## Abstract

In this paper, an absolute optical transfer delay measurement method based on pulse time signal, pseudo-random code phase, and microwave phase is proposed. By employing two-stage integer ambiguity resolution, not only can a measurement range of several hundred kilometers be achieved, but sub-picosecond level measurement accuracy can also be attained. A test system was built in the laboratory and experimentally verified on a fiber optic link. The experimental results verify an accuracy of ±0.1 ps in measuring an ultrahigh-accuracy optical delay line. In addition, long fiber is also tested, which proves that a measurement range of at least 100?km can be achieved. The refresh rate of the measurement results can reach 100?ms each time.

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## 1. Introduction

Highly accurate absolute optical transfer delay (AOTD) measurement has important applications in many fields, such as distributed detection systems [1–3] and precise navigation and positioning systems [4,5]. The current Global Navigation Satellite System (GNSS) has begun to make extensive use of laser interstellar links, which, in addition to interstellar communications, also require high-accuracy AOTD measurement in order to further improve the orbit determination accuracy of navigation satellites. At present, GNSS can realize orbit measurement with an accuracy of better than 3?cm based on ground station observation data [6]. If the sub-picosecond optical time delay measurement accuracy can be realized at the ultra-long distance of tens of thousands of kilometers between stars, the orbit determination accuracy can be further improved to better than 1 mm, which enables the GNSS to realize autonomous navigation and positioning without relying on ground stations. In addition, by improving GNSS orbit determination accuracy with high-accuracy AOTD measurement, the accuracy of mapping, ranging and positioning for daily use can also be improved from meter to centimeter level.

Time-domain-based AOTD measurement methods primarily measure the transfer delay of optical signals directly by calculating the transfer time of pulse signals [7,8]. This method is simple and intuitive, with a wide measurable range that can reach up to the scale of hundreds of kilometers. However, its disadvantages are also evident. The measurement accuracy is not high, only reaching the nanosecond level, and as the measurement range increases, the measurement accuracy decreases. This is because its measurement accuracy is closely related to the pulse width: a narrower pulse results in higher measurement accuracy. However, after long-distance transfer, the pulse width broadens due to dispersion, especially in optical fibers where the effect is more severe.

Frequency-domain-based AOTD measurement methods indirectly measure transfer delay through frequency measurements. Optical frequency domain reflectometry (OFDR) can achieve a distance resolution of 10 µm at a distance of several hundred meters [9], but when the distance increases to several kilometers, the resolution drops to the millimeter level. The proposed OFTD measurement methods based on phase-locked loops [10], based on mode-locked lasers [11], and based on free-running lasers [12] make it possible to achieve centimeter-level resolution at distances of tens of kilometers [13].

Due to advancements in phase discrimination technology, phase-based AOTD measurement methods have gained significant attention for their high measurement accuracy. These methods primarily include sine microwave phase measurement and pseudo-random code phase measurement. The AOTD measurement method based on the phase of sine signals typically employs multi-frequency signals to resolve the integer ambiguity problem [14,15]. The measurement accuracy is determined by the high-frequency signal, while the measurement range depends on the frequency interval. Generally, it can achieve sub-picosecond measurement accuracy at distances of tens of kilometers. Methods based on nonlinear frequency sweeping and phase derivation for distance measurement have achieved a time delay measurement accuracy of ±0.05 ps with a measurement range exceeding 37?km [16]. Benefiting from the autocorrelation characteristics of pseudo-random sequences, the AOTD measurement method based on pseudo-random code phase also exhibits high measurement accuracy [17,18]. The measurement accuracy is primarily determined by the code rate, while the measurement range is mainly determined by the code length. Limited by the storage depth of acquisition equipment, it can generally achieve ps-level measurement accuracy at measurement distances of tens of kilometers.

In addition, optical frequency comb (OFC) based techniques offer the possibility of highly accurate transfer delay measurements due to their ultra-high resolution [19–21]. The use of OFC and linear optical sampling (LOS) techniques have been verified to achieve fs-level delay measurement accuracy. By introducing a dual-comb LOS technique, where two optical pulses are sampled using a third OFC locked to one of the measured signals, a measurement accuracy of 82 fs can be achieved in a single LOS scan measurement [22]. The latest research results show that OFC based schemes also have the potential for long-range measurement [23]. The femtosecond-level time synchronization between remote microwave-clock-based timescales over a commercial fiber link of 205.86?km was successfully demonstrated.

