# Analytical Modeling of the Impact of Fiber Non-Linear Propagation on Coherent Systems and Networks

This is a continuation from the previous tutorial - ** laser mirrors and regenerative feedback**.

## 1. Why Are Analytical Models Important?

Analytical models of the impact of non-linear effects on system and network performance are important for several reasons: in the context of point-to-point (PTP) systems, they allow to explore design strategies efficiently, without resorting to lengthy computer simulations; in the context of networks, they can help in the optimization of the network architecture and layout, and can provide physical layer awareness for real-time control-plane tasks such as channel routing. In all contexts, they can be used for research purposes, to devise and theoretically test new and disruptive technologies.

### 1.1. What Do Professionals Need?

Ideally, system and network engineers, as well as researchers, would like to have analytical models that are:

- Easy to set-up and parameterize,
- Fast to compute,
- Accurate.

Unfortunately, no single model currently fully complies with all three of these requirements simultaneously. As a result, there are trade-offs that must be managed to provide the best compromise for each specific application.

On the other hand, recent progress in modeling has made it possible to work out acceptable solutions for a broad range of utilization scenarios, from PTP link design to real-time network optimization.

We will adopt a rather practical approach towards the goal of helping the reader find a suitable trade-off among the three modeling features listed above, for the different utilization scenarios.

As an important disclaimer that we make upfront, Section 4 focuses on the GN–EGN class of models, partly because it has been the subject of substantial research by the authors of this chapter, partly because it has recently enjoyed considerable attention by the technical community, and partly because of the practical difficulty of encompassing many different classes of models within a single book chapter. No claim of intrinsic superiority versus other models is implied.

As a counterbalance, Section 2 aims at providing some general theoretical background that should prove useful for the readers to carry out an informed model selection themselves. The idea is that, apart from the indications that we propose, the readers should be able to autonomously make or refine their choices regarding the identification of the most suitable models for their own applications, possibly outside of the GN–EGN model class.

As a whole, the field of research on modeling is currently extremely active. Which of the various model classes will emerge as a front-runner or eventual winner, it will probably become clear as the field progresses and new modeling efforts come to fruition. It may also be the case that different model classes are best suited for different types of problems or contexts of utilization.

## 2. Background

In this section we provide some background information on modeling approaches.

In general, one could call “non-linear fiber propagation model” any form of analytical description of the non-linear behavior of the optical fiber. In this respect, the well-known dual-polarization non-linear Schroedinger equation (DP-NLSE) is one such “model,” and a quite successful one.

Numerical integration of the DP-NLSE, or of the related Manakov equation (see Section 2.1.1), typically within a Monte-Carlo simulation environment, has been and still is one of the most powerful tools for the study and design of optical systems in the presence of non-linear effects.

However, in this chapter we are interested in “derived models.” They all assume that the DP-NLSE accurately represents the underlying physics of the fiber and try to obtain from it simpler results, if possible closed-form, using various approximations, mostly for the purpose of quantifying the system impact of the fiber non-linear behavior.

Many such models have been proposed over time. An extensive bibliography, encompassing several of the modeling efforts carried out in the last 25 years, is available at the end of this tutorial. Some of the listed papers contain, in turn, further extensive referencing, so that the interested reader can directly or indirectly find rather exhaustive orientation across this complex field.

Describing, classifying and discussing in detail all, or even just the majority, of non-linearity modeling efforts is far beyond the scope of a single book chapter. Instead, we collect in the following a list of the main approximations which have been used to derive such models from the DP-NLSE.

Discussing the underlying approximation is key to the understanding of the various models because, in essence, it is the employed approximations that define them and characterize their behavior and effectiveness.

We briefly discuss the impact and implications of the most common approximations and provide practical recommendations regarding their viability, pros and cons.

### 2.1. Modeling Approximations

The non-linear propagation models derived from the DP-NLSE typically exploit one or more of the following approximations:

- The Manakov equation approximation
- The single-polarization approximation
- The perturbation approximation
- The signal Gaussianity approximation
- The NLI additive-Gaussian-noise (AGN) approximation
- The locally-white NLI noise approximation
- The lossless fiber approximation
- The incoherent NLI accumulation approximation
- The noiseless propagation approximation
- The frequency-domain XPM approximation.

