At present, the stability of atomic frequency standards has reached an unprecedented level. Notably, cesium fountain clocks serving as primary frequency standards have exhibited frequency instability of a few ${10}^{-16}$ at one-day integration time^[1,2]. Nevertheless, traditional methods for frequency transfer over satellite links, which are commonly used to compare remote clocks, are limited by an instability of ${10}^{-15}$ at one-day integration time^[3]. This limitation is insufficient for effectively transferring the precision of modern atomic clocks. Furthermore, long-haul and ultra-stable radio frequency (RF) distribution is essential for numerous scientific and technical applications, including fundamental physics, radio astronomy, and very long baseline interferometry (VLBI)^[4–8].

Over the past decades, microwave frequency transmission^[9–20] via optical fiber has been extensively investigated due to factors such as the low attenuation, high reliability, and excellent potential for phase noise cancellation offered by optical fiber transmission. In previous fiber microwave frequency transfer experiments covering distances less than 100 km, the introduction of scramblers in the fiber transmission link, along with the use of the same wavelength laser for round-trip frequency transfer, has been effective in achieving long-term stability better than ${10}^{-18}/\mathrm{d}$^[21,22]. However, as the length of the fiber optic link increases to 100 km or more, the influence of backscattering and Rayleigh scattering in the fiber optic link necessitates the utilization of different wavelengths of carrier light in the round-trip link of fiber optic microwave transfer^[23,24]. The presence of environmental factors such as temperature fluctuations and vibrations can indeed introduce unavoidable asymmetric phase delays in wavelength division multiplexing (WDM) systems. This delay cannot be fully eliminated by the round-trip transmission link, consequently impacting the long-term stability of the system^[25–28]. Therefore, it is crucial to improve the long-term stability of a long-optic fiber microwave frequency transfer system by minimizing unavoidable asymmetric phase delay^[10,26].

In this paper, we propose a novel compensation technique for mitigating dispersion in fiber-optic microwave frequency transfer systems. The system can effectively reduce the influence of asymmetric phase delay in fiber links by adding an extra carrier light. Our main goal is to mitigate the adverse effects of optical fiber link dispersion on lasers employing diverse wavelengths, a factor that affects the long-term stability of microwave frequency transfer. Through rigorous implementation, the system achieves remarkable long-term stability of microwave frequency transfer spanning 100 km, surpassing the threshold of ${10}^{-18}$. This breakthrough implicates the feasibility of maintaining exceptional stability in the transmission of microwave signals over vast distances spanning hundreds of kilometers via optical fibers.

The architecture of the optic-fiber microwave frequency transfer system is depicted in Fig. 1, which incorporates a unique design employing three distinct optical carriers with wavelengths ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$. Specifically, in the direction from the transmitting end to the receiving end, ${\lambda }_1$ and ${\lambda }_3$ are utilized, while ${\lambda }_2$ is employed for the reverse transmission from the receiving end back to the transmitting end. Notably, the relationship between the wavelengths satisfies ${\lambda }_1-{\lambda }_2={\lambda }_2-{\lambda }_3$. The influence of fiber dispersion on the phase delay of fiber optic links can be expressed as $$\frac{\partial \tau (\lambda ,T)}{\partial \lambda }=L(T)·D(\lambda ,T)=\frac{1}{c}·L(T)·\frac{\partial n(\lambda ,T)}{\partial \lambda },$$(1)assuming ${\phi }_{p1}$ is the asymmetric phase delay introduced by the dispersion of the fiber link when the wavelength transitions from ${\lambda }_1$ to ${\lambda }_2$, and ${\phi }_{p2}$ corresponds to the asymmetric phase delay resulting from the transition from ${\lambda }_3$ to ${\lambda }_2$. We can observe that the magnitude of these two phase delays is equal but their signs are opposite. Specifically, ${\phi }_{p1}$ is caused by ${\lambda }_1-{\lambda }_2$ while ${\phi }_{p2}$ is caused by ${\lambda }_2-{\lambda }_3$. Therefore, the two-phase delays are equal in magnitude and opposite in sign, that is, $${\phi }_{p1}=-{\phi }_{p2}.$$(2)

