The uncertainty and instability of optical clocks have been demonstrated to be better than 10−18 [1–4]. Establishing an optical clock network through a satellite–ground link can significantly improve many existing applications, such as geodesy [5,6], global navigation satellite system (GNSS) [7,8], redefinition of the second [9–11], and research on fundamental physics [12–16]. When remote optical clocks are connected by time-frequency transfer techniques without degrading the clock signal, an optical clock network is formed. To achieve this, the accuracy of the time-frequency transfer must match that of the optical clock. Based on microwave links and traditional laser links, it is difficult for the satellite–ground time-frequency transfer to exceed the level of 10−17 within one day [17–21]. Currently, the technique [22] of combining optical frequency combs (OFCs) and linear optical sampling (LOS) has shown its potential in many applications [23–26], especially in coherent optical two-way time-frequency transfer (O-TWTFT) for a satellite–ground optical clock network. This scheme has demonstrated excellent transmission performance on long-distance free-space links [27–33]. However, for a real satellite–ground link, we need to consider the impact of relative motion and time delay.

Figure 1(a) shows the satellite–ground O-TWTFT. The Doppler effects and point-ahead effects caused by the relative motion and the delay effects between the satellite–ground transmission links, result in the breakdown of the time-of-flight reciprocity. Bergeron et al. used a moving platform to simulate a Doppler shift with a maximum radial velocity of 24 m/s, which corresponds to the velocity of satellite flight in 34,400 km high earth orbit, and achieved 10−19 instability [34]. Due to the relative motion, the satellite cannot capture the ground-to-satellite signal while the ground receives the signal from the satellite. To address this issue, a point-ahead mirror is placed on the transmitting path to aim at the position where the satellite will arrive beforehand, as shown in Fig. 1(b). The deflection of the point-ahead mirror produces a point-ahead angle, which represents the angular deviation between the transmitting and the receiving path, the uplink, and the downlink. This phenomenon is referred to as point-ahead effects. In addition, there is a time delay between when the ground-to-satellite signal and the satellite-to-ground signal reach the same location, which is referred to as delay effects. The higher the orbit, the larger the time delay. The delay for a medium earth orbit (MEO) or a geostationary earth orbit (GEO) is approximately 0.05 s or 0.1 s. In the satellite–ground link, due to atmospheric turbulence, the point-ahead effects and delay effects will cause spatial and temporal variations, respectively, which are atmospheric non-reciprocity effects. A theoretical analysis reported an even better expectation that point-ahead effects would be less than 2×10−17 at 1 s [35], and a preliminary experiment is conducted on the 2-km free-space link [36]. However, there has not been a sufficient experimental measurement of atmospheric non-reciprocity effects for O-TWTFT in scenarios closely resembling the satellite–ground situation, such as over long distances and under worse turbulence.

In this work, we simulate point-ahead effects and delay effects on the ground. An orthogonal polarization scheme is used to separate the transmitting and receiving paths of the optical transceiver. To eliminate the variation of time-of-flight caused by separate optical paths, we develop an integrated optical bench, which bonds all optical components onto a substrate. This not only improves optical path stability but also provides good temperature robustness. Then, we construct the two-way spatio-temporally separated links, which simulate atmospheric non-reciprocity effects. Based on these, we measure the effects for O-TWTFT over a 113-km free-space link. The results show that for short averaging time, the link instability deteriorates, worsened by spatial non-reciprocity effects, and increases from 5.0×10−15 at 1 s to 7.0×10−15 at 1 s. The temporal non-reciprocity effects further amplify the link instability, rising from 7.0×10−15 at 1 s to 1.5×10−14 at 1 s. For a long averaging time, the link instability is primarily impacted by temperature variations rather than the effects. Consequently, the impact of atmospheric non-reciprocity effects is less than 2.3×10−19 at 10,000 s.

