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Abstract
In the paper we present a solution for controlling the chain of bidirectional optical amplifiers, intended for long-haul fiber links used to distribute signals produced by optical atomic clocks. The solution is based on a dedicated two-channel noise detector, which allows independent measurement of the noise contributions related to interferometric signal fading and additive wideband noise. New signal quality metrics, based on two-dimensional noise detector, allows to distribute properly the needed gain among the chained amplifiers. Experimental results performed both in laboratory conditions and on a real 600 km-long link, confirming proper operation of proposed solutions, are presented.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The impressive development of optical atomic clocks displaying frequency accuracy and stability at the order of 10−18 [1–4] opens new perspectives in fundamental physics, spectroscopy, geodesy, astronomy, and other areas of science and top-level technologies [5–8]. The optical clocks, however, are extremely complex and expensive devices, still needing highly-competent specialists to operate them. In this situation long-distance distribution of the optical-domain frequency reference signal derived from optical clocks is the only option to increase their availability. Currently, the most promising and developed option for optical frequency distribution is using optical fibers operated bi-directionally, with a dedicated phase stabilization system arranged for reducing the phase/frequency fluctuations picked up along the fiber due to temperature variations and vibrations. Thanks to a high degree of correlation of the phase fluctuations imposed on the opposite-direction-propagating signals, they may be detected and canceled out to a high degree [9,10].
In case of long links (above approx. 200 km), the attenuation of the fiber becomes the limiting factor and have to be compensated. Generally, a few options are available: dividing the long link into several sections and retransmitting the signal with lasers phase-locked to the output of each section [11], using Brillouin amplifiers [12], Raman amplifiers [13] and Erbium-doped fiber amplifiers (EDFA). The last option is most widely used, frequently in combination with other solutions mentioned before [14,15]. In the experiments presented in this paper we used the chain of EDFAs, but the results would be applicable also to other bidirectional amplifiers, as Raman or semiconductor amplifiers.
The specific problem is that the EDFAs used for optical reference frequency distribution must be bidirectional. This is in contrast to regular EDFAs used in fiber-optic telecommunications, which contain optical isolators, and so are unidirectional. The problem with bidirectional EDFAs (and similarly any other bidirectional optical amplifiers) is that the parts of signal which are backscattered and/or reflected two times (generally - even times) are also amplified and interfere with the desired, directly propagating signal - see Fig. 1 and [16,17]. Moreover, the multiple-times backscattered and reflected parts of the signal are amplified by EDFAs many times, in contrast to the desired signal, which is amplified only once at each amplifier. For this reason, the ratio between the desired (directly propagating) signal and even-times backscattered and reflected destructive signals decreases proportionally to G2 for the double backscattering/reflection, proportionally to G4 for four-order backscattering/reflection, and so on. In case of bidirectional transmission of modulated signals generated by telecommunication-grade lasers of relatively low spectral coherence (i.e. with linewidth in range of a few megahertz or more), which spectrum is additionally broadened because of the modulation, the impact of even-times backscattering/reflections is manifested as yet another noise component, which produces at the photodiode output a stationary and relatively wideband additive noise [17–19]. Consequently, if some hardware detector estimating the signal quality is used for real-time amplifiers gain optimization [20,21], the impact of noise generated by even-times backscattering/reflections is naturally taken into account.
Fig. 1. Illustration of interference of multiple backscattered and reflected signals with the directly propagating one.
In case of the optical reference frequency transfer, the unmodulated laser of extremely low linewidth (below 1 Hz) is used, so its coherence length is many orders of magnitude higher that the link length, and so the even-times backscattering/reflections cause interferometric fluctuations of the signal strength, with total signal fading or even temporal phase reversal possible. It should be stressed that such events (fading and phase reversals) have generally different impact on optical reference frequency distribution than stationary wideband noise components, and so require different approach.
Unfortunately, the interferometric noise strongly depends on the various specific conditions, such as the lengths of consecutive fiber spans, location and magnitude of reflections (caused mainly by connectors) and lumped losses (connectors and splices), backscattering properties of particular fiber, magnitude of vibrations picked up along the fiber and so on. As many of these factors are difficult to be determined, and some of them may change in time, the theoretical calculation or simulation of the interferometric noise seems to be extremely challenging and potentially inaccurate. In this situation the reasonable solution is to perform a real-time measurement of the actual interferometric noise strength, and using the results when determining the optimal gains of amplifiers. It should be stressed that previously proposed algorithms for optimization of bidirectional optical amplifiers, as [17–20], and relevant noise detector [21] were developed for systems exploiting intensity-modulated laser sources of moderate coherence (MHz-wide spectral linewidth), and are unsuitable for processing the signals affected by the interferometric noise and suffering from fadings.
