## Abstract

Fiber-optic time and frequency synchronization technology demonstrates ultra-high synchronization performance and has been gradually applied in various fields. Based on frequency synchronization, this study addressed the problems of period ambiguity and initial phase uncertainty of the phase signal to realize the coherent transmission of the phase. An absolute phase marking technology was developed based on high-speed digital logic with zero-crossing detection and an optimized control strategy. It can realize picosecond-level absolute phase marking and provide a picosecond-level ultra-low peak-to-peak jitter pulse marking signal to eliminate phase period ambiguity and determine initial phase and transmission delay. Thus, by combining the high-precision phase measurement capability of the synchronized frequency signal and long-distance ambiguity elimination capability of the pulse-per-second signal, a high-precision remote coherent phase transmission over an optical fiber is realized. After frequency synchronization, the peak-to-peak jitter between the local and remote phase-marking signals can be only 3.3 ps within 10,000 s measurement time. The uncertainty of the coherent phase transmission is 2.577 ps. This technology can significantly improve the phase coherence of fiber-optic time and frequency transmission and provide a new approach to achieve peak-to-peak picosecond-level reference phase marking and high-precision fiber-optic remote coherent phase transmission. This demonstrates broad application prospects in coherence fields such as radar networking.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Time and frequency, which are considered to be among the basic physical quantities, are the basis in many application fields, including precise navigation and timing, clock-based geodesy, and fundamental constant measurement [1–6]. Time and frequency can be measured with the highest measurement and transmission accuracy using methods such as measurement via an optical clock [5–7] and fiber-based time and frequency transmission [8–13]. Fiber-based time and frequency transfer technology has been widely used since its inception. Fields, such as very long-baseline interference [14] and square kilometer array telescopes [15], use optical fibers to achieve radio frequency (RF) and time synchronization by modulating the RF signal and time signal (pulse-per-second, PPS) to continuous lasers [9,10,16–19]. Applications such as comparing optical clocks between different laboratories [20] use optical fibers to transmit ultra-stable optical frequency Refs. [11,20–22]. In addition, many large-scale scientific devices [23,24] use optical fibers to transfer femtosecond mode-locked laser pulses and achieve femtosecond-level time synchronization inside the system. Femtosecond laser pulse transmission can simultaneously provide stable optical frequency references, RF references, and ultra-low jitter time pulse information [12,25,26].

These applications predominantly focus on the transmission and synchronization of the time and frequency signals. The time and frequency references require to be distributed to the remote site separately or simultaneously, and recovered or synchronized at the remote site with high precision; that is, the phase difference between the frequency signals can remain stable. However, an increasing number of coherence applications have established the requirements for reference phase synchronization, such as radar networking [27,28] and distributed antenna interference [29]. They require stabilizing the phase difference, overcoming the phase period ambiguity with both transmission and generation, eliminating the initial phase uncertainty, and calibrating the transmission delay. Thus, the differences between frequency synchronization and coherent phase transmission are mainly reflected in two aspects. First, frequency synchronization does not consider the period ambiguity problem; second, the initial phase after frequency synchronization is uncertain, and must be resolved in coherent phase transmission. By marking the phase with a lower frequency (longer period) signal, the period ambiguity can be eliminated, and initial phase can be determined accurately.

In this paper, an absolute phase marking technology is proposed. It is realized by a high-speed digital logic circuit based on zero-crossing detection and an optimized control strategy. A marking pulse signal can be generated at the zero phase points of the frequency signal, and its generation position and repeat frequency can be selected by a coarse control signal. Hence, the phase-marking points strictly correspond to the pulse edges of the mark signal. Then, a fiber-optic remote coherent phase transmission scheme is proposed. It uses the absolute phase marking technology, which can not only overcome the phase ambiguity of the signal generation and transmission process, but also provide time reference signals with picosecond-level jitter, stabilize the phase difference, eliminate the initial phase uncertainty, and calibrate the transmission delay. Furthermore, all these processes are performed in the picosecond-level uncertainty. This paper provides an in-detail description regarding the principles and results. Section 2 explains the principle of absolute phase marking technology and provides experimental verification. The general principle of fiber-optic remote coherent phase transmission based on absolute phase marking is described in Section 3.1. The experiments for coherent phase transmission are designed in Section 3.2. The relative stability and uncertainty results of the coherent phase transmission are provided in Sections 3.3 and 3.4, respectively. In particular, the aspects of coherent phase transmission, which include providing a time reference signal with picosecond-level jitter, stabilizing the phase difference, eliminating the initial phase uncertainty, and calibrating the transmission delay, are realized step-by-step through the sections. Section 4 presents the conclusions of the study.