In the satellite–ground link, the separation of the uplink and downlink caused by point-ahead effects and delay effects can be described by spatio-temporal displacement. The larger the spatio-temporal displacement, the stronger the atmospheric non-reciprocity effects. As the spatio-temporal displacement increases, the difference in turbulence between the uplink and downlink becomes larger, resulting in larger phase variations between paths. The two beam paths become increasingly decorrelated, resulting in a decrease in transmission performance. The spatio-temporal displacement is given by [37], d(z)=X+θz+V(z)Δt.(1)Here, z is the optical propagation distance. X is the displacement between the transmitter and receiver apertures. A transceiver shares the same aperture for transmitting and receiving, X=0. θz is the spatial displacement that increases with z, and θ is the point-ahead angle. V(z)Δt is the temporal displacement, and it represents that the turbulence is displaced at V(z) by wind and by slewing of the satellite during the time delay Δt.

For the spatial displacement θz, θ is approximately 10–30 μrad between the ground and the MEO/GEO satellites. Compared to low earth orbit (LEO), the instability of long-term time-frequency transfer can be better evaluated in MEO or GEO because the higher the satellite orbit, the weaker the Doppler effects, and the longer the common-view time. z is considered as vertical atmospheric thickness, which is equivalent to 5–10 km of the horizontal atmospheric thickness. There is almost no atmosphere turbulence above the stratosphere, and atmospheric non-reciprocity effects can be ignored. Thus, the corresponding maximum separation distance θz is 50–300 mm. For the temporal displacement V(z)Δt, Δt is equal to the time-of-flight of the transmission links. For MEO and GEO satellites, the time-of-flight of the links is about 0.05–0.1 s, while through the atmosphere near the ground with an equivalent horizontal thickness of 5–10 km it is less than 40 μs.

To measure the atmospheric non-reciprocity effects of the satellite–ground link, we construct an equivalent spatio-temporal displacement in a 113-km free-space link. The spatial displacement is 174 mm, which remains constant over the 113-km free-space link and represents the average of the maximum value of θz. Then, the 113-km horizontal atmospheric link is 10 times the equivalent horizontal atmospheric thickness of the satellite–ground link. Therefore, the spatial atmospheric non-reciprocity effects can be equivalent to that of the satellite–ground link. For the delay of temporal displacement, the time-of-flight of the 113-km free-space link is approximately 380 μs, which is smaller than that of the satellite–ground link. To construct an equivalent delay, we shift the time labels of the one-way link by Δt with the raw data and combine the data of both sides with the same arrival time. Varying Δt allows us to achieve the same delay as the satellite–ground link. The time label offset of 0.05 s or 0.1 s corresponds to a delay at the MEO or GEO, respectively. Based on these, we construct an equivalent spatio-temporal displacement by two-way spatio-temporally separated links between two stationary terminals located 113 km apart. This is close to real satellite–ground link conditions, with worse turbulence and large attenuation due to long-distance transmission.

Based on comb-based O-TWTFT over a 113-km free-space link [32], we measure the atmospheric non-reciprocity effects. To construct the spatial displacement of the point-ahead effects between two stationary terminals, we separate the transmitting and receiving paths of the optical transceiver using an orthogonal polarization scheme and construct a separate two-way atmospheric link by using a semicircular aperture (SCA). To construct the temporal displacement of the delay effects over the free-space link, we shift the time labels of the one-way link with the experimental data. In addition, an independent fiber link is used to transmit the frequency of the ultra-stable laser (USL) from one terminal to another. As shown in Fig. 2(a), the free-space link between Nanshan (NS) and Gaoyazi (GY) is 113 km, and an independent fiber link connects two USLs. The primary mirror of Telescopes A and B is a Cassegrain reflector with an aperture of 400 mm and a focal length of 1600 mm. In Transceivers A and B, the OFC signal at wavelengths of 1545 nm (1545 signal) and 1563 nm (1563 signal) are transmitted to fiber collimators (FCs) via polarization-maintaining fibers. Three polarizing beam splitters (PBSs) are used to separate the transmitting and receiving paths, which require signals with high polarization contrast. The V-polarized signal passes through the transmitting path, while the H-polarized signal passes through the receiving path in Transceiver A, and vice versa in Transceiver B. The mirror (MIR) positioned in the transmitting path is used to simulate a point-ahead mirror that deflects the point-ahead angle. To achieve transmitting and receiving in the two OFCs signals independently, a dichroic mirror (DM1) combines them in the transmitting path and separates them in the receiving path. We fine-tune the concentricity and coaxiality of the two OFCs signals, and after calibration, they are better than 95%. To compensate for incoming wavefront distortions caused by atmospheric turbulence and enhance the coupling efficiency from free space into a single-mode fiber, we insert a deformable mirror (DMIR) in the receiving path.