In the following we introduce them one by one and discuss their implications. Before proceeding, we define the term “NLI.” The acronym stands for “non-linear interference.” Calling \(s_\text{WDM}(t)\) and \(s_\text{WDM}^\text{NL}(t)\) the received WDM signal in the absence and in the presence of fiber non-linearity, respectively, then NLI is the difference signal:

\[\tag{7.1}s_\text{NLI}(t)=s_\text{WDM}^\text{NL}(t)-s_\text{WDM}(t)\]

Rearranging the above formula, NLI can also be viewed as a disturbance created by non-linear effects which gets formally added to the WDM signal and degrades it.

#### 2.1.1. The Manakov Equation Approximation

The DP-NLSE is deemed to accurately account for the effect of the evolution of the state of polarization (SOP) of the signal on the generation of NLI. However, SOP variations along the fiber are a random process and this makes the DP-NLSE difficult to use, also because the characteristic lengths related to SOP evolution may be quite short (from a hundred meters down to a fraction of a meter).

The Manakov equation (ME) is an approximation of the DP-NLSE based on analytically averaging over the random evolution of the SOP along the fiber. As a result, the ME is a deterministic differential equation, which does capture the non-linear effect of one polarization onto the other, but averages over the fast dynamic of SOP variations.

More in detail, the SOP-evolution averaging procedure generates the Manakov-PMD (polarization-mode dispersion) equation. A simplified version of it, consisting of only the left-hand side of said equation, is the ME that is widely employed in analytical modeling and computer simulations. It entails the further approximation of neglecting both the linear and non-linear effects of PMD.

On the other hand, linear PMD is no longer a factor in modern coherent systems thanks to receiver digital signal processing (DSP), whereas non-linear PMD was assessed to be virtually negligible in typical transmission links. At present, the ME is generally regarded as quite accurate and effective.

Note that another polarization-related effect, polarization dependent loss (PDL), is indeed a source of penalty even in current DSP-assisted systems, both because of its linear and non-linear effects.

However, its linear impact is regarded as prevalent, especially in dispersion uncompensated systems, and is typically studied separately. Therefore, we consider PDL beyond our target level-of-detail for system-oriented non-linearity modeling and we disregard it henceforth.

In conclusion, the ME, that is adequately accounts for the main cross-polarization non-linear effects impacting practical coherent systems. For this reason, the GN–EGN models have been proposed based on it.

Practical recommendation: For system-oriented studies, the use of the Manakov equation for the modeling of fiber non-linear propagation provides a good compromise between accuracy and complexity and is therefore recommended.

#### 2.1.2. The Single-Polarization Approximation

This approximation assumes that propagation obeys the single-polarization (SP) NLSE. This is a scalar equation which completely neglects all polarization-related effects, either linear or non-linear.

The transmitted signal must be assumed single-polarization or, otherwise, it must be assumed that the two multiplexed signal polarizations propagate in a completely separate and independent way.

This approximation has been very popular over the years, especially prior to the coherent systems revolution, for various reasons.

- First, dealing with a scalar equation eases calculations and may lead to simpler final results.
- Secondly, before the advent of coherent systems, polarization multiplexing was not used and the transmitted intensity-modulation direct-detection (IM/DD) signals of the time were fundamentally scalar signals, detected through polarization-insensitive photodiodes.

As a result, the impact of polarization-mediated non-linear effects was minor and was typically neglected. PMD was a problem, but it was mostly studied separately as a linear effect.

On the other hand, today’s coherent systems are almost exclusively dual-polarization, and therefore cross-polarization effects should be properly accounted for. The sensible practical recommendation is then to use the Manakov equation to suitably account for non-linear polarization interactions.

In principle, this might lead to increased model complexity. However, at least in the case of the GN–EGN models, the final equations have identical complexity whether the single or dual-polarization derivation is used.

Practical recommendation: Do not use the single-polarization approximation for the modeling of the impact of NLI on dual-polarization coherent systems.

#### 2.1.3. The Perturbation Approximation

The vast majority of non-linear propagation models makes the assumption that non-linearity is relatively small, i.e., that it is a perturbation as compared to the useful signal. Thanks to this assumption, model derivation can exploit perturbation techniques, which allow to find approximate analytical solutions to the Manakov (or other) propagation equations.

