1. Introduction

The low-noise transfer of ultrastable continuous-wave (cw) optical frequencies is a highly active field of research [1–4]. It is a prerequisite for comparing the emerging new generation of atomic clocks based on optical transitions [5–9]. Following the latest reports, optical clocks can exhibit low instabilities and accuracies in the 10⁻¹⁸ range. Satellite frequency transfer using microwave carriers [10] would take more than one year to transmit a signal at this level of instability, if possible at all. Further applications of low-noise optical frequency transfer include relativistic geodesy [11, 12], remote spectroscopy and laser characterization [13], or remote synchronisation via ultrastable chirped frequency transfer [14].

Long distance transfer of a cw optical frequency via underground optical telecommunication fiber links has been demonstrated with 10⁻¹⁸ instability and accuracy, or better, for distances up to around 2000 km [1–4, 14, 15]. This is sufficient for comparing the most advanced optical clocks to date.

One challenge in long distance frequency transfer is the suppression or mitigation of the noise imprinted onto the signals phase by acoustic/mechanical and thermal perturbations to the fibers [15]. This can be done in real-time using interferometric stabilization [1–4, 14, 15] or in a two-way transfer scheme [16], possibly post-processing the recorded data [17].

Another challenge is the attenuation of the signal when transmitted over fiber lengths in excess of about 100 km. In the telecommunication wavelength window around 1550 nm, standard silica fiber exhibits an attenuation around 0.2 dB/km. Additional losses occur e.g. at splices and connectors, therefore the effective attenuation can be as high as 28. . . 30 dB per 100 km [2, 4].

In the context of ultrastable transfer of optical frequencies, different approaches to amplify or regenerate the signal are investigated and employed. These include the use of Er³⁺-doped fiber amplifiers (EDFA) as the most common one [1,2,4]. Other approaches investigate Raman amplification [18] or Brillouin amplification [19,20], where the transmission fiber itself is used as a gain medium, or phase-lock a repeater laser to the incoming signal [2]. Injection locking a slave laser to the incoming signal might also be a possibility [21]. In France and Germany, also mixed-amplifier links are operated, where EDFA and in-lab Brillouin pumps [1], or EDFA and repeater lasers [2] are employed to mitigate excess losses of the transfer.

Both the interferometric as well as the two-way approach to date require bidirectional, reciprocal transmission for achieving the highest performance [22]. Therefore, contrary to telecommunication applications, in ultrastable frequency transfer the amplifiers do not include optical isolators. When using broad-band amplifiers such as EDFA, typically this restricts the optimum gain to below 20 dB [2,4] to avoid spontaneous lasing due to Rayleigh back-scattering or back-reflections, which was also found for cascaded EDFA in optical radiofrequency transfer [23]. Thus the amplification does not fully match the attenuation mentioned above. Consequently, over long links excess losses will accumulate unless the fiber path between two adjacent EDFA is well below 100 km. Also, the occurrence of several signal failures or cycle slips in the detected signal per hour observed in optical frequency transfer [1,3,4] may be related to cascading bidirectional EDFA, in particular where the signal-to-noise ratio should support much smaller cycle slip rates [2].

Discontinuities of the measurement prolong the measurement time necessary to reach a given level of instability: for stabilized frequency transfer, the transfer instability for a discontinuous measurement, e.g. including dead-time, averages down with averaging time (τ) as $1/\sqrt{\tau }$ only. In the absence of cycle slips, for a continuous measurement dominated by white phase noise, the instability averages down as 1/τ (Allan deviation for unweighted averaging), or as τ^−3/2 (modified Allan deviation for triangular weighted averaging) [24,25]. Therefore we aim at long, cycle slip-free periods of continuous measurement.

Fiber Brillouin amplification [26, 27] relies on the interference between the signal light and blue detuned pump light. Consequently, the process depends on the relative polarization between pump and signal [20, 27]. The travelling intensity grating resulting from the interference in the fiber induces a corresponding phase grating by the process of electrostriction [26]. If conservation of momentum and energy in the medium is fulfilled, pump photons are coherently scattered off the grating phonons, amplifying the signal. The frequency offset between pump and signal is different for different materials and changes slightly with temperature and pressure. For silica this offset Brillouin frequency is around 11 GHz. Fiber Brillouin amplification, as opposed to Erbium-doped fibre amplifiers (EDFA), works only for counter-propagating pump and signal and has a narrow gain bandwidth of order 10 MHz [27, 28]. The narrow gain bandwidth requires a stabilization of the pump light’s frequency relative to that of the signal at around 11 GHz offset to much better than 10 MHz. Thus, together with the polarization dependence of the amplification it requires a more sophisticated amplifier setup compared to EDFA. Up to now this has restricted Brillouin amplification to laboratory applications. On the other hand, these features of fiber Brillouin amplification of narrow bandwidth and directionality allow to selectively amplify either the outgoing or the frequency-shifted return light. This allows exploiting small-signal gains in excess of 40 dB [19, 20].

