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

It has recently been shown that, in high bit-rate coherent optical links with no dispersion management (NDM), the nonlinear interference (NLI) is a zero-mean signal-independent additive circular complex-Gaussian noise already after a few spans [1,2]. Based on such a key observation, a nonlinear Gaussian model for NDM coherent communications has been proposed [35] and experimentally validated [68]. In wavelength division multiplexing (WDM), the nonlinear noise comes both from intrachannel nonlinearity and from interchannel nonlinearity. As we increase propagation distance, intrachannel nonlinearity eventually becomes dominant at baudrates in the range of tens of Gigabaud, such as those typically envisaged for modern coherent optical communications [9], unless some form of nonlinear equalization is employed [1013]. However up to date such nonlinear equalization techniques at long distance and high baudrate have unmanageable hardware implementation complexity.

Building on a well-established time-domain perturbation approach [1417], in this paper we extend the study of the nonlinear Gaussian model to the regime in which single-channel intrachannel four wave mixing (IFWM) is the dominant nonlinearity. Such a regime applies to both NDM and dispersion-managed (DM) long links at sufficiently large baudrates.

The paper, which is an extended version of [18], is organized as follows. Section 2 introduces the basics of the nonlinear Gaussian model, in which a single parameter, the NLI parameter aNL, completely determines nonlinearity. Section 3 proves that, within the applicability of the nonlinear Gaussian model, we can analytically derive the nonlinear threshold (NLT) at a target bit error rate (BER). Section 4 derives explicit analytical expressions of aNL for dual-polarization modulation both for NDM and for DM links under the assumption that IFWM is dominant. Section 5 finally compares the analytical NLT to the NLT derived by time-consuming Monte-Carlo simulations in [9]. The five appendices contain all the analytical derivations needed to support the results summarized in the main body of the paper. Table 1 provides a list of the main symbols used in the paper.

Tables Icon

Table 1. List of Main Symbols Used in the Paper

2. Nonlinear Gaussian Model

Consider a single-channel long-haul optical link with dual polarization (DP) coherent reception. Assume that both the amplified spontaneous emission (ASE) and the NLI are independent additive complex-Gaussian noises. After coherent reception with polarization demultiplexing and ideal linear electrical equalization, followed by matched filtering and ideal carrier estimation, the 2-dimensional sampled received complex field vector at sampling time t is: r(t)=PU(t)+nL(t)+nNL(t), where P [W] is the average signal power, U the normalized 2x1 complex signal vector, nL the ASE, and nNL the NLI vectors. The electrical signal-noise ratio (SNR) at the decision gate is

S=PNA+NNL
where: NA = E[|nL|2] = βN is the ASE power from both polarizations, where N is the number of spans and β = hνF(𝒢 – 1) depends on the in-line amplifiers noise figure F and gain 𝒢 = eαzA, being α the fiber loss coefficient and zA the span length; NNL = E[|nNL|2] = aNLP3 is the NLI power obtained from a first-order regular perturbation [3, 5]. The final relationship between Q-factor and SNR then depends on the modulation format [5, 6, 8].

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].

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]

P^NLT=1(3S0aNL)1/2
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 [3,4]. 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] that we wish to double-check with our theory.

4. Nonlinear Interference coefficient

We now describe a procedure to derive closed-form analytical expressions of the NLI coefficient aNL. Generalizing the work in [16, 17] to DP, Appendix 2 shows that, in absence of polarization mode-dispersion (PMD), the NLI field vector can be obtained from a first-order regular perturbation (RP1) as:

nNL(t)=jPΦNL(P)η(t1t2)U(t+t1)U(t+t1+t2)U(t+t2)dt1dt2
where: the nonlinear phase is ΦNL(P) ≜ PL < γG > with L the total link length, γ(s) 8/9 times the fiber nonlinear coefficient, G(s) the power gain at coordinate s, and the average < γG > is defined in Eq. (23); η(t1t2) is the time-kernel (where time is normalized to the symbol time T = 1/R, and R is the baudrate), which is the 2D-inverse Fourier transform of the frequency-kernel η̃ (ω1ω2) given explicitly in Appendix 2; U(t) is the received unit-power-normalized desired signal field, and stands for transpose conjugate.

