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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 7777–7791
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Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 7777-7791 (2012)
http://dx.doi.org/10.1364/OE.20.007777


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Abstract

By extending a well-established time-domain perturbation approach to dual-polarization propagation, we provide an analytical framework to predict the nonlinear interference (NLI) variance, i.e., the variance induced by nonlinearity on the sampled field, and the nonlinear threshold (NLT) in coherent transmissions with dominant intrachannel-four-wave-mixing (IFWM). Such a framework applies to non dispersion managed (NDM) very long-haul coherent optical systems at nowadays typical baudrates of tens of Gigabaud, as well as to dispersion-managed (DM) systems at even higher baudrates, whenever IFWM is not removed by nonlinear equalization and is thus the dominant nonlinearity. The NLI variance formula has two fitting parameters which can be calibrated from simulations. From the NLI variance formula, analytical expressions of the NLT for both DM and NDM systems are derived and checked against recent NLT Monte-Carlo simulations.

© 2011 OSA

1. Introduction

Table 1. List of Main Symbols Used in the Paper

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2. Nonlinear Gaussian Model

The main goal of this paper is to provide an approximate analytical expression of the NLI coefficient aNL, valid for dominant IFWM, in any DM or NDM link. Such an expression will be used to analytically cross-validate recent simulation results on nonlinear threshold in DP coherent transmissions [9

9. A. Bononi, N. Rossi, and P. Serena, “Transmission Limitations due to Fiber Nonlinearity,” Proc. OFC’11, paper OWO7 (2011).

].

3. Nonlinear Threshold

We define the constrained NLT at reference bit error rate BER0 (i.e., at its corresponding format-dependent SNR S0) as the transmitted power NLT yielding the maximum of the “bell-curve” S versus P, where the maximum value is constrained to S0. Maximization of Eq. (1) with ASE noise adjusted such that the top value is S = S0 yields [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,” Proc. OFC’11, paper OThO7 (2011).

]
P^NLT=1(3S0aNL)1/2
(2)
and depends only on S0 and aNL. It has been shown that the model [Eq. (1)], at the top S value, yields an SNR penalty with respect to linear propagation of 1.76 dB [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,” Proc. OFC’11, paper OThO7 (2011).

,4

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, 742–744 (2011). [CrossRef]

]. Appendix 1 reviews such results and extends them to prove that the 1dB NLT 1, i.e., the transmitted power needed to achieve S0 with 1 dB of SNR penalty, is ∼ 1.05 dB smaller than NLT. 1 corresponds to the NLT simulated in [9

9. A. Bononi, N. Rossi, and P. Serena, “Transmission Limitations due to Fiber Nonlinearity,” Proc. OFC’11, paper OWO7 (2011).

] that we wish to double-check with our theory.

4. Nonlinear Interference coefficient

For a linear digital modulation we have U(t)=k=skp(tk), where sk = [Xk,Yk]T is the vector of constellation symbols on polarizations X and Y at time k (T stands for transpose), and p(t) is the real, scalar common supporting pulse [19

19. For interleaved RZ (iRZ) we would need two different support pulses for each polarization, so here iRZ is excluded.

]. We are interested in the NLI at the sampling time of interest, say t = 0. We assume a Nyquist pulse p(0) = 1 and p(k) = 0 at any other integer k. Thus assuming uncorrelated symbols we have E[|U(0)|2] = E[|sk|2] = 1. When the time-kernel is much broader than the symbol time, a regime we call IFWM dominated, then we can approximate the supporting pulse with a delta function in each field term in the double integral of Eq. (3), and the NLI term simplifies to nNL(0) = cNLP3/2, with [16

16. X. Wei, “Power-weighted dispersion distribution function for characterizing nonlinear properties of long-haul optical transmission links,” Opt. Lett. 31, 2544–2546 (2006). [CrossRef] [PubMed]

, 17

17. A. Bononi, P. Serena, and M. Bertolini, “Unified Analysis of Weakly-Nonlinear Dispersion-Managed Optical Transmission Systems from Perturbative Approach,” C. R. Phys. 9, 947–962 (2008). [CrossRef]

