Optical time-frequency transfer across a free-space, three-node network

Optical clock networks hold great promise for a wide range of applications.^1–16 For example, networked clocks could aid the search for dark matter,^3–6 enable tests of relativity,^3,7–9 or provide the timebases necessary for very long baseline interferometry.¹⁰ Furthermore, networked optical clocks are required to probe the earth via relativistic geodesy^11–15 and to meet the criteria for the redefinition of the SI second.¹⁶ For clock networks to reach their full potential, high-precision methods of transferring time and frequency between clocks must be developed, not only for fiber links but also for free-space links. The development of frequency-comb-based optical two-way time-frequency transfer (O-TWTFT),^17–23 along with other methods of free-space optical time-frequency transfer,^24–26 therefore represents a crucial step toward future free-space networks of optical clocks.

Recently, comb-based O-TWTFT over turbulent air paths has been shown to support frequency comparisons below 10^{−18 21} and to enable full femtosecond time synchronization, even in the presence of motion.^22,23 The performance of this optical technique shows many orders of magnitude improvement over state-of-the-art, rf-based approaches to time and frequency transfer through free-space.^27,28 However, these demonstrations of comb-based O-TWTFT have so far been limited to point-to-point connections over a maximum span of 12 km.²⁰ More complex network geometries are required in order to synchronize multiple remote clocks to a single master clock or to connect two state-of-the-art optical clocks without a line-of-sight link.

Here, we demonstrate comb-based O-TWTFT across a three-node optical network. Physically, the network is arranged in the shape of a highly acute triangle [Fig. 1(a)], with nodes A and B co-located in a laboratory and node X located at a field site 14-km away. This folded network geometry is a convenience that allows straightforward evaluation of the network’s performance. Unfolded, the network has the topology needed to connect ultra-precise, laboratory-bound clocks that are separated by great distances or that lack line-of-sight connecting paths. This network transfers frequency with a fractional instability below 10⁻¹⁸ at a 200-s averaging time and transfers time with a time deviation below 1 fs at averaging times between 1 s and 1 h. This performance is achieved despite random walk noise in the free-running oscillator at the relay node, despite strong turbulence over the horizontal air path, despite 30-dB variations in received power, and despite field operation at temperatures between 1 °C and 28 °C. At this level of performance, comb-based O-TWTFT could support future deployed clock networks that connect high-performance clocks via lower-performance, fielded relay nodes.

The three-node network is a relay network, designed to transfer frequency and timing information from the master node (A) through the relay node (X) to the end node (B). To accomplish this, each node contains its own “optical clock” and its own O-TWTFT transceiver from which comb pulses are launched and in which comb pulses from local and remote clocks are detected and compared. A network diagram is shown in Fig. 2(a).

The O-TWTFT transceivers in the master and end nodes (A and B) are identical. Each one includes a free-space optical terminal,²⁹ which maintains the free-space link between local and remote nodes, a heterodyne receiver, and a real-time digital signal processor. The transceiver in the relay node (X) is slightly more complex. Because it is connected to both the master and end nodes, it includes two free-space terminals and two heterodyne receivers.

Here, each “optical clock” consists of a self-referenced fiber frequency comb³⁰ phase-locked to a 1535-nm cavity-stabilized laser that serves as an optical oscillator. These optical clocks are surrogates for full optical atomic clocks in which the cavity-stabilized laser is locked to an atomic transition with absolute frequency stability. Provided that there are no slips on the phase-locks of the comb (and this can be true for months³¹), the comb pulse train is coherent. Consequently, within each node, the comb pulse train provides a local optical timescale, with the individual comb pulses serving as “ticks” of the local clock and with the comb repetition frequency setting the spacing between ticks.

