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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 26 — Dec. 10, 2012
  • pp: B204–B216
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On the nonlinear threshold versus distance in long-haul highly-dispersive coherent systems

A. Bononi, N. Rossi, and P. Serena  »View Author Affiliations


Optics Express, Vol. 20, Issue 26, pp. B204-B216 (2012)
http://dx.doi.org/10.1364/OE.20.00B204


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Abstract

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

1. Introduction

In this paper, which is an extended version of [14

14. A. Bononi, N. Rossi, and P. Serena, “Nonlinear threshold decrease with distance in 112 Gb/s PDM-QPSK coherent systems,” in Proc. ECOC 2012 (Amsterdam, The Netherlands, 2012). Paper We.2.C.4.

], 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 1 (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 [15

15. J. C. Antona and S. Bigo, “Physical design and performance estimation of heterogeneous optical transmission systems,” C. R. Physique 9, 963–984 (2008). [CrossRef]

]) scales with symbol rate R at fixed bandwidth fill-factor η and at fixed distance N [10

10. A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper Th10E1.

, 12

12. A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. OFC 2011 (Los Angeles, CA, 2011). Paper OWO7.

]. We now wish to check by simulation how 1 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 1 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 [10

10. A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper Th10E1.

] 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 1 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 [10

10. A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper Th10E1.

]. In the DM case, an inline residual dispersion per span (RDPS) of 30 ps/nm and a straight-line rule precompensa-tion [16

16. Y. Frignac, J.-C. Antona, and S. Bigo, “Enhanced analytical engineering rule for fast optimization of dispersion map in 40 Gbit/s-based transmission systems,” in Proc. OFC 2004 (Los Angeles, CA, 2004). Paper TuN3.

, 17

17. A. Bononi, P. Serena, and A. Orlandini, “A unified design framework for single-channel dispersion-managed terrestrial systems,” J. Lightwave Technol. 26(22), 3617–3631 (2008). [CrossRef]

]
Dpre=DαN12RDPS
(3)
were used. In the coherent receiver we neglected laser phase noise and frequency offset. The standard digital signal processor consisted of a chromatic dispersion compensation block, of a data-aided polarization demultiplexer, and a 27 taps Viterbi and Viterbi phase estimator.

Fig. 1 Block diagrams of DU and DM simulated systems. TX block represents 19 WDM channels spaced by Δf = 50GHz, and NRZ-PDM-QPSK modulated at R = 28 Gbaud. In DM we use a residual dispersion per span of 30 ps/nm and distance-dependent pre-compensation (3). Propagation uses the SSFM with Manakov nonlinear step.

The objective of the simulations was to estimate the 1 dB NLT 1 versus distance when nonlinearities (NL) are selectively activated [10

10. A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper Th10E1.

]. The NLT 1 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 [18

18. A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal-noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol. 16(2), 73–85 (2010). [CrossRef]

]. 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 [18

18. A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal-noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol. 16(2), 73–85 (2010). [CrossRef]

]. 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 [19

19. M. Winter, C.-A. Bunge, D. Setti, and K. Petermann, “A statistical treatment of cross-polarization modulation in DWDM systems,” J. Lightwave Technol. 27(17), 3739–3751 (2009). [CrossRef]

] active (label “XPM”); 3) WDM with only cross-polarization modulation [19

19. M. Winter, C.-A. Bunge, D. Setti, and K. Petermann, “A statistical treatment of cross-polarization modulation in DWDM systems,” J. Lightwave Technol. 27(17), 3739–3751 (2009). [CrossRef]

] (label “XPolM”) active; 4) WDM with all nonlinearities active (label “WDM”).

Simulations were run using the open-source software Optilux [20

20.. “Optilux Toolbox,” [Online]. Available http://www.optilux.sourceforge.net

]. 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.

Fig. 2 1 dB NLT 1 at reference BER0 = 10−3 vs distance for 19-channel homogeneous 28 Gbaud NRZ PDM-QPSK system with Δf = 50 GHz over an Nx50 km SMF link. No PMD. (a) DU; (b) DM with 30 ps/nm RDPS and distance-dependent SLR precompensation (3); Solid lines: ASE noise loading at RX. Dashed lines: distributed ASE.