]
cNL=jL<γG>m,n,lsmslsnη((ml)(nl))
(4)
where the sum accounts for IFWM terms, i.e., runs over all integers m,n,l such that m+n–l = t = 0, with ml, nl. The above expression does not apply to intrachannel cross-phase modulation (IXPM) (m = l or n = l) or pure self-phase modulation (SPM) (m = n = 0); both such terms however tend to give a negligible contribution to the overall NL power with respect to IFWM as the time-kernel gets broader and broader, i.e., when IFWM is dominant. The NL power is PNLE[|nNL(0)|2] = E[|cNL|2]P3, hence we recognize that aNLE[|cNL|2], where the expectation is taken over the random symbols. Appendix 3 shows that, for any DP constellation with E[sk] = 0 and E[|sk|2] = 1, we get when IFWM is dominant
aNL=ηp8(L<γG>)2m=1n=1|η(mn)|2
(5)
where ηp=38 for DP, and ηp = 1 for SP transmission.

Since the time-kernel magnitude decreases for increasing τ and eventually vanishes after an effective time duration τM, we may then upper-bound the double sum as
Alim1τMln(τ)|η(τ)|2dτln(τM)1τM|η(τ)|2dτln(τM)0|η(τ)|2dτ.
(7)

What we need are expressions of both the kernel duration τM and of the above integral of the kernel magnitude that do not need the explicit time-kernel evaluation. We may choose τMμτrms for some positive multiplier μ of the rms width τrms2=τ2|η(τ)|2dτ/|η(u)|2du. If the parameter μ is chosen too large such that μτrms exceeds the actual time-kernel duration, then we just make the upper-bound of Eq. (7) looser. We will discuss the choice of parameter μ in the results Section. Now, for every optical link, both with and without dispersion management, a physically meaningful function is the power-weighted dispersion distribution (PWDD) J(c), representing signal power versus cumulated dispersion c [16

16. X. Wei, “Power-weighted dispersion distribution function for characterizing nonlinear properties of long-haul optical transmission links,” Opt. Lett. 31, 2544–2546 (2006). [CrossRef] [PubMed]

]. Appendix 4 shows that
DEN|η(τ)|2dτ=J2(c)dc2π
(8)
NUMτ2|η(τ)|2dτ=c2[J(c)+cJ(c)]2dc2π.
(9)

Thus aNL in Eq. (5) can be upper-bounded by the following expression depending solely on integrals of J(c), which are easy to evaluate for practical links:
aNLηp8(L<γG>)2DEN2ln(μNUMDEN).
(10)

5. Results

Fig. 1 (Left) aNL [mW−2] versus spans N from Eq. (10) (solid) and simulations (symbols). DP-QPSK on Nx100 km SMF links, R=28 Gbaud. (Right) 1dB NLT vs. symbol rate R for: theory 1 = NLT – 1.05 dBm, with NLT as in Eq. (2) (solid lines); simulations from [9] (symbols). DM30 = DM with 30 ps/nm RDPS.

Of course, one may play with the two fitting parameters to improve the prediction of aNL (and thus NLT) versus symbol rate. Focusing for instance on the NDM link, Fig. 2 (left) shows aNLversus R for a 20x100 km link. Symbols represent simulations, while the red line the theoretical aNL of Eq. (11) with the same (ηp,μ)=(388,6) parameters as in Fig. 1. We can decrease the gap to simulations by using the “optimized” parameters (ηp,μ)=(331,0.02) as shown by the magenta line, i.e., by pretending the time-kernel duration is smaller than its actual value. However this comes at the price of a reduced accuracy of the aNL versus spans N as shown in Fig. 2 (right).

Fig. 2 (Left) aNL versus symbol rate R for DP-QPSK 20x100 km NDM link. Symbols: simulations. Red line: theory (11) with (ηp,μ)=(388,6) as in Fig. 1. Magenta line: theory (10) with optimized (ηp,μ)=(331,0.02). (Right) aNL versus spans N for 28 Gbaud DP-QPSK NDM link. Symbols: simulations. Red and Magenta lines: theory.

We verified that the main reason of the inability of the model to correctly predict the shape of aNL versus symbol rate R (hence NLT versus R) over the wide range shown Figs. 1 and 2 stems from the key approximation [16

16. X. Wei, “Power-weighted dispersion distribution function for characterizing nonlinear properties of long-haul optical transmission links,” Opt. Lett. 31, 2544–2546 (2006). [CrossRef] [PubMed]

]
η(t1t2)p(t1m)p(t2n)p(t1+t2l)dt1dt2η((ml)(nl))
used to derive Eq. (4), which requires shorter and shorter pulses p(t) as (m – l) and (n – l) grow.