## 2. Absolute phase marking technology

The principle of the proposed absolute phase marking technology is illustrated in Fig. 1. First, we consider zero phase points of the continuous frequency signal (denoted by *F*) as the mark points. Then, *F* is converted into a square-wave signal with high-speed edges according to the mark points. Because every two mark points correspond to one period of *F* (denoted by *T _{f}*), the duration time of both the high and low levels of the square wave (denoted by

*W*) is equal to

_{m}*T*. If a certain period of

_{f}*F*corresponds to a high level of the square wave, the next period corresponds to the low level of the square wave, and the third period corresponds to the high level. Therefore, the repeated period of the square wave was twice that of

*T*. This indicates that the phase evolution law of

_{f}*F*has a one-to-one correspondence with the square wave. The phase has been marked completely, but arbitrary marking cannot be performed. Finally, a control signal (denoted by

*C*) was used to achieve arbitrary phase marking. The phase of

*F*is marked only when the rising edge of C arrives, and the phase-marking operation stops with the arrival of the falling edge of

*C*. The trigger level of the edges is equal to a 50% voltage difference between the high and low levels. The edge position, pulse width (denoted by

*W*), and repetition period (denoted by

_{c}*T*) of

_{c}*C*directly determine the phase-marking points and marking period (denoted by

*T*) in the result. Therefore, the absolute phase marking was completed, and the marking result can be described by the mark signal (denoted by

_{m}*M*) generated by the marking process.

Figure 1 also shows a type of phase-marking process. Three period phase changes of *F* is marked during the first *T _{c}* (denoted by p1, p2, and p3), and the output

*M*contains two sections of high level and one section of low level (denoted by m1, m2, and m3). Their duration time (denoted by

*W*) is equal to

_{m}*T*. While going to the second

_{f}*T*, another three continuous period phase changes are marked (denoted by p4, p5, and p6). The output

_{c}*M*is a double-pulse composite signal with a repetition period of

*T*, which is equal to the delay between the two sections marked phase points. Figure 1 further shows the transmission delay between the three signals. The transmission delay from

_{m}*C*to

*F*(denoted by

*τ*) is determined by the relative position of the rising edges of

_{cf}*C*and

*F*, which is a variable reflecting the phase-marking startup delay. The transmission delay from

*F*to

*M*(denoted by

*τ*) is determined by the technical principle, which is a constant reflecting the phase-marking output delay. When the control signal properties are modified, such as changing the position of the rising edge (that is, changing

_{fm}*τ*), narrowing the pulse width, and increasing the repetition period, the phase-marking process and result may change accordingly.

_{cf}We used the measurement system shown in Fig. 2 to verify the absolute phase marking technique. A high-precision frequency reference (Stanford Research Systems, FS725 Rubidium Frequency Standard) with 10 MHz outputs was used as the frequency source of the entire system. A digital pulse generator (Stanford Research Systems, DG645 and SRD1) was used to generate the control signal. A phase-locked frequency multiplier, mainly consisting of a 100 MHz oven controlled crystal oscillator (OCXO), phase-locked loop circuit, and 10-multiplier circuit, was used to generate a frequency signal of 1 GHz. Then, we used a high-speed logic module (HSLM) to perform the absolute phase-marking process. The HSLM is a T flip-flop with a reset function. When the reset pin is at a high level, it operates in a normal state, and its output switches to the opposite of its previous state according to the rising edge of the input clock. When the reset pin level is low, its output directly becomes low, regardless of the clock edge state. Therefore, we can use *F* as the clock input of the HSLM and *C* as the reset input of the HSLM to achieve the absolute phase-marking function and obtain *M* from the output of the HSLM. Finally, an oscilloscope (Keysight Technologies, MSOV334A) was used to test and analyze the frequency signal, control signal, and mark signal simultaneously, and was locked to the uniform 10 MHz frequency reference to decrease measurement errors. In addition, all cables used were fixed to avoid introducing additional delays during the entire measurement.

First, we set the properties of *C* as follows: *T _{c}* = 1 µs,

*W*= 1.75 ns, and

_{c}*τ*= 0.39 ns. The waveform traces of the three signals were simultaneously collected using the oscilloscope, as shown in Fig. 3. The marking period is 1 µs, which is equal to

_{cf}*T*. Special observations were made at the X position from −4 ns to 5 ns and from 1996 ns to 2005 ns.

_{c}*F*is a continuous sine signal with a frequency equal to 1 GHz, and

*C*is an irregular pulse signal with a pulse width of approximately 1.75 ns and rise time of approximately 100 ps (from 10% to 90%). Only one regular pulse of

*M*is included in these two X position ranges, which has a high level of 0.6 V, low level of 0 V, rise time of 30 ps, and pulse width of 1 ns equal to

*T*. In addition, the relative positions of the three signals in the two X position ranges are the same. The transmission delay can be accurately measured as

_{f}*τ*= 390.1 ps and

_{cf}*τ*= 544.7 ps, which were mainly caused by the inherent delay of the HSLM and cables. Furthermore, we used the oscilloscope to measure

_{fm}*T*,

_{f}*T*,

_{c}*T*, and

_{m}*W*over a long time. The average value and P-P jitter of

_{m}*T*are 999.8 ps and 1.1 ps, respectively, which indicate good phase properties of

_{f}*F*. The results of

*T*and

_{c}*T*are shown in Fig. 4 (a). The average value of

_{m}*T*and

_{c}*T*are both 1 µs, but the P-P jitter of the former is as high as 129.2 ps and that of the latter is 3.5 ps. Hence,

_{m}*M*shows a significantly precise repetition period, which benefits from the precise phase repetitiveness of

*F*(the precise

*T*). Figure 4 (b) shows the measured

_{f}*W*. The P-P jitter of

_{m}*W*is 3.3 ps, and the average value is 1.0005 ns, close to

_{m}*T*. These results demonstrate that the control signal strictly marks the phase of the frequency signal, and the phase-marking results can be accurately determined by measuring the properties of the mark signal.

