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
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 [3–5] and experimentally validated [6–8]. 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 , unless some form of nonlinear equalization is employed [10–13]. 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 [14–17], 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 , 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 . 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.
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: , 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 is3, 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 .
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 P̂NLT 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 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 P̂1, i.e., the transmitted power needed to achieve S0 with 1 dB of SNR penalty, is ∼ 1.05 dB smaller than P̂NLT. P̂1 corresponds to the NLT simulated in  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: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 , 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 . 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]Appendix 3 shows that, for any DP constellation with E[sk] = 0 and E[|sk|2] = 1, we get when IFWM is dominant
Expression (5) is simple, yet it requires the explicit evaluation of the time-kernel, which is analytically known only for lossless links  or for a single lossy span of infinite length . 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
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
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 τM ≜ μτrms for some positive multiplier μ of the rms width . 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 . Appendix 4 shows that
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:
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 ≳ 5:20]. Note the similarity of this expression with that of a Nyquist-WDM NDM system derived in  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. (, 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.
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 , 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.
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 , both at 20x100 km and at 120x50 km distance. Symbols refer to single-channel simulations taken from , solid lines to the formula P̂1 = P̂NLT – 1.05 [dBm] using Eq. (2) and the same (ηp,μ) 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 parameters as in Fig. 1. We can decrease the gap to simulations by using the “optimized” parameters 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).
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 Eq. (4), which requires shorter and shorter pulses p(t) as (m – l) and (n – l) grow.
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. 1–2. More work is needed on this issue to improve its accuracy.
Appendix 1: NLT at fixed distance N and fixed SNR
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 P̂NLT is called the NLT at S0, or the constrained NLT . 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 NA ≤ ÑA. The graph in Fig. 3 (right) is routinely used in system design . 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 , as shown in Fig. 3 (right).
Objective of this Appendix is to provide explicit expressions of SPm, SPM, P̂NLT, and the NLT P̂1 at SP = 1 dB.
i) Expressions of SPm, SPM and P̂NLT at NA ≤ ÑA
Inverting Eq. (13) at SNLT = S0 we get
From Eq. (1)Pm and PM are seen to solve the cubic equation . Cardan’s solutions (, p. 23) of the cubic equation y3 + py+q = 0 are discriminated by the value of the discriminant . When Q < 0 the cubic has 3 real roots, which can be expressed in trigonometric form asEq. (14) for all NA ≤ ÑA. In such a case, α = arcos(−NA/ÑA), with 90° < α ≤ 180°, and thus , so that , i.e. y1 is the largest positive solution, while y2 the smallest positive solution corresponds to the + sign in Eq. (15): and . Using Eq. (14), we rewrite the solutions explicitly as:
From Eq. (1), the sensitivity penalty is the ratio of the linear SNR P/NA and the nonlinear SNR S0: , hence finally the sought SP values are
ii) Expression of P̂ 1
We are now ready to answer the following question: at which power P̂1 does the SP w.r.t. S0 reach a value of 1 dB? From Eq. (17), letting SPm = 100.1 ≅ 1.26 and x = NA/ÑA, we look for the solution of equation , which is x1 ≅ 0.936. HenceFig. 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 (, Eq. (73))Eq. (19) we get: Eq. (21) and substituting into Eq. (20) one gets:
Now define the nonlinear phase (referred to nominal power P0) as ΦNL(P0) ≜ P0L < γG >, where L = NzA is the total link length, and
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:
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: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:Eq. (28). When we swap m ↔ n the constituent random variables (RV) and (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 m ↔ n, the RVs and (Type II) do change into new RVs.
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 is zero-mean, and so is cNL. ThereforeEq. (24) to P0):
The result for SP is obtained by using E[|X|2] = 1 and keeping only the Type I RV, hence
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 , and therefore . Figure 5 shows both estimated from SP transmission, and 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 is 6 dB larger than , as confirmed by the simulated aNL in Fig. 2 (left).
We start by recalling two important results that can easily be derived from :
- the PWDD J(c) is the inverse 1D Fourier transform of the frequency kernel considered as a function of the single variable w = ω1ω2: . Since η̃(w) is Hermitian, as per Eq. (25), then J(c) is real.
- 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: .
We thus can prove the following two results:
Proof: from Parseval’s theorem for Fourier pairs we haveEq. (31). Since the Fourier transform of η(τ) is real, then η(−τ) = η(τ)*, and thus |η(τ)|2 is even. Hence .
Proof: we know that time function η(τ) has real Fourier transform , since J(c) is real. Hence τη(τ) has transform and by Parseval’s theorem then .
Now, for ω > 0, and by the change c = 1/ω: , where . Similarly, for ω < 0, . Hence in general so , and thus by the change c = 1/ω we get Eq. (32).
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 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 ) 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 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-compensationFig. 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 and because the PWDDs of the various spans practically do not overlap. Hence
For large N, we get , hence
DM link with small RDPSFig. 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:
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