We show that the accumulation rate of nonlinearity in highly-dispersive long-haul coherent links can also be measured from the nonlinear threshold decrease rate, and provide simulations of such rates for both single- and cross-channel effects. We then show how the estimated rate can be used for the overall system design.
© 2012 OSA
A Gaussian nonlinear model (GNM) was recently introduced to justify the performance of dispersion uncompensated (DU) coherent links with orthogonal frequency division multiplexing signals [1,2] and with single-carrier signals [3–5]. According to that model, the nonlinear interference (NLI) is a signal-independent circular Gaussian noise, like the amplified spontaneous emission (ASE) noise. Thus performance just depends on the nonlinear signal to noise ratio (SNR)6], while a single-channel time-domain analytical model yielded a scaling law NNLI ∝ Nlog(kN) with k a system parameter . Since the variance NNLI of the NLI field scales as N1 if the span contributions fk are uncorrelated equal-variance RVs, or as N2 if the span contributions are the same RV, a pragmatic approximation to the distance scaling law assumes  8]. For the same system, a value 1 + ε ≅ 1.25 over the larger 5 < N < 50 can be read off the simulation results in . The decrease of ε in such systems when the range of N is extended is confirmed by the single-channel analysis and simulations in . Such WDM DU systems have a bandwidth fill-factor and the dominant nonlinear effect is single channel self-phase modulation . Recent experiments  performed on ultra-long DU SMF links with ∼50 km spans at variable η confirm that when η ≲ 0.6 the NLI accumulation exponent is 1 + ε ≅ 1.2 ÷ 1.6 (since single-channel nonlinear effects are dominant ), while at larger η (where cross-nonlinearity starts to dominate) the exponent decreases to 1 + ε ≅ 1.05. A recent simulation study on 28 Gbaud PDM-QPSK transmission over an Nx100 km SMF DU link  found that the NLI coefficient aNL can be approximated as the sum of single and cross-channel effects as aNL = αSPMN1.34 + αXPMN1.1 in the range 8 < N < 30, where αXPM increases for decreasing channel spacing Δf and eventually gives the dominant contribution as a unity fill-factor and thus the smallest spacing Δf = R is approached, thus indirectly confirming the above experimental findings.
In this paper, which is an extended version of , we want to add to the debate by taking a correlated but different point of view. We already showed by simulation how the nonlinear threshold (NLT) at 1dB of SNR penalty P̂1 (i.e., the channel power at which a target bit error rate BER0 = 10−3 is obtained with 1 dB extra SNR with respect to linear propagation ) scales with symbol rate R at fixed bandwidth fill-factor η and at fixed distance N [10, 12]. We now wish to check by simulation how P̂1 scales with distance N at a fixed symbol rate and fixed η. The reason is that, as we will see, the GNM theory for DU systems predicts that the plot of NLT versus N in a log-log scale decreases with slope −(1 + ε)/2 dB/dB. We wish to estimate the slope factor ε from our NLT simulations, and relate it to the ultimate system performance.
The paper is organized as follows. Sections 2-3 report on simulations of P̂1 versus distance (i.e., versus spans N) at a fixed symbol rate R = 28 Gbaud and Δf =50 GHz (η = 0.56) for a homogeneous WDM PDM-QPSK system over a Nx50 km SMF DU link, and we compare it to the one obtained with standard in-line dispersion management (DM). We use the nonlinear effects separation procedure  to estimate the slope at which the NLT due to individual self-and cross-nonlinear effects decreases at increasing N in both DU and DM systems. Section 4 finally provides a comprehensive view of the system implications of the 1dB NLT P̂1 and of the slope parameter ε.