At node X, a commercial, free-running, cavity-stabilized laser serves as the optical oscillator.³² This oscillator is compact, mobile, and robust enough to operate in the harsher environment of the field site. (Alternatively, for some applications, a quartz oscillator could serve as the reference oscillator, as demonstrated in Ref. 20.) Nodes A and B share a common optical oscillator: a cavity-stabilized laser located in a separate, temperature-controlled laboratory. Two parallel, fiber noise-canceled links transfer the frequency of this oscillator to the lab where nodes A and B are located. The choice of a common oscillator for clocks A and B is a convenience, eliminating the need for a separate truth data channel to monitor the frequency and time offsets between nodes A and B. To evaluate the performance of the network, we measure the frequency and time offsets of clocks B and X, relative to clock A, independently of any knowledge that nodes A and B share a common oscillator. Because nodes A and B do in fact share a common oscillator, their relative time offset is fixed, and consequently, any noise from the network itself appears as a frequency offset or wandering time offset of the clock at node B.

As described in Refs. 17–23, O-TWTFT relies on the bi-directional exchange of comb pulses between nodes. Here, the pulse repetition frequency of combs A and B is ∼200 MHz, while the repetition frequency of comb X is offset from the repetition frequency of combs A and B by 2.4 kHz. Within the heterodyne receivers of each node, the pulse train of the local comb is mixed with the incoming pulse train from the remote comb at the opposite end of the link. Because the repetition frequencies of the local and remote combs are offset, the interference pattern between local and remote pulse trains is a periodic waveform: a series of interferograms, illustrated in Figs. 2(b)–2(d), whose repetition frequency is equal to the difference between comb repetition frequencies. These interferograms are detected using a balanced photodetector and subsequently digitized by an analog-to-digital converter (ADC). A real-time signal processor then extracts the envelope of the digitized interferograms. The arrival time of each envelope maximum is localized with sub-sample accuracy and recorded.

Each ADC is clocked off the repetition frequency of the local comb, meaning that the local ADC clock cycle number is equivalent to the kth pulse of the local comb at the node. (Recall that each pulse of the local comb is equal to a single “tick” in the local timescale.) When a timestamp is recorded, it is recorded in the local timescale. Timestamps k_XA, recorded at node A, are recorded with respect to the local time of node A; timestamps k_XB are recorded with respect to the local time of node B; and timestamps k_AX and k_BX are recorded with respect to the local time of node X, where the first and second letters in the subscripts denote the remote and local nodes, respectively. (Note that because the arrival times of envelope maxima at each node are localized with subsample accuracy, these timestamps are non-integer.) Timestamps from pairs of nodes are then combined to compute the frequency and relative time offsets between nodes and, ultimately, across the network.

To obtain the time and frequency offsets between nodes, we first define the relationship between the unitless local time k within a node and a “universal time” t that is unknown but uniform across all three nodes. This universal time is related to each local timescale by $t=fr−1k−τt$⁠, where f_r is the fixed, nominal repetition frequency of the local comb and τ(t) is the slowly varying offset of the local clock. The instantaneous repetition frequency of the local comb is then given by $k̇=fr+δfrt=fr1+τ̇t$ and the fractional frequency offset of the comb by $δfrt/fr≡τ̇t$⁠, where the dot indicates a time derivative. The goal of O-TWTFT is to use a linear combination of timestamps to obtain the clock offset τ(t) of each node and, from this, both the relative time and fractional frequency offsets between nodes. The expressions for these quantities are derived in the supplementary material, and the results are summarized here.

We operated the three-node network during two measurement campaigns. Here, we report on a longer, 4-h measurement obtained during a winter campaign in December 2018 and two shorter, 1-h measurements obtained during a summer campaign in July 2019. In both winter and summer, the fielded relay node was subjected to significant ambient temperature fluctuations. During winter measurements, air temperatures varied between 0 °C and 10 °C, while during summer measurements, they varied between 25 °C and 38 °C. Below 0 °C, commercial electronic devices began to fail. At ambient air temperatures above 30 °C, the temperature control of the fielded cavity-stabilized laser was ineffective. Between temperatures of 1 °C and 28 °C, the network operated successfully, despite the large drifts in the ambient temperature of the field node during multi-hour measurements.