Based on the establishment of a laser communication link and by multiplexing this link, we propose a high precision optical transfer delay measurement scheme applicable to long distance at a lower cost than OFC. The scheme is based on a pulse time signal, a pseudo-random code signal and a sinusoidal microwave signal realized together. It is characterized by simple structure, low cost, high measuring accuracy and long measuring distance. First, a rough AOTD measurement is obtained by measuring the rising edge interval of the 1 pulse per second PPS (PPS) time signal before and after transfer. It has a wide measurement range but a relatively low accuracy in the sub-nanosecond order. Next, a correlation operation is performed on the pseudo-code before and after transfer to obtain a delay value with a small measurement range but higher accuracy. The rough AOTD measurement of 1?PPS is utilized to disambiguate the integer ambiguity of the pseudocode and improve the AOTD measurement accuracy to the picosecond order. Finally, the phase of the single-frequency sinusoidal microwave signal before and after transfer is measured. A time delay value with higher accuracy can be obtained from the phase difference. The pseudo-coded AOTD measurement is then utilized to disambiguate its integer ambiguity, further improving the accuracy of the AOTD measurement to the sub-picosecond order of magnitude. A fiber optic test system was set up in the laboratory using a 1?PPS time signal, a 250 Mbps pseudo-random code signal, and a 1?GHz sinusoidal microwave signal. We demonstrate that the AOTD measurement system can achieve measurement accuracy better than ±0.1 ps, a sampling rate of 100?ms, and a measurement range of more than 100?km in optical fiber.

## 2. Principle

Figure 1 illustrates the schematic diagram of the proposed AOTD measurement system. A continuous optical carrier signal is generated by a narrow linewidth, low-noise single-frequency laser and then transmitted to an electro-optic modulator (EOM). The modulation signals, including a 1?PPS, a pseudo-random binary sequence (PRBS), and a sinusoidal microwave, are all generated by an arbitrary waveform generator (AWG) and loaded onto the EOM in a time-division multiplexed manner to modulate the amplitude of the optical carrier. The modulated optical carrier passes through the link under test (LUT) and enters a photodetector, where the 1?PPS pulse signal, pseudo-random code signal, and sinusoidal microwave signal are demodulated and recovered in sequence. The demodulated and recovered signals, along with the untransmitted reference signals, are sent to their respective measurement modules to obtain corresponding delay measurement values. The PPS provides a large unambiguous range but with low measurement accuracy; the pseudo-random code offers a moderate unambiguous range and measurement accuracy; and the sinusoidal microwave signal provides the highest measurement accuracy but with the smallest unambiguous range. Therefore, the delay measurement of 1?PPS is used to resolve the integer ambiguity of the delay measurement of pseudo-random code, and the delay measurement of pseudo-random code is then used to resolve the integer ambiguity of the delay measurement of sinusoidal microwave. By utilizing this two-stage ambiguity resolution approach, both high measurement accuracy and a large unambiguous range can be achieved simultaneously.

The optical carrier signal generated by the laser can be expressed as:

where ${A}_{0}$ is the amplitude of the optical carrier and ${f}_{0}$ is the center frequency of the laser beam emitted by the laser. The initial phase is assumed to be 0 in order to simplify the equation.

The optical signal after amplitude modulation can be expressed as:

where*k*is the modulation depth of the amplitude modulation and

*S(t)*is the modulated signal.

This optical signal is transmitted through the LUT and can be expressed as:

where $\alpha $ is the link transfer loss and $\tau $ is the transfer delay. The modulated signal $S(t-\tau )$ is recovered by demodulation from the intensity of the optical carrier signal using direct detection.

The transmitted 1?PPS pulse signal and the untransmitted reference 1?PPS pulse signal are then measured by a time interval counter to obtain the transfer delay ${\tau}_{coarse\phantom{\rule{thickmathspace}{0ex}}}$. This delay measurement can be used to disambiguate the pseudo-random code's integer ambiguity *M*.

*m*is the code length of the pseudo-random code and ${t}_{0}$ is the chip width.

The pseudo-random code signals before and after transfer undergo an autocorrelation operation to obtain a delay value . After defuzzification with measurements of 1?PPS time signal, the true pseudo-random code transfer delay ${\tau}_{middle}$ is obtained.