2. Simulated DU and DM systems
Figure 1 shows the block diagram of the simulated DU and DM links. The transmitter consisted of 19 WDM non-return to zero (NRZ) 28 Gbaud PDM-QPSK channels, with Δf =50 GHz. All channels were modulated with 210 and 214 independent random symbols in the DM and DU cases, respectively. Each channel was filtered by a supergaussian filter of order 2 with bandwidth 0.9R. The state of polarization (SOP) of each carrier was randomized on the Poincaré sphere. The transmission line consisted of N spans of 50km of SMF (D =17 ps/nm/km, α =0.2 dB/km, γ =1.3 W−1km−1). Propagation used the vector split-step Fourier method (SSFM) with zero polarization mode dispersion and Manakov nonlinear step . In the DM case, an inline residual dispersion per span (RDPS) of 30 ps/nm and a straight-line rule precompensa-tion [16, 17]
The objective of the simulations was to estimate the 1 dB NLT P̂1 versus distance when nonlinearities (NL) are selectively activated . The NLT P̂1 is obtained from a series of BER Monte Carlo estimations (averaged over input polarization states) at increasing amplifiers noise figure until the target BER0 is obtained, as detailed in . In the DU case, noise was all loaded at the receiver since it is known that signal-noise nonlinear interactions are negligible at 28 Gbaud . In the DM case we calculated the NLT both with distributed noise and with noise loading. Using nonlinearity decoupling, we studied the following four cases: 1) single channel (label “SPM” in the following figures); 2) WDM with only scalar XPM  active (label “XPM”); 3) WDM with only cross-polarization modulation  (label “XPolM”) active; 4) WDM with all nonlinearities active (label “WDM”).
Simulations were run using the open-source software Optilux . Obtaining the curves presented in each subplot of the following Fig. 2 took approximately 3 weeks by running full time on an 8-core Dell workstation.
DU system. Figure 2(a) shows the corresponding NLT P̂1 versus distance, with span number N ranging from 5 to 320. We first remark that at all distances single-channel effects (SPM) dominate at this η = 0.56 bandwidth fill-factor . The effect of the comparable-size cross-nonlinearities XPM and XPolM on the overall NLT (WDM) is felt only at distances below 2000 km. The reason is that single-channel nonlinearity accumulates at a faster rate than cross-channel nonlinearity so that at longer distances it becomes largely dominant. Equation (18) predicts on this log-log plot a NLT decrease with a slope −(1 + ε)/2 dB/dB. From Fig. 2 (a) on the narrow-range up to 2000 km (40 spans) by least mean-square fitting data with straight lines we find εSPM ≅ 0.32, εXPM ≅ 0.22, εXPolM ≅ −0.08 and εWDM ≅ 0.16, while fitting on the wide-range (up to 320 spans) we measure εSPM ≅ 0.28, εXPM ≅ 0.08, εXPolM ≅ −0.08 and εWDM ≅ 0.22. We first note that the wide-range εWDM value is consistent with measurements in  at similar η, while the narrow-range εSPM is in agreement with . However, the most novel piece of information we learn from Fig. 2 is that the smaller slope of cross-nonlinearity observed in  is indeed an average of the XPolM and scalar XPM slopes, where scalar XPM seems to accumulate at a slightly faster rate. However, in DU links the rate difference is small. The negative measured εXPolM is ascribed to an insufficient pattern length, as discussed below.
DM system. It is instructive to also take a look at the NLT curves when the DU link is changed into a legacy DM link. Figure 2(b) shows both the unrealistic case of noise loading (solid lines) and the realistic case of ASE distributed at each amplifier, where nonlinear signal-ASE interactions are fully accounted for (dashed lines). We remark that pre-compensation is here changed at each value of N according to (3), hence the measured NLTs do not truly portray the noise accumulation with N in the same line. Figure 2(b) confirms that :
- scalar XPM plays a minor role in the PDM-QPSK constant intensity format, and is quite sensitive to signal-noise interactions, here manifested as nonlinear phase noise (NLPN);
- single channel (SPM) effects also play a minor role up to about 4000 km, and in that range are also quite sensitive to NLPN;
- the dominant nonlinearity is XPolM up to about 4000 km, but eventually SPM effects become dominant as they accumulate at a faster rate. XPolM is not impacted by signal-ASE interactions.