At Physikalisch-Technische Bundesanstalt (PTB), a field-able fiber Brillouin amplifier suited for remote operation is being developed [20] with support from the European Metrology Programme (EMRP). It is now operated on a 660 km loop of installed underground fiber, which is part of a new fiber connection between Braunschweig and Strasbourg.

In the first section we will describe the experimental setup of the loop measurement including the remote fiber Brillouin amplifier. We will then present measurements of the phase noise and instability observed in the transfer of an optical frequency, before finally discussing the results.

2. A remote fiber Brillouin amplifier used in a looped pair of underground fiber

At PTB a new instrument for in-fiber Brillouin amplification is being developed, which allows remote and largely autonomous operation. The remote Brillouin amplifier setup consists of an externally amplified, commercial extended cavity laser (containing a planar lightwave circuit with a Bragg grating) with a free-running linewidth of several kilohertz as a pump source. Its output is split into four branches which serve two directions into each of the two fibers, see Fig. 1. Two of the branches are shifted in frequency to accomodate the signal’s frequency shift on the returning path. The pump power injected into the fiber is around 10 mW. The four branches include electrically controlled polarization controllers for automated or remote optimization of the pump polarizations. For communication with the instrument we implemented an optical communication channel around 30 nm off the frequency-signal wavelength. Optimization of the polarization is done by maximizing the amplified signal’s intensity. On some days we observe fast fluctuations of the polarization, which we attribute to maintenance work along the link; they may cause a loss of lock of the remote amplifier and may warrant further development of the polarization optimization.

 

Fig. 1 Schematic sketch of the 660 km loop setup PTB-PTB, including one remote fiber Brillouin amplifier (FBA); EDFA: Erbium-doped fiber amplifier; FBA: fiber Brillouin amplifier; ω_1,2: acousto-optical modulator. The blue arrows indicate the Brillouin pump light injected into the underground fiber.

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Using a beat note around 11 GHz between the pump light and the signal coming in from the local end of the link, the pump laser here is phase locked to the signal (a frequency lock would in principle be sufficient). Locking can either be performed remotely, or automatically using a locking algorithm. For automated locking, the pump frequency is scanned to find the maximum of the amplified signal intensity followed by increasing the amplifier’s control loop gain for achieving the phase lock.

To realize the lock, we detect the beat note between the amplified light and the pump light using a fast photodiode. The photodiode output is amplified and divided by a factor of 320. The divided frequency is compared to a stable reference frequency source. Here, we use a chip-scale rubidium clock as a reference for the frequency source. A digital phase comparator produces an error signal, which is then used to control the pump laser’s frequency in a phase-locked loop.

The overall experimental setup is shown schematically in Fig. 1. The transfer laser is a continuous-wave (cw) fiber laser (Koheras Adjustik), phase-locked to an ULE-cavity stabilized master laser having sub-Hz linewidth. The transfer laser has a frequency of 194.4 THz, corresponding to ITU-channel 44. Its light is split into different paths for e.g. locking the local Brillouin pump laser to the outgoing signal light. The transmitted signal’s phase is stabilized actively using interferometric stabilization [1,2,4,15]: the setup forms a Michelson interferometer, where the link itself forms the long arm of the interferometer. The beat signal between the roundtrip light and the local light in the radio frequency (rf) range is detected and compared to the signal of a local rf oscillator. Here, the reference frequency is derived from a hydrogen maser. A phase-locked-loop suppresses the detected phase deviations by acting on the acousto-optical modulator “ω₁”, thus stabilizing the mean phase and frequency at the “remote” end. An acousto-optic modulator “ω₂” at the “remote” end applies a fixed frequency shift to the signal to distinguish it from light merely back-scattered along the link. At the same time, it shifts the signal’s frequency into the gain curve of the Brillouin pumps for the returning light.

The link itself consists of a pair of standard single-mode telecom fiber, forming part of a future fiber connection between Braunschweig and Paris. To allow convenient characterisation of the transfer, the two fibers are patched together in the federal state of Hassia between the cities of Kassel and Marburg, such that they form a loop starting and ending in the same laboratory at PTB. The length of this fiber loop is around 660 km.