Therefore, the crucial novelty of this work is proposing a practically measurable metric of the strength of the interferometric fading noise, the relevant detector hardware, and finally the way of incorporating this metric into the automated algorithm of controlling the gains of chained amplifiers, used for optical frequency transfer exploiting ultra-narrow-linewidth CW laser sources.
2. Concept and evaluation of a two-dimensional noise detector
Let us consider the generic solution for phase-stabilized, long distance distribution of an optical reference frequency generated by an optical clock, as depicted in Fig. 2. The ultra-stable CW optical signal referenced to the optical clock is launched into the fiber via an optical coupler OC1 and an acousto-optic modulator (AOM). The cancellation of the phase fluctuations imposed by the fiber is performed in a closed loop configuration, by sensing and cancelling the optical phase excursions observed by beating on photodiode PD1 the local signal with the one coming back from the link. The phase corrections are applied by changing the phase of the radio-frequency (RF) signal provided to the AOM, acting as a phase/frequency shifter. At the remote (user) end a narrow-linewidth clean-up laser is phase-locked to the incoming signal with some known frequency offset, usually in order of some tens of megahertz. For this purpose, also at the remote end, the beating between incoming signal and the clean-up laser is performed, using photodiode PD2. (In some cases the remote laser may be substituted with a mirror and another AOM [9], which however makes no substantial difference for our discussion.) In our case of long-haul systems, where the beat note at the PD1 and PD2 outputs are strongly corrupted by noise, the so-called tracking phase-locked-loop (PLL) follows each photodiode to reduce the impact of the noise [12].
Fig. 2. General scheme of a long-haul optical frequency transfer with fiber phase noise cancellation. OC stands for an optical coupler (50/50%), FM - Faraday mirror, AOM - acousto-optic modulator, PD - photodiode, and PC - polarization controller.
The fundamental stability and accuracy limitations of the frequency transfer result from the inherently limited bandwidth and efficiency of the cancellation loop used to suppress the fiber induced phase noise, uncompensated pieces of fiber in the local and remote modules, and the birefringence (polarization mode dispersion) of the optical path [10,22]. However, in case of improper gain settings of the amplifiers, additional severe stability and accuracy degradation arise due to so-called cycle slips occurring in the tracking PLLs [23,24]. The cycle slips manifest as the phase jumps of an integer multiple of an optical period observed at the PLLs outputs. The question how optimally set the gains of amplifiers located in between the fiber sections is at our focus in this work.
Generally, it may be noted that for too small total gain, the beating signal at the output of photodiodes PD1 and PD2 will be weak, thus strongly affected by noise of various origin, such as shot noise of the weak signal from the fiber, relative intensity noise (RIN) of the strong signal from the local laser, amplified spontaneous emission (ASE) noise generated in EDFAs, and electronic noise added mainly by amplifiers following PD1 and PD2. When noisy beat notes are provided to the tracking PLLs cycle slips may occur, which leads to degradation of both stability and accuracy of the entire frequency transfer system. The noise contributions mentioned above are generally wideband in terms of their power spectral density (PSD), close to white noise in the frequency range of our interest, which is a few megahertz around the beat frequency. Generally, the signal to noise ratio (SNR) related to this group of broadband noise sources may be improved by increasing the gains of EDFAs placed along the link [17].
Unfortunately, increasing the gains of EDFAs too much leads to abrupt manifestation of the abovementioned interferometric noise. Recalling, however, the fact that we are dealing with optical signals of extremely long coherence time, the observed interference manifests in slow fluctuations of the magnitude of the signals reaching the photodiodes, with the time-dynamics related to slow changes of phase (and also polarization) imposed by the fiber affected by environmental factors as temperature changes, strains and vibrations. This leads to an important observation that the PSD of the interferometric noise is located in a narrow bandwidth (some kilohertz) around the beat frequency. The important consequence is that the impact of the interferometric noise is different to that of the wideband noise components. As the interferometric signal fading or phase reversals are relatively long (hundreds of microseconds), the resulting cycle slips may be of huge magnitude, or even the permanent loss of PLL’s synchronization is possible. From the other hand, because of the generally slow nature of this events they occur rather rarely (in reasonable situation), so any attempt of their experimental characterization in particular conditions (as some set of amplifiers’ gains) takes days of observations. In this situation some metric describing the strength of the interferometric noise would be very useful for optimizing the chain of amplifiers in reasonable time.
