1. Introduction

The Gaussian Noise (GN) model has recently been shown to effectively predict the system performance of highly-dispersed coherent optical transmission systems, such as high baud-rate dispersion-uncompensated (DU) systems [1, 2]. In such a model, the GN reference formula (GNRF) provides a formally elegant and compact expression of the power spectral density (PSD) of the received nonlinear interference (NLI). However, the GNRF involves a double frequency integration which poses non-trivial numerical problems for multi-span wavelength division multiplexed (WDM) systems. Many of the numerical integration issues have been already addressed in [2]. Given the practical importance of developing an accurate GNRF numerical evaluator, however, for accuracy testing it proves quite useful to have exact expressions of the NLI PSD in special realistic cases. The case of rectangular per-channel input spectra has already served to clarify the integration regions and derive closed-form ’asinh’ NLI PSD approximations [2], and to obtain improved expressions of received NLI power in the single-channel/Nyquist-WDM case [3].

In this paper, we derive new exact single-integral expressions of the NLI PSD in the GNRF for both Nyquist and non-Nyquist WDM systems with input rectangular per-channel spectra. We provide explicit PSD formulas for both the single-channel interference (SCI) and the cross-channel interference (XCI) [2]. We formulate the GNRF in a generalized form that applies to any link configuration, be it with concentrated or distributed amplification, with or without inline compensation, and with possibly different spans. All link details are summarized by the kernel frequency function [4–6]. We verify the derived NLI PSD formulas for rectangular input spectra against split-step Fourier (SSF) simulations performed according to the key assumptions of the GNRF, i.e., that every input WDM channel is a low-power complex Gaussian process [6]. A discussion of how closely the GNRF approximates the NLI with digital input signals and specific modulation formats is beyond the scope of this paper, and is the topic of ongoing investigations [7,8]. Having verified that NLI PSDs are consistent with SSF simulations, we next study how SCI and XCI power scales with number of channels, number of spans, dispersion, signal bandwidth and frequency spacing, so that we can easily discriminate the dominant nonlinear effect. We finally derive a new upper-bound (UB) to the XCI power and check its accuracy.

The paper is organized as follows. Sec. 2 summarizes the needed background on the GNRF, and introduces the change of integration variables that leads to the new NLI PSD expressions for rectangular input spectra. Sec. 3 tackles the single-channel and Nyquist-WDM cases and also discusses the over-estimation when using the zero-frequency value of the NLI PSD to get the NLI power. Sec. 4 tackles the non-Nyquist XCI NLI and derives the new UB to the XCI power. Sec. 5 contains numerical examples of NLI scaling with major system parameters. Sec. 6 summarizes the main findings of the paper.

2. The GN reference formula

In dual-polarization transmission with uncorrelated signals with identical spectra on the two polarizations, the GNRF yields the NLI PSD as $G_{NLI}(f)=\frac{16}{27}I(f)$, with [1, 2, 5, 6]:

Now, for any link with or without dispersion management (DM) for which the GN model applies, one just has to calculate the link kernel function 𝒦 and then use (1) to get the NLI PSD. For multispan links, the kernel factorizes as

The trouble with the analytic formula (1) is that it involves a double frequency integration where the squared kernel |𝒦 (f₁f₂)|² is oscillating in frequency faster and faster as the number of spans increases because of the phased-array term and poses non-trivial integration problems [2]. Integration is eased by a suitable change of integration variables. In [2] the change to hyperbolic coordinates $u=-\frac{1}{2}\text{ln}(f_2/f_1)$, $v=\sqrt{f_1{f}_2}$ was proposed. The rationale was that the squared kernel is a function of v only, hence at fixed v, integration in the (f₁, f₂) plane follows the constant contour levels of |𝒦 (f₁f₂)|². With a similar rationale, we use here the equivalent but formally simpler change u = f₁, v = f₁f₂, whose Jacobian is J = u and whose inverse is f₁ = u, f₂ = v/u. With such a change, the double integral I(f) on the (f₁, f₂) plane becomes

When the input WDM signals have rectangular spectra, the square-bracketed integrals in Eq. (4) can be solved exactly. In the next sections we present single-integral formulas of the NLI PSD in such a case. These formulas allow to quickly determine whether single- or cross-channel nonlinearity dominates, and provide a case against which numerical double-integration routines of Eq. (1) can be checked for accuracy. Starting from these formulas, new simplified expressions are also derived at zero frequency. We present first the Nyquist-WDM case, and then the non-Nyquist case.