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]. 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, 17]

cNL=jL<γG>m,n,lsmslsnη((ml)(nl))
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
where ηp=38 for DP, and ηp = 1 for SP transmission.

Expression (5) is simple, yet it requires the explicit evaluation of the time-kernel, which is analytically known only for lossless links [16] or for a single lossy span of infinite length [20]. For practical lossy links of interest the time-kernel evaluation is a challenging numerical problem, and in this paper we seek an alternative procedure able to avoid its direct numerical computation. The idea is the following. We first approximate the double sum as

Alimm=1n=1|η(mn)|211|η(t1t2)|2dt1dt2=1ln(τ)|η(τ)|2dτ.

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.

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]. Appendix 4 shows that

DEN|η(τ)|2dτ=J2(c)dc2π
NUMτ2|η(τ)|2dτ=c2[J(c)+cJ(c)]2dc2π.

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).

The derivation in Appendix 3 clearly shows that this bound is valid for any zero-mean DP modulation format with independent polarization tributaries, at a given baudrate. In other terms, in the IFWM dominated regime both constant amplitude formats such as quadrature phase shift keying (QPSK), and variable amplitude formats such as quadrature amplitude modulation (QAM) at the same average power P do generate the same nonlinear power aNLP3. Appendix 5 derives closed-form expressions of the aNL upper-bound of Eq. (10) for several links of interest. For instance, for NDM links we obtain for N ≳ 5:

aNLηP(γα)2Nπ|𝒮|ln(4μ5(αzAN)2|𝒮|)
with span length zA, and fiber “strength” 𝒮β2αR2 [20]. Note the similarity of this expression with that of a Nyquist-WDM NDM system derived in [4] using a frequency-domain approach. The major difference is the Nlog(kN) scaling law in the IFWM-dominated regime, as opposed to the simpler N scaling when presumably cross-nonlinearities dominate. The Nlog(kN) scaling law is instead confirmed by the amplitude variance results of Mecozzi et al. ([15], Eq. (4)), which are based on the same RP1 time-domain approach and large-strength assumption as in this paper, although dealing with Gaussian shaped return-to-zero on-off keying modulation.

5. Results

Figure 1 (left) shows a plot of the aNL formula (10) versus number of spans N (solid), and numerically simulated values (symbols), for a single-channel 28 Gbaud DP-QPSK coherent transmission over single mode fiber (SMF, β2 = −21 ps2/km, α = 0.2 dB/km, γ = 1.26 W−1km−1) for an Nx100 km link, both NDM and DM with 30 ps/nm (DM30) of residual dispersion per span (RDPS) and no pre-compensation. For the NDM link we used the formula (11), while for the DM case the formula (10) and Eqs. (35)(36). In the theoretical curves we used the value μ = 6, which roughly matches the actual duration τM ≅ 6τrms of the time-kernel and gives the best fit of the shape of aNL versus N for both links, although for NDM links the dependence of aNL on μ is rather weak. Instead of the theoretical DP value ηp = 3/8, a smaller fitting factor ηp = 3/50 was used for DM, and ηp = 3/88 for NDM, in order to compensate for the upper-bounding in Eq. (7). We appreciate the match of theory and simulation, as well as the announced Nlog(kN) scaling law in the NDM case. The perceived NDM slope over a 50 span range is ∼ 1.25 dB/dB as in [1], although restricting the range to the first 15 spans gives ∼ 1.35 dB/dB, as we experimentally verified in a companion study [6, 7]. NLI grows faster in the DM case: aNL has an initial slope of ∼ 2 dB/dB and then bends at larger N.

 

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.