We note that the slope of SPM in the first 2000 km gives about εSPM ≅ 1, but such a slope decreases at larger distances, since the RDPS starts contributing enough cumulated dispersion to make the DM system look more similar to a DU system. Within the first ∼2000 km we also measure (with NLPN) εXPM ≅ 0.73 and εWDM ≅ 0.66. The fact that XPolM is less “resonant” than scalar XPM is seen in the lowering of the local XPolM slope after 2000 km. The intuitive reason of XPolM’s smaller slope is easily understood in a truly resonant DM map (i.e., with RDPS=0). In such a case the walkoff completely realigns the interfering pattern intensities, hence scalar XPM is identical at each span, i.e., it is truly resonant. Instead, the rotations induced by XPolM never bring back the final SOP to the same starting SOP at each span, hence XPolM is never truly resonant.
XPolM pattern dependence. The choice of the modulating pattern sequence length is of great importance in establishing the XPolM NLT. Note that in this study we did not use pseudo-random patterns, but completely random patterns which – at equal accuracy – allow the use of shorter pattern lengths and are thus more appropriate in simulations of DU systems [21, 22]. Figure 3 reports on the pattern length sensitivity of the XPolM NLT P̂1 vs. distance for the same DU and DM systems analyzed in Fig. 2. If the sequence length is too short, the XPolM P̂1 slope erroneously decreases at increasing distance. We found that a length of 210 symbols was sufficient for our DM systems, while for DU systems we used the longest 214 sequence length that allowed us to obtain the 1dB NLT curves in a feasible time, although still longer sequences may probably be needed in order to precisely estimate the XPolM NLT slope. Although the distinction between XPM and XPolM is fundamental in practice for DM systems, for DU systems it tends to get blurred, and since XPM and XPolM are correlated, in the end what matters is their joint variance, as recently studied in .
4. System implications of slope parameter and NLT
According to the GNM, the nonlinear SNR Eq.(1) uniquely determines the value of bit error rate or equivalently of Q-factor . Hence the surface of SNR versus power and spans contains all the needed information about global system performance. Figure 4 provides an example of the SNR surface in dB, SdB, versus power PdB and number of spans NdB (for any variable x, we will denote xdB = 10Log(x)). It is the objective of this section to summarize the analytical properties of such a surface and explain how the exponent ε and the P̂1 vs. N curves fit into such a global view. Section 4.1 describes the properties of the “vertical cuts” S vs. P at a fixed distance N, known as the “bell curves”. Section 4.2 describes the properties of the “horizontal cuts” P vs. N at a desired level S0, tackling the issue of maximum transmission distance. Note that according to the models in [6,7] the NLI coefficient aNL depends on the modulation format only through its symbol rate. Hence, by varying S0, one finds the performance of any zero-mean format at a given symbol rate.
4.1. Vertical cuts: the bell curves
We wish to summarize here the behavior of S versus P at a fixed distance N. More information can be found in [3,5]. We first notice from Eq. (1) two asymptotes, one at small power which is the linear SNR, and one at large power . The two asymptotes cross at the break-point power where ASE power equals nonlinear noise power. The linear asymptote has slope 1 dB/dB, while the nonlinear asymptote has slope −2 dB/dB. Figure 5 shows SNR versus P for fixed values of ASE power NA and nonlinearity aNL where the two asymptotes are indicated by dashed lines. Equation (1) can be rearranged asFig. 5.
4.1.1. Unconstrained nonlinear threshold
The unconstrained nonlinear threshold (NLT) is defined as the power PNLT that maximizes the bell curve. By setting we see that the NLT satisfies NA = 2(aNLP3), i.e., at the optimal power ASE noise is twice the nonlinear noise, and the SP is [3–5] and this is true for all GNM systems. Explicitly,
Since at NLT the penalty is 1.76 dB, it is interesting to ask at which power Py we reach a penalty of ydB. The answer is found by imposing SP = 10ydB/10 and rearranging Eq. (4): Py = PNLT · [2(10ydB/10 − 1)]1/3. For instance, is 0.95 dB below , as also marked in Fig. 5.