3. NLI PSD for single-channel and Nyquist-WDM systems

We tackle here the rectangular-spectrum single-channel case, or equivalently the WDM case where no bandwidth gaps are present between neighboring channels. The total power is P and the input PSD $G(f)=\frac{P}{2\delta }{\text{rect}}_{2\delta }(f+\delta )$ is a rectangular gate centered at f = 0 with total two-sided bandwidth 2δ. The integrand in Eq. (1) is non-zero only over the shaded domains in quadrants I through IV shown in Fig. 1 at several values of f. Note that, since |𝒦 (v)|² = |𝒦 (−v)|² for any kernel, Eq. (2), then the squared kernel is the same over the 4 quadrants, hence the integral in Eq. (1) over quadrants II and IV has always the same value.

 

Fig. 1 Domains over which integrand in Eq. (1) is non-zero when input PSD is a gate over f ∈ [−δ, δ]. Integration over domains I through IV yields the four lines in Eq. (4).

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We can now state our main result on SCI. Define the normalized double integral ℐ (f) ≜ I(f)/(P/(2δ))³, with I(f) as in Eq. (1). Using Eq. (4), we can prove that ℐ (f) can be exactly expressed as follows. If |f| < δ:

 

Fig. 2 NLI normalized spectra vs. frequency (normalized to bandwidth 2δ = 28GHz) for channels with rectangular input PSD at three dispersion values D (ps/nm/km) for 20×100km DU link. Kernel (3) had α = 0.2dB/km, ζ = 1, γ = 1.27(W km)⁻¹. Theory: cyan, yellow, dark green curves. SSF simulations: blue, red, green curves. (a)G_NLI(f)/P³ for single channel, theory Eqs. (5) and (6); (b)G_XCI(f)/P³ on central channel for M =15 channels at Δ = 50GHz, theory Eqs.(13) and (14).

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The next 3 subsections are introduced to explore the accuracy of the flat-PSD approximation used in Poggiolini’s approximations of the NLI power [2].

3.1. SCI PSD at zero frequency

From Eq. (5), the value at f = 0 is found as:

3.2. SCI PSD at zero dispersion

When dispersion tends to zero, any kernel becomes frequency-independent: 𝒦 (v) ≡ 𝒦 (0) at all v. In this case, it may be verified that Eqs. (5) and (6) tend to:

3.3. Accuracy of flat-PSD approximation for SCI and Nyquist WDM

For an M-channel Nyquist-WDM system with per-channel power P and bandwidth 2δ, the NLI PSD is given by $G_{NLI}(f)=\frac{16}{27}I(f)$, with $I(f)={\left(\frac{PM}{2\delta M}\right)}^3ℐ(f)$, and ℐ (f) explicitly given in Eqs. (5) and (6) where we substitute δ → Mδ. After receiver matched filtering with H(f) = 1 for |f| < δ and zero else (Appendix A), the total NLI power on the reference central channel can be conveniently written by introducing the average value $\overline{ℐ}\triangleq \frac{1}{2\delta }{\int }_{-\delta }^{\delta }ℐ(f)\text{d}f$ as:

For WDM computations, it is common practice to adopt the flat-PSD approximation, i.e., assume instead that ℐ (f) is flat at its ℐ (0) value over the receiver bandwidth 2δ, so that ℐ̄ is replaced by ℐ (0) in Eq. (10) [1–3]. Such an assumption leads to an over-estimation of the NLI power by an amount 𝒪 ≜ ℐ(0)/ℐ̄. For instance, at zero dispersion and assuming a Nyquist-WDM system with M channels, we get from Eq. (9): $\overline{ℐ}=\frac{1}{\delta }{\int }_0^{\delta }\left(3{\left(M\delta \right)}^2-f^2\right)\text{d}f={\delta }^2\left(3M^2-\frac{1}{3}\right)$, and ℐ (0) = 3(Mδ)², hence