_{f}Now, we set the other two different groups of the properties of *C* as follows: (a) *T _{c}* = 1 s,

*W*= 4.52 ns,

_{c}*τ*= 0.39 ns, and (b)

_{cf}*T*= 1 s,

_{c}*W*= 5.22 ns,

_{c}*τ*= 0.85 ns. Figure 5 shows the waveform traces acquired under the two settings. Currently, the marking period is 1 s, and the mark signal contains three pulses in one marking period. The first two pulse widths are 1 ns, but the third pulse width is less than 1 ns because the falling edge of

_{cf}*C*arrives and stops the marking process. As observed from the figure, the delay from the falling edge of

*C*to the falling edge of the current pulse of

*M*(denoted by

*τ*) represents the phase-marking shutoff delay, which is also constant owing to the technical principle. The waveform data shows that

_{cm}*τ*= 544.7 ps and

_{fm}*τ*= 875.4 ps for both the settings, and the third pulse width of (a) is 461.9 ps, but the third pulse width of (b) is 695.3 ps. These results demonstrate that modifying the properties of the control signal can achieve arbitrary phase marking effects.

_{cm}Therefore, we conclude that the properties of the control signal play a decisive role in the phase-marking process, and the phase-marking result can be determined by accurately measuring the mark signal. The pulse edges of the mark signal and marked phase zero points of the frequency signal strictly correspond. The measurements show that the period and pulse width of the mark signal have less than 3.5 ps P-P jitter for a measurement duration of more than 10,000 s. Therefore, the proposed absolute phase marking technology can achieve picosecond-level phase marking, and the generated mark signal with ultra-low time jitter can be used as a suitable time reference.

## 3. Fiber-optic remote coherent phase transmission based on absolute phase marking

#### 3.1 General principle

Based on the aforementioned absolute phase marking technology, we propose a fiber-optic remote coherent phase transmission system scheme. Figure 6 illustrates the general principle of the proposed scheme. To facilitate the analysis, we divided the system process into two steps. The first step, based on the existing fiber-optic joint time and frequency transfer technology, is the stable distribution of the signals. At the local site, the signal source outputs a frequency reference signal (continuous sine wave) and a control signal (PPS), which undergo electro-optical conversion (E/O) to form a modulated optical signal. The optical signal passes through a delay stabilization device and is sent to a remote site over the fiber. The modulated optical signal is demodulated at the remote site through opto-electrical conversion (O/E) to recover the frequency signal and control signal, which is used as the signal source of the remote site. The round-trip noise compensation method should be used to obtain stable signals. Therefore, the E/O/E method is also used to transmit the frequency and control signals back to the local site along the same fiber route. A delay measurement device was used to measure the delay change between the source and return signals at the local site. Half of the round-trip phase difference of the frequency signal (denoted by Δ*φ _{fr}*) is used as an error signal to drive the delay stabilization device in real time, making its delay change offset the delay fluctuation of the fiber link. Thus, a stable fiber transmission link was established. The frequency signals between both sites can have the same frequency and a stable phase difference, and the control signals can have a stable time delay. Thus, the signal distribution process is completed.

The second step, based on the absolute phase marking technology, is coherent phase transmission. After the first step, the two sites had relatively delay-stabilized frequency and control signals. We denote the control signal and frequency signal as *C _{L}* and

*F*at the local site, and

_{L}*C*and

_{R}*F*at the remote site. Two HSLMs are installed in each site to perform the phase-marking process and generate mark signals (denoted by

_{R}*M*and

_{L}*M*). By appropriately setting the initial control signal,

_{R}*M*and

_{L}*M*can be both 1 PPS, indicating that one phase period is marked within 1 s. Additionally, the mark signals can be used as time reference signals at the two sites. The reference plane of the coherent phase transmission is defined on the input location of the HSLMs at local and remote sites, as shown in Fig. 6. Owing to the physical time delay, synchronization can only occur between a certain period and other periods for the frequency signal. Figure 7 (a) shows the timing relationships between these signals. The marked phase points of

_{R}*F*and

_{L}*F*(denoted by

_{R}*φ*and

_{L}*φ*) are not yet synchronized, and

_{R}*M*and

_{L}*M*are not synchronized accordingly. The delay between

_{R}*C*and

_{L}*C*(denoted by Δ

_{R}*t*),

_{c}*φ*and

_{L}*φ*(denoted by Δ

_{R}*t*), and

_{f}*M*and

_{L}*M*(denoted by Δ

_{R}*t*), respectively, are theoretically equal.

