As the world’s largest radio telescope under construction, the Square Kilometre Array (SKA) consists of thousands of parabolic dishes and aperture arrays, which work together to form a one-square-kilometer collecting area. SKA will surpass any current telescope and help us further understand the universe. To ensure sufficient imaging fidelity, all of the telescopes should keep time and be frequency synchronized with stabilities superior to 2.3×10−12/s, 3.8×10−14/min, and 1.9×10−14/10min^[1]. In 2015, we proposed and demonstrated a precision reference frequency synchronization scheme via 1f–2f dissemination, according to the requirements and characteristics of SKA^[2]. Here, we name it a harmonic system. It features the phase-noise compensation performed at the client site to reduce space requirement and unnecessary complexity at central station. Considering our previous work along with the existing fiber-based time and frequency transfer schemes demonstrated by different groups^[3–17], they almost utilized fiber spools or urban telecommunication fibers, which basically run in buried cables. While for large-scale scientific and engineering facilities, such as SKA, on account of attractive infrastructure cost and simple laying, overhead fiber links will be adopted. Unfortunately, the overhead fiber links are affected by atmospheric temperature variation and mechanical stress. Consequently, they are much noisier than fiber spools and buried fiber^[18–25]. Taking the SKA South Africa site for an example, it is constructed in the desert region. From the meteorological parameters captured by the local weather station, rapid temperature change caused by a thunderstorm, gale, sunrise, and sunset is recorded frequently in the past. According to statistical analysis on the temperature data in the period from January 1, 2005 to March 31, 2011, the daily temperature fluctuation can reach 40°C^[26]. Therefore, a robust frequency dissemination system immune to strong temperature fluctuation is required.

After the performance test in the lab, the harmonic system was shipped to the SKA South Africa site to perform an outfield test on several overhead fiber links from the 19th to 27th September 2015. The frequency dissemination stability with phase-noise compensation via the 64 km overhead fiber is shown in Fig. 1. For easy comparison, the dissemination stability of 32 km overhead fiber without compensation is also shown in Fig. 1. During the field test, we did not measure the dissemination stability on 64 km overhead fiber without compensation, but it will be worse than that on 32 km overhead fiber^[27]. It can be seen that there is a bump on the Allan deviation plot of dissemination stability at the averaging time between 10 and 100 s, which had not shown up in previous lab tests using fiber spools. Through theoretical analysis and experimental verification, as described later, we find that the fiber-induced phase fluctuation due to strong temperature change cannot be completely compensated.

In this Letter, we analyze the reason why the previous harmonic scheme is not perfectly immune to temperature fluctuation and attribute this imperfection to limited isolation and nonlinear performance of the radio frequency (RF) components in the system. According to the analysis, we design a non-harmonic system and test its performance in an environment with temperature fluctuating 40°C per day, which is similar to the temperature fluctuation of the SKA South Africa site. The result of this test shows a frequency dissemination stability of 4.0×10−14/s and 3.0×10−16/104s. Furthermore, we carry out another test under the situation of rapid temperature change—20°C per hour—and obtain a frequency dissemination stability of 3.0×10−14/s and 3.0×10−16/104s. Based on these test results, we conclude that the non-harmonic frequency dissemination system is robust with both huge and rapid temperature fluctuation and, hence, is capable of operating in severe environments of large-scale scientific and engineering projects.

The schematic diagram of the harmonic reference frequency synchronization scheme is shown in Fig. 2. One transmitting site (TX) consists of several extensible disseminating channels, so that it has the ability to simultaneously collaborate with several receiving sites (RXs). Here, we just take one channel, for example, to explain the concept. At the TX, the 100 MHz frequency signal from an H-maser acts as the reference signal. A phase-locked dielectric resonant oscillator (PDRO) with a frequency of 2 GHz is phase-locked to it and can be expressed as V0=cos(ω0t+ϕ0).(1)

It is used to modulate the amplitude of a 1547.72 nm diode laser (Laser A). Then, the modulated laser light is divided into several equal light beams. After passing through a fiber coupler and an optical circulator, the light beam is disseminated from the TX to the corresponding RX via a fiber link. At the RX, another PDRO with a frequency of 1 GHz is phase-locked to a 100 MHz oven-controlled crystal oscillator (OCXO). The 1 GHz signal can be expressed as V1=cos(ω1t+ϕ1),(2)and is used to modulate the amplitude of a 1548.53 nm diode laser (Laser B). After passing through an optical circulator, the modulated laser light is coupled into the same fiber link and disseminated via the route RX–TX–RX. At the RX, the disseminated laser lights are separated from each other by a wavelength-division multiplexer (WDM). Two photodiodes (PDs) are used to recover the disseminated frequency signals, which can be expressed as V2=cos(ω0t+ϕ0+ϕp),(3)V3=cos(ω1t+ϕ1+ϕp′).(4)