Figure 3 shows the over-estimation 𝒪 for an M-channel Nyquist-WDM signaling over an N×100 km DU link when the per-channel spectra are rectangular of bandwidth 2δ, plotted versus (a) fiber dispersion D at 2δ = 28 GHz, and (b) versus 2δ at D = 17ps/nm/km. The top group of curves refer to single-channel (M = 1), the bottom group of curves to an M = 3 system. In Fig. 3(a) we see that at small D the over-estimation converges to the analytical value (11) (reported as a solid blue horizontal line), and as D → ∞ the over-estimation fades to 0 dB. For M = 1 we get the largest over-estimation ${𝒪}_{\mathit{max}}^{dB}~0.78$. When we increase the Nyquist-WDM channels to M ≥ 3 the over-estimation fluctuates around the zero-dispersion value Eq. (11) by less than 0.054 dB. The negative 𝒪^dB = −0.06 value at N = 20, D = 2.65 ps/nm/km is due to a dip at f = 0 of the NLI PSD such that the maximum of the PSD is away from f = 0. The curves in Fig. 3(b) provide a horizontal-axis rescaling of those in Fig. 3(a) because the NLI PSD Eq. (1) scales with the parameter Dδ². This is due to the fact that the kernel 𝒦 (v) in Eq. (2) is a function of the parameter β₂v, which is proportional to Df₁f₂. The take-away message from Fig. 3 is that for practical values of M ≥ 3 in Nyquist-WDM DU systems at any dispersion and per-channel bandwidth we can safely use ℐ (0) in place of the more demanding ℐ when evaluating the NLI power in Eq. (10).

 

Fig. 3 Over-estimation 𝒪 in an M-channel Nyquist-WDM N×100km DU system with each channel having rectangular spectrum of bandwidth 2δ. (a) versus fiber dispersion D at 2δ = 28GHz; (b) versus 2δ, at D = 17ps/nm/km. Blue lines: value at D = 0, Eq. (11).

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4. NLI PSD for Non-Nyquist WDM systems

We assume here a WDM system with a reference central channel, N_c channels to its left and N_c channels to its right on the frequency axis, hence M = 2N_c + 1 channels, with uniform frequency spacing Δ. The WDM comb has input PSD $G(f)={\sum }_{k=-N_c}^{N_c}S(f-k\mathrm{\Delta })$, where each lowpass equivalent channel envelope has power P and a rectangular PSD with bandwidth 2δ, namely $S(f)=\frac{P}{2\delta }{\text{rect}}_{2\delta }(f+\delta )$. Channels do not spectrally overlap, hence 2δ < Δ. The domain of integration in the double-integral in Eq. (1) is the set of islands depicted at f = 0 in Fig. 4 (Cfr. [2, Fig. 3]). Integration over the central red island yields the SCI and can directly be obtained from Eqs. (5) and (6). Integration over the on-axes yellow islands yields the XCI, and integration over all off-axes islands yields the multi-channel interference (MCI) [2]. Hence the NLI PSD may be decomposed as the sum of SCI, XCI, and MCI contributions: G_NLI(f) = G_SCI(f) + G_XCI(f) + G_MCI(f). Highly dispersed systems quite effectively suppress MCI (i.e., the classical FWM), so that XCI is the relevant cross-nonlinearity, and encompasses both scalar cross-phase modulation and cross-polarization modulation [16]. The XCI PSD is obtained from Eq. (1) as $G_{XCI}(f)=\frac{16}{27}{I}_{XCI}(f)$, with $I_{XCI}(f)\triangleq 2{\sum }_{m=1}^{N_c}\left(I_m(f)+I_{-m}(f)\right)$ (the term 2 is the degeneracy factor), and we defined

 

Fig. 4 Integration domain in Eq. (1) given at f = 0 by intersection of support of G(f + f₂) (horizontal stripes), G(f + f₁) (vertical) and G(f + f₁ + f₂) (tilted). Red island = SCI. Yellow islands = XCI. Blue islands = MCI. Red curves: contours of |𝒦 (f₁f₂)|².