It should be mentioned that there are also two additional destructive effects related to too high gains of bidirectional EDFAs, which also may result in cycle slips (or finally the loss of tracking PLLs synchronization). The first is chaotic lasing arising in a bidirectional EDFA above some threshold gain (usually between 15 dB and 25 dB), which is also caused by backscattering and/or reflections present at both sides of an amplifier. Because of phase and polarization fluctuations imposed by the fiber, this lasing is extremely random and occurs in the form of optical bursts of varying wavelengths. This lasing may directly corrupt the desired signal or may affect it by gain depletion related to optical bursts. From our perspective it is important that the impact of the lasing of EDFA is also visible in the beat note spectrum close to the beat frequency, similar to the interferometric noise. Yet another destructive phenomenon related to excessively high gains might be stimulated Brillouin scattering (SBS) [25], generating a frequency shifted backward-propagating signal at the expense of the power depletion of the original signal. For a highly coherent optical signal propagating in a real fiber, displaying relatively slow phase and polarization fluctuations, the intensity of SBS is also slowly fluctuating, which in turns causes fluctuation of the power reaching a photodiode producing the beat note signal, and so its low-frequency modulation.
The fact that all the noise contributions related to insufficient amplification are characterized by a PSD spread widely around the beat frequency, and the noise contributions arising from excessive amplification result mainly in close-to-carrier spectral components (which is illustrated in Fig. 3) gave us a hint to independently detect these two substantially different contributions and use them for optimizing the gains of the EDFAs. The proposed idea of the beat note noise detector is presented in Fig. 4.
Fig. 3. Schematic illustration of PSD of noise affecting the beat note in case of low and high gains of amplifiers.
The signal output from the photodiode (PD1 or PD2) amplifier is bandlimited with the bandpass filter (BPF1), centered on the beat frequency. In this situation the wideband noise is converted to a bandlimited one, centered at the same frequency as the beat note. Following the well-known theory of modulation, a sinusoidal signal corrupted by an additive bandlimited noise may be regarded as being amplitude and phase modulated by the so-called in-phase and quadrature components of this noise [26]. As the depth of the resulting amplitude modulation is directly related to the SNR, we employed an envelope detector (amplitude demodulator) followed with logarithmic amplifier for determining the SNR. The important fact is that at the envelope detector output, the two types of noise (related to low and high amplification regimes) are still distinguishable in the frequency domain, as they produce a low-frequency component and a relatively much wider component, respectively. To separate these two kinds of noise a low pass filter LPF and bandpass filter BPF2 are used. Then the output signals of the two filters are digitized with analog-to-digital converters (ADCs), and the standard deviation of both signals are calculated (using a microcontroller). Thanks to the logarithmic amplifier used, the readouts at both outputs of the detector are inversely proportional to the SNRs related to particular types of noise, but not dependent on the absolute magnitude of the beat note. Performing the amplitude noise detection only has also an important advantage that it is insensitive to the residual (not fully compensated) environmental phase noise of the link, which also affects the beat note PSD, but is not related to our problem of gain optimization.
The verification of the proposed detector was performed in the setup as in Fig. 2, with 2 × 200 km of fibers (on spools) and one bidirectional EDFA. Two similar detectors were connected to the beat note amplifiers following the PD1 and PD2 photodiodes, with the only difference in the respective BPF1 filters tuned to the corresponding beat notes (10 MHz and 70 MHz, respectively, 2 MHz bandwidth in both cases). LPF (10 kHz) and BPF (300 kHz ±70 kHz) are second-order LC ones. We use 12-bit ADCs integrated within STM microcontroller which calculates the standard deviations in 1 s intervals.
An example of signal fluctuations at BPF2 and LPF outputs, for both low and high gain EDFA regimes, are illustrated in Fig. 5. Comparing the upper and lower plots one may noticed that in case of high gain of the amplifier the output from LPF is strongly fluctuating, which indicates the intense fluctuations of the beat note magnitude, caused by the interferometric noise. The other (wideband) noise components in this case are low, which is reflected in small variation of the BPF2 output. Lower plot, taken with small gain, shows exactly opposite situation.
Fig. 5. Time-domain plots at LPF and BPF2 outputs in the noise detector for cases of low and high gain regime.