One possible perturbation approach consists of assuming that the signal propagates linearly from input to output, subject only to the action of fiber chromatic dispersion (CD) and loss/amplification; at each point along the fiber, this linearly-propagating signal excites fiber non-linearity and creates the NLI disturbance, which is calculated based on the employed propagation equation (the ME, the SP-NLSE or others) but is kept separate from the signal itself. At the end of the link, the sum of the linearly propagated signal and the NLI produced by it constitutes the overall non-linear fiber output signal (see Section A.4 for an example of this procedure).

The key feature of this method is that it leads to NLI computation formulas which are remarkably simple. This popular approach belongs to the class of regular perturbation (RP) methods and the one described above is its first-order version. Higher-order versions are possible whereby not only the linearly propagated signal but the produced NLI itself cooperate to create further NLI. To the purpose of obtaining system-impact models, first-order versions have however been used, with few exceptions.

Another perturbation method, based on truncated Volterra series, was proposed. Interestingly, it was shown that the RP method and the VS-method are equivalent, so in this context they can be unified as RP–VS methods. Other first-order perturbation method, which can be re-conduced or bear substantial similarities to the RP–VS methods. Further perturbation methods have also been proposed, such as the logarithmic perturbation (LP) method, a combination of the RP and LP method, the frequency-resolved LP (FRLP) method, the enhanced RP method, and still others.

Many simulations and experiments have shown the perturbation approximation to produce rather accurate results within the typical range of optimal system launch powers. Irrespective of the specific method, however, all perturbation techniques can be expected to break down at highly non-linear regimes.

For instance, first-order methods typically do not take into account that NLI noise is created at the expense of the WDM signal power, that is, the signal is assumed “undepleted.” In those systems where the power of the generated NLI is not small as compared to that of the WDM signal, neglecting signal power depletion may induce substantial model inaccuracy.

This effect was recently studied. It was found that a simple semi-heuristic correction can approximately account for such depletion, extending the range of usability of first-order perturbation methods. Caution should anyway be used in those cases where substantial signal-power depletion may occur.

Specifically regarding the GN–EGN model class, the derivation of these models is based on a first-order RP method.

Practical recommendation: The use of the perturbation approximation, in particular of first-order perturbation methods, to solve the fiber non-linear propagation equations appears adequate for typical system operating conditions.

#### 2.1.4. The Signal Gaussianity Approximation

According to this approximation, the transmitted signal is modeled as stationary circular Gaussian noise, whose power spectrum (or power spectral density, PSD) is shaped as the PSD of the actually transmitted WDM channels.

It allows to drastically simplify model derivation and strongly decreases the model final analytical complexity. One of its implications is that the model results are independent of the used transmission format, since this information is completely removed from the signal itself.

This approximation has been repeatedly used over the years. It is also one of the main approximations employed by the GN model. It is already found in what can be considered the first GN-model-class paper, dating back to 1993. It is instead removed in the EGN model, generalizing a procedure proposed for the XPM component of NLI (see Section 2.1.10). The EGN model is however quite substantially more complex than the GN.

One key feature of the error incurred by using the signal Gaussianity approximation is that the impact of NLI is never underestimated for QAM transmission formats. Rather, it is overestimated, to an extent that depends on several aspects, among which fiber type, span length, amplification scheme, transmission format, symbol rate, and others.

In typical EDFA-amplified PTP links, the NLI overestimation can lead to between 5% and 15% system reach underestimation, for 32 GBaud systems. Interestingly, the error due to the Gaussianity assumption tends to vanish when going towards large symbol rates (>32 GBaud).

It also tends to vanish for massively multi-carrier systems, such as OFDM, Nyquist-FDM or similar. See Section 4.4 for more details on this aspect. Nonetheless, the practical recommendation for PTP links is to avoid using the signal Gaussianity approximation, if possible. Either the very accurate, but more complex, EGN model, or an approximation of it should be used, as extensively discussed in Sections 4.2 and 4.3.

In the context of dynamically reconfigurable networks (DRNs), however, the situation is less clear-cut (see Section 4.5). In that environment neighboring channels can have different symbol rate, format, and accumulated dispersion. In addition, the re-routing process across the network may change the neighbors of a channel many times along a lightpath.