_{m}*C*and

_{L}*C*must be synchronized first to eliminate the large delay between

_{R}*φ*and

_{L}*φ*. The delay measurement device at the local site measures the delay between the source control signal (

_{R}*C*) and round-trip control signal (

_{L}*C*in Fig. 6) to obtain the absolute round-trip link delay (denoted by Δ

_{rt}*t*). Further, Δ

_{cr}*t*= Δ

_{c}*t*/2 under the assumption that the forward and backward transmission delays are the same. The delay control device then applies a delay of Δ

_{cr}*t*/2 to

_{cr}*C*such that the delayed local control signal (

_{L}*C*in Fig. 6) and

_{Ld}*C*are synchronized. The timing relationships are shown in Fig. 7 (b).

_{R}*φ*has moved

_{L}*N*periods when compared to Fig. 7 (a), where $N=\left\lfloor\Delta t_{c} / T_{f}\right]$, and “⌊ ⌋” is the round down operation. Therefore, Δ

*t*has been significantly eliminated, and only a small delay (denoted by

_{f}*τ*, equals the phase time difference between

_{f}*F*and

_{L}*F*) remains, where

_{R}*τ*= Δ

_{f}*t*−

_{c}*NT*<

_{f}*T*, and Δ

_{f}*t*=

_{m}*τ*. If

_{f}*τ*is eliminated, coherent phase transmission can be achieved. According to the measured Δ

_{f}*t*,

_{cr}*τ*can be calculated theoretically as

_{f}*φ*) as an auxiliary to improve the calculation accuracy of

_{fr}*τ*. Owing to the period ambiguity problem during phase measurement, the phase difference between

_{f}*F*and

_{L}*F*can be calculated according to Δ

_{R}*φ*as

_{fr}*ν*

_{0}is the frequency of

*F*and

_{L}*F*. Therefore, the true value of

_{R}*τ*must be determined using the measured Δ

_{f}*t*and Eq. (1). After the delay control device applies a delay of

_{cr}*τ*to

_{f}*F*, the delayed local frequency signal (

_{L}*F*in Fig. 6) and

_{Ld}*F*are phase-synchronized. Hence,

_{R}*φ*and

_{L}*φ*, and

_{R}*M*and

_{L}*M*are synchronized, as shown in Fig. 7 (c). Thus, the coherent phase transmission processes are completed, and the time interval between

_{R}*M*and

_{L}*M*can reflect the coherent phase transmission performance.

_{R}#### 3.2 Experimental setup

The experimental setup for the fiber-optic remote coherent phase transmission is shown in Fig. 8. First, a frequency reference (REF), phase-locked frequency multiplier (PLM), and digital pulse generator (DPG) constitute the signal source of the entire system, which mainly outputs a 1 GHz frequency signal and 1 PPS control signal. Then, we adopt the time and frequency simultaneous transmission method based on dense wavelength division multiplexing over the optical fiber to transmit frequency and control signals. At the local site, the frequency signal (*F _{L}*) is amplitude-modulated to laser diode-1 (LD-1) (internally modulated distributed feedback (DFB) laser, wavelength

*λ*

_{1}= 1548.51 nm), and the control signal (

*C*) is amplitude-modulated to LD-2 (electro-absorption modulated DFB laser, wavelength

_{L}*λ*

_{2}= 1550.12 nm). The two modulated optical signals are combined by an optical coupler (OC) and then pass through a polarization scrambler (PS), circulator (CIR), and an optical delay line (ODL), and finally input to the fiber link. The PS is used to eliminate the influence of the polarization mode dispersion [9]. At the remote site, the optical signal is sent from another CIR into a dense wavelength division multiplexer (DWDM) to separate two optical signals with different wavelengths. Then, the modulated signals are detected by two photodetectors (PDs). The PD-1 (New Focus 1611, bandwidth 1 GHz) demodulates the frequency signal (

*F*), while the PD-2 (New Focus 1811, bandwidth 100 MHz) demodulates the control signal (

_{R}*C*). After optical fiber transmission, the phase noise of the frequency signal is degraded [8,9,16]. However, phase noise is the fundamental factor causing timing jitter in high-speed digital devices and systems [30,31]. To achieve the best phase-marking performance, the phase noise must be cleaned, which is often achieved by phase-locking the transmitted frequency signal to an oscillator with ultra-low phase noise [8,9,16]. Therefore, the frequency signal output by PD-1 is sent to a phase noise cleanup device (the cleaner in Fig. 8). The cleaner is composed of a ten frequency divider, phase detector, an OCXO, and a ten frequency multiplier. The OCXO outputs a 100 MHz signal with ultra-low phase noise, and the phase lock bandwidth is approximately 50 Hz. Thus, phase noise above 50 Hz is cleaned, and outputs the phase-noise-cleaned 1 GHz frequency signal (

_{R}*F*). Then,

_{R}*F*is modulated to LD-3 (internal modulation DFB laser, wavelength

_{R}*λ*

_{3}= 1547.72 nm), and

*C*is modulated to LD-4 (electro-absorption modulated DFB laser, wavelength

_{R}*λ*

_{4}= 1549.32 nm) and sent back to the local site along the same fiber link. The local site detects the round-trip signal by another DWDM and two PDs, where PD-3 (same as PD-1) demodulates the frequency signal (

*F*) and PD-4 (same as PD-2) demodulates the control signal (

_{rt}*C*). The phase frequency detector (PFD, Analog Devices, HMC439) measures the phase difference between