6. Conclusions

Appendix 1: NLT at fixed distance N and fixed SNR

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,” Proc. OFC’11, paper OThO7 (2011).

,5

5. E. Grellier and A. Bononi, “Quality Parameter for Coherent Transmissions with Gaussian-distributed Nonlinear Noise,” Opt. Express 19, 12781–12788 (2011). [CrossRef] [PubMed]

] it is shown that the power that maximizes the SNR, called the unconstrained NLT, is obtained when ASE power is twice the nonlinear noise power. Hence explicitly the unconstrained NLT is
PNLT=(NA2aNL)13
(12)
and the corresponding maximum SNR value at NLT is
SNLTPNLT32NA=(33aNL(NA2)2)13
(13)
with an SNR penalty with respect to linear propagation of SP=10Log321.76dB.

Figure 3 (left) shows an example of the “bell curve” S versus P, where a reference SNR S0 was fixed, and the smallest and largest intersections of the SNR vs. P curve with the line at level S0 occur at power Pm and PM, respectively. The corresponding penalties are marked as SPm and SPM in the figure. The two intersections coincide at a specific value of ASE noise ÑA, and the corresponding power value NLT is called the NLT at S0, or the constrained NLT [21

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

]. Clearly, SP = 1.76 dB also at the constrained NLT. At NA > ÑA no intersections are found, i.e., the target SNR S0 is unachievable. Figure 3 (right) reports the sensitivity penalty values SPm and SPM at their respective powers Pm and PM as we vary the ASE noise over all achievable values NAÑA. The graph in Fig. 3 (right) is routinely used in system design [21

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

]. Since both N and S0 are fixed, we stress that the SP vs P points are actually obtained by using varying amounts of ASE noise. For each NA, the two corresponding (Pm, SPm) and (PM, SPM) points are found at the intersection of the SP curve with the unit slope straight line SLdB=PdBNAdB, as shown in Fig. 3 (right).

Fig. 3 (Left) example of SNR [dB] vs. P [dBm] “Bell” curve and its roots at S0 = 12 dB, along with graphical definition of constrained NLT NLT at 1.76 dB of penalty; (Right) SNR penalty SP=SdBS0dB at reference S0 vs. P [dBm]. It is shown in Eq. (18) that the constrained NLT at 1dB of penalty is below NLT by 1.05 dB.

Objective of this Appendix is to provide explicit expressions of SPm, SPM, NLT, and the NLT 1 at SP = 1 dB.

i) Expressions of SPm, SPM and NLT at NAÑA

Inverting Eq. (13) at SNLT = S0 we get
N^A=2(3S0)3/2aNL1/2.
(14)

From Eq. (1)Pm and PM are seen to solve the cubic equation P31S0aNLP+NAaNL=0. Cardan’s solutions ([22

22. G. A. Korn and T. A. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 2000).

], p. 23) of the cubic equation y3 + py+q = 0 are discriminated by the value of the discriminant Q=(p3)3+(q2)2. When Q < 0 the cubic has 3 real roots, which can be expressed in trigonometric form as
y1=2p3cos(α3)y2,3=2p3cos(α3±π3)
(15)
with α=arcos((q/2)2(p3)3). In our case p3=13S0aNL, q2=NA2aNL, so Q=1(3S0aNL)3+(NA/2)2aNL2, and using Eq. (14) Q=(N^A/2)2aNL2+(NA/2)2aNL2<0 for all NAÑA. In such a case, α = arcos(−NAA), with 90° < α ≤ 180°, and thus 30°<α360°, so that cos(α3+60°)>0, i.e. y1 is the largest positive solution, while y2 the smallest positive solution corresponds to the + sign in Eq. (15): PM=23S0aNLcos(α3) and Pm=23S0aNLcos(2πα3). Using Eq. (14), we rewrite the solutions explicitly as:
PM=3S0N^Acos(arcos(NA/N^A)3)Pm=3S0N^Acos(2πarcos(NA/N^A)3).
(16)

From Eq. (1), the sensitivity penalty is the ratio of the linear SNR P/NA and the nonlinear SNR S0: SPm,M=Pm,M/NAS0, hence finally the sought SP values are
SPM=3N^ANAcos(arcos(NA/N^A)3)SPm=3N^ANAcos(2πarcos(NA/N^A)3).
(17)