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We can prove that for any integer m > 0 the normalized double integral ℐ_m(f) ≜ (I_m(f) + I_−m(f))/(P/2δ)³ can be written as follows. Define $\left(\eta ,{\eta }_m^{+},{\eta }_m^{-}\right)\triangleq \left(\delta -\left|f\right|,m\mathrm{\Delta }+\eta ,m\mathrm{\Delta }-\eta \right)$ and $\left(\epsilon ,{\epsilon }_m^{+},{\epsilon }_m^{-}\right)\triangleq \left(\delta +\left|f\right|,m\mathrm{\Delta }+\epsilon ,m\mathrm{\Delta }-\epsilon \right)$. Then, if |f| < δ:

4.1. XCI PSD at zero frequency and new upper-bound

As a corollary, the value at f = 0 is found from Eq. (13) as follows. Define ${\mathrm{\Delta }}_m^{+}\triangleq m\mathrm{\Delta }+\delta$ and ${\mathrm{\Delta }}_m^{-}\triangleq m\mathrm{\Delta }-\delta$.Then,

 

Fig. 5 (a) The four integration domains in the (f₁, f₂)-plane whose integration yields the four v-integral terms in Eq. (15) in the order they appear (see also [13, Fig. 10]). Dashed curves are hyperbolas. (b) Domain 𝒟 delimited by hyperbola $v=f_1{f}_2=\delta {\mathrm{\Delta }}_m^{-}$ passing at point A, approximating squared kernel integral over domain 𝒟₁, and also 𝒟₃.

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4.2. Zero dispersion XCI PSD and flat-PSD approximation

At D = 0, integration over every XCI island (Cfr Fig. 4) produces the same result, hence the XCI PSD is just 4N_c times the SCI PSD in Eq. (9). Thus the estimation error of XCI power at D = 0 is again given by expression (11). If we define ℐ_XCI(f) ≜ I_XCI(f)/(P/2δ)³, similar conclusions regarding the use of ℐ_XCI(0) instead of ℐ̄_XCI in the flat-PSD approximation (Cfr. Eq. (10)) apply.

5. Single- and cross-channel NLI scaling properties

As briefly recalled in Appendix A, the NLI power obtained from the GNRF Eq. (1) enters in the calculation of the signal to noise ratio (SNR) for detection of digitally-modulated signals. Only input signal spectra matter in the GNRF, and not the modulation format [7, 8].

The NLI power can be expressed as P_NLI ≜ a_NLP³. We show next how the NLI coefficient a_NL scales with the main system parameters, since a_NL determines the scaling of both Q-factor [17] and nonlinear threshold [18]. In all the following figures, all analytical curves use the flat-PSD approximation (Sec. 3.3). Since SCI, XCI and MCI PSDs add up, the nonlinear coefficient is the sum of SCI, XCI and MCI contributions: a_NL = a_SCI + a_XCI + a_MCI. It is reported in all subsequent figures with the label “WDM”. For single-channel NLI we use

