise floor of the LOS measurement and ASE noise of
the in-line EDFAs, which relatively weakened the coherence of optical pulses
and caused a lager timing jitter in the microwave-clock-synchronization
scenarios than in the optical oscillator–based scenarios. Moreover, compare to
the result of
$2.8\times{10^{-15}}{\tau^{-{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}}}$
in Ref. [12], the trend of ${\tau^{-1}}$ comes from the flicker phase noise
caused by the technical noise of devices, so the delay-unsuppressed white-
phase noise especially with low frequency [48, 49] could not be detected.
Thus, we should optimize the microwave clock–based LOS measurement.
Nevertheless, this result, which is close to the optical clock time
synchronization, greatly closes the gap of microwave-microwave time
synchronization in the entire time-frequency network.

Probing fiber delay non-reciprocity. Similar to the probe of non-reciprocity
from multi-path effects [12] or motion platforms [19] in free space, many
impacts can affect the non-reciprocity in fiber-based OTWTT, which affects the
true clock-offset estimation results. A particular example of the recently
extensively concerned potential asymmetric delay attacks [37, 50, 39]. The
improvement of the clock-offset measurement enables us to sense the subtle
delay non-reciprocity of the fiber link and improves the security of long-haul
fiber transmission.

To verify this claim, we used an asymmetric delay attacking module at the
round-trip node to simulate the potential invasive attack, as shown in Fig. 4
(a). Two DWDMs were used to separate the forward and backward optical pluses.
An optical delay line (ODL) in the forward path was exploited to adjust the
relative time delays between forward and backward transmissions, and the
reciprocity was breakdown. As we discussed, the non-reciprocal propagation
time delay ${t_{{\rm{NR}}}}$ in Eq. (2) was set to zero before we started the
synchronization. If the non-reciprocal traveling time was stable, the
calculated time offset $\Delta{t_{{\rm{AB}}}}$ only reflected the offset
between the two clocks (because the link drift was canceled by the OTWTT
method), which can be compensated on the remote clock through closed-loop
feedback process. However, the term ${t_{{\rm{NR}}}}$ appeared when we
adjusted the ODL to perform asymmetric delay attacks, which made the estimated
clock offset deviate from the true value.

Here, we only set a common clock of two sites to probe the link variation to
show the ability of pulse measurement in the fiber and ignored the clock-
offset drift. When there is a clock offset, the link change can also be
obtained by establishing a clock-offset dynamics model to eliminate the
influence of the clock [37]. As Fig. 4 (b) shows, the in-loop time offset
$\Delta{t_{{\rm{AB}}}}$ suddenly changed when we applied a sub-picosecond-
level delay on one path, which caused a mismatch between synchronized clocks
and compromised the security. This type of asymmetric delay attack cannot be
identified and exceed the noise level of the measurement instruments. The time
wander of the system was approximately 100 fs in our system, which limits the
precision of probing fiber delay slips. This process can be optimized by
precisely controlling the temperature of laboratory devices.

This result indicates that the transmission of microwave clocks can identify
delay variations caused by sub-ps-level attacks, which is three orders of
magnitude better than the reported TIC scheme with the sub-nanosecond
precision [37]. This significant improvement helps us precisely identify the
sudden phase changes caused by unexpected factors in the link, which can more
securely evaluate the clock offset.

##  III Conclusion

In summary, we have demonstrated a femtosecond-level time synchronization
between remote microwave clock–based timescales over a commercial fiber link
of 205.86 km. The residual time deviation after synchronization was 6.23 fs at
1 s, and the instability was below $6\times{10^{-18}}$ for an averaging time
longer than 10,000 s with the dominate noise from the microwave clock–based
LOS measurement. This configuration can make the high time-resolution feature
of the optical frequency comb compatible with most existing fiber
infrastructures, which greatly promotes the application range of fiber-based
time-frequency networks. This result fully merges the bottleneck of time
synchronization based on microwave clocks in a hybrid time-frequency network
to achieving the interconnection between any microwave and optical clocks.
Thus, this method has the potential for microwave-based precise applications
such as positioning, navigation, timing and coherent sensing. Moreover, this
technique enhances the probe ability of subtle variations of fiber, which will
generate new possibilities for channel modeling and a secure time-transfer
network to resist potential delay attacks. An interesting extension of this
work will further investigate the quantum-limited properties [12] in the
microwave clock–based fiber scenarios.