All these factors produce a non-linear behavior which is varied and on average less distant from the signal-Gaussianity approximation. In addition, its use makes it possible to obtain very simple and powerful performance prediction tools, well-suited for real-time physical-layer awareness.

The combination of these aspects suggests that in DRNs the signal Gaussianity approximation may actually be recommended, in which case the GN model may be a possible option.

Practical recommendation: Avoid the signal Gaussianity assumption for PTP links, if practical. For DRNs, it may instead be an effective option, provided that one is fully aware of its limitations.

#### 2.1.5. The NLI Circular-Additive Gaussian-Noise Approximation

This approximation consists of assuming that the disturbance generated by non-linearity, that is, NLI, manifests itself as AGN at the output of the link, circular and independent of either the signal or ASE noise (for short CIAGN).

One of the key implications of this assumption is that the analytical assessment of the impact of non-linearity on system performance can be carried out by calculating the variance of NLI noise on the signal constellation, and then simply adding it to the noise variance due to ASE.

In particular, this approximation makes it possible to characterize a channel BER based on a modified “non-linear” OSNR:

\[\tag{7.2}\text{OSNR}_\text{NL}=\frac{P_\text{ch}}{P_\text{ASE}+P_\text{NLI}}\]

where \(P_\text{NLI}\) is a suitably calculated power of NLI (see Eq. 7.8). Due to its simplicity, Equation 7.2 has been used in many non-linearity modeling studies to assess the system impact of NLI.

It has been extensively validated in many practical system scenarios, where BER estimates based on it have been shown to come very close to Monte-Carlo simulation results, also when using the very accurate EGN model. Experimental results too, within the limitations of their error margins, have confirmed its validity.

In reality, NLI noise is only approximately CIAGN. Specifically, it has recently been pointed out that even in uncompensated systems NLI contains a “phase noise” component, which is non-additive.

Higher-order formats, such as PM-16QAM, tend to create more of it than lower-order formats, such as PM-QPSK. In addition, it has also been shown that a variable fraction of NLI may have a long correlation time, on the order of many tens, to hundreds of symbols.

On the other hand, the non-CIAGN and long-correlation components of NLI tend to be quite large only in special conditions. Specifically, they get larger the more the “ideal distributed amplification” condition is approached4 (see “lossless fiber approximation,” Section 2.1.7), whereas they are much smaller in conventional long-haul lumped-amplification systems. In such systems, they can typically be neglected and Equation 7.2 can be used to predict system performance with satisfactory accuracy.

This matter is however not settled. New results are steadily appearing, which should be closely monitored for potentially disruptive innovation.

Another aspect related to Eq. 7.2 is that, as already pointed out in Section 2.1.3, at high non-linear regimes WDM signal depletion may occur. A heuristic correction of Equation 7.2 is possible to approximately account for this phenomenon.

Practical recommendation: Equation 7.2 can typically be trusted, with caution to be exercised for system scenarios that depart substantially from conventional.

#### 2.1.6. The Locally-White NLI Noise Approximation

Provided that the previous approximation is accepted, then the NLI noise is fully characterized by its PSD. Such PSD is, in general, non-flat, even if looked at locally, over any single WDM channel.

On the other hand, it is generally not far from flat either. The typical shapes of NLI PSDs, found using the GN model, show that the error incurred assuming a NLI locally-white-noise (LWN) approximation, with constant PSD value equal to that of the channel center frequency, is modest.

In addition, it typically leads to overestimating noise slightly rather than underestimating it, that is, it is a conservative approximation. An indicative value of such overestimation was found to be 0.3 dB of NLI power for various typical system configurations. On the other hand, a similar investigation is not yet available regarding the EGN model, so this result should currently be taken with some caution.

The LWN approximation has the obvious substantial advantage of requiring the estimate of the NLI PSD at one specific frequency only, typically the center frequency of the channel-under-test (CUT). Besides reducing the computational burden, it also makes it easier to obtain closed-form formulas for the overall NLI power impinging on a channel (i.e., \(P_\text{NLI}\) in Eq. 7.2).

On the other hand, if very accurate predictions are needed, this approximation should not be used. In this case, the PSD of NLI must be evaluated at multiple frequencies within a CUT, as many as necessary to make the calculation of \(P_\text{NLI}\) accurate (see Section 4, Eq. 7.8).

