1. Introduction

Ultra Long Haul Transmission Systems may suffer recurrent outages due to constantly varying Differential Group Delay (DGD) in installed cables. Current engineering rules governing a system’s test and deployment procedure rely heavily on the assumed DGD temporal statistics. Since the instantaneous DGD values are frequency dependent, it is often believed that scrambling of DGD spectra over time leads to frequency-independent statistics for any given channel. In other words, the DGD at any given frequency is believed to sample the same Maxwellian distribution with the same mean value τ_mean. The validity of this assumption rests on a model in which dozens to hundreds of birefringent fiber sections undergo random reorientations over timescales of interests.

Various research groups have reported slow DGD dynamics in installed routes [1,2]. We have, in addition, established the existence of an underlying slowly varying (weeks-to-months timescale) long-term structure to DGD spectra, which is modified (on short timescales) in a reversible way by influence of ambient temperature on few exposed portions of the cable [3,4]. Interestingly, our study of the DGD in installed ULH systems showed rather fast dynamics, which arose mainly from a limited number of isolated polarization rotators rather than from continuous changes distributed along the fiber length [5,6]. These rotators, which we call “hinges”, can either be exposed sections of fiber such as at bridge attachments or some system components, DCMs in particular [5,6].

In this letter we present a statistical analysis of the data taken on a field-deployed trial 40Gb/s 986km system [5–8]. We show that as DGD spectra changes over time, some spectral variations remain: we observed that some channels, on average, experience a mean DGD almost twice as high as others. That is, as the DGD for a given channel varies in time, it constitutes a distinct statistical distribution, whose mean value <τ_ch>_time, is channel-specific and can differ between channels by a factor of 2. Thus, these channel-dependent means, themselves, form another distribution. In order to quantify this spread among different channels we introduce a new statistical measure – the standard deviation for the distribution of the time averaged channel means: σ(<τ_ch>_time). To relate this to the conventional model, a large number of hinges makes σ(<τ_ch>_time) tend to zero: each channel has the fiber’s mean, i.e. τ_mean. We propose a simple model showing that the existence of only a finite number of hinges leads to a non-zero σ(<τ_ch>_time) which decreases approximately as the inverse of the square root of the number of sections between hinges. This prediction is supported by experimental evidence. In addition, the impact of correlations between the gyrations of the individual hinges is studied numerically.

2. Experimental setup

An ULH system was deployed between two major cities, and the details of the installation and PMD measurements were published elsewhere [5–9]. The two end terminals were placed in switching offices, and five repeaters were installed in small unmanned buildings. These buildings had temperature-controlled environments with thermostat hysteresis bands of about 1–1.5°C. Correspondingly, sporadically occurring small (1–1.5°C) and periodic (1–3 hours) temperature variations caused DCMs in these buildings to act as strong polarization rotators (with a scope of rotation ≥π on Poincare Sphere) [5]. These temperature oscillations differed somewhat among the buildings in both amplitude and frequency. Thus, our system had several intermittently active hinges with distinct time signatures.

3. Results

Figure 1 shows the experimental probability density functions of τ_ch for two example channels 186.65 THz (○) and 188.15 THz (▾). The data shown were collected in a three day period over 562km: the system was looped back at the third repeater, and consecutively passed through repeaters # 1-2-3-2-1, for a total of five. During these measurements only the building which housed repeaters 1 and 3 showed strong inside temperature oscillations: thus we had a system with effectively only three active hinges. Rapid reorientation of the hinges insured that we assembled a few hundred statistically independent samples during the measurement interval. The plot clearly demonstrates that the DGD at each frequency exhibits a distinct statistical distribution. The time averaged DGD of these two channels is <τ_ch>_time=1.8 ps and 1.0 ps for 186.65 THz and 188.15 THz, respectively.

 

Fig. 1. Probability density of observations (symbols); Maxwellian distribution with matched means (thin lines).