Utilizing frequency division and mixing techniques, the system generates forward and backward signals with distinct frequencies, enabling highly precise measurements of link noise. The theoretical framework for phase compensation in this system is outlined as follows. At the local end, the reference signal $V_r$ can be expressed as a sinusoidal wave with a defined frequency and phase (neglecting its amplitude), $$V_r=\sin ({\omega }_{r}t+{\phi }_r).$$(3)

A voltage-controlled oscillator (VCO) generates a frequency signal $V_o$ that can be expressed as $$V_o=\sin ({\omega }_{o}t+{\phi }_o).$$(4)

The $V_o$ is frequency-divided by 2 to generate $V_1$ and $V_2$, $$V_1=V_2=\sin (\frac{{\omega }_o}{2}t+\frac{{\phi }_o}{2}).$$(5)

$V_1$ and $V_2$ are used to modulate the intensity of the two lasers with distinct wavelengths, which generates the forward signal. At the remote end, the incoming signal is detected by fast photodiodes (PD2 and PD3), resulting in electrical signals $V_3$ and $V_4$ detected by fast photodiodes (PD2 and PD3), which can be expressed as $$V_3=\sin (\frac{{\omega }_o}{2}t+\frac{{\phi }_o}{2}+\frac{{\phi }_p}{2}+{\phi }_{p1}),$$(6)$$V_4=\sin (\frac{{\omega }_o}{2}t+\frac{{\phi }_o}{2}+\frac{{\phi }_p}{2}+{\phi }_{p2}),$$(7)where ${\phi }_p$ is the symmetric phase delay introduced in the fiber link when the microwave frequency is ${\omega }_o$. The received microwave signal $V_{rx}$ is obtained using the up-mixing method for $V_3$ and $V_4$, which can be expressed as $$V_{rx}=\sin ({\omega }_{o}t+{\phi }_o+{\phi }_p+{\phi }_{p1}+{\phi }_{p2}).$$(8)

According to Eq. (2), it can be concluded that $$V_{rx}=\sin ({\omega }_{o}t+{\phi }_o+{\phi }_p).$$(9)

The $V_{rx}$ signal is frequency-divided by 4 to modulate the intensity of the third laser, which subsequently generates the backward signal. At the local end, the return signal $V_{\text{back}}$ is detected by the photodiode (PD1), which carries the round-trip phase fluctuations accumulated along the fiber. $V_{\text{back}}$ can be expressed as $$V_{\text{back}}=\sin (\frac{{\omega }_o}{4}t+\frac{{\phi }_o}{4}+\frac{{\phi }_p}{2}).$$(10)

After a series of electronic operations in the phase comparison system at the local end, an error signal $V_e$ is given by $$V_e=\sin (\frac{2{\omega }_o-5{\omega }_r}{4}t+\frac{2{\phi }_o+2{\phi }_p-5{\phi }_r}{4}).$$(11)

When the phase-locked loop(PLL) is closed, the loop filter controls the phase of the transmitted signal ensuring that $V_e=0$, and then there is the following equation: $$\{\begin{array}{c}{\omega }_o=\frac{5}{2}{\omega }_r\\ {\phi }_o=\frac{5}{2}{\phi }_r-{\phi }_p\end{array}.$$(12)

Therefore, the received signal at the remote end can be expressed as $$V_r=\sin (\frac{5}{2}{\omega }_{r}t+\frac{5}{2}{\phi }_r),$$(13)when the PLL is closed and locked to the reference signal $V_r$. This synchronization process effectively compensates for the link noise.

The experimental system (100 km) involves the construction from two 50 km fiber coils, each exhibiting respective losses of 10 and 11 dB. To compensate for these losses and enable the bidirectional transmission of signals over the entire 100 km span, a bidirectional erbium-doped fiber amplifier (Bi-EDFA) with a gain of approximately 17 dB is employed. To mitigate the impact of chromatic dispersion ($-17\mathrm{ps}/\mathrm{nm}·\mathrm{km}$), a dispersion compensation fiber (DCF) with a dispersion of approximately 1700 ps/nm and a loss of about 8 dB is inserted at the local end. The schematic diagram of the experimental system is shown in Fig. 2. Both the local and remote ends are located in the same laboratory. The reference frequency $V_r$, generated by an RF signal source (Keysight E8257D), is set at 4 GHz as the stable frequency reference for the system.