Practical recommendation: The LWN approximation is acceptable for approximate system performance assessment. It should be removed if high-accuracy predictions are needed.

#### 2.1.7. The Lossless Fiber Approximation

One approximation which has been used both in modeling efforts and in theoretical studies consists of assuming that the fiber is lossless or, equivalently, that ideal distributed amplification is present in the link, exactly canceling out loss, so that the signal power stays constant throughout the fiber. This approximation is often found together with the single-polarization approximation (see Section 2.1.2).

The reasons behind the use of this approximation have been varied. In certain cases it was that model derivation becomes easier, while it was perhaps assumed that the essential features of non-linearity generation would be preserved.

In other contexts, it was viewed as a limiting case of distributed Raman amplification, and hence considered a sufficiently plausible scenario where to estimate, or find bounds related to, the “ultimate” fiber capacity.

Very recent studies have however shown that the lossless fiber approximation creates propagation regimes whose features are markedly different from those of typical practical systems.

In particular, a lossless fiber leads to the creation of a large phase noise component within the NLI noise, which may exhibit long correlation time (tens to hundreds of symbols). In addition, it exacerbates the format-dependence of phase-noise generation.

These features are much less evident in conventional lumped-amplifications systems. In particular the strength of phase noise is less and its long-correlated component tends to decrease substantially.

Certain simultaneously forward- and backward-amplified all-Raman systems may exhibit a behavior which is, to some extent, similar to that of lossless systems. The features of completely lossless fiber systems are however substantially more extreme than even these special Raman systems.

Practical recommendation: The lossless fiber approximation should be avoided as it may produce results that diverge quite considerably, both quantitatively and qualitatively, from the actual behavior of realistic transmission systems.

#### 2.1.8. The Incoherent NLI Accumulation Approximation

This approximation assumes that the NLI produced in each span adds up incoherently, that is in power, at the receiver site. Specifically, defining as \(G_\text{NLI}^{(n)}(f)\) the PSD of NLI generated in the \(n\)th span, and assuming that it is linearly propagated till the end of the link, the incoherent accumulation approximation implies that the total PSD of NLI at the end of the link is:

\[\tag{7.3}G_\text{NLI}(f)\approx\boldsymbol{\sum}_{n=1}^{N_\text{span}}G_\text{NLI}^{(n)}(f)\]

In reality, the NLI contributions generated in each span should be added together coherently, at the field level, keeping both their amplitude and phase into account. This approximation, however, allows to greatly simplify the computation of the accumulation of NLI along a link. Its use is especially beneficial in the context of physical-layer aware DRNs, for various reasons that are explained in Section 4.5.2.

The accuracy issue for the incoherent approximation is complex. It is discussed in detail in Sections 4.3.4 and 4.3.5. Different behaviors can be obtained depending on the model it is used with (such as for instance the GN model, EGN model or others).

When applied to the GN model, it produces rather accurate estimates of system maximum reach for typical lumped-amplification PM-QAM systems, taking advantage of an error cancelation circumstance.

In other contexts it can yield less favorable results. As a rule of thumb, its accuracy is poor at very low span count and at very low, and especially single, channel count.

In essence, the incoherent accumulation approximation should be viewed as a practical heuristic tool for achieving drastic complexity reduction for certain specific modeling needs and application scenarios. As such, it needs targeted ad hoc validation. If properly tailored and used, it may prove very effective.

Practical recommendation: The incoherent accumulation approximation should not be employed for high-accuracy link design/analysis purposes, or for system research. It can however be a quite effective solution for specific applications (such as real-time management of DRNs), provided that its limitations are understood and pre-assessed in those scenarios.

#### 2.1.9. The Noiseless Propagation Approximation

This approximation consists of neglecting the NLI produced by ASE noise or by the interplay of ASE noise with the WDM signal. It can be equivalently stated by saying that NLI is generated only by the WDM signal propagating along the link.

This approximation relies on the observation that the OSNR of a coherent optical system cannot be too low, to ensure sufficiently low BER at the receiver. If the WDM signal is substantially stronger than ASE noise, then indeed NLI generation is almost exclusively due to the WDM signal itself and ASE noise can be neglected.