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Other channels (our experimental frequency range was 186.5–188.5 THz) had mean values between these two cases. All values of τ_ch are shown as a function of frequency in Fig. 2 (solid line). Figure 2 shows that our DGD spectra can be divided into several correlated spectral bands of ≈0.3 THz width. DGD values taken at frequencies further apart than this display distributions which may have noticeably different mean values <τ_ch>_time. We have also found that the standard deviation of the distribution sampled by each channel, σ_ch, is frequency dependent as well. A grey band between two dashed lines in Fig. 2 (<τ_ch>_time±σ_ch) is broader for higher mean values <τ_ch>_time.

 

Fig. 2. Time averaged DGD, <τ>_time, plotted as a function of frequency (solid). Grey band between two dashed lines indicates two standard deviation interval:<τ_ch>_time±σ_ch.

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We have observed similar behaviour in every set of DGD measurements with various system configurations, as long as there were temperature variations driving the hinges. When the temperature was stable, the measured DGD did not change appreciably. Figure 3 plots DGD values averaged over time <τ_ch>_time plotted as a function of frequency for five different system configurations (A, B, C, D, E). Configuration B denotes a folded link of 562 km total (data presented in Figs. 1 and 2), while all the rest correspond to different fiber sets over the same 986 km route. To achieve both low and high DGD values on the same route, two sets of fiber strands were chosen from the same conduit. The resulting PMD coefficients were 0.02ps/√km for configurations A and B, and 0.12ps/√km for configurations C, D, and E. Variations of mean DGD values <τ_ch>_time with optical frequency are evident for all five datasets. Note that while the DGD at any given time varies over optical spectrum, the average over time (shown in the Fig. 3) would, according to the usual model, be flat.

We surmise that our results can be explained within the framework of the hinge model: the PMD vector at each frequency is made of several fixed vectors (representing “dead” buried sections) connected by the active hinges. These fixed vectors are larger at some frequencies and smaller at others. For each frequency the resulting DGD is the magnitude of the vector sum of these randomly oriented vectors, each of fixed length. Then over the timescales for which the buried fiber can be considered “dead” [1–4], gyrating hinges produce a distribution of DGD values which, for each individual frequency, is similar to that of typical fixed-section PMD emulators. However, in contrast to typical emulators, frequency-dependent lengths of the individual sections result in frequency-dependent mean DGD values <τ_ch>_time.

 

Fig. 3. Time averaged DGD, <τ>_time, plotted as a function of frequency. Datasets A and B are taken on low DGD fiber; datasets C, D and E are taken on high DGD fiber.

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In fact, we devised the following procedure to estimate the standard deviation of the distribution of samples <τ_ch>_time over frequency, namely σ(<τ_ch>_time), as a function of the number of hinges. For each number of hinges N we created a set of 100,000 numerical “emulators”. Within this set, all emulators had the same fixed number of (N+1) different sections, but each section’s DGD was chosen by a random draw from the same Maxwellian distribution. Among 100,000 realizations of an emulator, each is a unique set of (N+1) different sections. It represents one channel or, more precisely, an independent frequency band in our experiment. Over time, we assume that all hinges are exercised, so that the mean DGD value <τ_ch>_time could be computed by using an analytical expression from [10] for each emulator realization. Then the normalized standard deviation for the 100,000 emulator realizations: ${\sigma }_{\mathit{norm}}=\frac{\sigma \left({〈\tau 〉}_{\mathit{time}}\right)}{{〈{〈\tau 〉}_{\mathit{time}}〉}_{\mathit{freq}}}$ was calculated numerically for each set of N-hinge emulators. Here, averaging in frequency was performed by ensemble averaging over the set of 100,000 emulator realizations. The estimates we obtained for this quantity, σ_norm, are plotted in Fig. 4 as black squares (▪). Also plotted as a dashed line a simple square root dependence for ${\sigma }_{\mathit{norm}}:{\sigma }_{\mathit{norm}}=\left(\sqrt{\frac{\mathit{3}\pi }{\mathit{8}}-\mathit{1}}\right)\frac{\mathit{1}}{\sqrt{N+\mathit{1}}}$. Naturally, for no hinges (a single-section emulator) the standard deviation σ_norm is equal to that of a Maxwellian distribution, and for an infinite number of hinges it asymptotically approaches zero.