3. Results

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

21. J.-C. Antona, E. Grellier, A. Bononi, S. Petitreaud, and S. Bigo, “Revisiting binary sequence length requirements for the accurate emulation of highly dispersive transmission systems,” in Proc. ECOC 2008 (Brussels, Belgium, 2008). Paper We.1.E.3.

, 22

22. E. Grellier, J.-C. Antona, A. Bononi, and S. Bigo, “Revisiting binary sequence length requirements to accurately emulate optical transmission systems in highly dispersive regime,” SPIE 7136, 713613 (2008). [CrossRef]

]. Figure 3 reports on the pattern length sensitivity of the XPolM NLT 1 vs. distance for the same DU and DM systems analyzed in Fig. 2. If the sequence length is too short, the XPolM 1 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 [6

6. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-Linear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30(10), 1524–1539 (2012). [CrossRef]

].

Fig. 3 XPolM NLT 1 vs. distance for same 19-channel WDM DU (a) and DM (b) systems as in Fig. 2, for several values of the number of random symbols in the SSFM.

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 [5

5. E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express 19(13), 12781–12788 (2011). [CrossRef] [PubMed]

]. 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 1 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

6. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-Linear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30(10), 1524–1539 (2012). [CrossRef]

,7

7. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express 20(7), 7777–7791 (2012). [CrossRef] [PubMed]

] 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.

Fig. 4 Qualitative example of 2-dimensional SNR surface [dB] versus both power P [dBm] and number of spans N (logarithmic scale).

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

3. G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance prediction for WDM PM-QPSK transmission over uncompensated links,” in Proc. OFC 2011 (Los Angeles, CA, 2011). Paper OThO7.

,5

5. E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express 19(13), 12781–12788 (2011). [CrossRef] [PubMed]

]. We first notice from Eq. (1) two asymptotes, one at small power SLPNA which is the linear SNR, and one at large power SR1aNLP2. The two asymptotes cross at the break-point power PB=(NAaNL)13 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 as
SP=SLS=1+aNLP3NA
(4)
where SP is the sensitivity penalty in linear units, and in dB it is the distance of the SNR curve from the linear asymptote SL, as seen in Fig. 5.

Fig. 5 SNR versus P at distance N. At 1dB above NLT, ASE has same power as NLI.

4.1.1. Unconstrained nonlinear threshold

The unconstrained nonlinear threshold (NLT) is defined as the power PNLT that maximizes the bell curve. By setting dSdP=0 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 (32)dB~1.76dB [3

3. G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance prediction for WDM PM-QPSK transmission over uncompensated links,” in Proc. OFC 2011 (Los Angeles, CA, 2011). Paper OThO7.

5

5. E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express 19(13), 12781–12788 (2011). [CrossRef] [PubMed]

] and this is true for all GNM systems. Explicitly,
PNLT=(NA2aNL)13
(5)
and this is exactly 1 dB below the break-point PB. The corresponding maximum SNR is:
SNLT=(33aNL(NA2)2)13PNLT32NA.
(6)

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, P1dB is 0.95 dB below PNLTdB, as also marked in Fig. 5.

4.1.2. Constrained NLT

Given a desired SNR level S0, it was shown in ([7

7. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express 20(7), 7777–7791 (2012). [CrossRef] [PubMed]

] Appendix 1 and Fig. 3) that the bell curve reaches its top level at value S0 when ASE power is N^A=2(3S0)3/2aNL1/2 and signal power is
P^NLT=32S0N^A=13S0aNL
(7)
which we call the constrained NLT at S0. When NA < A, the SNR S exceeds S0 over the whole range Pm < P < PM, where
PM=3S0N^Acos(arcos(NA/N^A)3)Pm=3S0N^Acos(2πarcos(NA/N^A)3).
(8)

When instead NA > A, the target S0 is unachievable. The penalty corresponding to the extremes Pm, PM is found as SPm,M=Pm,MNAS0, since S0NA is the required minimal power to achieve S0 in the linear regime. Figure 3(b) in [7

7. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express 20(7), 7777–7791 (2012). [CrossRef] [PubMed]

] shows a plot of SPm,M versus power. When PmPM 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 y (i.e., either Pm when ydB < 1.76, or PM when ydB > 1.76.) achieving SP = ydB. We call y the constrained NLT at ydB of penalty. Defining
c(ydB)P^NLT/P^y
(9)
we already showed in [7

7. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express 20(7), 7777–7791 (2012). [CrossRef] [PubMed]

] that c(1) = 1.27, hence the constrained 1dB NLT 1 is 1.04 dB below NLT (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/A ∈ [0, 1], and look for the solution of the nonlinear equation SP=3xcos(2πarcos(x)3) when ydB ≤ 1.76, and of SP=3xcos(arcos(x)3) 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).