As a check, when NA = ÑA the angle α = π, cos(π/3) = 1/2 and we obtain the known value SPm,M=32, and the NLT explicit value is P^NLT=32S0N^A=1(3S0aNL)1/2, which can more directly be obtained by substituting Eq. (14) into Eq. (12).

ii) Expression of 1

We are now ready to answer the following question: at which power 1 does the SP w.r.t. S0 reach a value of 1 dB? From Eq. (17), letting SPm = 100.1 ≅ 1.26 and x = NAA, we look for the solution of equation 100.1=3xcos(2πarcos(x)3), which is x1 ≅ 0.936. Hence
P^NLTP^1=32S0N^A3S0N^Acos(2πarcos(x1)3)1.273
(18)
which means 1 is 10log10(1.273) ≅ 1.05 dB below the NLT at 1.76 dB penalty. This result is also sketched in Fig. 3 (right).

Appendix 2: Regular Perturbation Solution

Using such definitions, multiply and divide Eq. (22) by L < γG >, thus finally obtaining the Manakov dispersion-managed NLSE (M-DM-NLSE) in the form:
U˜(z,ω)z=jΦNL(P0)γ(z)G(z)ejC(z)ω1ω2L<γG>U˜(z,ω+ω1).U˜(z,ω+ω1+ω2)U˜(z,ω+ω2)dω12πdω22π.
(24)

If one adds at the receiver a post-compensating fiber with accumulated normalized dispersion ξpost, one finally has the RP1 field at the receiver as:
r˜(ω)=P0elnG(L)+jξtotω22[U˜(0,ω)+U˜NL(ω)]
(27)
where ξtotC(L) +ξpost. For a “power-transparent” line G(L) = 1, and typically for coherent systems ξtot = 0, so that after chromatic dispersion compensation at the receiver we have r˜(ω)=P0[U˜(0,ω)+U˜NL(ω)], and in the time domain r(t)=P0U(0,t)+nNL(t) where nNL(t) is the inverse Fourier transform of P0U˜NL(ω) and thus has the expression reported in Eq. (3). Note that the reference power P0 can be freely chosen to simplify the analysis.

Appendix 3: Power of NL term

We need to evaluate aNL = E[|cNL|2], where we rewrite cNL in Eq. (4) as:
cNL=jL<γG>m,nsmsm+nsnη(mn)=jL<γG>m,n(Xm(XnXm+n*+YnYm+n*)Ym(XnXm+n*+YnYm+n*))η(nm)
(28)
and the summation runs over all signed non-zero m,n integer pairs, which we visualize as points on the (m,n) plane. Each point corresponds to a pair of RVs, one per polarization, as given by the big parenthesis in Eq. (28). When we swap mn the constituent random variables (RV) XnXmXn+m* and YnYmYn+m* (Type I) remain unchanged: they represent the same RV, which in the double summation in Eq. (28) must be counted df = 2 times, and the double summation for them then runs on half the (m,n) plane, i.e., for instance on the pairs (m,n) below and on the bisectrix m = n (actually the points at which m = n have degeneracy df = 1, but we will disregard this subtlety for very broad time-kernels, and use df = 2 even for them), except the axes m = 0 and n = 0 which collect the IXPM and pure SPM terms. The situation for Type I RVs is summarized in Fig. 4. On the contrary, when we swap mn, the RVs XmYnYm+n* and YmXnXm+n* (Type II) do change into new RVs.

Fig. 4 Colored dots represent IFWM points on (m,n) plane when infinitely many precursors and postcursors are taken into account. They all have degeneracy factor df = 2, except those on the m = n line (partially degenerate) which have degeneracy df = 1. Points on axes (IXPM) and (0,0) point (pure SPM) should not be included in IFWM count.

