Optical atomic clocks have been developed rapidly in recent years^[1-4]. The distribution of these high-precision clock reference signals is significant and has many important applications, such as geodesy, fundamental physics, timekeeping, global navigation, and astronomy^[5-9].

Optical fiber links offer an ideal solution for users to coherently transfer stable optical frequency signals to remote sites, particularly for long-distance transmission and multi-user access to high-performance time and frequency signals^[10-16]. Currently, remote ultra-stable laser comparison has been realized through optical frequency transmission via the field fiber link with 2220 km length by Schioppo et al. in 2022^[17].

Unfortunately, the environmental temperature variations and mechanical perturbations along the fiber links will dominate the fluctuations of transfer delay of the fiber links and introduce additional phase noise to the optical frequency signal. To overcome this problem, a local site active noise compensation technique was demonstrated by Ma et al. in 1994, which generally obtained the phase error from a beat note signal of round-trip and reference light to feed a servo system^[18]. Based on this local phase noise cancellation scheme, various noise compensation links have been demonstrated by several research groups and have achieved remarkable results with point-to-point structures^[19-24]. A point-to-multi-point access structure via the fiber topology model at local site compensation was demonstrated by Grosche in 2014^[25]. After that, more flexible solutions using the branching method for optical frequency transfer with remote site compensation have been proposed^[26-29]. In these schemes, the error signal is obtained by beating the single trip light against the third trip light in the fiber link, and the phase noise is compensated at the remote site. So, the remote time base has an influence on the precision of the transferred signal. In contrast, Refs. [28,29] have achieved time-base-free optical frequency distribution at the remote site, avoiding the time-base noise accumulation with the growth of users. In their system, to eliminate the influence of the time base, an acousto-optic modulator (AOM) is equipped out of the transfer loop at the remote site while its thermal noise and the additional mismatch of optical fiber cannot be compensated, which will affect the transmission performance in an unstable external environment. Otherwise, Xue et al. demonstrated a novel remote site passive compensation technique for transferring the optical frequency through a branching fiber link. In their experiment, a high transmission instability was achieved to 3.7×10−19 at 10,000 s integration time with the same time base^[27]. In addition, the noise floor and transfer performance are effectively optimized by precise temperature control (∼10mK) of the transfer system at local and remote sites. However, the temperature control will increase the cost and complexity of the system. Hence, the most effective and simplest method is to design a remote site compensation-based optical frequency transfer system with a 10 MHz time-base-free and temperature-insensitive structure by properly designing the fiber optics system.

In this paper, we demonstrate a novel time-base-free optical frequency transfer method via a 150 km fiber link with remote site compensation. In our experiment, we utilize special choosing radio frequencies (RFs) and a commercial 10 MHz oscillator to realize time-base independence at the remote site. By optimizing the optical path design and inserting the phase compensation device into the compensation loop, the sensitivity of the transmission system to temperature changes can be further reduced, and the transmission performance can be improved. The induced phase noise of the fiber and the thermal noise of the optical devices are compensated by the active noise cancellation technique. By comparing the transferred optical frequency signal against the reference light, the 150 km fiber link relative instability is 7.42×10−15 at 1 s integration time and scales down to 1.07×10−18 at 10,000 s integration time as the 1/τ slope. Meanwhile, the frequency uncertainty is a few 10−19 in this link. Otherwise, this scheme has an optimized mismatch of the interferometer and a better noise floor without temperature control in the system, showcasing its potential for ultra-stable optical frequency signal transfer among point-to-point or point-to-multi users. The paper is organized as follows. First, the principle of time-base-free at remote site compensation is illustrated. Then, the experimental setup and the results are demonstrated including the instability in the time domain, the phase noise in the frequency domain, and the frequency transfer accuracy. Finally, the results are analyzed.

In this remote compensation scheme, the laser source is a commercial narrow-linewidth fiber laser (Koheras Basik E15) operating at 1550.12 nm with a linewidth of 0.1 kHz. Figure 1 shows the diagram of the experimental setup for optical frequency signal transfer through the fiber link. The initial electric field E0 of the light from the laser can be expressed as E0=A0cos(ω0t+ϕ0),(1)where A0 is the amplitude, ω0 is the angular frequency, and ϕ0 is the phase of this transfer light. Then, the light passes through a 1×2 fiber optical coupler 1 (OC1) and a 2×2 OC2 to acousto-optic modulator 1 (AOM1). Here, the light is first frequency-shifted +50 MHz, and the AOM1 driving frequency is from direct digital synthesizer 1 (DDS1), which is synchronized with the local rubidium clock time base. Then the electric field becomes E′=A′cos[(ω0+ωAOM1)t+ϕ0+ϕAOM1],(2)where ωAOM1=2π×50MHz and ϕAOM1 are the driving angular frequency and phase of AOM1, respectively. Then, the transferred light is coupled into the fiber link with a power of ∼5mW. For compensating fiber link loss (∼0.2dB/km), two bidirectional erbium-doped fiber amplifiers (Bi-EDFAs) are installed in the link.