We have also computed the same quantity σ_norm from the experimental data sets A, B, C, D, and E. The results are plotted vs. the putative number of hinges in Fig. 4 as open circles (○). We assume that the number of hinges was equal to the number of repeater sites with active thermal fluctuations, resulting in 3 hinges for B; 6 hinges for C and E; and 10 hinges for A and D configurations (two circles almost on top of each other in Fig. 4). While the field results show the same trend as the numerical estimates, the numbers derived from the experimental data are somewhat smaller than those from the numerical data. We attribute the discrepancy to either a limited experimental frequency range or an ambiguity in number of hinges: it is possible that some bridges along the route acted as hinges as well, which could cause us to underestimate the number of hinges.

The above simulation was based on PMD emulator statistics [10], which assumes freely rotating hinges uniformly covering a full 4π steradians. We have also used a conventional retarder plate model to study the effect of hinges which are not random, but rather have a periodic functional dependence on time and a limited scope of rotation. We simulated different routes with the same mean DGD (averaged over frequency). Each route consisted of (N+1) spans (200 sections in each span), connected by N hinges, where N ranged from 1 to 10. Hinges were modeled as Stokes-space rotators about fixed axes, whose angle of rotation α evolved as a predetermined function of time. We present cases, first with angle α_k=1.5π sin[2πft] for strongly correlated hinges and second, for more decorrelated hinges α_k=1.5π sin[2πf_kt+(k-1)π/8]2πt/500, where k is the hinge number and f_k was 1/47 (1/29) for even (odd) hinges.

 

Fig. 4. Normalized standard deviation (with frequency) of DGD averaged in time: σ_n(<τ>_time). Numerical simulation for correlated (Δ); decorrelated (▾); completely random (▪) hinges. Experimental data in open circles (○). Dashed line is $\propto \frac{\mathit{1}}{\sqrt{N+\mathit{1}}}$.

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We then calculated the DGD as a function of optical frequency and time, and again computed the normalized standard deviation σ_norm. The results are plotted in Fig. 4 as empty triangles (Δ) for the first case, correlated hinges, and filled triangles (▾) for the second case, decorrelated hinges. Interestingly, the statistics of a system with correlated hinges is almost independent of the number of hinges beyond N=2. As hinges become decorrelated the statistics of σ_norm approaches that for completely random ones. We thus infer that many deterministic but decorrelated hinges have the same effect as a smaller number of random ones. In a series of trials with different time dependence for α_k, we found that the two curves (Δ,▾) in Fig. 4 did not depend significantly on the exact functional form used for the respective α_k’s.

The fact that we observed channel-dependent DGD temporal statistics inevitably suggests a channel-dependent outage probability in real field-deployed systems. In fact, our hinge model can serve as a basis for a new analysis of outage probabilities. Recently, we numerically demonstrated that a significant number of channels in systems with hinges would be outage-free for long time periods, while a smaller fraction of channels would experience frequent outages [11,12].

4. Conclusions

The DGD statistics of a link with a finite number of hinges was shown to differ from conventional Maxwellian statistics. Based on statistical analysis of data taken on a field deployed telecom system, we demonstrated that the observed DGD values, measured over time on individual channels, comprise distinct statistical distributions, similar to the finite hinge model predictions. We introduced a new measure to characterize the variation of the mean DGD between various channels. We experimentally observed the dependence of these variations on the number of hinges and also verified this dependence in simulations based on the hinge model. Finally, we also discussed the effects of correlation between rotations of different hinges.