We construct a two-way separated atmospheric link with 174 mm spatial displacement by partially blocking the signal in the transceiver. As shown in Fig. 2(a), an SCA is positioned in the transmitting path of Transceiver A, allowing only the left half of the 1545 signal to pass through, while the right half is being blocked. The left half of the SCA aligns with the lower half of Telescope A, while the right half aligns with the upper half, as the signal’s orientation is flipped to pitch as it travels from the fast steering mirror (FSM) to the telescope. In Transceiver B, an SCA is positioned in the receiving path with the same attitude as that of the transmitting path of Transceiver A. The signal is also flipped from Telescope B to the FSM, so optical interference occurs between the signal passing through the telescope’s lower half and the local signal. Therefore, from Telescope A to Telescope B, the 1545 link is established in the lower half of both telescopes. Similarly, two SCAs are positioned on the receiving path of Transceiver A and the transmitting path of Transceiver B. This ensures that only the right half of the 1545 signal passes through, while the left half is being blocked. From Telescope B to Telescope A, the 1545 link is established in the upper half of both telescopes. Based on these, the two-way 1545 link is separated. The calculated spatial displacement is 174 mm, determined by the telescope’s 400 mm aperture and a central circular blocking diameter of 70 mm. In addition, an additional 6 dB loss is added to the received power because half of the signal is lost at each SCA. To confirm the spatial non-reciprocity of the two-way link, we cover the lower or upper half of the telescopes and measure the received power of the 1545 signal in Transceivers A and B (see Appendix A). In addition to constructing the two-way separated 1545 link (the non-reciprocal 1545 link), we also establish the two-way overlapping 1563 link (the reciprocal 1563 link), while the 1563 link is used as a reference for comparison to eliminate the drift of the USLs.

To simulate point-ahead effects, we separate the transmitting and receiving paths, resulting in the time-of-flight of two paths that differ and vary with temperature. Thus, we need to consider the appropriate method for establishing the separated optical paths. Traditional optical instruments typically consist of optical-mechanical structures where the optics are first fixed in mechanical fixtures before being installed on an aluminum experimental platform. However, the coefficient of thermal expansion (CTE) of the aluminum is approximately 2×10−5/°C, leading to poor structural stability. The gravitational wave detection satellite LISA Pathfinder, launched in 2015, used a highly stable optical bench in its core unit [38]. The optical bench is an integrated optical technology that bonds low CTE materials together, ensuring excellent stability and robustness against temperature variations. As shown in Fig. 2(a), our optical bench comprises a fused silica substrate with a CTE of 5.5×10−7/°C, along with the DM, PBS, MIR of the same CTE. Figures 2(b) and 2(c) show their pictures: the base size of the substrate is 170mm×150mm×30mm for Transceiver A and 190 mm×140mm×30mm for Transceiver B. Precise positioning of all optics is achieved, and they are bonded to the substrate using ultraviolet-curing adhesive. Since adjustments cannot be made after integration, all optics must exhibit extremely high positioning and pointing accuracy prior to bonding. To achieve this, a high-accuracy hexapod with a multi-dimensional, high-precision automatic adjustment mechanism is employed. Additionally, laser beam profilers, calibrated collimators, and external detection systems are employed to facilitate proper installation and accurate alignment of the optics. Moreover, we implement temperature control for the optical bench with a temperature variation of 1°C peak-to-peak. To verify the path stability of the separate transmitting and receiving paths, we conduct simulations (see Appendix B). The results demonstrate a total asymmetric optical path variation of approximately 73 nm on both optical benches with a temperature change of 1°C, corresponding to a time-of-flight of 0.24 fs.