_{rt}*F*and

_{L}*F*(round-trip phase difference, Δ

_{rt}*φ*). Then, a proportion integration differentiation controller (PID) generates an error signal and feeds it back to the ODL to cancel the fluctuation of Δ

_{fr}*φ*(with the same feedback bandwidth of the cleaner). Therefore, the transmission delay of the link is stabilized. Next, the delay between

_{fr}*C*and

_{L}*C*(round-trip time delay, Δ

_{rt}*t*) is measured by a time interval counter (TIC, Stanford Research Systems, SR620), and the local DPG generates

_{cr}*C*according to Δ

_{Ld}*t*. According to Δ

_{cr}*t*and Δ

_{cr}*φ*, the electrical delay line (EDL) delays

_{fr}*F*to generate

_{L}*F*. Finally, the local HSLM completes the phase-marking process and generates the mark signal

_{Ld}*M*. At the remote site, another DPG reshapes the control signal to adapt

_{L}*C*to the marking condition, and then the remote HSLM completes the phase-marking process and generates the mark signal

_{R}*M*.

_{R}In the experiment, we set up optical fiber links of different lengths to study the performance of the coherent phase transmission system. The shortest link is a 1 m fiber patch cord, and the longest link consists of 150 km cascaded fiber spools. When the fiber length increases, the attenuation of the optical power also increases. Particularly, when the fiber link is longer than 90 km, a bidirectional erbium-doped fiber amplifier (Bi-EDFA) is added to compensate for the power loss. Here, the ODL length is 10 km (with 18 ns dynamic range). Hence, the Bi-EDFA is placed at half the total length of the fiber link and ODL to compensate for the power loss in both directions symmetrically. When the fiber link is shorter than 40 km, the ODL length is 5 km (with a dynamic range of 9 ns). The entire system was situated in an ordinary laboratory environment without an enclosed control. The performance was evaluated at the reference plane via phase analysis (phase difference measurement and phase noise measurement) and time analysis (time interval measurement), and the measurement equipment uniformly used the frequency reference of the REF to reduce measurement errors.

#### 3.3 Phase marking and its relative stability between local and remote sites

When the system reaches the closed-loop state, where the propagation delay is stabilized, we first evaluated the relative stability of the phase marking between local and remote sites. Based on previous studies [13,32], the relative stability of frequency synchronization (commonly expressed through Allan deviation) can reach the order of 10^{−14} at 1 s and 10^{−17} at 10,000 s, and the P-P jitter of the relative delay between local and remote control signals can be stabilized on the order of 100 ps. To facilitate the evaluation of relative phase-marking stability, we delayed *C _{L}* appropriately (the delay parameters are discussed in the calibration section) such that the time interval between

*M*and

_{L}*M*(called the marking time difference) is within 1 ns. We used an oscilloscope (MSOV334A) to measure the marking time difference under different fiber lengths, with a 1 s measurement interval and more than 10,000 s measurement duration. Figure 9 (a) shows the statistical analysis results of the obtained marking time difference data, which reflects the relative stability of the phase marking between local and remote sites, including the P-P jitter, time deviation (TDEV) at an averaging time of 1 s, and root-mean-square (RMS) jitter. The abscissa represents the total length of the fiber link and ODL; for example, 10 km indicates that the optical fiber link is 5 km and the ODL is 5 km. Under different fiber lengths, the P-P jitter does not exceed 3.5 ps, and the TDEV at 1 s and RMS jitter do not exceed 500 fs, showing ultra-high and consistent stability. The inset in Fig. 9 (a) shows the marking time difference fluctuation under the longest 150 km fiber link and 10 km ODL, with only a 3.3 ps P-P jitter within 10,000 s measurement time. Moreover, we used a signal source analyzer (AnaPico, APPH40G) to measure the absolute phase noise of the frequency signals under a 100 km fiber link and 10 km ODL, as shown in Fig. 9 (b). When compared with the source signal (

_{R}*F*), the phase noise of the frequency signal detected by PD-1 has significantly deteriorated at frequency offsets greater than 1 kHz, while the phase noise is significantly optimized after cleaner. This indicates the influence of optical fiber transmission on the phase noise of the frequency signal and cleanup effect of the cleaner. The inset in Fig. 9 (b) shows the relative stability of the 1 GHz frequency signal after cleaner, with 2.2×10

_{L}^{−14}at 1 s and 1.6×10

^{−17}at 10,000 s, proving a fine closed-loop state. The results also show that the mark signals can be used as ultra-low jitter time references in a time-frequency system.