4.1.2. Constrained NLT
When instead NA > N̂A, the target S0 is unachievable. The penalty corresponding to the extremes Pm, PM is found as , since S0NA is the required minimal power to achieve S0 in the linear regime. Figure 3(b) in  shows a plot of SPm,M versus power. When Pm ≡ PM we know the SP with respect to S0 is 1.76 dB. For any other SP value ydB, it is also interesting to find the power P̂y (i.e., either Pm when ydB < 1.76, or PM when ydB > 1.76.) achieving SP = ydB. We call P̂y the constrained NLT at ydB of penalty. Defining7] that c(1) = 1.27, hence the constrained 1dB NLT P̂1 is 1.04 dB below P̂NLT (while we have seen that the unconstrained P1 is 0.95 dB below the unconstrained PNLT).
The whole c(ydB) curve can be obtained as follows. Let SP = 10ydB/10 and x = NA/N̂A ∈ [0, 1], and look for the solution of the nonlinear equation when ydB ≤ 1.76, and of when ydB > 1.76. Once x is found, calculate c(ydB) = 3/(2x · 10ydB/10). The result is plotted in Fig. 6. In blue we show the dB value of the curve c(ydB), which is zero at ydB = 1.76, while in green we show the curve x(ydB).
4.2. Horizontal cuts
ASE in dual polarization systems accumulates as NA = βN, where β = hνFGBRX depends on photon energy hν at signal frequency ν, amplifiers noise figure F, gain G, and two-sided receiver noise bandwidth BRX. We now assume for aNL a simplified distance scaling law as in Eq. (2) and want to check the impact of the exponent factor ε.
Figure 7 shows a set of horizontal cuts of the SNR surface S(N,P) at levels S0 spaced by 1 dB. In the figure we assumed ε = 0.22. The analytical shape of each contour is obtained fromEq. (10) are the powers PM (top branch up to the red circle) and Pm (bottom branch) whose analytical expression is given in Eq. (8). Each contour level has two asymptotes, obtained by suppressing either ASE or NLI in Eq. (10):
4.2.1. Unconstrained NLT
Consider the magenta-line locus of points of maximum reach (N0, P0) at each S0 level, marked with red circles in Fig. 7. Information on how (N0, P0) change with S0 is useful to understand the effect of the modulation format: at a given symbol rate, in general different formats will require different SNR S0 to reach a desired target Q-factor Q0 or BER value BER0. It is thus important to understand by how much maximum distance and optimal power change as the modulation format is changed. The points (N0, P0) are such that Pm ≡ PM, i.e., they correspond to a distance N0 where power P0 coincides with P̂NLT, the constrained NLT at S0. Therefore each red circle corresponds to the top of the “vertical cut” at N0. In other words, the magenta-line is the “crest” of the S surface, or equivalently the locus of the unconstrained NLT PNLT as N varies. As such, given N0, we have from Eq. (5), Eq. (6):
The first equation in Eq. (14) shows that the magenta line has a slope of . With the value for SPM-dominated DU systems of ε = 0.22 found by simulations in Fig. 2(a), this slope is −0.07 dB/dB, which is hardly noticeable in lab experiments, so that common wisdom has that in DU systems the unconstrained NLT does not depend on distance, as experimentally verified by many authors, see e.g. [11, 24].
The crest is obtained when ASE power is 2 times the NLI power. However, one can similarly prove that the locus of points where ASE power is k times the NLI power at coordinate Nk satisfiesFig. 4 is composed of parallel straight-lines.