Download Full Size | PPT Slide | PDF

Figure 1 (right) shows the 1dB NLT at BER0 = 10−3 versus baudrate for a DP-QPSK format for both NDM, and a DM30 link with straight-line rule (SLR) pre-compensation [20], both at 20x100 km and at 120x50 km distance. Symbols refer to single-channel simulations taken from [9], solid lines to the formula 1 = NLT – 1.05 [dBm] using Eq. (2) and the same (ηp) fitting factors as in Fig. 1 (left). While for DM links theory only captures the general trend versus R with obviously major discrepancies at lower R where IFWM is not dominant, the match in NDM links (optimized at 28 Gbaud through the fitting factors ηp as in Fig. 1 (left)) seems more reasonable. Notwithstanding the numerical discrepancies observed in Fig. 1 (left), which may be large for practical design purposes, the analytical NLT curves are of great theoretical importance, as they provide a first model able to confirm the general trends of NLT observed in simulations, and quickly predict the NLT qualitative trends as we vary the main system parameters.

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.

Download Full Size | PPT Slide | PDF

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]

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

We provided a time-domain model of NLI in IFWM dominated links, which reasonably models NDM links, as well as high baudrate DM links. Such a model provides a quick qualitative tool to compare transmission link parameters in terms of their impact on received SNR. The model has two fitting parameters which may be optimized to best fit simulations, although it has difficulties in reproducing the correct behavior of aNL versus symbol rate over the wide range shown in Figs. 12. More work is needed on this issue to improve its accuracy.

Appendix 1: NLT at fixed distance N and fixed SNR

In [3,5] 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
and the corresponding maximum SNR value at NLT is
SNLTPNLT32NA=(33aNL(NA2)2)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]. 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]. 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.

Download Full Size | PPT Slide | PDF

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.

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

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).

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
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

We will generalize here the scalar case time-domain analysis presented in [16, 17], which extends previous analytical work based on Gaussian supporting pulses [14, 15]. The propagation equation of a DP single channel in the retarded normalized time frame t (physical time normalized to the symbol interval T = 1/R), can be described in absence of PMD by the Manakov-Nonlinear Schroedinger equation (M-NLSE) in engineering notation as ([23], Eq. (73))

A(z,t)z=g(z)2A(z,t)+jβ2(z)R222A(z,t)t2jγ(z)|A2|A(z,t)
where A = [Ax, Ay]T is the signal field envelope on the two polarizations (in W), γ(z) is 8/9 times the nonlinear coefficient, g(z) is the net gain/attenuation coefficient per unit length, β2(z) is the dispersion coefficient, and such parameters are z–varying functions, with span k ending at coordinate kzA, k = 1,..., N. The function G(z)=e0zg(s)ds is the power gain from 0 to z. Since the nonlinear term may be also written as |A|2A = A(AA), by taking the Fourier transform of Eq. (19) we get:
A˜(z,ω)z=g(z)jω2β2(z)R22A˜(z,ω)jγ(z)A˜(z,ω+ω1)A˜(z,ω+ω1+ω2)A˜(z,ω+ω2)dω12πdω22π
where for any function g(t) we define its Fourier transform (with engineering sign) as g˜(ω)=g(t)ejωtdt, where ω = 2πf, and f is the frequency normalized to the baud rate. The input modulated field Ã0(ω) may be pre-chirped to give A˜(0,ω)A˜0(ω)ejω22ξpre, where ξpre ≜ −LpreβpreR2 is the normalized cumulated dispersion in the pre-compensation fiber of dispersion coefficient βpre and length Lpre. Now make the change of variable
A˜(z,ω)=P0elnG(z)+jC(z)ω22U˜(z,ω)
where P0 is a reference normalizing power, and C(z)ξpreR20zβ2(s)s˙ is the normalized cumulated dispersion up to z. Differentiating Eq. (21) and substituting into Eq. (20) one gets:
U˜(z,ω)z=jγ(z)P0G(z)ejC(z)ω1ω2U˜(z,ω+ω1).U˜(z,ω+ω1+ω2)U˜(z,ω+ω2)dω12πdω22π.

Now define the nonlinear phase (referred to nominal power P0) as ΦNL(P0) ≜ P0L < γG >, where L = NzA is the total link length, and

<γG>1L0Lγ(s)G(s)ds.

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.