#### 3.4 Calibration and the uncertainty of coherent phase transmission

To achieve a high-precision coherent phase transmission, a further calibration procedure is required to eliminate period ambiguity and initial phase uncertainty. According to the general principle, the time delay between *C _{L}* and

*C*must be calibrated first. Owing to the system asymmetry and influence of the external components located outside the entire loop, the time delay between

_{R}*C*and

_{L}*C*(called the control time difference) should be

_{R}*t*is the group delay difference caused by chromatic dispersion, and Δ

_{λ}*t*

_{0}is the inherent delay caused by the inconsistent devices and cables at the two sites. The two terms are almost unchanged when the environmental temperature of the system changes relatively slowly. In the system experiment of a 25 km optical fiber link and 5 km ODL, the system is switched off and restarted, and the time intervals from shutdown to restart range from 2 to 30 min. We measured the round-trip delay Δ

*t*and control time difference Δ

_{cr}*t*under each closed-loop state, with a measurement time interval of 1 s and a duration longer than 300 s. After calculating the average values of each measurement, we calculated the difference between the mean values of Δ

_{c}*t*and Δ

_{c}*t*/2, and the results are shown in Table 1. In 10 measurements, the value of Δ

_{cr}*t*−Δ

_{c}*t*/2 is between 104749 and 104788 ps, which is mainly due to the trigger delay of the DPG outside the loop located at the remote site. The trigger delay was approximately 98 ns. The results represent the range of Δ

_{cr}*t*+Δ

_{λ}*t*

_{0}. Because the maximum difference between these measurements is 39 ps, we may consider the average of the fourth column of the table as the value of Δ

*t*+Δ

_{λ}*t*

_{0}; thus, Eq. (3) becomes where Δ

*t*and Δ

_{c}*t*are given in units of ps. By calculating this delay value and using the DPG to apply it to

_{cr}*C*,

_{L}*C*and

_{Ld}*C*will be synchronized. The calculated delay value is denoted by

_{R}*d*. The calibration uncertainty is the uncertainty of Δ

_{c}*t*+Δ

_{λ}*t*

_{0}. This calibration is insufficient for high-precision time synchronization. However, because of the principle of phase marking technology, the control signal’s rising edge position only needs to be ensured between the two phase zero points of the frequency signal. Hence, the synchronization between

*C*and

_{Ld}*C*allowed is relatively rough here. The maximum allowable synchronization uncertainty was 500 ps (or the allowable time difference was 0 ± 500 ps) with the appropriate setting.

_{R}Table 1 also shows the round-trip phase time difference (*τ _{fr}*), phase time difference between

*F*and

_{L}*F*(

_{R}*τ*), and marking time difference (Δ

_{f}*t*). The phase time difference is calculated as

_{m}*φ*is the phase difference measured by the PFD, and

*v*

_{0}= 1 GHz. According to the 6×10

^{−5}rad resolution of Δ

*φ*, the resolution of the phase time difference measured indirectly by Eq. (5) is approximately 10 fs, which is much higher than the 100 fs resolution of direct measurement by the oscilloscope. Owing to the principle of the “round-trip phase stabilization method,”

*τ*is almost unchanged. However,

_{fr}*τ*and Δ

_{f}*t*sometimes change by approximately 500 ps, which can be determined from the value of Δ

_{m}*t*. As shown in Table 1, the value of Δ

_{cr}*t*increases by 999 ps from measurement 4 to 5, indicating that the round-trip delay of the system only increases one period of the frequency signal. Hence,

_{cr}*τ*remains unchanged, but Δ

_{fr}*t*increases by 510 ps,

_{c}*τ*increases by 499.71 ps, and Δ

_{f}*t*increases by 499.1 ps, indicating that the one-way delay of the system increases by half a period. From measurement 5 to 6, the value of Δ

_{m}*t*increases by 1004 ps, while Δ

_{cr}*t*increases by 509 ps,

_{c}*τ*decreases by 499.90 ps, and Δ

_{f}*t*increases by 499.9 ps. When compared with measurement 4, the round-trip delay increases by two periods, while

_{m}*τ*returns to the state of measurement 4, and other one-way parameters increase one period because they do not exhibit period ambiguity problem. The results show that the two round-trip parameters Δ

_{f}*t*and

_{cr}*τ*can be combined for coherent phase transmission.

_{fr}The control signal was calibrated and verified under the 25 km optical fiber link. In a calibration experiment, the measured Δ*t _{cr}* = 290506136 ps, Δ

*t*= 145357907 ps, and Δ

_{c}*t*= 145357408 ps before calibration. At the local site, the delay value that the DPG applies to

_{m}*C*is

_{L}*d*= 290506136÷2 + 104768 = 145357836 ps, 71 ps smaller than Δ

_{c}*t*. In practice, because the resolution of the DPG was 5 ps, the delay was set to 145357835 ps. After calibration, the measured Δ

_{c}*t*= 487 ps and Δ

_{c}*t*= 405.4 ps. The large time delay between the mark signals is eliminated after the calibration, and the remaining small delay is determined by the phase difference between the frequency signals. However, the value of Δ

_{m}*t*here is different from the expected value, which is theoretically equal to 71 ps. This is because of a certain error between the DPG delay control and true delay value described in its manual. As aforementioned, the synchronization accuracy of the control signals need not to be very high. To analyze the allowable range of the control time difference in reality, we set the delay value with a step of 100 ps based on

_{c}*d*, and then measured the Δ

_{c}*t*and Δ

_{c}*t*calibrated according to each delay value. As shown in Fig. 10 (a), Δ

_{m}*t*remains approximately 405 ps when the setting delay is from

_{m}*d*−500 ps to

_{c}*d*+300 ps. Correspondingly, the control time difference is from 232 ps to 1012 ps, and when the setting delay is out of that range, the value of Δ