Here, ϕp represents the phase fluctuation induced by the 50 km fiber dissemination for the 2 GHz signal V0, and ϕp′ represents that of the 50 km fiber round-trip dissemination for the 1 GHz signal V1. The one-way accumulated phase fluctuation of the 2 GHz frequency signal is the same as the round-trip phase fluctuation of the 1 GHz frequency signal, namely ϕp=ϕp′. By frequency mixing down V2 and V3, we can get a 1 GHz signal: V4=cos[(ω0−ω1)t+ϕ0+ϕp−ϕ1−ϕp′].(5)

Then, V4 is frequency mixed with V1, and the generated DC signal can be expressed as V5=cos[(ω0−2ω1)t+ϕ0+ϕp−2ϕ1−ϕp′].(6)

As an error signal, V5 is used to feedback-control the phase of the OCXO by a phase-locked loop (PLL). With the relations of ω0=2ω1, and ϕp=ϕp′, V5 can be expressed as V5=cos(ϕ0−2ϕ1).(7)

When the PLL is closed, the OCXO at the RX is phase-locked to the reference signal at the TX, namely, ϕ0=2ϕ1. Thus, the reference signal is recovered at the RX.

Here, to simulate the ambient temperature fluctuation of the SKA South Africa site, the fiber spool is placed in a temperature-controlled box, and its temperature fluctuation is set to be 10°C, 20°C, 30°C, and 40°C per day, respectively, as Fig. 3(a) shows. The corresponding relative frequency stability between the 100 MHz reference signal and the recovered 100 MHz signal is shown in Fig. 3(b). We can see from the results that when the ambient temperature of the fiber link fluctuates, a bump shows up on the Allan deviation plot at the averaging time between 10 and 1000 s. The position of bump varies according to different temperature fluctuations.

Even though the measured stability can fulfill the SKA requirement, we still made an analysis on the bump and solved it. In the previous discussion, the frequency mixing between V2 and V3 is considered to be an ideal process. That is to say, the output signal V4 just contains the ideal signal [see Eq. (5)]. Nevertheless, influenced by the limited isolation and nonlinear performance of the frequency mixer in the system, components of the output signal V4 are complex. Owing to the desired output signal with the frequency of 1 GHz, the leakage of V3 with the same frequency cannot be filtered. It would limit the frequency dissemination stability. The frequency mixer we used (Marki T3-03) features a high third-order intercept point (IP3) that it can reduce the unwanted nonlinear effect, but the influence still exists. The frequency signals V2 and V3 in Taylor’s series can be expressed as $$\begin{gathered} V2=cos(ω0t+ϕ0+ϕp)+a1cos(2ω0t+2ϕ0+2ϕp) \\ +a2cos(3ω0t+3ϕ0+3ϕp)+⋯, \end{gathered}$$(8)$$\begin{gathered} V3=cos(ω1t+ϕ1+ϕp′)+b1cos(2ω1t+2ϕ1+2ϕp′) \\ +b2cos(3ω1t+3ϕ1+3ϕp′)+⋯, \end{gathered}$$(9)where a and b are coefficients. The components of these two signals are frequency mixed with each other. Through a 1 GHz band pass filter, the output signal V4 can be expressed as $$\begin{gathered} V4=cos[(ω0−ω1)t+ϕ0+ϕp−ϕ1−ϕp′] \\ +ξcos(ω1t+ϕ1+ϕp′) \\ +ζcos[(3ω1−ω0)t+3ϕ1+3ϕp′−ϕ0−ϕp] \\ +⋯, \end{gathered}$$(10)where ξ and ζ are small coefficients. The first term is the ideal signal, the second is the leakage of V3, and the third is produced by the third harmonic of V3 frequency mixing with V2. Since other higher order harmonics and products are much smaller, we just take these two influence factors into consideration. Considering ϕp=ϕp′, V5 can be expressed as $$\begin{gathered} V5=cos(ϕ0−2ϕ1) \\ +ξcos(ϕp) \\ +ζcos(2ϕ1+2ϕp−ϕ0). \end{gathered}$$(11)

The continuous variation of ϕp will lead to periodic variation of the second and third terms of V5 in Eq. (11). Thus, the phase ϕ1, which is tunable by changing the control voltage of the OCXO, will also vary periodically when the PLL is closed, instead of being completely equal to 12ϕ0 as expected. Consequently, the Allan deviation plot of the frequency dissemination stability in Fig. 3(b) has a bump on it. Indeed, in our experiment, we find the power of the third-harmonic component is one-hundred times smaller than that of the 1 GHz leakage component. Here, we will mainly consider the 1 GHz leakage in the analysis. The phase item ϕ1 will be disturbed at the same period with cos(ϕp).