We will restrict the analysis to common DP coherent modulation formats, for which E[sk] = 0. Moreover, we choose E[|sk|2] = 1 to be the unit power of the normalized constellation symbols sk = [Xk,Yk]T. Symbols are assumed to be uncorrelated in time. Tributary symbols Xk,Yk are also zero-mean uncorrelated and have the same power E[|Xk|2] = E[|Yk|2] = 1/2. Then the RV Smnsmsm+nsn is zero-mean, and so is cNL. Therefore
aNL=Var[cNL]=(L<γG>)2m,nVar[Smn]|η(mn)|2
since the RVs adding up to build cNL are uncorrelated. Taking into account the degeneracy factor df we thus get
aNL(L<γG>)2=2((m,n):mn18df2|η(mn)|2+(m,n)18|η(mn)|2)
where the first sum is on Type I RVs and accounts for the “self-polarization” variance, while the second sum is on Type II RVs and accounts for the variance due to cross-polarization crosstalk between X and Y. The factor 2 accounts for the contribution to NL variance from the two polarizations, and 1/8 = (E[|X|2])3 = E[|X|2](E[|Y|2])2 is the variance of both Types of RVs. Using the fact that the magnitude square of the kernel |η(mn)|2 is the same on the 4 quadrants of the (m,n) plane, the above further simplifies to
aNLDP(L<γG>)2=2(28df2+48)m=1n=1|η(mn)|2
(29)
where we added the superscript DP for clarity. The per-component aNLpc in DP (such that aNLpcP03 is the NL variance on each component and P0 = P/2 is the per-component power) is obtained using E[|X|2] = 1, E[|Y|2] = 1 (i.e., normalizing the M-DM-NLSE of Eq. (24) to P0):
aNLpc(L<γG>)2=(2df2+4)m=1n=1|η(mn)|2.
(30)

The result for SP is obtained by using E[|X|2] = 1 and keeping only the Type I RV, hence
aNLSP(L<γG>)2=(2df2)m=1n=1|η(mn)|2.

We clearly see from Eqs. (29) and (30) that variance coming from Type I RVs (the one present also in SP transmission) is twice that due to cross-polarization, hence aNLpc=32aNLSP, and therefore aNLDP=38aNLSP. Figure 5 shows both aNLSP estimated from SP transmission, and aNLpc from DP transmission of a single QPSK modulated channel in a 20x100 km single-mode fiber (SMF) NDM link. We observe the convergence of the gap to the value 3/2 predicted by theory already at 28 Gbaud. When convergence is reached, we are in the “IFWM dominated regime”. Note that aNLpc is 6 dB larger than aNLDP, as confirmed by the simulated aNL in Fig. 2 (left).

Fig. 5 aNL versus baudrate in 20x100 km SMF NDM coherent link with SP- and DP-QPSK single channel transmission. aNLpc is the per-component NLI coefficient in DP, with aNLpc=4aNLDP. The figure shows convergence of aNLpc to the value 32aNLSP theoretically predicted when IFWM is dominant.

Appendix 4

In this Appendix we prove formulas (8)(9).

We start by recalling two important results that can easily be derived from [16

16. X. Wei, “Power-weighted dispersion distribution function for characterizing nonlinear properties of long-haul optical transmission links,” Opt. Lett. 31, 2544–2546 (2006). [CrossRef] [PubMed]

]:
  1. the PWDD J(c) is the inverse 1D Fourier transform of the frequency kernel considered as a function of the single variable w = ω1ω2: J(c)=𝒡1[η˜(w)]η˜(w)ejwcdwdπ. Since η̃(w) is Hermitian, as per Eq. (25), then J(c) is real.
  2. the time-kernel η(τ) seen as a function of the single variable τ = t1t2 can be obtained as the following inverse 1D Fourier transform: η(τ)=𝒡1[1|ω|J(1ω)].

We thus can prove the following two results:
  1. |η(τ)|2dτ=J2(c)dc2π
    (31)

    Proof: from Parseval’s theorem for Fourier pairs we have
    |η(τ)|2dτ=|1|ω1|J(1ω)|2dω2π=J2(1ω)dωω212π
    and after the change of variable c = 1/ω we finally get Eq. (31). Since the Fourier transform of η(τ) is real, then η(−τ) = η(τ)*, and thus |η(τ)|2 is even. Hence 0|η(τ)|2dτ=12J2(c)dc2π.

  2. τ2|η(τ)|2dτ=c2[J(c)+cJ(c)]2dc2π.
    (32)

    Proof: we know that time function η(τ) has real Fourier transform V(ω)=1|ω|J(1ω), since J(c) is real. Hence τη(τ) has transform jdV(ω)dω and by Parseval’s theorem then τ2|η(τ)|2dτ=(dV(ω)dω)2dω2π.