The noiseless approximation is widely used in modeling efforts and system investigations. It makes modeling easier and allows certain useful results to be obtained in closed-form, such as the optimum channel launch power.

On the other hand, a recent study has pointed out that non-negligible loss of predictive accuracy may be incurred when neglecting ASE-generated NLI, even in systems that do not operate at extremely low OSNRs.

For instance, in a PM-QPSK system with 15 channels (32 GBaud, spacing 33.6 GHz, non-zero dispersion-shifted fiber as in Table 7.1, operating at BER = 2 × 10\(^{-2}\)), neglecting ASE-generated NLI caused an overestimation of the maximum reach by over 6%.

In this system the OSNR due to ASE alone was about 8 dB at the receiver, at maximum reach, over a bandwidth equal to the symbol rate. This result shows that even apparently sizeable OSNRs do not guarantee that NLI generation is completely unaffected by ASE.

The reach overestimation error further increases for systems capable of operating at even lower OSNRs, which can be the case either because their FECs can tolerate higher BERs or because intrinsically more robust, such as those using PM-BPSK.

In these cases, ASE-generated NLI must necessarily be accounted for to avoid large reach overestimation errors. Both rigorous and heuristic methods have been proposed, for instance.

Practical recommendation: The noiseless propagation approximation is sufficiently accurate as long as the ASE-only OSNR at the receiver (over a bandwidth equal to the symbol rate) is larger than about 9–10 dB. For lower OSNRs, ASE noise should be accounted for in NLI generation.

#### 2.1.10. The XPM Approximation

The XPM approximation consists of taking into account only the “cross-phase-modulation” contribution among all the possible NLI-generating processes. Specifically, XPM accounts exclusively for the non-linear distortion induced on the CUT by the power fluctuations of each single interfering (INT) channel in the WDM comb, individually.

Neglected are all effects involving the mutual non-linear interaction of two different INTs or three INTs, affecting the CUT, often referred to as FWM. Note that in the absence of dispersion, XPM would manifest itself as pure phase noise on the CUT, which is why the XPM acronym refers to “phase.”

Note that XPM does not include single-channel non-linearity, which is the non-linear effect of a channel onto itself (also called self-channel interference, SCI, or SPM, self-phase modulation).

For the purpose of this discussion, though, we intend the “XPM approximation” as neglecting certain inter-channel non-linear effects, whereas we assume that single-channel ones are separately and correctly accounted for.

In the context of the GN–EGN model class, the XPM approximation was proposed. To distinguish it from XPM approximations applied to other model classes, we call it frequency-domain (FD) XPM. This denomination appears appropriate the proposed model based on the XPM approximation is introduced as “frequency-domain analysis.”

A visually intuitive way of depicting the FD-XPM approximation is in relation to Figure 7.1. This figure will be exhaustively explained in Section 3.2, so the reader interested in the theoretical details should refer to that section. Here we simply point out that Figure 7.1 represents the two-dimensional frequency plane where the integral equations of the GN and EGN model must be evaluated.

In particular each closed lozenge or triangle is a specific integration sub-domain, which contributes some of the NLI power falling on the center channel of a WDM comb. To account for all contributions, all such sub-domains (or “islands”) should be considered. Note that the actual analytical form of the integrals also changes according to the specific islands.

The FD-XPM approximation consists of considering the contribution of the X1 islands only, while neglecting all others. Specifically, the X1 regions are those straddling the \(f_1\) and \(f_2\) axes. This approach clearly simplifies the model calculations considerably. On the other hand, the majority of the NLI contribution islands is neglected. Yet, despite this circumstance, the FD-XPM approximation may be sufficiently accurate because the X1 islands, although a minority in number, often contribute most of the total NLI.

For certain system scenarios, though, the FD-XPM approximation may underestimate NLI considerably. This happens specifically when fiber dispersion is small, channel spacing is Nyquist or quasi-Nyquist, and the symbol rate is low, these three conditions re-enforcing one another.

As an indicative rule for Nyquist or quasi-Nyquist systems, derived from considerations on the relative strength of the XPM and FWM contributions present in the GN–EGN model, a substantial NLI underestimation error may be incurred by the FD-XPM approximation when \(\beta_2R^2\lt1/100\) (in units km\(^{-1}\)), with \(R\) the symbol rate and \(\beta_2\) the fiber dispersion coefficient (see Section 4.2 for more detailed symbol definitions).

