The field of fiber-based frequency dissemination continues to advance with ever-increasing performances and fiber lengths. Bidirectional fiber links have demonstrated the capacity to transfer optical frequency with instabilities below ${10^{- 15}}$ at 1 s down to the low ${10^{- 20}}$ level for the long term, enabling the remote comparison of the best performing atomic clocks. Optical fiber links are particularly important for remote optical frequency standards comparison in the roadmap of redefinition of the second [1–6]. It also finds applications in a variety of emerging scientific fields, such as chronometric geodesy [7–9] and spectroscopy [10].

In order to fully exploit the possibilities of fiber-based frequency dissemination for metrological applications, the few existing fiber links should be up-scaled into a global fiber network, which is feasible only with a sustainable approach. Renting dedicated fiber especially for long distance connections can be very costly and raises the issue of manpower cost for the maintenance and supervision of the network. One alternative solution is to use one dedicated channel in a standard telecommunication fiber network. While it requires some modifications to realize the bidirectional transmission of the metrological signals, it has the advantage that the fiber link is shared with data traffic optimizing the use of resources. An example is the REFIMEVE$+$ network in France. Separated fibers are used as up-link and down-link for the unidirectional propagation of the digital data, and unidirectional optical amplifiers are installed to overcome the transmission loss. The propagation of bidirectional metrological signals, relying on a single fiber for both up-link and down-link, is enabled by inserting optical add–drop multiplexers and bidirectional amplifiers at relevant network nodes [11]. Another alternative solution is to use two dedicated channels of separated fibers in a standard telecommunication network. Each fiber can be used to transfer the metrological signal in a single direction [12–14]. This method can be integrated into a standard telecommunication network with little to no modifications and thus can take full advantage of the unidirectional nature of existing long-haul fiber networks. While this unidirectional topology is expected to show a lower stability than a bidirectional configuration, its significantly lower cost and its ability to access networks blocked for bidirectional transmission (e.g., in submarine cables) make it an attractive alternative.

In this Letter, we report on the longest experimental data for an optical frequency transfer based on a unidirectional configuration and investigate the major limiting factors. We present our experimental setup and methods followed by a detailed analysis of the 183-days-long data set. We assess the performances of our unidirectional setup in the long run and discuss its suitability for atomic clock comparisons. The main limitation of the unidirectional configuration is the asymmetry between the forward and backward propagation of the signal. Unlike in a single fiber, where the fiber-related noise of the forward and backward signals demonstrates a high correlation (99.9999999%), the correlation between two adjacent fibers is significantly lower (99.99%) due to the physically different optical paths for two propagation directions [15]. Here we consider two aspects contributing to the asymmetric noise and discuss the limitations of a unidirectional scheme due to optical amplifiers and fiber length asymmetry.

Figure 1 shows a schematic of our experimental setup, as previously reported in detail in [15]. An ultra-stable laser is split and injected into two 43 km adjacent fibers (i.e., in the same cable) linking the laboratory Systèmes de Référence Temps-Espace (SYRTE) and the laboratory Laboratoire de Physique des Lasers (LPL) [16], denoted by up-link and down-link. These fibers are mainly in underground technical ducts in Paris and its suburbs and are subject to noise related to human activities. They are expected to be noisier than buried fibers but less noisy than aerial fibers [17,18]. The round-trip signal (RT1) transmitted on the up-link is detected on the photodiode (PD1), and is used to actively compensate for the fiber noise through a feedback system (phase-locked loop). Due to the active noise compensation on the up-link, the optical phase at point A is copied to point D and can consequently be regarded as a virtual ultra-stable laser source at LPL, which is injected into the down-link. This configuration suppresses noise resulting from differential frequency drifts between two lasers usually used for the up-link and down-link in a unidirectional configuration [19]. Then three beat notes (RT2, OWB, OWF) are detected on PD2. Round-trip 2 (RT2) is the round-trip signal transmitted on the down-link. One-way backward (OWB) is the beat note between the virtual laser source transmitted backward on the down-link and the reference signal at SYRTE. One-way forward (OWF) is the beat note between the laser source at SYRTE transmitted forward on the down-link and the virtual reference signal at LPL, reflected back and measured at SYRTE. All beat notes are separated, filtered, tracked, then simultaneously recorded by a dead-time free frequency counter operated in $\Pi$-type and $\Lambda$-type with 1 s gate time.

 

Fig. 1. Experiment setup. AOM, acousto-optic modulator; PD, photodiode; PLL, phase-locked loop; bi-EDFA, bi-directional erbium-doped fiber amplifier; p-FM, partial Faraday mirror; PC, polarization controller.