Now, for ω > 0, dV(ω)dω=ddω(1ωJ(1ω)) and by the change c = 1/ω: dV(ω)dω=ddc(cJ(c))dcdω=[J(c)+cJ(c)](1ω2), where J(c)ddcJ(c). Similarly, for ω < 0, dV(ω)dω=[J(c)+cJ(c)](1ω2). Hence in general dV(ω)dω=[J(1ω)+1ωJ(1ω)](sgn(ω)ω2) so (dV(ω)dω)2=[J(1ω)+1ωJ(1ω)]21ω4, and thus by the change c = 1/ω we get Eq. (32).

Appendix 5

NDM link without pre-compensation

Fig. 6 PWDD versus normalized cumulated dispersion on SMF fiber link (D = 17 ps/nm/km, α = 0.2 dB/km) and transmission at R = 28 Gbaud, corresponding to strength 𝒮 = 0.35, for (Left) N = 20 span NDM system with span length zA = 100 km (Right) N = 120 span DM system at 30 ps/nm RDPS and span length zA = 50 km.

For large N, we get kNk+1k+1, hence
τrmsNDM(2N45ξs4+2N23ξs2𝒮2+2N2ξs𝒮32𝒮2)12=15(αzAN)2𝒮
and the time-kernel width is seen to scale with N2. Note that we ignored the delta terms in J′(c).

DM link with small RDPS

Assuming RDPS≪ D/α (D is fiber dispersion) we can neglect the ripples in the PWDD [16

16. X. Wei, “Power-weighted dispersion distribution function for characterizing nonlinear properties of long-haul optical transmission links,” Opt. Lett. 31, 2544–2546 (2006). [CrossRef] [PubMed]

,17

17. A. Bononi, P. Serena, and M. Bertolini, “Unified Analysis of Weakly-Nonlinear Dispersion-Managed Optical Transmission Systems from Perturbative Approach,” C. R. Phys. 9, 947–962 (2008). [CrossRef]

] and use the smooth approximation (we assume 𝒮 > 0)
JDM(c)={1ecξpre𝒮ξinifξpre<c<ξpre+ξinecξpreξin𝒮ecξpre𝒮ξinifc>ξpre+ξin
(34)
with normalized pre-compensation ξpre=|βpre|R2=DpreD/α𝒮 (where βpre [ps2] and Dpre [ps/nm] are the pre-compensation parameters) and total in-line dispersion ξin = s, with normalized per-span residual dispersion ξs=|βs|R2=RDPSD/α𝒮 (where βs [ps2] and RDPS [ps/nm] are the residual dispersion parameters per span). For instance, Fig. 6 (right) shows the true J(c) for a 120x50 km SMF DM system with 30 ps/nm RDPS and SLR pre-compensation, such as the one whose NLT is reported in Fig. 1 (right); JNDM(c) in Eq. (34) represents the smooth average of the true J(c). Dropping for brevity the subscript DM:
(J+cJ)2={[1ecξpre𝒮(1c𝒮)]2ξin2ifξpre<c<ξpre+ξin(1c𝒮)2e2(cξpre)𝒮(1eξin𝒮)2ξin2ifc>ξpre+ξin
and thus NUM=1ξin2{ξpreξpre+ξinc2[1ecξpre𝒮(1c𝒮)]2dc2π+(1eξin𝒮)2ξpre+ξinc2(1c𝒮)2e2(cξpre)𝒮dc2π}. Long calculations lead to
NUM=12πξin26𝒮{eξs𝒮[6(ξin+ξpre)4+12(ξin+ξpre)3𝒮+30(ξin+ξpre)2𝒮2+54(ξin+ξpre)𝒮3++51𝒮4]+[3ξin4+6ξpre4+12ξpre3𝒮+30ξpre2𝒮2+54ξpre𝒮3+51𝒮4+2ξin3(6ξpre+𝒮)++3ξin(2ξpre+𝒮)(2ξpre2+𝒮2)+3ξin2(6ξpre2+2ξpre𝒮+𝒮2)]}.
(35)
Similarly, DEN=12πξin2{ξpreξpre+ξin[1ecξpre𝒮]2dc+(1eξin𝒮)2ξpre+ξine2(cξpre)𝒮dc}, leading to
DEN=ξin𝒮(1eξin𝒮)2πξin2
(36)
independently of ξpre. Note that DEN is always ≥ 0.