For instance, two examples of a 15-channel, 32 GBaud PM-QPSK quasi-Nyquist system are shown, operating over either NZDSF or LS fiber (see Table 7.1 for fiber parameters). The resulting values of \(\beta_2R^2\) are \(1/200\) and \(1/400\). After 20 spans (span length 100 km) the FD-XPM approximation underestimates NLI by 1.7 and 2.7 dB, respectively.

According to the \(\beta_2R^2\) rule above, the FD-XPM approximation should be increasingly critical for decreasing symbol rates. Provided that the symbol rate is low enough, inaccuracy should be observable over any fiber, including high-dispersion ones.

This is indeed what is found when analyzing the problem of assessing the symbol rate minimizing the generation of NLI, for a constant spectrally efficiency. This topic is dealt with in detail in Section 4.4.

When trying to address it using the FD-XPM approximation, an increasing NLI estimation inaccuracy is found when decreasing the symbol rate. Incidentally, this feature of the FD-XPM approximation makes it also inadequate for dealing with those OFDM or Nyquist-FDM systems whose per-subcarrier rate is low.

Another form of XPM approximation is used by the FRLP (frequency-resolved logarithmic-perturbation) model. This model bears some similarities with the EGN model both because it adopts a first-order perturbation method and because it assumes that the CUT is decomposed into elementary frequency components, as it is done in the EGN model and in general in the SpS methods (see Section 3).

However, it departs substantially from the EGN model because it uses a logarithmic-perturbation (LP) approach rather than a RP one, and because the effect of non-linearity on the CUT is expressed in the form of a time-dependent transfer function. Yet another version of the XPM approximation is used in a time-domain (TD) first-order RP model (TD-XPM).

These further XPM models are quite effective in depicting certain specific features of NLI, and in particular non-linear phase noise (see Section 2.1.5), for which they even allow to find simple closed-form results. Contrary to the FD-XPM approximation, there is currently no specific targeted investigation of the possible errors incurred by these further XPM models, when going towards low \(\beta_2R^2\) products and low channel spacing. Since they explicitly neglect FWM, they may suffer from similar limitations, but this topic has not been explicitly explored as yet.

Practical recommendation: The XPM approximation should be used with caution, as it may substantially underestimate NLI at low symbol rates and/or low dispersion. In general, when used within any model, the accuracy of the XPM approximation should be verified when \(\beta_2R^2\lt1/100\) (km\(^{-1}\)). In addition, if the overall NLI is needed, single-channel effects must be somehow re-introduced, since XPM does not include them.

## 3. Introducing the GN-EGN Model Class

In this section, we provide background information on the GN–EGN model class. We start out with a brief overview of the literature on the GN model and on various prior modeling efforts which are related to the GN model. We then address its early validation and the subsequent observation of its limitations that have eventually led to the formalization of an enhanced-GN (EGN) model.

### 3.1. Getting to the GN Model

Numerous models were proposed prior to the GN model, which bear substantial similarities to the GN model itself. They all rely on first-order perturbation approaches.

To the best of our knowledge, the earliest of these models dates back to 1993. It was based on directly postulating that all non-linearity was produced by FWM acting among the WDM signal spectral components, assumed “incoherent.” This latter assumption is equivalent to the signal-Gaussianity approximation of Section 2.1.4.

Though limited to single-polarization, ideal-distributed amplification and a rectangular overall WDM spectrum, the derived equations were essentially the same as those of the GN model for such idealized system scenario.

In 2003, it was shown that results similar could also be derived using a different perturbation approach. Equations similar to the single-polarization GN model were also independently derived using the truncated VS approach in frequency domain.

More recently, a derivation approach was taken up again, based on ideally slicing up the signal spectrum into discrete spectral components. This “spectral slicing” (SpS) approach naturally lends itself to describing OFDM systems, and in fact it was first used to model NLI limited to OFDM.

There, the generated NLI PSD was found through first-order perturbation dual-polarization analytical FWM formulas, applied to the OFDM subcarriers. This modeling effort obtained what can be viewed as a specialized version of the GN model for OFDM.