When the fiber link output light arrives at the remote site, it is first frequency shifted by −152MHz through AOM2, in which the electric field E′′ of the −1st diffracted signal is $$\begin{gathered} E′′=A′′cos[(ω0+ωAOM1−ωAOM2)t+ϕ0 \\ +ϕAOM1−ϕAOM2+ϕfiber]. \end{gathered}$$(3)

At this time, the noise from the fiber link ϕfiber and the effects of AOM2 are added to the transmitted light. AOM3 is a specially designed device, and the 1×2 structure has two output arms, that the zeroth arm directly passes the light without frequency shift, while the first arm light is driven by DDS3 with a frequency of +102 MHz. Here, the zeroth arm light and two Faraday mirrors (FMs) form an active phase-locked loop with the 1×2 OC3. Assuming the fiber link phase noise in forward and backward directions is equal to the disturbance from the environment, the electric fields of the single trip and the third trip are expressed as $$\begin{gathered} E′′′=A′′′cos[(ω0+ωAOM1−ωAOM2)t+ϕ0 \\ +ϕAOM1−ϕAOM2+ϕfiber+ϕAOM30th], \end{gathered}$$(4)$$\begin{gathered} E1=A1cos[(ω0+3ωAOM1−3ωAOM2)t+ϕ0 \\ +3ϕAOM1−3ϕAOM2+3ϕfiber+3ϕAOM30th]. \end{gathered}$$(5)

Photodetector 2 (PD2) extracts the phase noise by beat notes E′′′ and E1. Consequently, this term of PD2 carries twice the phase noise given by Eq. (6), and the beat note frequency is 204 MHz. Within the feedback bandwidth, the fiber-induced phase noise can be eliminated, $$\begin{gathered} Iac∝2A′′′A1cos[(2ωAOM1−2ωAOM2)t \\ +2ϕAOM1−2ϕAOM2+2ϕfiber+2ϕAOM30th]. \end{gathered}$$(6)

DDS4 generates another 204 MHz RF signal whose electric field is EDDS4=ADDS4(ωDDS4t+ϕDDS4). Then the signal is mixed with Iac, and the error signals flit to the servo system. The servo system makes ωAOM1−ωAOM2=ωDDS4, and ϕAOM1−ϕAOM2+ϕfiber+ϕAOM30th=ϕDDS4 achieves the link phase noise compensation.

When the transfer link is compensated, the single trip light of the remote site at the AOM3 first arm is $$\begin{gathered} Est=A3cos[(ω0−ωDDS4+ωAOM31th)t+ϕ0 \\ −ϕDDS4+ϕAOM31th]. \end{gathered}$$(7)

Note that here ωAOM31th and ϕAOM31th depend on a remote time base, so make ωAOM31th=ωDDS4 and ϕAOM31th=ϕDDS4. Consequently, Eout=Est is Eout=Aoutcos[(ω0)t+ϕ0].(8)

We can see that Eout is not affected by the local or remote time base from Eq. (8).

We have shown the diagram of the experimental configuration in Fig. 1. The transfer link is 150 km and consists of two fiber spools: one span is 100 km, another span is 50 km, and the single trip loss is more than 30 dB. Two Bi-EDFAs with a gain of 16 dB are placed at the link to compensate for the fiber loss. To achieve a heterodyned beat note signal at PD1, we up mix the drive frequency of AOM3 to 107 MHz by mixer 2 with a signal generator (RS SMB100A) that locks on the local rubidium clock. With this configuration, we obtain a 5 MHz beat note signal by combining the AOM3 first arm light and the local laser reference light with the 1×2 OC5. Because of the complex RF signals in the system, a DC-5 MHz low-pass filter (LPF) 1 is applied after PD1 and a 100–110 MHz band-pass filter (BPF) 2 before the SMA interface of AOM3. Meanwhile, a tracking oscillator filter (TOF) is equipped after the PD2 to amplify and stabilize the signal amplitude with a center frequency of 204 MHz. Mixer 1 down mixes the DDS4 signal and the in-loop signal to obtain an error for the servo system. Here, these two signals are both divided by 204 to 1 MHz with two programmable digital frequency dividers. To guarantee that the system is working properly, a DC-100 kHz LPF2 is applied after the mixer. In addition, the drive amplitude should be taken into account and carefully adjusted to the zeroth and first diffracted light powers of the AOM3.