While the local and the remote setup are collocated in the same laboratory with active climate control, care has been taken to put the local interferometer and the remote optics for generating the beatnote into separate, thermally insulated (no active temperature control) housings onto separate aluminum supports. This approximates a situation where the ends are not collocated, as in the operation of a point-to-point link.

At Kassel University, at a distance (fiber length) of around 250 km from PTB (geographical distance around 130 km) the remote Brillouin amplifier is installed and operated in a server room of the computing centre. The pump light is phase-locked to the incoming signal at an offset around 11 GHz, corresponding to the Brillouin shift given by the silica fiber.

The setup for remote fiber Brillouin amplifier setup performs a total of four amplifications of the outgoing and returning signal in the fiber loop. At the “remote” end (at the lab in PTB), a fifth, in-lab Brillouin pump amplifies the “remote” signal. Here the pump is offset-locked locally to the signal laser. At the “remote” end the transferred light is combined with that of the signal laser to obtain the remote beat signal for out-of-loop characterisation of the frequency transfer. Also, the setup comprises in-lab bidirectional EDFAs as signal boosters at the local and remote end, because of considerable on-campus losses and excess losses on the fiber PTBKassel. The beat frequencies are each tracked by two tracking oscillators (bandwidth larger than 10 kHz) in parallel and are measured by dead-time free totalizing K&K counters [29]. To characterize the frequency transfer, we measure the phase noise of the transferred light relative to the outgoing signal light, as well as its frequency instability and the deviation from the expected frequency offset given by the frequencies of the acousto-optical modulators.

The fibers at the Kassel location were chosen such that the Brillouin shifts in all directions are the same to within less than 10 MHz. A central control unit of the amplifier allows for remote access as well as for largely autonomous operation of the amplifier setup. The autonomous operation includes the active mitigation of temperature changes by acting on the pump laser using control software developed in-house.

3. Instability and phase noise of the transferred optical frequency

To characterize the optical transfer over the 660 km loop, we first measure the phase noise of the “remote” beatnote both for the case of active stabilization and for the unstabilized case, in which the frequency of acousto-optical modulator “ω₁” is fixed. The results are shown in Fig. 2, where the phase noise was obtained from the phase data taken with a K&K counter in Π-mode at a gate time t of 2 ms. The resolution bandwidth for the phase noise is given by the measurement time. Here, it is about 1 mHz.

 

Fig. 2 The figure shows the phase noise (panel (a)) of the “remote” beat frequency for the loop being unstabilized and stabilized, respectively, as well as the corresponding short-term instabilities (Allan deviation and modified Allan deviation, panel (b)). The data are obtained using a K&K frequency counter in Π-mode. Also shown is the expected phase noise for the stabilized case as calculated from the phase noise of the unstabilized loop according to [15]. The dashed line is a guide to the eye, while the dotted line is the phase noise corresponding to a laser with 40 mHz linewidth (assuming purely white frequency noise).

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The signal-to-noise ratio (100 kHz resolution bandwidth) is around 32 dB for the “remote” beat and around 27 dB for the roundtrip inloop beat, supporting cycle slip rates smaller than 10⁻⁴ [2].

Shown in Fig. 2(a) is the noise of the unstabilized link. Between 0.1 Hz and about 10 Hz the phase noise of the link falls off approximately as 300/f², where f is the Fourier frequency. At around 10 Hz to 20 Hz the phase noise exibits a broad maximum, which we attribute to infra-sound perturbations of the fiber, arising from road traffic [30], or from air pipes in climate controls. Beyond this maximum the noise falls off more steeply.

In the stabilized case a servo bump of the link stabilization is visible at around 45 Hz, which is lower in frequency than expected from the path delay alone. This effect is still under investigation. The phase noise is suppressed within this bandwidth, achieving approximately white phase noise below 10 Hz, close to the calculated limit [15]. Closer inspection reveals that between around 0.1 Hz and around 6 Hz the phase noise increases with Fourier frequency with a slope of about 0.6 rad²/(Hz · Hz). Recently, a similar behaviour of phase noise increasing with frequency was observed for a link in Italy [4]. Also shown in the plot is the phase noise of an idealized laser, i.e. neglecting a 1/f³ behaviour at small Fourier frequencies ([31], eq. 9). Here we give the example of a laser with 40 mHz Lorentzian linewidth, corresponding to the linewidth obtained when stabilizing a laser to a monocrystal silicon cavity [32]. This illustrates, that on the present link, at least for Fourier frequencies smaller than about $\frac{1}{5}{\text{s}}^{-1}$, a 40 mHz-Laser could be disseminated with high fidelity, provided the link is stabilized actively. This could be used to disseminate an ultrastable optical reference, e.g. for stabilizing a local clock laser. Correspondingly, see. Fig. 2(b), for averaging times τ larger than 2 s the instability contribution of the frequency transfer would be smaller than e.g. the instability of an optical clock signal averaging down as $3.4\times {10}^{-16}/\sqrt{\tau }$ (τ measured in seconds) [9]. Note, that the unstabilized link shows a phase noise comparable to that of a laser with a linewidth of order 100 Hz.