If the field terms in the integrand in Eq. (24) are approximated as z-independent, then Eq. (24) can be integrated on the link [0,L] to yield the first-order regular perturbation (RP1) solution:

U˜(L,ω)=U˜(0,ω)jΦNL(P0)0Lγ(s)G(s)ejC(s)ω1ω2dsL<γG>.U˜(0,ω+ω1)U˜(0,ω+ω1+ω2)U˜(0,ω+ω2)dω12πdω22π
with U˜(0,ω)A˜0(ω)/P0 the initial condition. Define now the (scalar) frequency-kernel as
η˜(w)0Lγ(s)G(s)ejC(s)wds0Lγ(s)G(s)ds
so that the RP1 solution writes as Ũ(L,ω) = Ũ(0) + ŨNL(ω), with
U˜NL(ω)jΦNL(P0)η˜(ω1ω2)U˜(0,ω+ω1)U˜(0,ω+ω1+ω2)U˜(0,ω+ω2)dω12πdω22π.

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)]
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)
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.

Download Full Size | PPT Slide | PDF

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
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.

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.

Download Full Size | PPT Slide | PDF

Appendix 4

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

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

  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

    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.

    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

We compute in this Appendix the closed-form expression of the upper-bound Eq. (10) for NDM and DM links. In these calculations, the term NUM=c2[J(c)+cJ(c)]2dcdπ is the most critical: if the PWDD J(c) has a discontinuity at c0 ≠ 0, then J′(c) has a term δ(c – c0) which causes NUM (and thus the kernel width τrms=NUM/DEN) to diverge to infinity. In such a case, which occurs in most links of interest, the double sum Alim in Eq. (6) still converges, and we verified that the value τrms=NUM/DEN obtained by neglecting the Dirac deltas in J′(c) still provides a meaningful time scale for measuring the width of the time-kernel.

NDM link without pre-compensation

Consider an NDM link composed of N identical spans of length zA1α, with fiber dispersion parameter β2 [ps2/km]. The fiber strength is defined as 𝒮=β2αR2 [20]. Assuming 𝒮 > 0, the PWDD is [17]:

JNDM(c)=1Nk=0N1Js(ckξs)
where Js(c)=1𝒮ec𝒮U(c), with U(c) the unit step function, and the normalized cumulated dispersion per span is ξs = −β2zAR2 = αzA𝒮. For instance, Fig. 6 (left) shows JNDM(c) for a 20x100 km SMF NDM system such as the one whose NLT is reported in Fig. 1 (right). Dropping the subscript NDM for brevity, we get J(c)=1Nk=0N1Js(ckξs) and (J(c)+cJ(c))2=1N2k=0N1(Js(ckξs)+cJs(ckξs))2 because the PWDDs of the various spans practically do not overlap. Hence
NUM=k=0N10c2(Js(ckξs)+cJs(ckξs))2dc2πN2=24ξs4+22ξs2𝒮2+21ξs𝒮3+𝒮48πN2𝒮3
where 1=k=0N1k=(N1)N2, 2=k=0N1k2=(N1)N(2N1)6, and 4=k=0N1k4=(N1)N(2N1)(3N23N1)30. Similarly,
DEN=k=0N10(Js(ckξs))2dc2πN2=14πN𝒮.

 

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.

Download Full Size | PPT Slide | PDF

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 [16,17] and use the smooth approximation (we assume 𝒮 > 0)

JDM(c)={1ecξpre𝒮ξinifξpre<c<ξpre+ξinecξpreξin𝒮ecξpre𝒮ξinifc>ξpre+ξin
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)]}.
Similarly, DEN=12πξin2{ξpreξpre+ξin[1ecξpre𝒮]2dc+(1eξin𝒮)2ξpre+ξine2(cξpre)𝒮dc}, leading to
DEN=ξin𝒮(1eξin𝒮)2πξin2
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]  

References

  • View by:
  • |
  • |
  • |

  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]

2011 (2)

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]

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

2010 (1)

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]

2009 (1)

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]

2008 (4)

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]

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

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

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]

2006 (2)

2000 (2)

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]

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]

1998 (1)

Antona, J. C.

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

Antona, J.-C.

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).

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]

Bayvel, P.

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]

Behrens, C.

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]

Bertolini, M.