References and links

1.

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical Characterization of PM-QPSK Signals after Propagation in Uncompensated Fiber Links,” Proc. ECOC’10, paper P4.07 (2010).

2.

P. Ramantanis and Y. Frignac, “Pattern-dependent nonlinear impairments on QPSK signals in dispersion-managed optical transmission systems,” Proc. ECOC’10, paper Mo.1.C.4 (2010).

3.

G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, and F. Forghieri, “Performance Prediction for WDM PM-QPSK Transmission over Uncompensated Links,” Proc. OFC’11, paper OThO7 (2011).

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, 742–744 (2011). [CrossRef]

5.

E. Grellier and A. Bononi, “Quality Parameter for Coherent Transmissions with Gaussian-distributed Nonlinear Noise,” Opt. Express 19, 12781–12788 (2011). [CrossRef] [PubMed]

6.

F. Vacondio, C. Simonneau, L. Lorcy, J.-C. Antona, A. Bononi, and S. Bigo, “Experimental characterization of Gaussian-distributed nonlinear distortions,” Proc. ECOC’11, paper We.7.B.1 (2011).

7.

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of coherent systems,” Opt. Express (to be published). [PubMed]

8.

E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, and F. Forghieri, “Experimental Validation of an Analytical Model for Nonlinear Propagation in Uncompensated Optical Links,” Proc. ECOC’11, paper We.7.B.2 (2011).

9.

A. Bononi, N. Rossi, and P. Serena, “Transmission Limitations due to Fiber Nonlinearity,” Proc. OFC’11, paper OWO7 (2011).

10.

E. Ip and J. M. Kahn, “Compensation of Dispersion and Nonlinear Impairments Using Digital Backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008). [CrossRef]

11.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. Savory, “Mitigation of Fiber Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron . 16, 1217–1226 (2010). [CrossRef]

12.

K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra Series Approach for Optimizing Fiber-Optic Communications Systems Design,” J. Lightwave Technol. 16, 2046–2055 (1998). [CrossRef]

13.

Y. Gao, F. Zhang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282, 2421–2425 (2009). [CrossRef]

14.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of Intrachannel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett. 12, 392–394 (2000). [CrossRef]

15.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “System Impact of Intra-Channel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett . 12, 1633–1635 (2000). [CrossRef]

16.

X. Wei, “Power-weighted dispersion distribution function for characterizing nonlinear properties of long-haul optical transmission links,” Opt. Lett. 31, 2544–2546 (2006). [CrossRef] [PubMed]

17.

A. Bononi, P. Serena, and M. Bertolini, “Unified Analysis of Weakly-Nonlinear Dispersion-Managed Optical Transmission Systems from Perturbative Approach,” C. R. Phys. 9, 947–962 (2008). [CrossRef]

18.

A. Bononi, P. Serena, and N. Rossi, “Modeling Nonlinearity in Coherent Transmissions with Dominant Interpulse-Four-Wave-Mixing,” Proc. ECOC’11, paper We.7.B.4 (2011).

19.

For interleaved RZ (iRZ) we would need two different support pulses for each polarization, so here iRZ is excluded.

20.

A. Bononi, P. Serena, and A. Orlandini, “A Unified Design Framework for Single-Channel Dispersion-Managed Terrestrial Systems,” J. Lightwave Technol. 26, 3617–3631 (2008). [CrossRef]

21.

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

22.

G. A. Korn and T. A. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 2000).

23.

C. R. Menyuk and B. Marks, “Interaction of Polarization Mode Dispersion and Nonlinearity in Optical Fiber Transmission Systems,” J. Lightwave Technol. 24, 2806–2826 (2006). [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 29, 2011
Revised Manuscript: December 7, 2011
Manuscript Accepted: December 8, 2011
Published: March 21, 2012

Virtual Issues
European Conference on Optical Communication 2011 (2011) Optics Express

Citation
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, 7777-7791 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7777