At the local end, the microwave signal $V_o$ is frequency-divided into two signals ($V_1$ and $V_2$) for transmission. These signals are then modulated onto carrier lasers with distinct wavelengths (1550.92 and 1552.52 nm) using Mach–Zehnder modulators (MZMs). After undergoing polarization scrambling (PS) to mitigate polarization-dependent loss, the signals enter the input port of a fiber optic circulator (FOC) that incorporates the wavelength division multiplexing (WDM) functionality. At the local end, the optical carriers with different wavelengths are separated using WDM. Two fast photo detectors (FPD2 and FPD3) are utilized to convert the optical signals back to microwave signals $V_3$ and $V_4$. To circumvent the impact of the third-order intermodulation in the mixer, an auxiliary source is introduced. This auxiliary source mixes with $V_3$ to generate $V_5$ through up-conversion and mixes with $V_4$ to yield $V_6$ via down-conversion. Finally, $V_5$ and $V_6$ are mixed to obtain the received 10 GHz signal ($V_{rx}$). Subsequently, $V_{rx}$ is further frequency-divided to 2.5 GHz using a low noise frequency divider (Analog Devices, HMC365). This 2.5 GHz signal serves as the backward signal and is then modulated onto carrier lasers (1551.72 nm) using an MZM.

At the local end, the 2.5 GHz backward signal is detected by FPD1, and the resulting output signal $V_{\text{back}}$ encapsulates the round-trip phase fluctuations accumulated over the fiber. The microwave module and phase-locked loop circuit depicted in Fig. 1 are instrumental in compensating for delay disturbances within the fiber optic link.

In the measurement of frequency transfer instability (see Fig. 2), the received 10 GHz signal at the remote end undergoes frequency division by a factor of 2.5, resulting in a 4 GHz signal. This 4 GHz signal is then mixed with a 4 GHz reference signal using the heterodyne method, which downconverts it to a DC voltage $V(t)$. This voltage is precisely measured by a digital multimeter with 8½ digits resolution (model 3458 A from Keysight). Notably, the frequency division by 2.5 is achieved through the utilization of the second harmonic generated by a frequency divider designed for a division factor of 5. A phase shifter is employed to precisely align the phases of the two inputs to the mixer, ensuring they are in quadrature. In this configuration, the resulting DC voltage $V(t)\approx 0$ is directly proportional to the phase fluctuations around the quadrature point. The propagation delay fluctuations (commonly referred to as the normalized phase fluctuations) within the compensation link are then derived from this measurement: $$x(t)\approx \frac{1}{{\omega }_r}·\frac{V(t)}{V_{\mathrm{pp}}/2},$$(14)where ${\omega }_r$ denotes the angular frequency of the reference signal and $V_{\mathrm{pp}}$ signifies the peak-to-peak voltage of $V(t)$ when the phase difference between the two mixer inputs varies by $2\pi$. The frequency transfer instability, expressed as the Allan deviation, is subsequently determined based on the measurement of $x(t)$.

Figure 3 demonstrates the temporal evolution of the propagation delay for both the previous scheme (black trace)^[17] and the new scheme (red trace) over a period of six days. Clearly, after several days of testing with the previous scheme, the fiber optic link exhibited a delay fluctuation of approximately 1 ps. However, with the implementation of the new scheme, the delay fluctuations in the fiber link were significantly reduced to below 0.3 ps. This significant reduction indicates that the new scheme offers more than three times the long-term stability compared to the previous one.

Figure 4 presents the fractional frequency instability of the 100 km fiber link, comparing the results obtained with the previous scheme (black squares)^[17], the new scheme (red circles), and the system noise floor achieved using an optical attenuator to shorten the link (blue triangles). The Allan deviation analysis reveals a frequency instability of $4.4\times {10}^{-15}$ at 1 s integration time and $7.7\times {10}^{-19}$ at one-day integration time for the new scheme. This performance is notably better than the ${10}^{-18}$ instability achieved at one-day integration time with the previous scheme. This improvement validates the effectiveness of our novel microwave frequency transfer system and its ability to minimize undesired fiber propagation effects.