Because the beam paths between nodes A and X and between nodes X and B were within 100 m of the ground [Fig. 3(a)], the entirety of the 28-km free-space path across the network was subject to very high levels of atmospheric turbulence. (In fact, a close-to-ground path of only a few kilometers experiences similar levels of path-averaged turbulence as do beam paths between ground and mid-Earth-orbit.³⁴) Along both links, the resulting beam spread, scintillation, and optical time-of-flight variations were significant.

At each node, the comb light launched from the free-space optical terminal had a 1/e² Gaussian beam diameter of 4 cm. From diffraction alone, the beam diameter should increase to 70 cm at the far end of the 14-km path. However, due to strong atmospheric turbulence, the actual received beam diameter was about 2 m, as shown in Fig. 3(b) (Multimedia view). From this, we estimate that the path-averaged turbulence structure constant C_n² was ∼4 × 10⁻¹⁴ m^−2/3.³⁵ 

Additionally, turbulence-induced scintillation caused strong fluctuations in the received optical power coupled into the single-mode fibers. Approximately 4 mW of comb light in a 1.5-THz optical bandwidth was launched from each free-space terminal. At the opposite end of each link, received powers varied over more than 30 dB from below the lower detection threshold of ∼2 nW to above the upper detection threshold of ∼2 µW, as shown in Fig. 3(c). Roughly, 25% of the received interferograms fell below the lower threshold, while less than 1% exceeded the upper threshold, above which the photodetectors could saturate. During these signal fades, the time offset of the network is not updated, but when the link is re-established the time offset is immediately updated. In other words, signal fades do not cause a phase slip, or reset, in the time offset measurement.

Finally, turbulence also caused the time of flight across the free-space links to vary. Over short timescales of 1 s, time of flight across the 14-km link varies by ∼100 fs to ∼500 fs. Over longer timescales, changes in air temperature, and to a lesser extent, in humidity and pressure,³⁶ can cause even larger time-of-flight variations. During the 4-h winter measurement, the time of flight varied by 120 ps, which is consistent with the measured power spectral densities given in Ref. 37 for time-of-flight variations across turbulent air paths. Because the network geometry is folded, time-of-flight variations along the two links are correlated at frequencies below ∼1 Hz. This correlation does not result in common-mode noise rejection, but rather in a doubling of noise at Fourier frequencies below 1 Hz and, therefore, an increase in the effective noise level of time-of-flight variations.^38,39

As described in Refs. 18 and 21, O-TWTFT is used for both time and frequency transfer between clocks. Although the frequency-transfer performance of O-TWTFT may be deduced from its time-transfer performance, we consider the two cases separately for two reasons. First, the frequency transfer performance of the network is significant on its own due to its potential to enable comparisons between state-of-the-art optical atomic clocks. Second, to transfer time through a relay node with a free-running oscillator, explicit knowledge of fractional frequency offset at the relay node is needed.

From the time derivatives of (1) and (2) in Sec. II [or explicitly from (S20) and (S23) in the supplementary material], we calculate the measured fractional frequency offsets of both the relay node, $τ̇Xt=δfrXt/frX$⁠, and the end node, $τ̇Bt=δfrBt/frB$⁠, relative to their nominal values. These frequency offsets are shown in Fig. 4. Because the true frequency offset of the clock at the end node (B) is zero, the difference between the measured fractional frequency offset of the end node and its true value of zero is used as a direct measure of network performance.