Figure 2(b) shows the corresponding plots of the fiber link frequency instability (Allan deviation and modified Allan deviation). In the stabilized case, for averaging times longer than 0.1 s (10 Hz) it falls off as a/τ as is expected e.g. for white phase noise, with a = 4.5 × 10⁻¹⁴ s and τ being the averaging time. Also shown is the modified Allan deviation. For averaging times between around 0.1 s and around 20 s it falls off almost as 1/t². These averaging times roughly correspond to the Fourier frequency range, within which the phase noise shown in panel 2(a) slowly increases with frequency. Previously we observed a τ⁻²-behaviour of the modified Allan deviation also on a different link connecting PTB and Garching [3].

Figure 3 shows measurements of the frequency instability for longer averaging times, this time taken with the counters operating in Λ-averaging mode with 1 s gate time [25].

 

Fig. 3 Shown are the fractional frequency instabilities; all data are taken on a deadtime-free K&K-counter operated in Λ-averaging mode with 1 s gate time. Panel (a) shows two measurements of the noise floor (modified Allan deviation) for the loop being short-cut in the lab; panel (b) and (c) show the frequency instabilities for two measurements of the optical frequency transfer over the 660 km loop; the time between the measurements is several weeks. The measurements (modified Allan deviation) after about 100 s reach the noise floor at around 3 × 10⁻¹⁹. Also shown in panel (b) is the frequency instability of the unstabilized loop as obtained from the correction signal of the stabilization. In panel (c), the dashed line indicates the modified Allan deviation of the inloop beat frequency, where we added a frequency offset of 1 Hz to a single point of the frequency data.

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The noise floor, see Fig. 3(a), was measured by optically short-cutting the stabilized link in the lab, i.e. connecting the output of the local acousto-optic modulator to the input of the remote EDFA using optical attenuators. It is around 1.6 × 10⁻¹⁷ at an averaging time of 1 s and reaches a floor in the 10⁻²⁰-range for a few hundreds of seconds of averaging time. We attribute the variation in the 10⁻²⁰ range to differential temperature fluctuations in the local and the remote optical setup. Therefore, we regard observations of instabilities below 10⁻¹⁹ as fortuitous in the present setup, where the temperature of the optical setups is not controlled actively.

The measurements of the transfer instability via the 660 km loop, Figs. 3(b) and 3(c), were taken over a period of around 2.5 days, Fig. 3(b), and over a period of around six days, Fig. 3(c), respectively. The measurements are separated by about one month. No data points were removed within the measurement periods. Double tracking the beatnotes did not reveal any cycle slips, and also the instability curves are consistent with a cycle slip rate smaller than 10⁻⁴/s.

Figure 3(b) shows the instability of the free-running link as given by the correction signal of the link stabilization (crosses). It undulates in the 10⁻¹⁵ range with an overall mean of the fractional frequency offset of here −1.7 × 10⁻¹⁴. The instability curve indicates processes with periods of around 500 s and around one day, respectively. We attribute the slow perturbation to daily temperature variations along the link, while the shorter-period process might by due to cycles of the climate control e.g. at Kassel University.