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The four beat notes exhibit the round-trip fiber noise on the up-link and down-link, as well as the forward and backward one-way fiber noise on the down-link. Here we focus on the two-way unidirectional (TWU) observables, which can be obtained from these four beat notes. Following the same notations as in [15], the optical phase of the TWU observables are defined as

The experiment ran almost continuously for six months with the last month overlapping an international clock comparison among National Physical Laboratory (NPL, UK), SYRTE (France), and Physikalisch-Technische Bundesanstalt (PTB) (Germany) [20]. As a result, we have obtained a 183-day data set from January 1 to July 2, 2017. Figure 2(a) shows the phase evolution of the four beat notes, obtained from the $\Lambda$-type frequency data and converted into a variation of the unidirectional propagation delay, denoted by time error. During ${\rm MJD} = 57754 \sim 57849$, we measured only three of the four free running one-way phase signals ($\frac{1}{2}{\phi _{{\rm RT1}}}$, $\frac{1}{2}{\phi _{{\rm RT2}}}$, ${\phi _{{\rm OWB}}}$). During ${\rm MJD} = 57849 \sim 57858$ (gray), we reconstructed and upgraded our unidirectional setup by slightly changing the frequency of (acousto-optic modulator) AOM3 and optimizing the parameters of the tracking devices. Consequently, during ${\rm MJD} = 57859 \sim 57936$, we recorded all four beat notes, including OWF. We note that the four free running phases are nearly identical and follow the same trend exhibiting a phase evolution of ${\sim}15\;{\rm ns} $ over the total recorded time span.

 

Fig. 2. Phase evolution of (a) four beat notes: RT1, RT2, OWB, OWF and (b) three unidirectional two-way observables: TWU1, TWU2, TWU3. The considered period: 01/01/2017–02/07/2017.

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The three unidirectional two-way observables (${\phi _{{\rm TWU1}}}$, ${\phi _{{\rm TWU2}}}$, ${\phi _{{\rm TWU3}}}$) shown in Fig. 2(b) are obtained through post-processing the free running one-way signals according to Eq. (1). They reject the correlated free running fiber noise between the up-link (RT1) and the down-link (RT2, OWF, OWB) with correlation of 99.99% [15] and describe the residual phase difference between the up- and down-links. They exhibit a total phase evolution of ${\sim}100\;{\rm ps} $ over 183 days. These residual uncorrelated phase noises may be dominated by the fiber length asymmetry or the different environmental sensitivity of the two adjacent fibers, which we will discuss later.

Figure 3 shows the optical phase noise power spectrum density (PSD) for the free running one-way fiber noise (for the sake of clarity, we show only $\frac{1}{2}{\rm RT2}$) and for the unidirectional two-way observables TWUs. We find that the free running one-way fiber noise approximately follows a power-law dependence as ${\sim}h\!/\!{f^3}$ for $f {\lt} {10^{- 2}}$ and ${\sim}h\!/\!{f^2}\;{\rm Hz}$ for $f{\gt} {10^{- 2}}\;{\rm Hz}$. The TWUs show a power-law dependence as ${\sim}h\!/\!{f^2}$ over all the Fourier frequencies characteristic of white frequency noise, indicating that this unidirectional configuration does not permit working in the phase coherent regime. We observe excess noise around ${10^{- 3}}\;{\rm Hz}$ typical of the air conditioning’s cycle time. Both the one-way and two-way PSDs exhibit sharp peaks at $f \approx 2 \times {10^{- 5}}\;{\rm Hz}$ and ${10^{- 5}}\;{\rm Hz}$, corresponding to perturbations with a periodicity of half a day and one day, respectively. These two peaks are most likely caused by diurnal temperature fluctuations. Another much wider peak is observed corresponding to a $\simeq \!20$ day periodicity. Seasonal fluctuations are not observed, as a four times longer data set would be required.

 

Fig. 3. Power spectral densities (PSD) of the free running fiber noise (showing only RT2 for clarity) and the unidirectional two-way observables.

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Fig. 4. Long-term fractional frequency instabilities of a unidirectional two-way scheme in terms of modified Allan deviation (MDEV) ($\Lambda$-data).

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Figure 4 shows the fractional frequency instability of the free running one-way and unidirectional two-way observables over 183 days of data ($\Lambda$-type) in terms of modified Allan deviation (MDEV). The instability of the RT2 signal clearly shows two peaks at $\tau = 20 000 \;{\rm s}$ and 400,000 s, which is explained by diurnal temperature variations and a longer $\simeq\! 20$ days periodic fluctuation related to external temperature variations in field. For TWU1 and TWU2, the MDEVs start from $8.0 \times {10^{- 16}}$ at 1 s integration time and decrease to $1.0 \times {10^{- 17}}$ and $6 \times {10^{- 18}}$ at 1,000,000 s, respectively. It is noteworthy that the respective instabilities of $1.5 \times {10^{- 17}}$ and $1.3 \times {10^{- 17}}$ have been reached at an integration time of 4,000,000 s. To the best of our knowledge, this is the longest instability data ever reported for an optical fiber link. We point out that an instability at the level of mid-${10^{- 17}}$ can be reached within an hour of measurement with this unidirectional two-way scheme and be kept for the very long term. Compared to a bidirectional scheme [15,16], its stability is degraded due to the lower noise correlation, and this level of performance is still not sufficient for an optical clock comparison. However, it is already far beyond the most advanced satellite techniques used for intercontinental Cs fountain clock comparison, and could therefore enable a faster and more accurate computation of the International Atomic Time (TAI).