Fig. 6 cNLT/y [dB] (left axis) and x = NA/A (right axis) versus penalty ydB.

4.2. Horizontal cuts

ASE in dual polarization systems accumulates as NA = βN, where β = hνFGBRX depends on photon energy 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 from
S0=PβN+αNLN1+εP3
(10)
which yields the explicit contour equation P versus N. The two solutions of the cubic Eq. (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):
  1. a lower (or linear) asymptote which in dB is a straight line with positive slope 1dB/dB:
    PL=LT1N
    (11)
    where we defined the “linear threshold”
    LT1=βS0
    (12)
    as the power achieving S0 at 1 span in the linear regime;
  2. a higher asymptote:
    PH=(S0aNL)1/2
    (13)
    which, in the simplifying assumption Eq. (2), in dB is a straight line with negative slope −(1 + ε)/2. Only the asymptotes for the lowest contour level are marked in Fig. 7.

Fig. 7 Set of horizontal cuts of SNR surface S(N,P) at levels S0 = [10.12 : 1 : 15.12] dB, for a DU Nx50km SMF link at 28Gbaud. Parameters: αNL = 3.95 · 10−4 [mW−2], ε = 0.22, F = 13 dB, BRX = 32.5 GHz. Locus of maxima (P0, N0) for varying S0 shown as dash-dotted magenta line with slope −ε/3 ≅ −0.07 dB/dB.

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 PmPM, i.e., they correspond to a distance N0 where power P0 coincides with NLT, 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):
P0=(β2αNLN0ε)13;S0=(33αNLN03+ε(β2)2)13P032βN0.
(14)

The first equation in Eq. (14) shows that the magenta line has a slope of ε3dB/dB [11

11. O. V. Sinkin, J.-X. Cai, D. G. Foursa, H. Zhang, A. N. Pilipetskii, G. Mohs, and Neal S. Bergano, “Scaling of nonlinear impairments in dispersion-uncompensated long-Haul transmission,” in Proc. OFC 2012 (Los Angeles, CA, 2012). Paper OTu1A.

]. 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

11. O. V. Sinkin, J.-X. Cai, D. G. Foursa, H. Zhang, A. N. Pilipetskii, G. Mohs, and Neal S. Bergano, “Scaling of nonlinear impairments in dispersion-uncompensated long-Haul transmission,” in Proc. OFC 2012 (Los Angeles, CA, 2012). Paper OTu1A.

, 24

24. V. A. J. M. Sleiffer, M. S. Alfiad, D. van den Borne, S. L. Jansen, M. Kuschnerov, S. Adhikari, and H. De Waardt, “A comparison of 43-Gb/s POLMUX-RZ-DPSK and POLMUX-RZ-DQPSK modulation for long-haul transmission systems,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper Mo.2.C.4.

].

The second equation in Eq. (14) shows that the “crest” of surface SdB versus N0 is a straight line with slope 3+ε3~1dB/dB as for linear systems, as experimentally verified in [25

25. M. Salsi, C. Koebele, P. Tran, H. Mardoyan, S. Bigo, and G. Charlet, “80×100-Gbit/s transmission over 9,000km using erbium-doped fibre repeaters only,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper We.7.C.3.

].

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 satisfies
Pk=(β2αNLNkε)13;S=Pkk+1kNA=((k+1)3αNLNk3+ε(βk)2)13
(15)
which proves that the whole SdB surface sketched in Fig. 4 is composed of parallel straight-lines.