The filled symbols represent the result of applying the Allan deviation formalism to the Λ-type data, corresponding to an unweighted averaging of the in itself Λ-weighted 1 s values. We refer to the resulting deviation as Λ-Allan deviation (Λ-ADEV) to avoid confusion with an Allan deviation obtained from Π-type frequency data [25]. The Λ−ADEVs fall off as 5 × 10⁻¹⁶s/τ as expected e.g. for white phase noise. For the first measurement, a flicker floor around 1 × 10⁻¹⁹ seems to be reached after about 3000 s of averaging time τ; for the second measurement the Λ−ADEV averages down to around 10⁻²⁰. Also shown is the modified Allan deviation (ModADEV) for the data (open symbols), corresponding to overlapping Λ-type weighting. It falls off as 1/τ² for averaging times up to 20 s, as discussed previously [3]. This also indicates the averaging time over which triangular weighted averaging as opposed to un-weighted averaging is advantageous in terms of instability. At an averaging time of around 100 s it reaches an instability (ModADEV) of 3 × 10⁻¹⁹, and of 1 × 10⁻¹⁹ for an averaging time of around 400 s. The unweighted mean of the fractional frequency offset of the first measurement (Fig. 3(b) is −2.1 × 10⁻²¹ for the inloop beat and 1.1 × 10⁻¹⁹ for the “remote” beat. For the second measurement (Fig. 3(c), the unweighted means are 1.1 × 10⁻²¹ and −2.2 × 10⁻²⁰ for the inloop and “remote” beat, respectively. In both cases, the mean of the “remote” beat frequency offset is slightly larger than the last point of the Λ−ADEV for unweighted averaging. However this is to be expected from the variations of the current noise floor of the setup without the long fiber link. The variable noise floor, which for averaging times beyond around 300 s is below 10⁻¹⁹, as stated above is likely to arise from out-of-loop noise uncorrelated between the local and the “remote” optical setup.

For comparison in Fig. 3(c) we also show the modified Allan deviation of the inloop beat frequency trace (>500.000 data points), where we intentionally introduce a 1 Hz frequency offset to a single data point. Note that this degrades the ModADEV significantly. A comparison to the ModADEV of the “remote” beat frequency indicates, that the “remote” beat frequency is unlikely to contain more than one such incoherent frequency deviation over the measurement period of several days.

4. Summary & outlook

This paper presents for the first time results from long distance ultrastable optical frequency transfer using a fiber Brillouin amplifier installed along the link as the only means of remote amplification. The link is formed by a pair of fibers forming a 660 km loop, where the roundtrip inloop signal experiences a total of five Brillouin amplifications. The fibers are part of a future connection Braunschweig-Strasbourg-Paris. We obtain continuous measurement intervals of several days without observing cycle slips of the detected signal.

We find that the phase noise we observe is slightly larger than noise we have observed on a different, 920 km link between Braunschweig and Garching, including an “infra-sound” bump between 10 Hz and 20 Hz. Between 10 mHz and 10 Hz the phase noise of the free-running link is observed to fall off as approximately 300/f² rad²/Hz. From the observed phase noise in the stabilized case we infer, that for Fourier frequencies below around 0.2 Hz the spectral content of e.g. a laser with a linewidth of 40 mHz could be transferred via such a link to be used at the “remote” end.

For the transfer of an optical continuous-wave frequency we find an instability better than 10⁻¹⁹ and an accuracy of 1 × 10⁻¹⁹. Due to a τ⁻²-behaviour of the modified Allan deviation at short averaging times, it reaches 3 × 10⁻¹⁹ within 100 s of averaging time. Below 10⁻¹⁹ we are currently limited by noise uncorrelated between the local and the “remote” end, e.g. because of differential temperature fluctuations.

The observed performance of the optical frequency link is sufficient to compare state-of-the-art optical clocks, as will be required for a future redefinition of the SI-second, or for relativistic geodesy using the terrestrial gravitational redshift of around 10⁻¹⁶/m [11, 12]. For the latter, transportable clocks only have to give a reproducible frequency for measuring differences of the gravitational potential via a fiber link. Furthermore, with a link such as this one, we can conduct time-dependent geodetic measurements: for detecting fluctuations of the gravitational potential difference between two places over time we only require sufficiently stable frequency sources and a link between them. From tidal variations of the geoid, variations of the gravitational red-shift are expected at the 10⁻¹⁷ level [33]. In the future, using a long-distance fiber link to compare two cavities that have e.g. an instability of 10⁻¹⁷ at all times from 1s to 12 hours would allow testing for tidal variations of the gravitational potential difference exceeding 0.1 m. Lasers stabilized to cryogenic silicon cavities at present seem the best contenders for such frequency sources, due to their low thermal noise floor (currently near 5 × 10⁻¹⁷ [32]) and inherently low drift [34].

Independent from their actual implementation, measurements of the gravitational potential difference and its time dependence are important for a meaningful realization of an extended earthbound timescale: The results presented in this paper demonstrate that remote Brillouin amplification is well-suited to serve present and future needs of ultrastable optical frequency transfer. Compared to EDFA as the traditional means of amplification, it allows operating at larger gain and thus allows a larger distance between the amplifiers. Here, the maximum distance bridged between amplifications was around 250 km.

Currently a second remote Brillouin amplifier is being set up to investigate the performance of a link containing a chain of remote Brillouin amplifiers. This will extend the length of the loop to around 1000 km.