The readouts of the wideband and narrowband noise indicators versus the gain of the amplifier are presented in Fig. 6. As it may be noticed, the readout of the wideband noise indicator ${\sigma _{BP}}$ is monotonically decreasing with increasing EDFA gain, as expected. For very low gain (below 10 dB) we observed slightly different readouts for detectors installed at the local and remote side, which is probably related to different beat note frequencies and different bandpass filters used. In case of low gain, cycle slips were observed for a readout of the wideband noise detector above approximately 20 (arbitrary units). The narrowband noise indicator ${\sigma _{LP}}$ reacts mainly to the interferometric (fading) noise and amplifier lasing occurring for high gain (as expected), but is also slightly sensitive to the noise related to small gain, which is an undesired cross-sensitivity. Fortunately, this cross-sensitivity is many times smaller than the sensitivity of the wideband noise indicator, and it proved to cause no problem for our optimization algorithm. In case of too high gain, malfunction of the system (cycle slips or even loss of the PLLs lock) was observed for a readout of the narrowband noise indicator above approximately 15 a.u. As the interferometric noise is slow and somehow chaotic (non-stationary), the readout of the narrowband noise indicator is also fluctuating, and it should be averaged for at least some tens of seconds.
It should be stressed, however, that the maximum acceptable values of the wideband and narrowband noise indicators mentioned above are strictly related not only to a particular realization of the detector, but to a particular design of the local and remote devices of optical frequency distribution system as well, as its specific design may be more or less sensitive to noisy beat notes, compared to our implementation. Therefore, a similar characterization should be performed for each particular hardware solution.
3. Gains tuning algorithm
The proposed beat note noise detector opens a way towards an on-line automated optimization of the chain of amplifiers, in the way that both wideband noise and close-to-carrier noise can be simultaneously mitigated by tuning the gains of all amplifiers appropriately. The detector may be located either on the local or remote side of the link, and a computer or just a microcontroller can run the algorithm controlling the amplifiers, which should be remotely accessible (e.g. via the internet network).
The goal function $GF$, used in the algorithm, is defined as:
which value is to be minimized. The term related to ${\sigma _{LP}}$ contains a weight constant W, which gives the flexibility to trim the relative significance of the readouts of both detector outputs. The square power applied to both noise terms is intended to highly suppress situations where one of the terms is much higher than the other, which would be risky in case of some unexpected changes in the link, for example arising from excess attenuation or reflection. It should be stressed that the particular formula of the GF proposed in (1) is somehow arbitral (and intuitive) one, and basing on the described two-dimensional noise detector also the GFs of different shape might be considered.The proposed optimization algorithm is shown in Fig. 7. In each round of the algorithm gain of each amplifier is changed up or down by 1 dB, and the change is preserved if it improves the value of the goal function. Thanks to our detector, which allows predicting whether the gains are generally too low or too high, in the first attempt the gain is changed accordingly, which means the algorithm compares terms $W{\sigma _{LP}}$ and ${\sigma _{BP}}$ before deciding the direction. When the change actually improves the value of the goal function, it is preserved, and if not, the change in opposite direction is also checked. Then this procedure is applied successively to each amplifier in the chain, returning to the first one after reaching the last amplifier, and so on. It should be stressed that in case when the initial gains of amplifiers are far from optimal, i.e. the link performance is dominated by noise contributions related to either insufficient or too high gains, the proposed algorithm deliberately applies the gain changes in the proper direction, which shortens the optimization time, reduces cycle slips occurrence, and eliminates the risk of losing lock of the entire transfer system. In the presented version the algorithm runs endlessly, but it could also be terminated manually or automatically after a few rounds, when consecutive rounds do not lead to substantial changes of noise detector readouts and gains of amplifiers.
4. Experimental results
The experimental verification of the proposed solution was performed in two steps. In the first one all the equipment (local/remote devices, amplifiers, phase/frequency counter, computer running the algorithm) and sections of fiber on spools were located in the same laboratory at AGH University. This way we were able to easily modify various details of the setup, especially we could manipulate with the reflective connectors (i.e. provoke the changes of reflections) and observe the on-line reaction of our gain controlling algorithm. In the second step we arrange the experiments in a real environment of fully outdoor fiber link, being a part of the operational Polish PIONIER network. The source laser, local/remote modules and phase/frequency counter were located in the Poznan Supercomputing and Networking Center in Poznan, and the experiments were controlled remotely.
In the first step we arranged a link consisting of four fiber sections and three amplifiers, as depicted in Fig. 8. The first and last 150 km-long sections were on spools, and the inner 60 km-long ones were fibers deployed in an urban environment in Krakow, with many patches and relatively high attenuation. Most of the optical connectors along the link were low-reflection APC ones, but at both sides of A1 and one side of A3 there were PC connectors which were deliberately loosely fixed to provoke substantial reflections. The signal source we used was a narrow-linewidth laser stabilized to an external optical etalon. At the local side we installed the noise detector described above and a computer running the optimization algorithm. Amplifiers were controlled via an Ethernet network.