The 1545 signal through the two-way separated atmospheric link has a time-of-flight of T1545,A from Transceiver B to Transceiver A, and a time-of-flight of T1545,B from Transceiver A to Transceiver B, T1545,A=TLink+T1NR,s+T2NR+ΔTAB,(2)and T1545,B=TLink−T1NR,s−T2NR−ΔTAB.(3)TLink is the common time-of-flight across the link. T1NR,s is the spatial time-of-flight noise caused by the 174 mm spatial displacement. T2NR is the noise of the separated transmitting and receiving paths of the 1545 signal in the transceivers. ΔTAB is the time difference between the two USLs. The time difference of the non-reciprocal 1545 link is T1545,A−T1545,B2=T1NR,s+T2NR+ΔTAB.(4)Similarly, for the reciprocal 1563 link, T1563,A=TLink+T3NR+ΔTAB,(5)and T1563,B=TLink−T3NR−ΔTAB.(6)T3NR is the noise of the separated transmitting and receiving paths of the 1563 signal in the transceivers. The time difference of the 1563 link is T1563,A−T1563,B2=T3NR+ΔTAB.(7)The time difference ΔTAB is eliminated by combining two time-frequency transfer links. The time-of-flight noise is given by (T1545,A−T1545,B)−(T1563,A−T1563,B)2=T1NR,s+T2NR−T3NR.(8)The time noise of T2NR and T3NR is minimized with the assistance of the optical bench, so we no longer consider T2NR and T3NR. The spatial time-of-flight noise T1NR,s can be measured. Simultaneously, we construct the temporal displacement of delay effects by shifting the time label of one side of the non-reciprocal 1545 link by Δt with the raw data. Equation (4) can be written as T1545,A(t+Δt)−T1545,B(t)2=T1NR,s,t+ΔTAB(t+Δt)+ΔTAB(t)2.(9)T1NR,s,t is the time-of-flight noise caused by spatio-temporal displacement. But this causes a delay of ΔTAB(t+Δt) between the two USLs. If we use the same method to eliminate the time difference of the USLs with a delay, we also need to shift the time label of the 1563 link by Δt. Equation (7) is written as T1563,A(t+Δt)−T1563,B(t)2=T1NR,t+ΔTAB(t+Δt)+ΔTAB(t)2.(10)We still eliminate the time difference of the USLs by combining the 1545 link and the 1563 link, but the T1NR,t caused by the temporal displacement also appears on the 1563 link. We cannot measure T1NR,s,t by using the 1563 link as a reference for comparison. Therefore, an independent fiber link is used to transmit the frequency of the USLs. We compensate for the phase noise in the fiber link to enable the transfer of the USL frequency from NS to GY. Based on this, we calculate ΔTAB by comparing the transmitted frequency and the frequency of the local USL. By shifting the time label of ΔTAB by Δt, we obtain ΔTAB(t+Δt), and this is not affected by atmospheric delay effects. Therefore, we can measure T1NR,s,t by combining the non-reciprocal 1545 link and the fiber link.

To measure the atmospheric non-reciprocity effects of the satellite–ground link, we construct an equivalent spatio-temporal displacement in a 113-km free-space link. First, we establish two free-space links for two-way time-frequency transfer and an independent fiber link for frequency transfer. One of the free-space links is a two-way spatio-temporally separated link. Then, we measure four arrival times at two terminals by dual-comb interference and linear sampling. The sampling rate is approximately 3 kHz, determined by the difference between the repetition rate of the combs. Based on these, the T1NR,s of the spatial displacement can be measured by combining the non-reciprocal 1545 link and the reciprocal 1563 link. The T1NR,s,t of the spatio-temporal displacement can be measured by combining the non-reciprocal 1545 link and the fiber link.

During the experiment, the wind speed and the refractive index structure parameter Cn2 [39] are measured at Nanshan and Gaoyazi. Wind speed varies between 0.4 and 2.5 m/s, while Cn2 varies between 2×10−16 and 9×10−16m−2/3. Cn2 is a measure of the strength of the fluctuations in the atmospheric refractive index, which describes the turbulence strength.