Considering the mechanical tension coefficients of fiber length and refractive index, the effective temperature coefficient of phase time delay is 76 pskm·°C^[28]. For the signal frequency of 2 GHz and the fiber length of 50 km, the temperature coefficient of phase delay should be dϕpdT=15.2πrad/°C.(12)

Taking the temperature fluctuating 40°C/day as an example, the temperature changing rate is dTdt=11080°C/s (increasing or decreasing 40°C in 12 h). According to Eq. (12), the phase delay changing rate can be calculated as dϕpdt=dϕpdT·dTdt=15.2π·11080=0.014πrad/s.(13)

Correspondingly, the period of cos(ϕp) is 2π0.014π=140s, which is the same with the disturbance period of ϕ1. Consequently, the horizontal position of the bump on the Allan deviation plot should be at half of the period, which is 70 s. As to the temperature fluctuating 10°C, 20°C, and 30°C per day, the horizontal positions should be 280, 140, and 93 s, respectively. These calculated results agree with the test results shown in Fig. 3(b).

The schematic diagram of the non-harmonic precision reference frequency synchronization scheme is shown in Fig. 4.

The TX remains the same. Namely, the expression of V0 is the same as Eq. (1). After fiber dissemination from the TX to the RX, the expression of V2 is also the same as in Eq. (3). At the RX, the 1 GHz signal from the PDRO of the previous scheme is replaced by two signals, 1fa and 1fb, from frequency synthesizers with a frequency of 1GHz+130Hz and 1GHz−130Hz, respectively. Both of them are phase-locked to the same OCXO. Then, V1 can be expressed as V1=cos[(ω1+130Hz)t+ϕ1],(14)and V3 can be expressed as V3=cos[(ω1+130Hz)t+ϕ1+ϕp′].(15)

By frequency mixing down V2 and V3, the down-conversion signal can be expressed as $$\begin{gathered} V4=cos[(ω0−ω1−130Hz)t+ϕ0+ϕp−ϕ1−ϕp′] \\ +ξcos[(ω1+130Hz)t+ϕ1+ϕp′] \\ +ζcos[(3ω1−ω0+390Hz)t+3ϕ1+3ϕp′−ϕ0−ϕp]. \end{gathered}$$(16)

The second term is the leakage of V3, and the last term is produced by the third harmonic of V3 mixing with V2. The frequency signal 1fb is used for frequency mixing, and it can be expressed as V6=cos[(ω1−130Hz)t+ϕ6],(17)where ϕ6 can be considered the same as ϕ1 with negligible difference. Then, V6 is mixed with V4. An error signal is obtained: $$\begin{gathered} V5=cos(ϕ0+ϕp−2ϕ1−ϕp′) \\ +ξcos(260Hz×t+ϕp′) \\ +ζcos(520Hz×t+2ϕ1+3ϕp′−ϕ0−ϕp). \end{gathered}$$(18)

Through a low pass filter, the last two terms can be filtered. In this non-harmonic scheme, the relation of phase fluctuation compensation becomes (1+1.3×10−7)ϕp=ϕp′. Then, V5 can be expressed as V5=cos(ϕ0−2ϕ1−1.3×10−7ϕp).(19)

When the PLL is closed, it will be ϕ0=2ϕ1+1.3×10−7ϕp.(20)

A little part of ϕp is brought into the error signal. Taking the temperature fluctuating 40°C as an example, the residual phase error 1.3×10−7Δϕp is 1.9×10−14s, which can be neglected.

With the ambient temperature of the fiber link fluctuating 40°C/day, which is similar to the temperature fluctuation of the SKA South Africa site, we measure relative frequency stabilities between 100 MHz reference signal and recovered signal after 50 km dissemination. The result is shown in Fig. 5. The bump on the Allan deviation plot of dissemination stability at the averaging time between 10 s and 1000 s almost diminished, and relative frequency stabilities of 4.0×10−14/s and 3.0×10−16/104s are obtained. For easy comparison, the dissemination stability of the harmonic system with the temperature fluctuating 40°C/day is also shown in Fig. 5.

Furthermore, to test the performance of this non-harmonic system under rapid temperature fluctuation, we conduct another experiment, in which the ambient temperature of the fiber spool varies 20°C/h. There is also no bump on the Allan deviation plot, and the relative frequency stability reaches 3.0×10−14/s and 3.0×10−16/104s, as shown in Fig. 5.

For the proposed scheme, a small part of the phase noise induced by the fiber link is left intentionally uncompensated. They are the reasons why the short term stability of the non-harmonic system is worse than the harmonic system, as shown in Fig. 5. However, for the non-harmonic system, the advantages are that it is not affected by the limited isolation and nonlinear performance of RF components in the system anymore. Also, it is more suitable for practical applications with strong temperature fluctuation.

In conclusion, we design a non-harmonic synchronization scheme and prove it to be immune to strong temperature fluctuations and, hence, is capable of operating in severe environments of large-scale scientific and engineering projects.

This work is being carried out for the SKA Signal and Data Transport (SaDT) consortium as part of the SKA project. The SKA project is an international effort to build the world’s largest radio telescope, led by the SKA Organization with the support of 10 member countries. Fourteen institutions from eight countries are involved in the SaDT consortium, led by the University of Manchester.