Monitoring of the frequency transfer stability was realized by beating the signals from the master laser and the remote clean-up laser. The beat note phase evolution was registered using K&K FXE 07 phase/frequency counter. The cycle slips were detected by differentiating the trace of phase recorded as averaged value over consecutive 1 s intervals. Differences equal or higher than one period was interpreted as slips (see Fig. 9(c)).
Fig. 9. Results of optimization process shown as time evolution of: ${\sigma _{LP}}$ and ${\sigma _{BP}}$ (a), amplifiers’ gains (b) and cycle slips registered in the first hour of the experiment (c).
After initial experiments we decided to take $W = 1.5$, which means that the readout of the close-to-carrier (interferometric) noise is taken with higher weight than the wideband one. This was motivated by the fact that when we increased ${\sigma _{LP}}\; $ and ${\sigma _{BP}}$ (alternatively), we observed that cycle slips started for lower readouts in case of dominating impact of the interferometric noise. Additionally, the cycle slips in this case appeared to be more severe, more frequently causing loss of the system lock. However, changing the value of W in the range between 1 and 2 did not make a significant difference in the operation of the algorithm in the meaning that in all cases the final values of gains obtained after a few rounds of the algorithm were similar, as well as the final values of ${\sigma _{LP}}\; $ and ${\sigma _{BP}}$. It should be stressed, however, that the proper value of constant W strongly depends on specific implementation of the noise detector and the local/remote modules, and so should be tuned within particular hardware environment.
We also observed that the readout of the close-to-carrier noise detector was changing randomly up to approximately ±25% for consecutive observation periods (of 1 s), which we assume is related to the non-stationary nature of the interferometric noise. This randomization of the readouts causes some wander in the algorithm, i.e. some back and forth changes of amplifiers’ gain may be observed. To reduce this effect we applied 100 s averaging of the close-to-carrier noise readouts, which reduces the wander (and also the algorithm speed, which is however not critical). Even after applying this averaging some wandering of the gains was still observed, mainly in the form that for certain periods the gains of some amplifiers were decreasing, while simultaneously the others were increasing, and at some point this trend reversed without any identifiable external reason. Therefore we additionally applied a slight modification of the goal function to introduce a small “gravity” which gently pulls the gains of amplifiers towards a moderate value, and so mitigates the unnecessary random wandering of gains. The modified goal function was:
where ${G_i}$ are gains of amplifiers taken in [dB], M is the moderate value of gain (“a gravity center”), and S is a factor determining the strength of this “gravity”. We assumed $M = 15\; dB$ and $S = 1$. It should be stressed however, that the observed gains wander does not affect the efficiency of the algorithm substantially (i.e. does not lead the system to high noise regions and cycle slips), and the described counter-measure was proposed mainly for improving the clarity of observed behavior and its interpretation.Next we launched a long-term experiment with continuous recording of transfer stability, cycle slips and noise detector readouts, lasted for 5.5 days. At the beginning of the we set the gains of all the amplifiers equal to 16 dB, which resulted in the proper synchronization of the entire system, but with frequent cycle slips. Then we started the algorithm and registered the readouts of the beat note noise detector, the gain changes applied by the algorithm, and cycle slips. The results are presented in Fig. 9. From time to time we also touched and applied some force to the loose PC connectors at both sides of A1, to provoke a change of its reflections, which is marked with four dashed-line vertical arrows between plot (a) and (b). At the beginning of the optimization process the beat note noise was dominated by the wideband component, and so the algorithm started by increasing gains of all amplifiers, but soon the gains started to differentiate. After the first few rounds of the algorithm the cycle slips vanishes (see Fig. 9(c), where the first hour of the experiment is zoomed). Then, in the whole 5.5 days-long experiment we observed only two additional cycle slips. Both occurred in the hours of most intense human activity in the laboratory where the spools were located, so they probably were provoked by extraordinary loud acoustic perturbations affecting the spools.
The explanation of the observed unequal gains is as follows: around A1 we deliberately placed two loose PC connectors which produce reflections and thus strong interferometric noise, and so the gain of A1 ended up relatively smaller than the others. On the other hand all connectors around A2 were APC ones, and additionally the outdoor links connected to A2 display excess attenuation because of many patches (characteristic for an urban area), and so the backscattering as well as reflections reaching this amplifier are low, and the algorithm may safely increase the gain of A2 even above 25 dB without provoking significant interferometric noise. When we maneuvered with the loose PC connectors and so decreased or increased the interferometric noise, the gains of amplifiers were subsequently corrected by the algorithm, to minimize the goal function in changed conditions.