Because the clock at node X is free-running, its fractional frequency offset changes in a non-stationary manner over the course of both the 4-h winter measurement and the two 1-h summer measurements. Drift rates in the fractional frequency offset of this free-running clock reach up to 14 Hz/s, with the result that the fractional frequency offset of the relay node reaches more than −4 × 10⁻¹⁰ by the end of the 4-h measurement [Fig. 4(a)]. Despite this, the fractional frequency offset of the clock at end node (B), measured across the network via the relay node (X), remains consistent with its true value of zero [Fig. 4(b)]. Specifically, the mean of the fractional frequency offset measured during the 4-h run is −3 × 10⁻²⁰ ± 4.4 × 10⁻¹⁹, where the uncertainty is taken from the last point in the modified Allan deviation [Fig. 4(c)]. The consistency of this measured mean with its true value demonstrates the network’s frequency-transfer accuracy, while the low uncertainty demonstrates the network’s precision.

The fractional frequency instabilities of clock X and of the network (i.e., of clock B) are both dominated by white phase noise at averaging times below 1 s. [Figure 4(c) shows only the modified Allan deviations for the longer 4-h measurement, since the instabilities of the summer measurements are very similar.] Unsurprisingly, at longer averaging times, the fractional frequency instability of the free-running clock in the fielded relay node is limited by random walk frequency noise from the local cavity-stabilized laser. We attribute this to variation in the ambient temperature at the field site. As a result, after dropping to a minimum of 2.5 × 10⁻¹⁴ at 0.4 s of averaging, the instability of clock X is driven up to almost 10⁻¹⁰ at averaging times over 1 h. Note that this does not represent a fundamental limitation on the frequency stability available at the relay node. Although clock X was allowed to be free-running here, its frequency offset could have been tracked in real time and either fed back to the oscillator or used to generate a real-time correction to the local time at node X. In this way, the entire network—clocks A, B, and X—could achieve the same frequency instability.

The fractional frequency instability of the network, shown in Fig. 4(c), shows that O TWTFT successfully tracks the high relative instability of the optical oscillator at the relay node and enables precise frequency measurements at the end node. At an averaging time of 10 s, the fractional frequency instability of the network is below 1 × 10⁻¹⁶. At just under 200 s of averaging time, this instability drops below 1 × 10⁻¹⁸. Between 900 s and 3500 s (∼1 h) of averaging time, the instability reaches a flicker floor of ∼6 × 10⁻¹⁹, over eight orders of magnitude below the instability of the relay node. This flicker floor could likely be reduced with additional thermal management of short lengths of out-of-loop fiber in the relay node. Nevertheless, we note that the network instability reported here is below that required to support comparisons of state-of-the-art atomic optical clocks.

For comparison, Fig. 4(c) also shows the instability of previous O-TWTFT measurements obtained using a 4-km point-to-point link.²¹ At averaging times below 100 s, the three-node network has an instability less than a factor of 2 higher than that of previous point-to-point links. At averaging times above 900 s, the relay network’s instability increases to about five times that of the point-to-point link due to thermal drifts in fiber within the O-TWTFT transceiver in the fielded node. Overall, the instability of the three-node relay network is only slightly higher than that of the 4-km, point-to-point link, indicating that we incur negligible performance penalties from the longer link distance and from the use of a fielded relay node.

The computation of timing offsets in the three-node network differs from earlier computations for point-to-point networks^17–21 since we must now correctly combine measurements from all three nodes and handle the large frequency drifts of the fielded local oscillator at the relay node. During phase-continuous operation—i.e., during measurements in which none of the three clocks experiences a phase-slip, as is the case here—the time offsets of clocks X and B, relative to clock A, may be obtained using (1) and (2) from Sec. II C. The relative time offsets of both clocks include arbitrary initial values that depend on the choice of references plane in the transceivers. Here, we ignore these initial values, as the choice of reference planes is application-specific. Instead, we set the initial time offsets of both clock B and clock X to zero and measure their subsequent variation. Once again, since here we expect a fixed time offset at node B, any measured timing noise is interpreted as the time-transfer noise of the relay network.