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References

  1. A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, F. Forghieri, “Statistical Characterization of PM-QPSK Signals after Propagation in Uncompensated Fiber Links,” Proc. ECOC’10, paper P4.07 (2010).
  2. P. Ramantanis, Y. Frignac, “Pattern-dependent nonlinear impairments on QPSK signals in dispersion-managed optical transmission systems,” Proc. ECOC’10, paper Mo.1.C.4 (2010).
  3. G. Bosco, A. Carena, R. Cigliutti, V. Curri, P. Poggiolini, F. Forghieri, “Performance Prediction for WDM PM-QPSK Transmission over Uncompensated Links,” Proc. OFC’11, paper OThO7 (2011).
  4. P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical Modeling of Non-Linear Propagation in Uncompensated Optical Transmission Links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011). [CrossRef]
  5. E. Grellier, A. Bononi, “Quality Parameter for Coherent Transmissions with Gaussian-distributed Nonlinear Noise,” Opt. Express 19, 12781–12788 (2011). [CrossRef] [PubMed]
  6. F. Vacondio, C. Simonneau, L. Lorcy, J.-C. Antona, A. Bononi, S. Bigo, “Experimental characterization of Gaussian-distributed nonlinear distortions,” Proc. ECOC’11, paper We.7.B.1 (2011).
  7. F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, S. Bigo, “On nonlinear distortions of coherent systems,” Opt. Express (to be published). [PubMed]
  8. E. Torrengo, R. Cigliutti, G. Bosco, A. Carena, V. Curri, P. Poggiolini, A. Nespola, D. Zeolla, F. Forghieri, “Experimental Validation of an Analytical Model for Nonlinear Propagation in Uncompensated Optical Links,” Proc. ECOC’11, paper We.7.B.2 (2011).
  9. A. Bononi, N. Rossi, P. Serena, “Transmission Limitations due to Fiber Nonlinearity,” Proc. OFC’11, paper OWO7 (2011).
  10. E. Ip, J. M. Kahn, “Compensation of Dispersion and Nonlinear Impairments Using Digital Backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008). [CrossRef]
  11. D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, S. Savory, “Mitigation of Fiber Nonlinearity Using a Digital Coherent Receiver,” IEEE J. Sel. Top. Quantum Electron. 16, 1217–1226 (2010). [CrossRef]
  12. K. V. Peddanarappagari, M. Brandt-Pearce, “Volterra Series Approach for Optimizing Fiber-Optic Communications Systems Design,” J. Lightwave Technol. 16, 2046–2055 (1998). [CrossRef]
  13. Y. Gao, F. Zhang, L. Dou, Z. Chen, A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282, 2421–2425 (2009). [CrossRef]
  14. A. Mecozzi, C. B. Clausen, M. Shtaif, “Analysis of Intrachannel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett. 12, 392–394 (2000). [CrossRef]
  15. A. Mecozzi, C. B. Clausen, M. Shtaif, “System Impact of Intra-Channel Nonlinear Effects in Highly Dispersed Optical Pulse Transmission,” IEEE Photon. Technol. Lett. 12, 1633–1635 (2000). [CrossRef]
  16. X. Wei, “Power-weighted dispersion distribution function for characterizing nonlinear properties of long-haul optical transmission links,” Opt. Lett. 31, 2544–2546 (2006). [CrossRef] [PubMed]
  17. A. Bononi, P. Serena, M. Bertolini, “Unified Analysis of Weakly-Nonlinear Dispersion-Managed Optical Transmission Systems from Perturbative Approach,” C. R. Phys. 9, 947–962 (2008). [CrossRef]
  18. A. Bononi, P. Serena, N. Rossi, “Modeling Nonlinearity in Coherent Transmissions with Dominant Interpulse-Four-Wave-Mixing,” Proc. ECOC’11, paper We.7.B.4 (2011).
  19. For interleaved RZ (iRZ) we would need two different support pulses for each polarization, so here iRZ is excluded.
  20. A. Bononi, P. Serena, A. Orlandini, “A Unified Design Framework for Single-Channel Dispersion-Managed Terrestrial Systems,” J. Lightwave Technol. 26, 3617–3631 (2008). [CrossRef]
  21. J. C. Antona, S. Bigo, “Physical design and performance estimation of heterogeneous optical transmission systems,” C. R. Phys. 9, 963–984 (2008). [CrossRef]
  22. G. A. Korn, T. A. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 2000).
  23. C. R. Menyuk, B. Marks, “Interaction of Polarization Mode Dispersion and Nonlinearity in Optical Fiber Transmission Systems,” J. Lightwave Technol. 24, 2806–2826 (2006). [CrossRef]

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