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]

Bigo, S.

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

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).

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]

Bononi, A.

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

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]

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]

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

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).

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]

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).

Bosco, G.

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]

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).

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).

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).

Brandt-Pearce, M.

Carena, A.

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]

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).

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).

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).

Chen, Z.

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]

Cigliutti, R.

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).

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).

Clausen, C. B.

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]

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]

Curri, V.

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]

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).

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).

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).

Dou, L.

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]

Forghieri, F.

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]

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).

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).

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).

Frignac, Y.

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).

Gao, Y.

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]

Grellier, E.

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

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]

Hellerbrand, S.

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]

Ip, E.

Kahn, J. M.

Killey, R. I.

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]

Korn, G. A.

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

Korn, T. A.

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

Lorcy, L.

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]

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).

Makovejs, S.

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]

Marks, B.

Mecozzi, A.

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]

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]

Menyuk, C. R.

Millar, D. S.

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]

Nespola, A.

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).

Orlandini, A.

Peddanarappagari, K. V.

Poggiolini, P.

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]

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).

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).

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).

Ramantanis, P.

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).

Rival, O.

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]

Rossi, N.

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).

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

Savory, S.

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]

Serena, P.

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]

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]

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

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).

Shtaif, M.

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]

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]

Simonneau, C.

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]

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).

Taiba, M. T.

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).

Torrengo, E.

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).

Vacondio, F.

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).

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]

Wei, X.

Xu, A.

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]

Zeolla, D.

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).

Zhang, F.

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]

C. R. Phys. (2)

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]

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

IEEE J. Sel. Top. Quantum Electron (1)

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]

IEEE Photon. Technol. Lett (2)

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]

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]

IEEE Photon. Technol. Lett. (1)

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]

J. Lightwave Technol. (4)

Opt. Commun. (1)

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]

Opt. Express (2)

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]

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

Opt. Lett. (1)

Other (9)

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

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).

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

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).

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

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).

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).

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).

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).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
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.

Fig. 2
Fig. 2

(Left) aNL versus symbol rate R for DP-QPSK 20x100 km NDM link. Symbols: simulations. Red line: theory (11) with ( η p , μ ) = ( 3 88 , 6 ) as in Fig. 1. Magenta line: theory (10) with optimized ( η p , μ ) = ( 3 31 , 0.02 ) . (Right) aNL versus spans N for 28 Gbaud DP-QPSK NDM link. Symbols: simulations. Red and Magenta lines: theory.

Fig. 3
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 S P = S d B S 0 d B 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.

Fig. 4
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.

Fig. 5
Fig. 5

aNL versus baudrate in 20x100 km SMF NDM coherent link with SP- and DP-QPSK single channel transmission. a N L p c is the per-component NLI coefficient in DP, with a N L p c = 4 a N L D P . The figure shows convergence of a N L p c to the value 3 2 a N L S P theoretically predicted when IFWM is dominant.

Fig. 6
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 �� = 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.

Tables (1)