We use power spectral density (PSD) to analyze the T1NR,s, as in Refs. [36, 37]. Figure 3(a) shows the comparison of the measured PSD between the links with and without T1NR,s. Evident time-of-flight noise is observed in the frequency range of 0.01–100 Hz, with values ranging approximately between 0.3×10−19 and 1.3×10−19s2/Hz. Above 100 Hz, both curves show similar noise levels, which can be attributed to the system noise floor. Below 0.01 Hz, the PSDs of both curves exhibit a continuous increase in noise, which is primarily influenced by temperature fluctuations of air conditioning and is no longer the function of atmospheric turbulence [36]. Figure 3(b) shows the comparison of time deviation (TDEV) between the links with and without T1NR,s. TDEV is used to evaluate the time instability in two-way time-frequency transfer. Both curves have the same starting point of about 19.3 fs at 0.002 s. Between 0.2 and 100 s, T1NR,s causes a maximum time deviation of 1.1 fs. But after 50 s, the TDEV is predominantly affected by thermal drift, with both curves experiencing the thermal effects.

The impact of the spatio-temporal displacement on frequency instability is also evaluated by modified Allan deviation (MDEV), as shown in Fig. 4. For spatial displacement, T1NR,s worsens the link instability from 5.0×10−15 at 1 s to 7.0×10−15 at 1 s, which compares with the link without T1NR,s. After an averaging time of more than 100 s, the dominant factor influencing the MDEV is thermal drift. Eventually, the link instability with and without T1NR,s reaches 2.3×10−19 at 10,000 s. It is also observed that the instability, when comparing two free-space links, is lower than when comparing the fiber link with the free-space link. This may be attributed to the presence of common-mode noise between the 1545 link and the 1563 link. For T1NR,s,t, we add a time delay of 0.05, 0.1, and 0.3 s to the time label of one side of the non-reciprocal 1545 link. At the averaging time of 0.2–800 s, compared to the links without T1NR,s,t, a delay of 0.3 s worsens the link instability. For example, the link instability is worsened from 7.0×10−15 to 1.5×10−14 at 1 s averaging time. However, at longer averaging times, the instability of the time-delay link becomes almost identical to that of the link without delay. Thus, the T1NR,s,t of the spatio-temporal displacement primarily affects the transmission performance within short averaging times.

Instability is an important parameter for transmission performance, and the systematic fractional offset is equally important. Specifically, we need to consider the impact of spatio-temporal displacement on systematic offset. We obtain the systematic offset by calculating the non-zero slope of the measured link’s residual time offsets, as shown in Fig. 5. The fractional offset is 4.0×10−19(1σ) with a weighted average of 4.1×10−19. Compared to the same system [32], the displacement of atmospheric non-reciprocity effects does not obviously worsen the fractional offset.

We simulate point-ahead effects and delay effects between two stationary terminals located 113 km apart and construct an equivalent spatio-temporal displacement. Then, we measure the spatio-temporal displacement of the atmospheric non-reciprocity effects for optical two-way time-frequency transfer. The result shows that the effects have a negative effect on the transmission performance at short averaging times, but the long-term effects are less than 2.3×10−19 at 10,000 s. To simulate point-ahead effects and delay effects of a real satellite–ground link on the ground, a feasible scheme is proposed. We design an optical transceiver with separate transmitting and receiving paths, achieved by an orthogonal polarization scheme. To reduce the impact of separating the transmitting and receiving paths, we develop an optical bench. The equivalent spatio-temporal displacement is constructed by two-way spatio-temporally separated links. The atmospheric non-reciprocity effects we measured can be equivalent to the satellite–ground link.

In addition to atmospheric non-reciprocal effects, another effect caused by satellite motion is the Doppler effect. It is a challenge for comb-based O-TWTFT because of the relevant transfer asymmetry. One potential solution for addressing the Doppler effect is to use the precise satellite orbit determination. A previous study [34] has investigated the low-velocity Doppler effect over the 2–4-km free-space link. For a real satellite–ground link conditions, the larger Doppler velocity still needs to be experimentally investigated.

Acknowledgment. Yuan Cao and Qi Shen are supported by the Youth Innovation Promotion Association of CAS.