At the same time, Fig. 4 also shows the system noise floor achieved by utilizing an optical attenuator instead of the fiber optic link. It is observable that the medium- and long-term instability of the new scheme does not align with the results achieved by the system noise floor. This discrepancy in long-term stability can be attributed to residual polarization mode dispersion (PMD) effects or amplitude modulation to phase modulation (AM/PM) conversion in photodiodes and mixers^[21]. However, despite these residual effects, the frequency transfer instability of the new scheme is significantly superior to the performance of cesium fountain clocks.

To further highlight the long-term stability enhancement of the new scheme over the previous one, we conducted experiments utilizing a 100 km optical fiber link placed in a high- and low-temperature chamber. In the first experiment, we employed 1550.92, 1551.72, and 1552.52 nm lasers to carry out round-trip signal transmission through the fiber link using the previous scheme. In contrast, the second experiment utilized the new scheme proposed in this paper to transmit the round-trip signal over the same fiber link. During the temperature control experiment, the temperature of the optical fiber spools was varied from 20°C to 50°C. Due to uneven heating of the 100 km fiber spools during the test, only a consistent section of data from the middle was selected to plot the graph showing the phase delay change with temperature, as depicted in Fig. 5. This comparison aims to clearly demonstrate the improved stability of the new scheme under varying temperature conditions.

As evident from Fig. 5, during the heating process, the propagation delay exhibits opposing temperature slopes in the two experimental groups of the previous scheme. Specifically, the slopes are 0.061 and $-0.052\mathrm{ps}/°\mathrm{C}$ respectively. However, in the new scheme, the slope of the transmission delay with respect to temperature shows a significant decrease, approximately 0.011 ps/°C. This reduction in slope indicates a significant improvement in the long-term stability of the new scheme when compared to the previous one.

According to Eq. (14) in Ref. [25], the relationship between the optical fiber link delay and temperature can be expressed as $$\delta \tau =\frac{\partial \mathrm{\Delta }\tau }{\partial T}=\mathrm{\Delta }\lambda ·L·(k+D·\alpha ).$$(15)

In the case of a commercial G652 optical fiber, with $\alpha =5.6\times {10}^{-7}/°\mathrm{C}$, $D=17\mathrm{ps}/(\mathrm{nm}·\mathrm{km})$, and $k=-1.45\times {10}^{-3}\mathrm{ps}/(\mathrm{km}·\mathrm{nm}°\mathrm{C})$ at approximately 1550 nm^[25], microwave frequency transmission over a 100 km fiber optic link employs two adjacent channels (e.g.,  C31 and C32) within the ITU standard wavelength grid for round-trip transmission. This configuration results in an optical wavelength difference of 0.8 nm. Based on our calculations, the phase delay jitter of the 100 km fiber optic link is determined to be $\delta \tau =-0.056\mathrm{ps/}°\mathrm{C}$.

The data slopes observed in the two sets of experiments conducted under the previous scheme are found to be opposite in direction, yet both are approximately equal in magnitude to $\delta \tau$. This consistency validates our theoretical calculations. Further analysis reveals that the slope of the propagation delay with respect to temperature variation in the new scheme is approximately equal to the sum of the slopes obtained from the two experiments under the previous scheme. This finding demonstrates that the new scheme effectively eliminates the influence of asymmetric phase delay due to temperature changes, thereby enhancing the long-term stability of the link. The primary reason why the slope of the new scheme cannot be reduced to zero is attributed to the inability to achieve perfect offset among the wavelengths of the three lasers used in the experiment, as well as the presence of wavelength drift in the lasers themselves.

In conclusion, a novel compensation technique is introduced for mitigating dispersion in fiber-optic microwave frequency transfer systems. By introducing an additional optical signal, the proposed approach effectively mitigates the effects of dispersion induced by temperature fluctuations. This enhancement significantly contributes to the improvement of fiber microwave frequency transfer stability over long distances. Specifically, over a 100 km fiber link, the frequency transfer stability of the new scheme achieves $4.4\times {10}^{-15}$ at 1 s integration time and $7.7\times {10}^{-19}$ at one-day integration time, which is much better than ${10}^{-18}$ stability threshold at one-day integration time. In summary, the proposed scheme holds great potential for advancing the reliability and performance of long-distance optical fiber microwave frequency transfer systems.