In the second step we performed experiments with a real fiber link being a part of the Polish PIONIER network, with the intension to verify our solution in a real telecom network environment. The link was arranged in a loop configuration, going from Poznan to Torun and back to Poznan, which gives a total length of 603 km, with nine potential points of amplification. The experimental setup was similar as depicted in Fig. 8, but a fiber noise cancellation setup of slightly different design was used, and the analyzed beat note had 30 MHz nominal frequency this time. Initially, we installed amplifiers in all (nine) accessible locations. In this case the beat note noise (both ${\sigma _{LP}}\; $ and ${\sigma _{BP}}$) was however very small for gains varied over a relatively wide range, so this case occurred to be not very challenging for the optimization procedure being under examination. Therefore, in the next step we eliminated four amplifiers, causing highly unequal and irregularly located spans of both short and long fiber lengths, namely: 96.5 km - A1 - 142 km - A2 - 63 km - A3 - 63 km - A4 - 64 km - A5 - 174.5 km. During preparations to the experiment we lost remote access to the amplifier A3 located at Torun University, which was related to a networking issue. Its gain was 20 dB just by chance, and we had no means to change it. Thus A3 was excluded from the optimization algorithm and operated with its constant gain of 20 dB. The initial gains of A1, A2, A4 and A5 were chosen as 18 dB, which (in contrast to the first experiment) resulted in dominant, very high interferometric noise. Cycle slips occurred frequently before starting the algorithm. An additional change was that this time we resigned from the “gravity” term in the goal function, i.e. we returned to the initial version given by (1).(Just to demonstrate that this term might be useful but is not essential for successful optimization.) After launching the algorithm the cycle slips vanished in ten minutes, i.e. within the first round of the algorithm, when the gains of all four controllable amplifiers were decreased by 1 dB each. The experiment lasted for seven days, and within this period four additional cycle slips occurred, which, however, were not correlated with any noticeable events in the registered record of the algorithm operation, and was probably caused by some extreme mechanical stresses affecting the fiber link.
The results of the experiment are illustrated in Fig. 10, where the evolution of noise detector readouts and amplifier gains is plotted for the first 30 hours of the experiment. From Fig. 10(a) it may be observed that at the very beginning ${\sigma _{LP}}$ rapidly decreased at the cost of some small increase of ${\sigma _{BP}}$, and then the values did not change substantially. A relatively long period of noticeably higher noise may be, however, identified between hours 18 and 23. It may also be noticed that in this period the algorithm reacted to this situation and modified the gains, mainly of A4 and A5. The worsening of the beat note noise in this period may possibly be attributed to changes of the signal polarization along the fiber, which would affect (increase) the interferometric noise during this period. From Fig. 10(b) it may be also concluded that the algorithm leads to systematically unequal amplifier gains - generally the gain of A1 was permanently kept highest (ignoring the out-of-control gain of A3), and the gain of A4 was consistently lowest. In addition, the stability of the frequency transfer is shown in Fig. 10(c) in form of the modified Allan deviation. The red trace is calculated from the data recorded at the beginning of the experiment, before starting the algorithm, when the frequent cycle slips were present. The blue one is calculated from the data starting ten minutes after launching the algorithm (with four additional cycle slips, observed within the entire seven-days measurement, excluded from the data). It may be noticed that by running the gain tuning algorithm the stability was improved substantially, and only then it reached the expected level (see [27], [28] for reference); the modified Allan Deviation drops below 10−18 for 103 s observation time, and below 10−19 for observation longer than 104 s.
Fig. 10. Results of the in-field experiment: evolution of ${\sigma _{LP}}$ and ${\sigma _{BP}}$ (a), evolution of amplifiers’ gains (b) and frequency transfer stability (c).
5. Conclusion
The proposed solution of assessing the beat note quality allows us to quantitatively and separately characterize the two types of noise corrupting the beat note signal and thus overall performance (stability and accuracy) of the optical reference frequency distribution exploiting the ultra-narrow linewidth laser sources. The key observation underlying the described two-dimensional noise detector is that noise related to too high gains of the bidirectional amplifiers is located spectrally close to the beat note frequency, whereas the other types of noise, which manifests for insufficient gains, is wideband, so the two types may be distinguished. The described detector may be used in a few different arrangements: as a stand-alone device used for checking the quality of the beat note signals, as a continuously working link quality detector providing data for the operator’s supervisory system, or may be engaged in the manual or automated amplifiers’ gain tuning, as presented in this work. A minor drawback of the implemented detector is the observed small cross-sensitivity of the narrowband noise detector on the wideband noise. In the current design the narrowband noise detector comprises a very simple second-order LC low-pass filter with 10 kHz cut-off frequency, and in future work we plan to use a higher order filter and optimize its bandwidth, or place the ADC directly at the output of the envelope detector and realizing the filters in digital domain.