To measure the performance of the compensation link, we use a no-dead-time frequency counter (FXE K+K) with the ∏-type mode and a gate time of 1 s to record the out-loop data and calculate the overlapping Allan deviation (OADEV) in the time domain. As shown in Fig. 2, the relative frequency instability of the light after the compensated fiber link with the rubidium clock time base is 8.43×10−15 at 1 s integration time, and it decreases and reaches 1.16×10−18 at 10,000 s integration time (blue triangle curve) with the 1/τ slope. The red dot curve illustrates the OADEV result of the 10 MHz commercial time base in which the 1 s integration time is 7.42×10−15 and scales down to 1.07×10−18 at 10,000 s integration time, also as the 1/τ slope. From these two instability results, it becomes evident that, when the transfer link is compensated, the remote site time-base does not affect the transfer precision, as the transfer results are very similar and not affected by the time base. As a comparison, the black square curve illustrates the OADEV of a commercial 10 MHz oscillator as the remote time base. The pink triangle curve is the free-running link when the remote compensation system is off. In our experiment, it is observed that the stability of the optical frequency is improved by approximately 4 orders of magnitude at 10,000 s integration time by the remote site compensation system. The green triangle curve shows the noise floor of the 1 m transfer link by active phase noise cancellation. Here, we find that the noise floor reached the 10−20 level at 2000 s integration time, which is more than 2 orders of magnitude in the short term (1 s) and 1 order of magnitude in the long term (4000 s) lower than the compensation link in this system, while the OADEV slope is limited by the temperature variation-induced length fluctuation of the interferometer mismatch fiber in the measurement system.

To analyze the phase noise of this link in the frequency domain, a phase noise analyzer (Rohde & Schwarz FSWP8) is utilized to measure the phase noise power spectral density (PSD) for both compensated and free-running configurations over the 150 km optical signal transfer, as depicted in Fig. 3. When the link is free running, the green curve illustrates that the phase noise PSD Sfiber(f) agrees with the type of Eq. (9)^[22] below 200 Hz as indicated by the yellow dashed line, mainly limited by the flicker phase noise, Sfiber(f)=h0/f2,(9)where f is the Fourier frequency and h0 is dependent on environmental disturbances of fiber links and equals 10 in this 150 km transfer system. Once the compensation system is engaged in the remote site by the users, the relative phase noise is effectively canceled down by approximately 50 dB at 1 Hz, shown as the red curve. The compensated phase noise PSD is close to 10−4rad2/Hz in the low-frequency range of between 0.5 Hz to 10 Hz, which means that the compensated fiber link is mainly constrained by the white phase noise after stabilizing the transfer link. Meanwhile, according to the results of the red curve, it can be seen that the feedback bandwidth of the system is about 167 Hz (1/8τ)^[29]. The theoretical phase noise Stheoretical(f) limitation from free running is calculated by^[22,29]Stheoretical(f)=13(2πfτ)2Sfiber(f)(10)and plotted as the blue curve in Fig. 3, and it has a good agreement with the compensated link.

For further evaluation of this compensation system, we also evaluate the accuracy of the optical frequency transfer. Figure 4(a) shows the frequency deviations of the recorded data by the FXE K+K counter with a 1 s gate time and Π-type mode. The data are measured over successive 51,000 s (green point, left axis). We can obtain 103 points by averaging the data every 500 s, as shown in Fig. 4(a) (orange points, right axis). Histograms and Gaussian fits of the frequency deviation are also shown in Fig. 4(b). To better show the details of the distribution, the frequency deviation axis has been enlarged by 1000 times. According to the Gaussian fits in Fig. 4(b), the calculated results demonstrate that the mean frequency is shifted by −7.9μHz (−4.1×10−20), and the standard deviation of the 500 s data points is 2.90 mHz (1.5×10−17), which is more than 2 orders of magnitude smaller than the OADEV at 1 s as expected for this Π-type evaluation.

In conclusion, we demonstrate a remote time-base-free technique for a 150 km coherence optical frequency transfer link, and the fiber-induced optical phase noise is compensated at the remote site. In this scheme, an ultra-stable optical frequency reference can be reproduced using a commercial 10 MHz oscillator as the time-base instead of a high-performance clock by the remote user. In addition, to optimize the impact of the asymmetric part of the fiber on the performance of the system, we use a 1×2 AOM in the loop in which the zeroth light is used to build a servo loop and the first diffracted light is launched to users. Based on this experimental setup, the OADEV of the relative optical frequency transfer instability is 7.42×10−15 at 1 s integration time and decreases as the 1/τ slope to 1.07×10−18 at 10,000 s integration time. Benefiting from the time-base-free technique, the noise floors at 1 s and 10,000 s are improved by almost 2 and 1 order of magnitude, respectively. The frequency uncertainty of the light after transferring through the fiber is a few 10−19. With this phase noise compensation setup, once the system is locked, the phase noise can be effectively suppressed by approximately 50 dB at 1 Hz. The results demonstrate significant potential for constructing a flexible, convenient, and cost-effective point-to-point or point-to-multi-user topology structure in this setup. This alternative method of optical frequency transfer complements existing local or remote site compensation systems and is compatible with cascaded transfer links.