Tables Icon

Table 1 List of Main Symbols Used in the Paper

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

S = P N A + N N L
P ^ N L T = 1 ( 3 S 0 a N L ) 1 / 2
n N L ( t ) = j P Φ N L ( P ) η ( t 1 t 2 ) U ( t + t 1 ) U ( t + t 1 + t 2 ) U ( t + t 2 ) d t 1 d t 2
c N L = j L < γ G > m , n , l s m s l s n η ( ( m l ) ( n l ) )
a N L = η p 8 ( L < γ G > ) 2 m = 1 n = 1 | η ( m n ) | 2
A lim m = 1 n = 1 | η ( m n ) | 2 1 1 | η ( t 1 t 2 ) | 2 d t 1 d t 2 = 1 ln ( τ ) | η ( τ ) | 2 d τ .
A lim 1 τ M ln ( τ ) | η ( τ ) | 2 d τ ln ( τ M ) 1 τ M | η ( τ ) | 2 d τ ln ( τ M ) 0 | η ( τ ) | 2 d τ .
D E N | η ( τ ) | 2 d τ = J 2 ( c ) d c 2 π
N U M τ 2 | η ( τ ) | 2 d τ = c 2 [ J ( c ) + c J ( c ) ] 2 d c 2 π .
a N L η p 8 ( L < γ G > ) 2 D E N 2 ln ( μ N U M D E N ) .
a N L η P ( γ α ) 2 N π | 𝒮 | ln ( 4 μ 5 ( α z A N ) 2 | 𝒮 | )
η ( t 1 t 2 ) p ( t 1 m ) p ( t 2 n ) p ( t 1 + t 2 l ) d t 1 d t 2 η ( ( m l ) ( n l ) )
P N L T = ( N A 2 a N L ) 1 3
S N L T P N L T 3 2 N A = ( 3 3 a N L ( N A 2 ) 2 ) 1 3
N ^ A = 2 ( 3 S 0 ) 3 / 2 a N L 1 / 2 .
y 1 = 2 p 3 cos ( α 3 ) y 2 , 3 = 2 p 3 cos ( α 3 ± π 3 )
P M = 3 S 0 N ^ A cos ( arcos ( N A / N ^ A ) 3 ) P m = 3 S 0 N ^ A cos ( 2 π arcos ( N A / N ^ A ) 3 ) .
S P M = 3 N ^ A N A cos ( arcos ( N A / N ^ A ) 3 ) S P m = 3 N ^ A N A cos ( 2 π arcos ( N A / N ^ A ) 3 ) .
P ^ N L T P ^ 1 = 3 2 S 0 N ^ A 3 S 0 N ^ A cos ( 2 π arcos ( x 1 ) 3 ) 1.273
A ( z , t ) z = g ( z ) 2 A ( z , t ) + j β 2 ( z ) R 2 2 2 A ( z , t ) t 2 j γ ( z ) | A 2 | A ( z , t )
A ˜ ( z , ω ) z = g ( z ) j ω 2 β 2 ( z ) R 2 2 A ˜ ( z , ω ) j γ ( z ) A ˜ ( z , ω + ω 1 ) A ˜ ( z , ω + ω 1 + ω 2 ) A ˜ ( z , ω + ω 2 ) d ω 1 2 π d ω 2 2 π
A ˜ ( z , ω ) = P 0 e ln G ( z ) + j C ( z ) ω 2 2 U ˜ ( z , ω )
U ˜ ( z , ω ) z = j γ ( z ) P 0 G ( z ) e j C ( z ) ω 1 ω 2 U ˜ ( z , ω + ω 1 ) . U ˜ ( z , ω + ω 1 + ω 2 ) U ˜ ( z , ω + ω 2 ) d ω 1 2 π d ω 2 2 π .
< γ G > 1 L 0 L γ ( s ) G ( s ) d s .
U ˜ ( z , ω ) z = j Φ N L ( P 0 ) γ ( z ) G ( z ) e j C ( z ) ω 1 ω 2 L < γ G > U ˜ ( z , ω + ω 1 ) . U ˜ ( z , ω + ω 1 + ω 2 ) U ˜ ( z , ω + ω 2 ) d ω 1 2 π d ω 2 2 π .