5.1. Scaling with symbol-rate

Figure 6 shows the NLI coefficient versus channel bandwidth 2δ ≡ R for a 20×100 km DU link over standard single-mode fiber (SMF) with D = 17 ps/nm/km, for an M = 81 channel WDM system at a bandwidth efficiency $\eta =\frac{2\delta }{\mathrm{\Delta }}=1$ (Nyquist-WDM, Fig. 6(a)) and η = 0.56 (e.g., Δ = 50GHz at R = 28Gbaud, Fig. 6(b)). Fixing η implies increasing the channel spacing Δ as R is increased [16]. We see that the NLI coefficient is always decreasing with R, and a decrease as 1/R² is forecast by the XCI UB (20). Besides the SCI, the SCI UB, the XCI and the XCI UB (whose color code was already introduced), we also report in blue solid line the approximation a_NL ≅ a_SCI + a_XCI (label XCI+SCI). In the Nyquist-WDM case, Fig. 6(a), the WDM a_NL coefficient is analytical (circles) and is obtained from Eq. (18) using for ℐ (0) Eq. (7) where δ is replaced by Mδ. In the non-Nyquist case we do not have an exact analytical WDM formula. Thus in Fig. 6(b) at η = 0.56 the WDM a_NL curve (squares) was obtained through a fast-Fourier-transform (FFT) based direct numerical evaluation of the double frequency integral (1) that yields the whole NLI PSD [9]. The XCI+SCI curve lays slightly above the WDM squares at larger R because of the flat-PSD over-estimation in Eqs. (18) and (19), and increases to match the WDM square at R = 5Gbaud because MCI starts to be important at smaller R. Blue and green crosses report the closed-form ’asinh’ expressions for WDM and single-channel [2, eq. (13),(22)], obtained by using an effective bandwidth B_WDM = 2δM^2δ/Δ, and we note the excellent match with the XCI+SCI curve. In the Nyquist-WDM case, Fig. 6(a), the effect of MCI is more clearly visible because of the noticeable difference of the theoretical WDM (circles) and XCI+SCI curves at symbol-rates below 10Gbaud. Crosses again report the closed-form ’asinh’ expressions [2, eq. (13), (22)] with an almost perfect match with the exact WDM circles. From Fig. 6 we learn that in this large WDM system on a 20×100km SMF DU link the XCI always dominates the SCI. We also note that the SCI UB over the shown range is always within 0.5dB from the true SCI value (the ’asinh’ approximation is here poor below 20GHz [2]), while the XCI bound for Nyquist-WDM is within 0.5dB of the true XCI value at symbol-rates above 23Gbaud, and above 16Gbaud for the non-Nyquist η = 0.56 system.

 

Fig. 6 NLI coefficient [dB(mW⁻²)] versus channel bandwidth 2δ ≡ R [GHz] for SMF DU link with N = 20 spans, span length z_A = 100km, M = 81 channels, at a bandwidth efficiency $\eta =\frac{2\delta }{\mathrm{\Delta }}=1$ (a); and η = 0.56 (b). Crosses: [2, eq. (13), (22)].

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5.2. Scaling with bandwidth efficiency

Figure 7 illustrates the use of the analytical formulas to quickly determine the dominant nonlinear effect in a very long 120×50km SMF DU link with either 11 or 81 WDM channels, as the symbol-rate is fixed to 28 Gbaud and the channel spacing ranges from 28GHz (Nyquist-WDM case) up to 50GHz. While the SCI is independent of Δ, the XCI clearly decreases as the spacing Δ is increased. We see that at Δ = 50GHz (η = 0.56) and 11 channels the SCI is the dominant effect (Cfr [16, Fig. 1], where the number of channels was M = 11 at R ≥ 15Gbaud), but as the channel count is increased to 81 the SCI and XCI give equal nonlinear contributions. For the Nyquist-WDM case Δ = 50GHz, XCI is the dominant effect at both channel counts. Again, note the tightness of the XCI UB, which improves at increasing channel count. We also reported both the XCI+SCI (blue curve) and the WDM (blue squares) obtained by the FFT-based numerical integrator [9] that calculates the whole NLI PSD. The XCI+SCI curve is within 0.3dB of the WDM ’true value’ (blue squares) because of the flat-PSD approximation in Eqs. (18) and (19) and not because of the presence of MCI, as we could verify by removing the flat-PSD approximation and using the complete analytical NLI PSD calculated through Eqs. (5) and (6) and Eqs. (13) and (14). We verified that the flat-PSD approximation only gives an over-estimation of the SCI curve (shown as the M = 1 curve in Fig. 3), while the XCI curve has essentially no over-estimation. As predicted in Section 3.3, the over-estimation disappears in the Nyquist-WDM case, as confirmed by the coincidence of the WDM square with the analytical WDM value that uses Eq. (18) (blue circles at Δ = 28GHz in Fig. 7). Blue and green crosses also report the closed-form ’asinh’ expressions [2, eq. (13), (22)].

 

Fig. 7 NLI coefficient [dB(mW⁻²)] vs. channel spacing Δ [GHz], at fixed channel bandwidth 2δ ≡ R, for 120×50km SMF DU link, and M = [1, 11, 81] channels. Crosses: [2, eq. (13),(22)].