_{c}*t*changes. Figure 10 (b) shows the time fluctuation data of Δ

_{m}*t*within 20,000 s of the measurement time. Its P-P jitter is 217 ps; hence, the allowable control time difference at any time should be from 232−217÷2 ps to 1012 + 217÷2 ps, that is, from 123.5 ps to 1120.5 ps, and the range size is 997 ps. Therefore, the actual allowable range of the control time difference is close to 1 ns, but the center is not at zero time difference. According to the phase-marking principle, this phenomenon is caused by a delay between the control signal and frequency signal (

_{c}*τ*). First,

_{cf}*τ*at the local site is not equal to

_{cf}*T*/2. The transmission delays of the control signal and frequency signal are not equal, and thus,

_{f}*τ*at the remote site changes by a certain delay value. Therefore, Δ

_{cf}*t*is limited to 0 ± 500 ps, but adjusting

_{c}*τ*by the DPGs at both sites can change the range of Δ

_{cf}*t*. Thus, the calibration of the control signal is relatively rough. At a later stage, we operate according to Eq. (4) and do not establish strict requirements regarding the control time difference after calibration to be close to zero.

_{c}According to the above calibration method of the control signal, delays before and after calibration under 10 times system shutdown and restart operations are measured, and the results are shown in Table 2. The difference between the calculated *d _{c}* and measured Δ

*t*before calibration was less than 100 ps. The Δ

_{c}*t*after calibration was between 480 ps and 539 ps, indicating good synchronization consistency of the control signals. A comparison of the marking time difference before and after calibration is observed, proving the validity of the calibration. After calibration, Δ

_{c}*t*is close to two values, 400 ps and 900 ps, which is approximately half the period of the frequency signal, and its variation law strictly corresponds to the variation of

_{m}*τ*. However, because of the inconsistency delay of the cable and instrument between the PFD-based measurement and oscilloscope-based measurement, the absolute values of

_{f}*τ*and Δ

_{f}*t*are not equal, and can be calibrated by uniformly using the oscilloscope. Moreover,

_{m}*τ*has several ps deviations between different measurements with similar values, resulting in the deviation of Δ

_{f}*t*, while

_{m}*τ*does not change more than 100 fs, which is the asymmetry problem caused by inconsistent system states. The problem occurred in all time and frequency transfer systems based on the round-trip method [8–10,18].

_{fr}The final step is the calibration of the frequency signal to eliminate *τ _{f}*. This is for phase synchronization, and the mark signals are automatically synchronized through absolute phase marking. According to Eq. (1) and (2), we can calculate

*τ*but require additional measurement and evaluation, making the calibration work complex. Based on the aforementioned experimental results, we can first observe the values of all delay parameters in a certain closed-loop system state and set them as the reference values. Then, we are required to calculate the round-trip delay variation relative to the reference value to obtain the present one-way values. If measurement 5 in Table 2 was used as the reference values, the Δ

_{f}*t*of measurement 1 increased by 3029 ps. That is, the delay was 3

_{cr}*T*longer. Hence,

_{f}*τ*increases by 1.5

_{f}*T*, and correspondingly, Δ

_{f}*t*should also increase by the same amount, which should change from 900.8 ps to 2400.8 ps. However, the 2 ns increment is eliminated by the control signal calibration such that the expected Δ

_{m}*t*is changed to 400.8 ps, which is 500 ps less than the reference value (900.8 ps). The actual measured value of the Δ

_{m}*t*is 400.1 ps, which is only 0.7 ps less than the expected value. Measurement 2 was identical. In addition, the Δ

_{m}*t*of measurements 6, 7, 8, and 10 relatively decreases by approximately 1 ns, and thus, Δ

_{cr}*t*decreases by approximately 500 ps. While the relative change of Δ

_{m}*t*is an even multiple of

_{cr}*T*, Δ

_{f}*t*is relatively unchanged, such as measurements 3, 4, and 9. Therefore, we monitored Δ

_{m}*t*and calculated the relative period change to identify whether to delay

_{cr}*F*by half of

_{L}*T*. The relative period change is calculated as

_{f}*t*represents the reference value of Δ

_{ref}*t*, “$[\; ]$” represents the rounding operation. This study uses measurement 5 in Table 2 as the reference values uniformly; hence, Δ

_{cr}*t*= 290507121 ps. When Δ

_{ref}*N*is odd, we delay

*F*by half of

_{L}*T*, and when Δ

_{f}*N*is even, we do not take any action.