4.2.2. Constrained NLT

Using Eq. (2) in Eq. (7) we have
P^NLT=NLT1N1+ε2
(16)
where we defined the effective 1-span constrained NLT power as
NLT1(3S0αNL)12.
(17)

Since the higher asymptote Eq. (13) can be also expressed as PH=3NLT1N1+ε2, then it is clear that the dB-dB plot of NLT versus N is a straight line parallel to PH but lowered by 3~2.38dB, as shown by the magenta dash-dotted line in Fig. 8. Similarly, from Eq. (9) and using Eq. (16) we also have
P^y=NLT1c(ydB)N1+ε2
(18)
where c(ydB) is shown in Fig. 6. A similar scaling law was already reported in [23

23. Y. Ye, L. N. Binh, E. Zhou, T. Wu, S. Zhang, and X. Xu, “A simple criterion for performance estimation of 112Gb/s PDM-QPSK WDM system over uncompensated links,” in Proc. OFC 2012 (Los Angeles, CA, 2012). Paper JW2A.45.

] for DU systems with ε = 0. Hence for instance 1 is a straight line parallel to NLT 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 1 vs. N “WDM” curve in Fig. 2(a).

Fig. 8 Horizontal cut of SNR surface S(N, P) at level S0 = 10.12 dB (BER0=10−3 for the PDM-QPSK receiver used in simulations) for DU Nx50km SMF link at 28 Gbaud, for 2 values of ASE factor β spaced by 3dB. Parameters: αNL = 3.95 · 10−4 [mW−2], ε = 0.22 chosen such that the blue line 1 is the least-mean-squares fit of the simulated “WDM” 1dB NLT (at the same S0 = 10.12 dB) in Fig. 2(a). Other parameters F = [13, 16] dB, BRX = 32.5 GHz. Upper and lower asymptotes PH and PL shown as dashed black lines. Locus of maxima (P0, N0) for varying β (i.e. NLT) shown as dash-dotted magenta line parallel to PH, shifted downwards by 10Log3=2.38dB (cfr Eq. (17)). The 1dB NLT 1 (blue, solid) is parallel to NLT, and located c =1.04 dB below (c values at other penalties are found in Fig. 6). Maxima are 10Log(3/2) = 1.76 dB from lower asymptote (green arrows). At N = 1, have PLLT1 and NLTNLT1.

Note that Eq. (16) can be rearranged as
NP^NLT(N)=NLT1N1ε2.
(19)

The left-hand side is often called the integrated NLT, and its slope versus N was measured in ([26

26. E. Grellier, J.-C. Antona, and S. Bigo, “Revisiting the evaluation of non-linear propagation impairments in highly dispersive systems,” in Proc. ECOC 2009 (Vienna, Austria, 2009). Paper 10.4.2.

], 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 [8

8. F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express 20(2), 1022–1032 (2012). [CrossRef] [PubMed]

].

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 [5

5. E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express 19(13), 12781–12788 (2011). [CrossRef] [PubMed]

], and it makes sense, since the 1 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 P0=32βS0N032PL. Figure 4 in [27

27. V. Curri, P. Poggiolini, G. Bosco, A. Carena, and F. Forghieri, “Performance evaluation of long-haul 100 Gb/s PM-QPSK transmission over different fiber types,” IEEE Photon. Technol. Lett. 22(19), 1446–1448 (2010). [CrossRef]

] confirms this fact. The explicit expression of the constrained maximum distance N0 is obtained from the two relations in Eq. (14) as
N0=((3S0)3αNL(β2)2)13+ε(NLT132LT1)23+ε
(20)
and its associated “optimal” power as
P0=(β2)1+ε3+ε(3S0)ε3+εαNL13+ε(NLT1)23+ε(32LT1)1+ε3+ε.
(21)

If we look instead at the coordinates (N0A,P0A) of the point where the asymptotes cross, we find N0A/N0=(32/3)23+ε and P0A/P0=(2(2/3)ε)13+ε. 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.

Equations (20),(21) have two important consequences:

i) Sensitivity. By differentiating Eq. (20), Eq. (21) we immediately get
{N0dBS0dB=33+εN0dBβdB=23+εN0dBαNLdB=13+ε{P0dBS0dB=ε3+εP0dBβdB=1+ε3+εP0dBαNLdB=13+ε
(22)
i.e., the sensitivity to changes in S0 (e.g. modulation format or coding), β (e.g. amplifiers noise figure) and αNL (e.g. fiber parameters such as nonlinearity and dispersion). Such changes need not be small, since the surface is made of straight lines. For instance, from the top left relation we see that if a more powerful code lowers the SNR S0 by 1dB (without appreciably increasing the symbol rate), maximum distance is extended by 33+ε dB and the most beneficial case is ε = 0. The top right relation tells us that P0 is very weakly dependent on SNR S0, hence on modulation format. We can also use the middle relations Eq. (22) to conclude that in Fig. 8 when β is doubled (+3dB), the optimal power P0 is increased by 3+3ε3+ε dB (1.13 dB in figure), while maximum distance N0 is decreased by 63+ε 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 [28