##  IV Methods

Asynchronous locking system. In our experiment, the repetition frequencies of
two combs were phase-locked to a microwave clock to guarantee the coherence of
the pulse-envelope phase, while the CEO frequency was free running. The reason
is that the CEO frequency has negligible influence on the envelope measurement
at an appropriate sampling rate [34].

To maintain the pulse-envelope phase synchronization between two combs with a
small frequency offset, we developed an asynchronous locking system, which
contained a high-precision single-input two-output DDS, and two fiber-loop
optical-microwave phase detectors (FLOM-PDs) [51]. The single-input two-output
DDS was used to generate two electrical signals with repetition frequencies of
1 GHz and 1.00001 GHz, which were synchronized to the 10-MHz microwave clock
and served as the references of two combs. To maintain the synchronization
between the outputs of the DDS and two combs, we locked the repetition
frequencies of two combs to the outputs of the DDS using the FLOM-PD, where
the 10-th harmonic of the respective combs were used to improve the locking
stability. The overall locking-induced time deviation of the system was 45 fs
at 1 s.

Eliminating of the ambiguity. Instead of the modulated laser scheme to remove
the ambiguity range, we used the MZM to directly encode time markers on the
intensity of the optical pulses. The MZM was driven by the digital-to-analog
converters (DAC) synchronized to the local clock. High control voltage was
supplied to suppress the intensity of the optical pulse to label data “0”,
whereas no control voltage was supplied to maintain the intensity of the
optical pulses to label data “1”.

Since we collected interferograms with a refresh rate of 1 kHz, the modulated
frequency of MZM was set to 1 kHz to label symbols on the initial part of the
interferograms of every second to represent the timestamp. Therefore, one
labeled interferogram corresponds to $10^{5}$ labeled optical pulses. Here,
the timestamp was encoded by a pseudorandom binary sequence (PRBS) with 10
bits (corresponding to 10 ms duration) per second. The frame structure of the
PRBS is as follows: The first and last bits were fixed as “0”, and the middle
8 bits used pseudo-random numbers to simulate the timestamps.

The next step was to recognize the frames. First, once the measurement voltage
of the reference interferogram was 0, the frame header recognition began.
Second, at each site, after identifying all pseudo-random numbers and
receiving the last “0” symbol, we labeled the next interferogram as the first
pulse and recorded the time on the FPGA. The target interferogram was
similarly measured. The synchronization was maintained until the symbol “0” of
the next frame was received, and the measurement time difference was
maintained during the frame recognition process.

Out-of-loop verification. To verify the time offset of the clocks after
synchronization, the out-of-loop measurement of the time offset
$\Delta{T_{{\rm{AB}}}}$, which is independent from the in-loop calculated time
offset $\Delta{t_{{}_{{\rm{AB}}}}}$, is necessary. Here, we also filtered an
output on both combs at each site to generate the out-of-loop reference
interferogram that is independent from the in-loop signals to verify the
synchronization results. The out-of-loop reference interferograms at two sites
contained the timing information of two clocks and could directly eliminate
uncertainty from the photoelectronic detection. Then, the out-of-loop
verification was performed by measuring the relative time interval of two out-
of-loop reference interferograms from two sites using an ADC and transmitted
to an computer with a real-time data refresh rate of 1 s, which is independent
of the FPGA processor used for synchronization. Therefore, the PSD is
available only for Fourier frequencies below 0.5 Hz.

##  V DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  VI CODE AVAILABILITY

The code used in this study is available from the corresponding author upon
reasonable request.

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##  VII Acknowledgments

This work was supported by the National Natural Science Foundation of China
(Grants Nos. 62201012), and the National Hi-Tech Research and Development
(863) Program.

##  VIII AUTHOR CONTRIBUTIONS

Z.C., D.Y., G.L. and Y.Z. performed the experiment and overall data analysis.
Z.C., D.Y. and Y.Z. performed the theoretical analysis. Z.C., D.Y., and G.L.
processed the data. Z.C. and H.G. contributed to all parts of the work. Z.C.
and D.Y. conceived the experiment, and Z.C., S.Y. B.L. and H.G. supervised the
project. All of the authors were involved in the discussion and
interpretations of the results. Z.C. wrote the manuscript, and D.Y., G.L.,
Y.Z., S.Y., B.L. and H.G. provided revisions.

##  IX COMPETING INTERESTS

The authors declare no competing interests.