During every measurement, the time offset of clock X wanders away from its initial value of zero, reaching nearly 4 µs at the end of the longer 4-h measurement [Fig. 5(a)]. This is due to random frequency walk noise in the free-running oscillator at the relay node. Nevertheless, the relative time offset measured at the end node (B) is close to zero, as expected. Over the course of 4 h, it varies by less than 5 fs [Fig. 5(b)]. This level of variation is not surprising; we expect up to 7 fs/°C of variation due solely to temperature changes in non-common fibers within the heterodyne receivers at node X, which are temperature-controlled to within ∼1 °C.

The time deviations of the measured time offsets of the relay and end nodes—i.e., of clocks X and B [Fig. 5(c)]—illustrate the scale of potential time offsets throughout the network. The time deviation of the free-running clock at the relay node briefly dips below 10 fs at 0.2 s of averaging, but random frequency walk noise drives the time deviation up by more than seven orders of magnitude at longer averaging times. In contrast, the time deviation of the network falls below 1 fs at a 1-s averaging time and remains below 1 fs out to 1 h of averaging time.

As in the case of frequency transfer, we note that the large time deviation at the relay node does not imply that accurate timing information is inaccessible at that node. Since the time and frequency offsets of clock X are continuously tracked and available via the O-TWTFT measurements, the measured time offset could be used to feed back to the oscillator frequency in real time or to provide corrections to the time at node X. More significantly, the time and frequency offsets at the end node may be precisely measured through the relay node, without doing either, showing that frequency comparisons between ultra-stable oscillators, e.g., optical atomic clocks, may be made across a network that includes a significantly less stable oscillator.

We have demonstrated the first deployment of a free-space, three-node, optical network capable of supporting state-of-the-art optical clocks and oscillators. Using two separate 14-km free-space optical links, the network connected two lab-based optical timescales through a fielded relay node. The fielded node operated in an uninsulated trailer through a wide range of environmental conditions. Nevertheless, the network transferred frequency with a fractional instability of 8 × 10⁻¹⁷ at a 10-s averaging time and below 1 × 10⁻¹⁸ at averaging times above 200 s. In the time domain, the network had a time deviation below 1 fs for averaging times between 1 s and 1 h. This time-frequency transfer was accomplished despite 120 ps of time-of-flight fluctuations across the two free-space links, strong environmental perturbations at the fielded relay node, random walk frequency noise at the fielded relay node approaching fractional values of 10⁻¹⁰, and resulting timing wander of nearly 4 µs at the fielded relay node.

This demonstration of a three-node network for optical time and frequency transfer across open air paths is an initial step toward the deployment of free-space optical clock networks. The network geometry used here could be directly translated to the connection of state-of-the-art atomic optical clocks without a direct line-of-sight between them. It is noteworthy that the overall time and frequency transfer performance of the network was not appreciably degraded by timing noise at the field site—timing noise that will be typical of a fielded oscillator, whether it be a compact cavity-stabilized laser or quartz oscillator. This result is significant because it opens the possibility of extending the range of future free-space optical clock networks with additional relay nodes.

In summary, the work here is the first demonstration of ultrahigh precision optical time-frequency transfer over a multi-node free-space network, the first demonstration with a fielded node, and the longest-range eye-safe operation to date, spanning a total of 28-km of strongly turbulent air with transmit powers of only 5 mW. These advances show that frequency comparisons between ultra-stable oscillators, e.g., optical atomic clocks, may be made across multi-node free-space networks and that these networks may contain nodes with less stable oscillators. The successful operation of this three-node network therefore represents a concrete step toward future networks that include compact airborne or space-borne O-TWTFT systems.

See the supplementary material for the general solution for computing timing offsets in a three-node optical clock network (Sec. I), the on-the-fly calibration method used to measure the asynchronicity of interferogram arrival times at the three nodes (Sec. II), and the relationship between the general solution for timing offsets in a three-node network and the specific solution for offsets in our particular network (Sec. III).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

We thank Mick Cermak for technical assistance, and we thank David Leibrandt and Kevin Cossel for comments on this manuscript. We acknowledge funding from the Defense Advanced Research Projects Agency Defense Sciences Office and the National Institute of Standards and Technology.