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5.3. A flat input spectrum is optimal

Figure 8 illustrates the use of the new analytical formulas to quickly verify that the best supporting pulse in a digital optical transmission is the one that achieves the broadest input spectrum [22] (Cfr discussion at end of Appendix A). Figure 8(a) shows the input spectrum for a WDM transmission with fixed per-channel power, with channel spacing Δ = 50GHz, per-channel symbol-rate R = 5Gbaud, and a variable pulse bandwidth (i.e., per-channel PSD bandwidth), ranging from the symbol-rate (2δ = R, green PSD), up to the no-spectral overlap limit (2δ = Δ, red PSD). Figure 8(b) shows the NLI coefficient versus channel bandwidth 2δ over the range delimited by the two extreme cases (green, red) depicted in Fig. 8(a). We observe that both SCI and XCI monotonically decrease as the channel bandwidth increases up to its top limit Δ. Hence we conclude that in order to minimize nonlinearity and thus maximize signal-to-noise ratio one needs to choose 2δ = Δ, i.e., a flat input WDM spectrum (red spectrum in Fig. 8(a)). As the channel bandwidth is increased, we also note that XCI decreases more slowly than SCI, since clearly the gap between channels is decreasing to zero thus favoring XCI. However, notwithstanding closer channel spectra, the frequency enlargement of per-channel input PSD brings about a global decrease also for XCI; from Eq. (20) it is clear that at fixed spacing Δ and symbol-rate R, the XCI power decreases almost as a_XCI−UB ∝ 1/(2δ)². For all realistic kernels we find a minimum when a flat input WDM spectrum is achieved. Although our numerical study is based on rectangular spectra, a simple theoretical explanation of the optimality of a flat input spectrum is proposed in [9]. As a last observation, please note the excellent accuracy of both the SCI and XCI upper-bounds in this numerical experiment.

 

Fig. 8 (a) Input WDM spectra at increasing per-channel bandwidth 2δ. (b) NLI coefficient vs. channel bandwidth 2δ, at fixed symbol-rate R = 5Gbaud, for SMF 20x100km DU link, M = 81 channels, Δ = 50GHz spacing. Blue circle at Nyquist-WDM case 2δ = 50GHz gives the analytical WDM value.

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5.4. Scaling with number of spans

Finally, Fig. 9 illustrates the use of the analytical formulas to explain the different accumulation rates of SCI and XCI as the number of spans N increases [18, 20]. As in [18], we consider transmission of a 19-channel WDM system with per-channel bandwidth/symbol-rate 2δ = R = 28Gbaud over an N × 50km SMF DU link. The top row of figures refers to the Nyquist-WDM case (η = 1) with channel spacing Δ = 28GHz, and the bottom row to the case Δ = 50GHz (η = 0.56). The left-column figures show the NLI coefficient a_NL[dB(mW⁻²)] versus N (in log scale), while blue and green crosses report the closed-form ’asinh’ expressions for WDM and single-channel, respectively [2, eq. (13),(22)]. The right-column figures show the local slope (1 + ε) [dB/dB] of the left-column a_NL curves [2, 18]. An explicit formula of the SCI slope parameter ε can be derived from the a_NL formula for DU links in [11] as

 

Fig. 9 (a),(c): NLI coefficient versus spans N (log scale) at fixed bandwidth/symbol-rate 2δ = R = 28Gbaud, for SMF DU link with span length z_A = 50km, and M = 19 channels. (b),(d): local slope [dB/dB] of log-log plots in (a),(c) versus N. Dash-dotted: average slope. Black dashed: ε-fit Eq. (21). Crosses: [2, eq. (13), (22)].

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6. Conclusions

We provided new semi-analytic formulas of the PSD of the received nonlinear interference in coherent optical links for which the GN model applies. These formulas can be used to both discriminate single- and cross-channel nonlinearity, and to check the accuracy of numerical routines that solve the GNRF double frequency integral (1). Such numerical routines are useful for instance in mixed-fiber links where simple ’asinh’ formulas [2] do not exist. A novel bound for XCI was obtained that clearly reveals the major XCI scaling laws, and can find use in path routing in future elastic optical networks. We tested in numerical case studies the accuracy of both the bound and the ’asinh’ formulas [2] against the new exact analytical formulas.