28. P. Poggiolini, “The GN model of non-Linear propagation in uncompensated coherent optical systems,” J. Light-wave Technol. (2012), Early Access. [CrossRef]

] 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
N0AN=(PH(N)PL(N))23+ε.
(23)

Hence the maximum extra distance N0A/N 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 NLT132LT1, 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 NLT (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 NLT (N) at S0 at given distance N as
N0N=(P^NLT(N)32PL(N))23+ε
(24)
which only depends on the local power budget P^NLT(N)/(32PL(N)).

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:
N0,AN0,B=((P^NLT,A(N)P^NLT,B(N))/(LT1,ALT1,B))23+ε
(25)
where LT1,ALT1,B is the ratio of sensitivities (at BER0) of the two formats in the linear regime. Clearly, a format A that has x dB larger NLT than B, but also x dB worse sensitivity than B, achieves the same maximum distance.

5. Conclusions

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

8. F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express 20(2), 1022–1032 (2012). [CrossRef] [PubMed]

, 11

11. O. V. Sinkin, J.-X. Cai, D. G. Foursa, H. Zhang, A. N. Pilipetskii, G. Mohs, and Neal S. Bergano, “Scaling of nonlinear impairments in dispersion-uncompensated long-Haul transmission,” in Proc. OFC 2012 (Los Angeles, CA, 2012). Paper OTu1A.

, 13

13. O. Rival and K. Mheidly, “Accumulation rate of inter and intra-channel nonlinear distortions in uncompensated 100G PDM-QPSK systems,” in Proc. OFC 2012 (Los Angeles, CA, 2012). Paper JW2A.52.

]. 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.

Acknowledgments

The first author gladly acknowledges fruitful discussions with O. Rival, M. Salsi, F. Vacondio, E. Grellier, S. Bigo, and G. Charlet of Alcatel-Lucent.

References and links

1.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. S. Pak, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 4228–4236, (2008). [CrossRef] [PubMed]

2.

X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express 18(18), 19039–19054 (2010). [CrossRef] [PubMed]

3.

G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance prediction for WDM PM-QPSK transmission over uncompensated links,” in Proc. OFC 2011 (Los Angeles, CA, 2011). Paper OThO7.

4.

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23(11), 742–744 (2011). [CrossRef]

5.

E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express 19(13), 12781–12788 (2011). [CrossRef] [PubMed]

6.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-Linear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30(10), 1524–1539 (2012). [CrossRef]

7.

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express 20(7), 7777–7791 (2012). [CrossRef] [PubMed]

8.

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express 20(2), 1022–1032 (2012). [CrossRef] [PubMed]

9.

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. Tapia Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper P4.07.

10.

A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper Th10E1.

11.

O. V. Sinkin, J.-X. Cai, D. G. Foursa, H. Zhang, A. N. Pilipetskii, G. Mohs, and Neal S. Bergano, “Scaling of nonlinear impairments in dispersion-uncompensated long-Haul transmission,” in Proc. OFC 2012 (Los Angeles, CA, 2012). Paper OTu1A.

12.

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. OFC 2011 (Los Angeles, CA, 2011). Paper OWO7.

13.

O. Rival and K. Mheidly, “Accumulation rate of inter and intra-channel nonlinear distortions in uncompensated 100G PDM-QPSK systems,” in Proc. OFC 2012 (Los Angeles, CA, 2012). Paper JW2A.52.

14.

A. Bononi, N. Rossi, and P. Serena, “Nonlinear threshold decrease with distance in 112 Gb/s PDM-QPSK coherent systems,” in Proc. ECOC 2012 (Amsterdam, The Netherlands, 2012). Paper We.2.C.4.

15.

J. C. Antona and S. Bigo, “Physical design and performance estimation of heterogeneous optical transmission systems,” C. R. Physique 9, 963–984 (2008). [CrossRef]

16.