The important feature of the proposed noise detector is that it is based on envelope (amplitude) detection, and so is insensitive to the residual (not fully compensated) environmental phase noise of the link, which also affects the beat note PSD, but is not related to our problem of gains optimization.
The two-dimensional detector allows to perform gains optimization in a “smart” way, i.e. the first actions of the algorithm after its initialization or some change in the link (such as varying reflections or increasing attenuation) are performed always in the right direction, i.e. the gains are reduced or increased according to the currently dominating noise type. This reduces the time of reaction, and, moreover, eliminates unnecessary cycle slips, amplifier lasing or even loss of the entire transfer system lock It should be pointed out, however, that before starting the optimization the beat note of the correct frequency must be present at the input of the detector, so all the phase/frequency controlling loops involved in the fiber phase noise cancellation system must be locked. This means that one cannot start the optimization with completely arbitrary values of the gains. A reasonable starting point may be equal gains sufficient for system locking, but not significantly higher.
It should be also noted that in the presented experiments we used our proprietary bidirectional EDFAs with a true gain control, which allowed us to apply precisely 1 dB gain steps. As some other designs of bidirectional EDFAs use indirect gain control by changing the pump-laser current or power, an open question is how this can affect the proposed optimization algorithm.
Acknowledgments
This work was supported in part by the TiFOON 18SIB06 Project through the EMPIR Programme co-financed by the Participating States and through the European Union’s Horizon 2020 Research and Innovation Program and in part by the Polish National Science Center under the grant 2019/33/N/ST7/01373.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015). [CrossRef]
2. T.L. Nicholson, S.L. Campbell, R.B. Hutson, G.E. Marti, B.J. Bloom, R.L. McNally, W. Zhang, M.D. Barrett, M.S. Safronova, G.F. Strouse, W.L. Tew, and J. Ye, “Systematic evaluation of an atomic clock at 2 × 10–18 total uncertainty,” Nat. Commun. 6(1), 6896 (2015). [CrossRef]
3. N. Huntemann, C. Sanner, B. Lipphardt, Chr. Tamm, and E. Peik, “Single-ion atomic clock with 3 × 10–18 systematic uncertainty,” Phys. Rev. Lett. 116(6), 063001 (2016). [CrossRef]
4. E. Oelker, R. B. Hutson, C. J. Kennedy, L. Sonderhouse, T. Bothwell, A. Goban, D. Kedar, C. Sanner, J. M. Robinson, G. E. Marti, D. G. Matei, T. Legero, M. Giunta, R. Holzwarth, F. Riehle, U. Sterr, and J. Ye, “Demonstration of 4.8 10−17 stability at 1 s for two independent optical clocks,” Nat. Photonics 13(10), 714–719 (2019). [CrossRef]
5. S. Kolkowitz, I. Pikovski, N. Langellier, M. D. Lukin, R. L. Walsworth, and J. Ye, “Gravitational wave detection with optical lattice atomic clocks,” Phys. Rev. D 94(12), 124043 (2016). [CrossRef]
6. C. Lisdat, G. Grosche, N. Quintin, et al., “A clock network for geodesy and fundamental science,” Nat. Commun. 7(1), 12443 (2016). [CrossRef]
7. C. Sanner, N. Huntemann, R. Lange, C. Tamm, E. Peik, M. S. Safronova, and S. G. Porsev, “Optical clock comparison test of Lorentz symmetry,” Nature 567(7747), 204–208 (2019). [CrossRef]
8. C. Clivati, R. Aiello, G. Bianco, C. Bortolotti, P. De Natale, V. Di Sarno, P. Maddaloni, G. Maccaferri, A. Mura, M. Negusini, F. Levi, F. Perini, R. Ricci, M. Roma, L. S. Amato, M. S. de Cumis, M. Stagni, A. Tuozzi, and D. Calonico, “Common-clock Very Long Baseline Interferometry using a coherent optical fiber link,” Optica 7(8), 1031–1037 (2020). [CrossRef]
9. L.-S. Ma, P. Jungner, J. Ye, and J. L. Hall, “Delivering the same optical frequency at two places: accurate cancellation of phase noise introduced by optical fiber or other time-varying path,” Opt. Lett. 19(21), 1777–1779 (1994). [CrossRef]