According to the above calibration method of the frequency signal, the delays before and after the calibration under 12 times system shutdown and restart operation are measured, and the time interval between the last shutdown and the next restart is from 2 to 30 min. However, the interval between measurement 10 shutdown and measurement 11 restart reaches 10 h, and the next interval reaches 5 h. The results are shown in Table 3. When Δ*N* is even, EDL does not delay *F _{L}*, and

*τ*and Δ

_{f}*t*remain unchanged; when Δ

_{m}*N*is odd, EDL delays

*F*by 500 ps, and both

_{L}*τ*and Δ

_{f}*t*increase by approximately 500 ps. After calibration, the

_{m}*τ*is between 660.80 ps to 666.73 ps; hence, the maximum difference between the 12 measurements is 5.93 ps, which corresponds to the maximum phase difference of 0.0373 rad. The Δ

_{f}*t*is between 899.3 ps to 906.4 ps after calibration, and the maximum difference between the 12 measurements is 7.1 ps, indicating good coherent phase transmission consistency. To reflect the relationship between the marked phase difference and marking time difference, an oscilloscope was used to remeasure the calibrated

_{m}*τ*, as shown in the last column of Table 3. Here, Δ

_{f}*t*−

_{m}*τ*is between 4.4 ps to 5.5 ps. Because the HSLMs performing the phase-marking process at the two sites are not completely consistent, the phase-marking transmission delay

_{f}*τ*is different, which affects the consistency of Δ

_{fm}*t*and

_{m}*τ*. The experimental device shown in Fig. 2 was used to measure the transmission delays of two HSLMs;

_{f}*τ*= 544.7 ps (local HSLM) and

_{fmL}*τ*= 549.3 ps (remote HSLM) were obtained. Therefore, according to the principle of phase marking, Δ

_{fmR}*t*−

_{m}*τ*should be equal to

_{f}*τ*−

_{fmR}*τ*= 4.6 ps, which is consistent with the data in the table.

_{fmL}After the above calibration, the marked phase difference between two sites is maintained to be approximately a constant value. The constant value can be eliminated or adjusted by applying an appropriate delay to the frequency signal according to the user requirements at any time. Therefore, we have achieved coherent phase transmission by applying absolute phase marking technology. We can evaluate the uncertainty of the coherent phase transmission based on the measured values in Table 3. The average values of the 12 measurements of *τ _{f}* (using the 7th column data measured by PFD for accurate evaluation) and Δ

*t*are 662.96 ps and 902.6 ps, respectively. Under a confidence probability of 99.9%, the

_{m}*t*-distribution factor is 4.318; hence, the calculated uncertainties of type A are 2.577 ps and 2.65 ps, respectively. Because the value of the measurement resolution is sufficiently small, type B uncertainty is negligible. Therefore, type A is the total uncertainty. The uncertainty of coherent phase transmission evaluated by phase time difference is approximately 2.577 ps, and the uncertainty of coherent phase transmission evaluated by marking time difference is approximately 2.65 ps. Thus, the measured

*τ*and Δ

_{f}*t*in Table 3 can be expressed as 662.96 ± 2.577 ps and 902.6 ± 2.65 ps, respectively. This indicates that the system achieved a satisfactory and consistent coherent transmission performance. Moreover, the measurement results of Δ

_{m}*t*also show an excellent performance of the mark signals, which can be used as suitable time references. Thus, high-precision remote time synchronization over optical fiber can be realized simultaneously.

_{m}## 4. Conclusion

Reference phase synchronization is becoming increasingly significant in many applications. Therefore, we propose and demonstrate a technical scheme that can simultaneously achieve absolute phase marking and picosecond-level fiber-optic remote coherent phase transmission. The scheme uses the potential of high-speed digital communication circuits in high-precision analog information processing, and combines the characteristics of pulse time signal and continuous frequency signal. The former is used to eliminate long-distance ambiguity and the latter has high measurement control precision. First, through high-speed digital logic based on zero-crossing detection and optimized control, the absolute phase marking for the frequency signal is realized, transferring the excellent phase performances of the frequency signals to mark signals. After frequency synchronization and absolute phase marking, the relative frequency stability of the 1 GHz frequency signal can reach the order of 10^{−14} at 1 s and 10^{−17} at 10,000 s, and the relative phase-marking stability of picosecond-level P-P jitter is achieved under different fiber lengths. Particularly under a 150 km fiber link, the P-P jitter can be only 3.3 ps within 10,000 s measurement time, and TDEV can be as low as 438 fs at 1 s. Then, by measuring the round-trip time delay of the control signal, the periodic changes of the phase difference between the frequency signals are identified to eliminate the ambiguity of periodic signals. Combining with the phase difference measurement, the control signal calibration and frequency signal calibration are completed to remove the initial phase uncertainty. The coherent phase transmission over a 25 km fiber link is realized. The coherent transmission uncertainty evaluated by the phase time difference is approximately 2.577 ps, while the coherent transmission uncertainty evaluated by marking time difference is approximately 2.65 ps, indicating a fairly good coherent phase transmission consistency. The proposed scheme provides a new approach for realizing reference phase marking and remote coherent phase transmission. It can significantly improve the phase coherence of fiber-optic time-frequency transmission and has broad application prospects in fields such as radar networking and coherent array detection. It also demonstrates the excellent performance of the application of high-speed digital communication technology in high-precision analog signal transmission, and provides a method for simultaneous remote synchronization of time, frequency, and phase over optical fiber.

## Funding

National Natural Science Foundation of China (61875214); Strategic Priority Research Program of the Chinese Academy of Sciences (XDB43000000); Youth Innovation Promotion Association of the Chinese Academy of Sciences (YIPA2019251); National Key Research and Development Program of China (2020YFC2200300).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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