Y. Frignac, J.-C. Antona, and S. Bigo, “Enhanced analytical engineering rule for fast optimization of dispersion map in 40 Gbit/s-based transmission systems,” in Proc. OFC 2004 (Los Angeles, CA, 2004). Paper TuN3.

17.

A. Bononi, P. Serena, and A. Orlandini, “A unified design framework for single-channel dispersion-managed terrestrial systems,” J. Lightwave Technol. 26(22), 3617–3631 (2008). [CrossRef]

18.

A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal-noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol. 16(2), 73–85 (2010). [CrossRef]

19.

M. Winter, C.-A. Bunge, D. Setti, and K. Petermann, “A statistical treatment of cross-polarization modulation in DWDM systems,” J. Lightwave Technol. 27(17), 3739–3751 (2009). [CrossRef]

20..

“Optilux Toolbox,” [Online]. Available http://www.optilux.sourceforge.net

21.

J.-C. Antona, E. Grellier, A. Bononi, S. Petitreaud, and S. Bigo, “Revisiting binary sequence length requirements for the accurate emulation of highly dispersive transmission systems,” in Proc. ECOC 2008 (Brussels, Belgium, 2008). Paper We.1.E.3.

22.

E. Grellier, J.-C. Antona, A. Bononi, and S. Bigo, “Revisiting binary sequence length requirements to accurately emulate optical transmission systems in highly dispersive regime,” SPIE 7136, 713613 (2008). [CrossRef]

23.

Y. Ye, L. N. Binh, E. Zhou, T. Wu, S. Zhang, and X. Xu, “A simple criterion for performance estimation of 112Gb/s PDM-QPSK WDM system over uncompensated links,” in Proc. OFC 2012 (Los Angeles, CA, 2012). Paper JW2A.45.

24.

V. A. J. M. Sleiffer, M. S. Alfiad, D. van den Borne, S. L. Jansen, M. Kuschnerov, S. Adhikari, and H. De Waardt, “A comparison of 43-Gb/s POLMUX-RZ-DPSK and POLMUX-RZ-DQPSK modulation for long-haul transmission systems,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper Mo.2.C.4.

25.

M. Salsi, C. Koebele, P. Tran, H. Mardoyan, S. Bigo, and G. Charlet, “80×100-Gbit/s transmission over 9,000km using erbium-doped fibre repeaters only,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper We.7.C.3.

26.

E. Grellier, J.-C. Antona, and S. Bigo, “Revisiting the evaluation of non-linear propagation impairments in highly dispersive systems,” in Proc. ECOC 2009 (Vienna, Austria, 2009). Paper 10.4.2.

27.

V. Curri, P. Poggiolini, G. Bosco, A. Carena, and F. Forghieri, “Performance evaluation of long-haul 100 Gb/s PM-QPSK transmission over different fiber types,” IEEE Photon. Technol. Lett. 22(19), 1446–1448 (2010). [CrossRef]

28.

P. Poggiolini, “The GN model of non-Linear propagation in uncompensated coherent optical systems,” J. Light-wave Technol. (2012), Early Access. [CrossRef]

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

ToC Category:
Transmission Systems and Network Elements

History
Original Manuscript: September 26, 2012
Revised Manuscript: November 6, 2012
Manuscript Accepted: November 6, 2012
Published: November 29, 2012

Virtual Issues
European Conference on Optical Communication 2012 (2012) Optics Express

Citation
A. Bononi, N. Rossi, and P. Serena, "On the nonlinear threshold versus distance in long-haul highly-dispersive coherent systems," Opt. Express 20, B204-B216 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-26-B204