4.2.2. Constrained NLT
Since the higher asymptote Eq. (13) can be also expressed as , then it is clear that the dB-dB plot of P̂NLT versus N is a straight line parallel to PH but lowered by , as shown by the magenta dash-dotted line in Fig. 8. Similarly, from Eq. (9) and using Eq. (16) we also haveFig. 6. A similar scaling law was already reported in  for DU systems with ε = 0. Hence for instance P̂1 is a straight line parallel to P̂NLT but lowered by ∼ 1.04 dB as reported in blue solid line in Fig. 8. The parameters NLT1 and ε used in the figure were obtained by straight-line fitting the simulated P̂1 vs. N “WDM” curve in Fig. 2(a).
Note that Eq. (16) can be rearranged as
The left-hand side is often called the integrated NLT, and its slope versus N was measured in (, Fig. 1) to be ∼0.31 dB/dB for a 50 GHz spaced 28 Gbaud PDM-QPSK system over an Nx100 km DU SMF link, consistently with the experimentally estimated value ε ∼ 0.37 .
Figure 8 shows that the magenta dash-dotted line is also the locus of maximum-reach points of coordinates (P0, N0) as the β factor (i.e., amplifiers noise figure for instance) is varied. This was already proven in , and it makes sense, since the P̂1 is obtained in simulations by varying the received ASE until 1 dB of penalty is measured. Similarly, the locus of maximum distance points (P0, N0) as the αNL factor is varied with β held constant is a straight line parallel to the linear asymptote and 1.76 dB above it, since from Eq. (14) we have . Figure 4 in  confirms this fact. The explicit expression of the constrained maximum distance N0 is obtained from the two relations in Eq. (14) as
If we look instead at the coordinates of the point where the asymptotes cross, we find and . Hence if we judge the maximum distance from the asymptotes we over-estimate the true N0 by 2.76/(1 + ε/3) dB and the true P0 by (3 − 1.76ε)/(3 + ε) dB, as seen in Fig. 8.Eq. (22) to conclude that in Fig. 8 when β is doubled (+3dB), the optimal power P0 is increased by dB (1.13 dB in figure), while maximum distance N0 is decreased by dB (1.86 dB in figure). The relations in the last line tell us that a 1dB increase in αNL brings about a modest decrease by less than 1/3 dB in both N0  and P0. It is the same decrease for both N0 and P0, since the optimal (N0, P0) in this case slides along a 1 dB/dB straight line parallel to the lower asymptote, as we already remarked.
ii) Power budget. It is simple to verify from the asymptotes that
Hence the maximum extra distance predicted from the asymptotes only depends on the “power budget” PH(N)/PL(N), corresponding to the vertical dB spread of the contour at S0 between the asymptotes at N. Similarly, Eq. (20) tells us that maximum reach N0 at S0 only depends on the power budget , i.e., the vertical spread of the contour at S0 at N = 1 span from 1.76 dB above the lower linear asymptote PL(1) = LT1 up to the magenta dash-dotted line P̂NLT (1) = NLT1. Also, by using the second form of Eq. (20) and Eq. (19) one can estimate the extra distance ratio N0/N from the measured value of the constrained NLT P̂NLT (N) at S0 at given distance N as
Finally, by checking the constrained NLT ratio of two different formats A and B at a common BER0 (hence at possibly different S0,A, S0,B) one can estimate by how much the best format extends the maximum reach from:
We have shown that in dispersion uncompensated systems where the nonlinear Gaussian assumption holds, the nonlinear interference accumulation rate 1 + ε can also be measured through the nonlinear threshold decrease rate with distance, and we have provided the accumulation rates of the individual self- and cross-channel nonlinear effects, thus corroborating and complementing recent simulation and lab results [8, 11, 13]. We have then shown how the estimated value ε (when all nonlinear effects are taken into account) as well as the 1dB nonlinear threshold can be used to predict the ultimate transmission performance.
The first author gladly acknowledges fruitful discussions with O. Rival, M. Salsi, F. Vacondio, E. Grellier, S. Bigo, and G. Charlet of Alcatel-Lucent.
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