10. N. R. Newbury, P. A. Williams, and W. C. Swann, “Coherent transfer of an optical carrier over 251 km,” Opt. Lett. 32(21), 3056–3058 (2007). [CrossRef]
11. O. Lopez, A. Haboucha, F. Kéfélian, H. Jiang, B. Chanteau, V. Roncin, C. Chardonnet, A. Amy-Klein, and G. Santarelli, “Cascaded multiplexed optical link on a telecommunication network for frequency dissemination,” Opt. Express 18(16), 16849–16857 (2010). [CrossRef]
12. O. Terra, G. Grosche, and H. Schnatz, “Brillouin amplification in phase coherent transfer of optical frequencies over 480 km fiber,” Opt. Express 18(15), 16102–16111 (2010). [CrossRef]
13. C. Clivati, G. Bolognini, D. Calonico, S. Faralli, F. Levi, A. Mura, and N. Poli, “Distributed Raman Optical Amplification in Phase Coherent Transfer of Optical Frequencies,” IEEE Photonics Technol. Lett. 25(17), 1711–1714 (2013). [CrossRef]
14. K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, “A 920-Kilometer Optical Fiber Link for Frequency Metrology at the 19th Decimal Place,” Science 336(6080), 441–444 (2012). [CrossRef]
15. N. Chiodo, N. Quintin, F. Stefani, F. Wiotte, E. Camisard, C. Chardonnet, G. Santarelli, A. Amy-Klein, P.-E. Pottie, and O. Lopez, “Cascaded optical fiber link using the internet network for remote clocks comparison,” Opt. Express 23(26), 33927–33937 (2015). [CrossRef]
16. M. van Deventer and O. Koning, “Bidirectional transmission using and erbium-doped fiber amplifier without optical isolators,” IEEE Photonics Technol. Lett. 7(11), 1372–1374 (1995). [CrossRef]
17. Ł. Śliwczyński and J. Kołodziej, “Bidirectional Optical Amplification in Long-Distance Two-Way Fiber-Optic Time and Frequency Transfer Systems,” IEEE Trans. Instr. Meas. 62(1), 253–262 (2013). [CrossRef]
18. Ł. Śliwczyński, P. Krehlik, and K. Salwik, “Modeling and optimization of bidirectional fiber-optic links for time and frequency transfer,” IEEE Trans. Ultrason. Ferroelect., Freq. Cont. 66(3), 632–642 (2019). [CrossRef]
19. L. Wang, Y. Liu, W. Jiao, L. Hu, J. Chen, and G. Wu, “Fast and on-line link optimization for the long-distance two-way fiber-optic time and frequency transfer,” Opt. Express 30(14), 25522–25535 (2022). [CrossRef]
20. K. Salwik, Ł. Śliwczyński, P. Krehlik, and J. Kołodziej, “Optimization of time and frequency fiber-optic links exploiting bi-directional amplifiers, based on real-time performance measurement,” Opt. Fiber Technol. 62, 102465 (2021). [CrossRef]
21. K. Salwik, Ł. Śliwczyński, and P. Krehlik, “Monitoring of phase jitter in fibre optic time and frequency transfer systems,” Metrol. Meas. Syst. 25(3), 487–497 (2018). [CrossRef]
22. D. Xu, O. Lopez, A. Amy-Klein, and P.-E. Pottie, “Non-reciprocity in optical fiber links: experimental evidence,” Opt. Express 29(11), 17476–17490 (2021). [CrossRef]
23. G. Ascheid and H. Meyr, “Cycle slips in phase-locked loops: A tutorial survey,” IEEE Trans. Commun. 30(10), 2228–2241 (1982). [CrossRef]
24. D. Calonico, E. K. Bertacco, C. E. Calosso, C. Clivati, G. A. Costanzo, M. Frittelli, A. Godone, A. Mura, N. Poli, D. V. Sutyrin, G. Tino, M. E. Zucco, and F. Levi, “High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link,” Appl. Phys. B 117(3), 979–986 (2014). [CrossRef]
25. A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2(1), 1–59 (2010). [CrossRef]
26. S. Haykin, Communication systems, 4th edition (John Wiley & Sons Ltd, 2004).
27. S. Koke, E. Benkler, A. Kuhl, and G. Grosche, “Validating frequency transfer via interferometric fiber links for optical clock comparisons,” New J. Phys. 23(9), 093024 (2021). [CrossRef]
28. P. Krehlik, Ł. Śliwczyński, Ł. Buczek, H. Schnatz, and J. Kronjäger, “Optical multiplexing of metrological time and frequency signals in a single 100 GHz-grid optical channel,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 68(6), 2303–2310 (2021). [CrossRef]