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References

  1. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. S. Pak, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16(20), 4228–4236, (2008). [CrossRef] [PubMed]
  2. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010). [CrossRef] [PubMed]
  3. G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance prediction for WDM PM-QPSK transmission over uncompensated links,” in Proc. OFC 2011 (Los Angeles, CA, 2011). Paper OThO7.
  4. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011). [CrossRef]
  5. E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express19(13), 12781–12788 (2011). [CrossRef] [PubMed]
  6. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-Linear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol.30(10), 1524–1539 (2012). [CrossRef]
  7. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express20(7), 7777–7791 (2012). [CrossRef] [PubMed]
  8. F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express20(2), 1022–1032 (2012). [CrossRef] [PubMed]
  9. A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. Tapia Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper P4.07.
  10. A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper Th10E1.
  11. O. V. Sinkin, J.-X. Cai, D. G. Foursa, H. Zhang, A. N. Pilipetskii, G. Mohs, and Neal S. Bergano, “Scaling of nonlinear impairments in dispersion-uncompensated long-Haul transmission,” in Proc. OFC 2012 (Los Angeles, CA, 2012). Paper OTu1A.
  12. A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. OFC 2011 (Los Angeles, CA, 2011). Paper OWO7.
  13. O. Rival and K. Mheidly, “Accumulation rate of inter and intra-channel nonlinear distortions in uncompensated 100G PDM-QPSK systems,” in Proc. OFC 2012 (Los Angeles, CA, 2012). Paper JW2A.52.
  14. A. Bononi, N. Rossi, and P. Serena, “Nonlinear threshold decrease with distance in 112 Gb/s PDM-QPSK coherent systems,” in Proc. ECOC 2012 (Amsterdam, The Netherlands, 2012). Paper We.2.C.4.
  15. J. C. Antona and S. Bigo, “Physical design and performance estimation of heterogeneous optical transmission systems,” C. R. Physique9, 963–984 (2008). [CrossRef]
  16. Y. Frignac, J.-C. Antona, and S. Bigo, “Enhanced analytical engineering rule for fast optimization of dispersion map in 40 Gbit/s-based transmission systems,” in Proc. OFC 2004 (Los Angeles, CA, 2004). Paper TuN3.
  17. A. Bononi, P. Serena, and A. Orlandini, “A unified design framework for single-channel dispersion-managed terrestrial systems,” J. Lightwave Technol.26(22), 3617–3631 (2008). [CrossRef]
  18. A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal-noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol.16(2), 73–85 (2010). [CrossRef]
  19. M. Winter, C.-A. Bunge, D. Setti, and K. Petermann, “A statistical treatment of cross-polarization modulation in DWDM systems,” J. Lightwave Technol.27(17), 3739–3751 (2009). [CrossRef]
  20. “Optilux Toolbox,” [Online]. Available http://www.optilux.sourceforge.net
  21. J.-C. Antona, E. Grellier, A. Bononi, S. Petitreaud, and S. Bigo, “Revisiting binary sequence length requirements for the accurate emulation of highly dispersive transmission systems,” in Proc. ECOC 2008 (Brussels, Belgium, 2008). Paper We.1.E.3.
  22. E. Grellier, J.-C. Antona, A. Bononi, and S. Bigo, “Revisiting binary sequence length requirements to accurately emulate optical transmission systems in highly dispersive regime,” SPIE7136, 713613 (2008). [CrossRef]
  23. Y. Ye, L. N. Binh, E. Zhou, T. Wu, S. Zhang, and X. Xu, “A simple criterion for performance estimation of 112Gb/s PDM-QPSK WDM system over uncompensated links,” in Proc. OFC 2012 (Los Angeles, CA, 2012). Paper JW2A.45.
  24. V. A. J. M. Sleiffer, M. S. Alfiad, D. van den Borne, S. L. Jansen, M. Kuschnerov, S. Adhikari, and H. De Waardt, “A comparison of 43-Gb/s POLMUX-RZ-DPSK and POLMUX-RZ-DQPSK modulation for long-haul transmission systems,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper Mo.2.C.4.
  25. M. Salsi, C. Koebele, P. Tran, H. Mardoyan, S. Bigo, and G. Charlet, “80×100-Gbit/s transmission over 9,000km using erbium-doped fibre repeaters only,” in Proc. ECOC 2010 (Turin, Italy, 2010). Paper We.7.C.3.
  26. E. Grellier, J.-C. Antona, and S. Bigo, “Revisiting the evaluation of non-linear propagation impairments in highly dispersive systems,” in Proc. ECOC 2009 (Vienna, Austria, 2009). Paper 10.4.2.
  27. V. Curri, P. Poggiolini, G. Bosco, A. Carena, and F. Forghieri, “Performance evaluation of long-haul 100 Gb/s PM-QPSK transmission over different fiber types,” IEEE Photon. Technol. Lett.22(19), 1446–1448 (2010). [CrossRef]
  28. P. Poggiolini, “The GN model of non-Linear propagation in uncompensated coherent optical systems,” J. Light-wave Technol. (2012), Early Access. [CrossRef]

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