This measurement can be used to further de-ambiguate the integer ambiguity *N* of sinusoidal microwave signals.

The phase change $\mathrm{\Delta}\theta $ of the sinusoidal microwave signal before and after transfer is obtained using phase discriminator measurements. Since a microwave phase discriminator is used, the measurement range of the phase is only [0, 2$\pi $). Therefore, the time-delay measurement result of the pseudo-random code is used to disambiguate the phase ambiguity of 2$\pi $. The time-delay measurement result of a sinusoidal microwave signal can be expressed as:

Table 1 summarizes the properties of the three measurement signals used above and describes the role they each play in the final measurement result. The selection of the specific parameters of the three measurement signals needs to consider the processing capability of the hardware in addition to the matching relationship between them.

## 3. Results and discussion

An experiment is performed based on the setup shown in Fig. 1. The amplitude of the 1?PPS time signal used is 3.3?V, the pulse width is 50 µs, and the width of the rising edge is 10?ns; the amplitude of the pseudo-random code signal used is 2?V, the code rate is 250 Mbps, and the code length is 2^{11}; and the amplitude of the sinusoidal microwave signal used is 2?V, and the frequency is 1?GHz. The optical carrier generated by the NKT-X15 Narrow Linewidth Laser is amplitude modulated by the EOM (iXblue, MAXN-LN-10). The phase-discriminating voltage output from the discriminator is captured by a 6½-digit high-precision digital multimeter (KEYSIGHT, 34465A) with a sampling rate of 100?ms each time and a phase-discriminating accuracy of up to 6 × 10^{−4?}rad @ 1?GHz. The optical fiber model used is a G.652D, with a typical refractive index in the 1550?nm band of n = 1.467. A rubidium clock (FS725 Rubidium Frequency Standard, Stanford Research Systems) capable of outputting 10?MHz frequency standard signals and 1?PPS time signal was used as the reference source for the entire system.

According to Eq. (7), the measurement accuracy of the proposed method is related to the frequency of the microwave signal. Figure 2 shows the relationship between the measurement accuracy and period of ambiguity with the modulated microwave frequency. The left axis reflects the measurement accuracy and the right axis reflects the period of ambiguity. From the figure, it can be seen that as the microwave frequency increases, the measurement accuracy increases, but the period of ambiguity range is gradually decreasing. The black solid squares and black solid dots in the figure mark the relevant parameters at 1?GHz and 10?GHz frequencies, respectively. When the microwave frequency is 1?GHz, the measurement accuracy is up to 0.095 ps and the period of ambiguity range is 1.00?ns; when the microwave frequency is increased to 10?GHz, the measurement accuracy can be improved to 0.010 ps and the period of ambiguity range is reduced to 0.10?ns.

Before an AOTD measurement can be made on the LUT, the systematic difference of the test system must be measured. All subsequent measurements need to be calibrated by subtracting the delay value from this systematic difference. Without connecting the LUT, the fiber optic patch cords between A and B in Fig. 1 are directly connected to each other via flanges, and the delay in the part of the figure within the dashed box on the light green background is measured to obtain the systematic difference. After calibrating the system difference, connect the fiber optic patch cords at A and B to the LUT, and measure the time delay of the part inside the faint yellow background dashed box in the figure, which is the AOTD of the LUT.

Figure 3 shows the results of the systematic difference. The blue dotted line graph in the figure represents the systematic difference obtained from the measurements using the 250 Mbps pseudo-random code. The measured values fluctuate within the range of 43.6834-43.6843?ns, with a peak-to-peak jitter of 0.9 ps, a mean value of 43.6838?ns, and a standard deviation of 0.3 ps. The red dotted graph represents the systematic deviation measured using a 1?GHz sinusoidal microwave signal, and the magnified view of the first 20 systematic deviation measurements is shown in the upper left. The measured values fluctuate within the range of 43.68375-43.68381?ns, with a peak-to-peak jitter of 0.06 ps, a mean value of 43.68378?ns, and a standard deviation of 0.01 ps. It can be seen that the mean value of the microwave measurements differs from that of the pseudo-random code measurements by 0.05 ps, which is within the accuracy range of the pseudo-random code measurements. It can be concluded that the delay based on microwave measurement obtained after defuzzification is reliable. Therefore, 43.68378?ns is taken as the systematic difference of the test system.