U ˜ ( L , ω ) = U ˜ ( 0 , ω ) j Φ N L ( P 0 ) 0 L γ ( s ) G ( s ) e j C ( s ) ω 1 ω 2 d s L < γ G > . U ˜ ( 0 , ω + ω 1 ) U ˜ ( 0 , ω + ω 1 + ω 2 ) U ˜ ( 0 , ω + ω 2 ) d ω 1 2 π d ω 2 2 π
η ˜ ( w ) 0 L γ ( s ) G ( s ) e j C ( s ) w d s 0 L γ ( s ) G ( s ) d s
U ˜ N L ( ω ) j Φ N L ( P 0 ) η ˜ ( ω 1 ω 2 ) U ˜ ( 0 , ω + ω 1 ) U ˜ ( 0 , ω + ω 1 + ω 2 ) U ˜ ( 0 , ω + ω 2 ) d ω 1 2 π d ω 2 2 π .
r ˜ ( ω ) = P 0 e ln G ( L ) + j ξ tot ω 2 2 [ U ˜ ( 0 , ω ) + U ˜ N L ( ω ) ]
c N L = j L < γ G > m , n s m s m + n s n η ( m n ) = j L < γ G > m , n ( X m ( X n X m + n * + Y n Y m + n * ) Y m ( X n X m + n * + Y n Y m + n * ) ) η ( n m )
a N L = Var [ c N L ] = ( L < γ G > ) 2 m , n Var [ S m n ] | η ( m n ) | 2
a N L ( L < γ G > ) 2 = 2 ( ( m , n ) : m n 1 8 d f 2 | η ( m n ) | 2 + ( m , n ) 1 8 | η ( m n ) | 2 )
a N L D P ( L < γ G > ) 2 = 2 ( 2 8 d f 2 + 4 8 ) m = 1 n = 1 | η ( m n ) | 2
a N L p c ( L < γ G > ) 2 = ( 2 d f 2 + 4 ) m = 1 n = 1 | η ( m n ) | 2 .
a N L S P ( L < γ G > ) 2 = ( 2 d f 2 ) m = 1 n = 1 | η ( m n ) | 2 .
| η ( τ ) | 2 d τ = J 2 ( c ) d c 2 π
| η ( τ ) | 2 d τ = | 1 | ω 1 | J ( 1 ω ) | 2 d ω 2 π = J 2 ( 1 ω ) d ω ω 2 1 2 π
τ 2 | η ( τ ) | 2 d τ = c 2 [ J ( c ) + c J ( c ) ] 2 d c 2 π .
J N D M ( c ) = 1 N k = 0 N 1 J s ( c k ξ s )
N U M = k = 0 N 1 0 c 2 ( J s ( c k ξ s ) + c J s ( c k ξ s ) ) 2 d c 2 π N 2 = 2 4 ξ s 4 + 2 2 ξ s 2 𝒮 2 + 2 1 ξ s 𝒮 3 + 𝒮 4 8 π N 2 𝒮 3
D E N = k = 0 N 1 0 ( J s ( c k ξ s ) ) 2 d c 2 π N 2 = 1 4 π N 𝒮 .
τ rms N D M ( 2 N 4 5 ξ s 4 + 2 N 2 3 ξ s 2 𝒮 2 + 2 N 2 ξ s 𝒮 3 2 𝒮 2 ) 1 2 = 1 5 ( α z A N ) 2 𝒮
J D M ( c ) = { 1 e c ξ pre 𝒮 ξ in if ξ pre < c < ξ pre + ξ in e c ξ pre ξ in 𝒮 e c ξ pre 𝒮 ξ in if c > ξ pre + ξ in
( J + c J ) 2 = { [ 1 e c ξ pre 𝒮 ( 1 c 𝒮 ) ] 2 ξ in 2 if ξ pre < c < ξ pre + ξ in ( 1 c 𝒮 ) 2 e 2 ( c ξ pre ) 𝒮 ( 1 e ξ in 𝒮 ) 2 ξ in 2 if c > ξ pre + ξ in
N U M = 1 2 π ξ in 2 6 𝒮 { e ξ s 𝒮 [ 6 ( ξ in + ξ pre ) 4 + 12 ( ξ in + ξ pre ) 3 𝒮 + 30 ( ξ in + ξ pre ) 2 𝒮 2 + 54 ( ξ in + ξ pre ) 𝒮 3 + + 51 𝒮 4 ] + [ 3 ξ in 4 + 6 ξ pre 4 + 12 ξ pre 3 𝒮 + 30 ξ pre 2 𝒮 2 + 54 ξ pre 𝒮 3 + 51 𝒮 4 + 2 ξ in 3 ( 6 ξ pre + 𝒮 ) + + 3 ξ in ( 2 ξ pre + 𝒮 ) ( 2 ξ pre 2 + 𝒮 2 ) + 3 ξ in 2 ( 6 ξ pre 2 + 2 ξ pre 𝒮 + 𝒮 2 ) ] } .
D E N = ξ in 𝒮 ( 1 e ξ in 𝒮 ) 2 π ξ in 2

Metrics