In order to better understand the limitations of a unidirectional scheme and assess its potential for frequency dissemination on a larger scale, we set up a testbed in the laboratory duplicating the unidirectional scheme described above, but using fiber spools for the up- and down-links.

In a long-haul telecommunication fiber network, for instance the REFIMEVE$+$ network, cascaded optical amplifiers are used to counteract the transmission loss along the fiber link. When employing the unidirectional scheme in long-haul fiber networks, the effect of amplifiers should be considered. In order to study the pure effect of amplifiers, instead of fiber spools, attenuators are inserted into the link, simulating the transmission loss of the fiber followed by amplifiers. Since a bidirectional propagation in each fiber is necessary to detect the roundtrip signals measuring the fiber noise, we use bidirectional amplifiers. This imposes an ${\sim}10\;{\rm dB}$ gain limit to prevent oscillations in the link. To keep the optical budget the same after each 10 dB amplifier, two 5 dB attenuators are inserted before each amplifier. The performance of the unidirectional scheme is evaluated as a function of inserted amplifiers, whereby the up- and down-links have the same number of amplifiers. As expected, the optical signal to noise ratio (OSNR) degrades with increasing the number of amplifiers due to their spontaneous emission. The OSNR is ${\sim}60\;{\rm dB}$ in a bandwidth of 100 GHz without amplifiers and decreases to ${\sim}20\;{\rm dB}$ with six amplifiers. However, the instabilities are not influenced, as shown in Fig. 5 (solid). So, we conclude that the amplifiers will not be the main limiting factor for this unidirectional configuration.

 

Fig. 5. Fractional frequency instabilities of unidirectional two-way observable TWU1 with increasing the number of amplifiers (solid) and with increasing the fiber length asymmetry between up-link and down-link (dashed).

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Additionally, we analyze the influence of fiber length asymmetry between the up-link and down-link by using 12 fiber spools with length of around ${\sim}25\;{\rm km} $. The absolute length of each fiber spool is estimated, and the length asymmetries of the fiber spool pairs range from 0.3 to 657.5 m. Fig. 5 (dashed) displays the MDEV of TWU1 for four pairs of fiber spools with low, medium, and high length asymmetry. They are at a level of a few ${10^{- 16}}$, and this instability does not increase with the length asymmetry. Moreover, it varies from one day to another for the same pairs of fiber spools. Given that we see no clear dependence of the fiber noise on length asymmetry, we consider that length asymmetry is not the limiting factor for the unidirectional frequency transfer. Future research on polarization might extend the investigation of the limiting factor.

The stability of the unidirectional scheme on a fiber link is at least 10 times smaller than the combined uncertainties of the best atomic fountain clocks for all time scales of integration [21,22]. This fiber-based setup is complementary to a satellite-based measurement, and is already far beyond the most advanced GPS and two-way carrier phase capabilities (see, for instance, [23–25] and references therein). Indeed, this unidirectional scheme does not require a modification of the telecommunication network amplification chain along the link. On a regional scale, as demonstrated here, the unidirectional scheme allows one to reach an instability as low as $4 \times {10^{- 17}}$ at 1000 s integration time. Moreover, due to the robustness of the self-redundant measurements, we believe this unidirectional configuration over a telecommunication fiber network fits very well the requirements of comparing frequency standards routinely and for the long term.

In the future, this unidirectional configuration could also be applied to submarine links. Submarine fiber links show lower noise levels than links on land [18]. An extrapolation or prediction of such unidirectional topology for a submarine link was reported in [14,18]. For a submarine fiber link of length up to 7000 km, the ultimate instability can be expected to be at the level of $1 \times {10^{- 16}}$ within a few hours of measurement. Therefore, combining land-based fiber links and submarine fiber links would make atomic frequency standards comparison over fiber links possible on an intercontinental scale.

In conclusion, we present an optical frequency transfer using a unidirectional scheme over two adjacent fibers and show its performance for a long continuous operation of six months. To the best of our knowledge, this is the longest data set for an optical frequency comparison on fiber links, exceeding previously reported data in the literature by a factor of three [26]. We show the effectiveness of the noise compression with this unidirectional configuration and demonstrate a frequency instability of ${\lt}2.0 \times {10^{- 17}}$ at 4,000,000 s integration time. The best stability point is as low as $6.0 \times {10^{- 18}}$ at 1,000,000 s integration time. This unidirectional two-way method is not yet sufficient to compare optical clocks, but gives the possibility to perform faster comparisons of international primary standards over fiber links instead of satellite links. It also provides a very cost-effective configuration for disseminating frequency standards over telecommunication networks for a reasonable performance reduction when the traditional fully bidirectional frequency transfer techniques are not directly applicable.