To evaluate the accuracy of the proposed scheme, the fiber optic patch cords at points A and B were connected to a motorized variable optical delay line (General Photonics MDL-002) with a resolution of less than 1 fs and an accuracy of ±0.01 ps, which was used as a reference for the measurement accuracy of the system.

Figure 4(a) shows the time delay of the measurements when the MDL is varied in 1 ps steps. It can be seen that the measured value of the system agrees well with the set value of the MDL. Figure 4(b) shows the deviation of the system measurements from the set value of the MDL, which fluctuates within the range of -0.05-0.09 ps. Therefore, the measurement accuracy of the system is considered to be better than ±0.1 ps.

To verify the measurable range of the system, fiber optic patch cords at points A and B were connected to a 100?km fiber. Figure 5 shows the AOTD measurements obtained using 1?PPS time signal, 250 Mbps pseudo-code signal and 1?GHz microwave signal, respectively. Figure 5(a) shows 10 AOTD measurements obtained using 1?PPS time signal, the measurements fluctuate in the range of 490145.27-490145.86?ns with a peak-to-peak jitter of 0.59?ns and an average of 490145.56?ns. Figure 5(b) shows 100 AOTD measurements obtained using 250 Mbps pseudo-code signals with a peak-to-peak jitter of 0.0012?ns. It is clear that there is integer ambiguity in these measurements, and the value of integer ambiguity is calculated to be M = 59. The AOTD measurements after removing the ambiguity fluctuate within the range of 490145.5560-490145.5573?ns, with a mean value of 490145.5566?ns. There is a deviation of 3.4 ps from the mean value measured directly from the 1?PPS time signal that is within the accuracy of the 1?PPS time delay measurement. Therefore, we consider the pseudo-random code based AOTD measurement after removing the ambiguity to be reliable. We used its mean value of 490145.5566?ns to remove ambiguity for the 1?GHz microwave signal-based measurements shown in Fig. 5(c), and computed the integer ambiguity value N = 490145. As can be seen in Fig. 5(c), the peak-to-peak jitter of the microwave signal-based AOTD measurements is reduced to 0.62 ps. Therefore, we use the measurements obtained after two-stage integer ambiguity resolution as the final AOTD measurements.

Figure 6 shows the final accurate AOTD measurement results obtained after two-stage integer ambiguity resolution, with a refresh rate of 100?ms and a continuous measurement time exceeding 30 minutes. A delay drift in the range of 46 ps was observed, which is due to the susceptibility of long fibers to environmental changes. Notably, this delay drift shows a sinusoidal trend with a frequency of about 2?mHz. The reason for this is that the air conditioning was turned on in the laboratory, which caused the room temperature to show a cyclic oscillatory variation centered at 25 °C. The results of this test show that the proposed system has a large dynamic range of at least 100?km. Meanwhile, in order to avoid the influence of environmental factors such as temperature and vibration on the test system as much as possible, we have taken temperature control and vibration isolation measures for the fiber-optic devices and electrical devices in the system. The blue curve in Fig. 6 shows the delay measurement results of the back-to-back system. It can be seen that the delay of the system under temperature control and vibration isolation remains stable, and the fluctuation during the measurement is within ±0.1 ps.

It should be noted that since the measurable unambiguous range of a 1?PPS time signal is 1 s, the theoretically measurable dynamic range is 200,000?km in optical fiber and 300,000?km in space. However, in the actual application process, there are still many difficulties in realizing such ultra-long-distance measurements. Firstly, the serious power attenuation has to be solved, secondly, the establishment of a stable transmission link for ultra-long distances is very challenging, in addition, the transfer signal itself will be interfered by the link noise, and will face many problems such as distortion, spectrum expansion and stability degradation, etc. Therefore, the actual measurement effect of ultra-long distance needs to be analyzed and experimentally verified. Therefore, more detailed analysis and experimental verification of the actual measurement effect at ultra-long distances are needed.

## 4. Conclusion

In this paper, a simple and reliable high-accuracy AOTD measurement method over a large range is proposed. The method combines the advantages of pulse time signal, pseudo-random code signal and sinusoidal microwave signal in time-delay measurements, which not only achieves measurement accuracy on the sub-picosecond scale, but also meets the measurement range of tens of thousands of kilometers. By calibrating the systematic differential measurements, AOTD measurement can be realized. In addition, the refresh rate of the measurement results can reach the sub-second level.