The SpS approach was independently exploited to address generic WDM systems. In these papers, SpS was used early in the derivation and then it was removed through a transition to continuous spectra. A first-order regular perturbation method applied to the Manakov equation was used. The dual-polarization general form of the GN model was obtained as a result.

Two further papers proposed detailed re-derivations of the GN model. Specifically, used a variation on the SpS approach while was based on a modified version of the first-order RP method, called enhanced-RP.

Both independently confirmed the GN model equations and provided insightful extensions and generalizations. Various follow-up papers have been published on the GN model, providing further generalizations and numerous approximate closed-form solutions to the GN model reference integral equation formula.

#### 3.1.1. GN Model Analytical Derivation

Having introduced the GN model theoretical and bibliographical background, it would be in order to provide the model formulas and an outline of the model derivation analytical steps.

Since, however, the GN model has recently evolved into the more accurate EGN model, of which it is one of the constituents and with which it shares most of the analytical derivation, we elect to address these aspect in a unified way, for both the GN and EGN models.

The model formulas will be introduced in Section 4.3, whereas an outline of the main derivation steps is provided in Section A.4. We therefore go directly to the topic of the accuracy tests on the GN model, since the observation of some of the GN model accuracy limitations was essential for motivating its evolution into the EGN model.

#### 3.1.2. GN Model Simulative and Experimental Tests

The GN model was proposed specifically to address uncompensated transmission (UT) multi-channel systems, operating at typical commercial system symbol rates (>10 GBaud), over fibers whose dispersion was not too low.

It was deemed inaccurate for DM systems or for UT systems operating near zero-dispersion, and caveats were also put forth regarding its use for single-channel systems. For this reason, its initial simulative validation was targeted at UT multi-channel conditions, at a minimum dispersion value of \(D\) = 3.8 ps/(nm km), corresponding to that of a typical long-haul non-zero dispersion-shifted fiber (NZDSF).

Figure 7.2 shows a comprehensive set of maximum reach results at 32 GBaud (see the figure caption for more system details), where markers are simulations and lines are predictions based on the GN model.

The solid lines refer to the “incoherent” GN model, which makes use of the incoherent NLI accumulation approximation (Section 2.1.8) whereas the dashed ones do not use such approximation.

In both cases the GN model performance is rather good. It holds up well from 1 bit/(sHz) spectral efficiency (SE) and 20,000 km maximum reach to 6 bit/(sHz) SE and 200 km maximum reach, across four transmission formats, three different fibers and six channel spacings.

Similar good agreement was found also when changing the symbol rate and various other parameters, at Nyquist spacing. Simulative tests were run independently by groups not involved in the development of the GN model, who also found good model accuracy.

The GN model has also enjoyed extensive experimental confirmation. Two of these experiments were specifically designed to test the model over different fiber types. In particular, the latter addressed seven fiber types, including a dispersion-compensating fiber (DCF) used as transmission fiber. It was based on a 22-channel WDM PM-16QAM system, running at 15.625 GBaud. Overall, a good match between predictions and experiments was found (see Figure 7.3).

In particular, a 152-channel WDM signal using quasi-Nyquist transmission (32 GBaud, 33 GHz spacing) with PM-QPSK, PM-16QAM or PM-64QAM was transmitted over 60-km spans of high-performance PSCF, and subjected to numerous tests.

Figure 7.4 shows the measured and GN-model predicted Q factor on the center channel (ch. 77) versus distance, for various formats. The results of this experiment also agreed with some of the GN-model predicted general system features, at least within the accuracy of the obtained experimental measurements.

For instance, the prediction that the optimum launch power should be format-independent agreed with the experimental results. In a similar set-up, the GN model prediction regarding the extent of the benefit obtainable through single-channel digital BP (backward-propagation) were experimentally confirmed to be rather accurate.

Despite the ample validation, certain discrepancies between the GN model predictions and certain simulative results were observed. For instance, a paradox is present in Figure 7.2: the GN model curves appear somewhat less accurate than the incoherent GN model ones.

This is puzzling, since the GN model is a less approximate model that the incoherent GN model. It has later been understood that the incoherent GN model benefits from two approximations canceling each other error’s out. In any case, the paradox clearly signaled the problem of some inaccuracy being intrinsic to the GN model.

Issues with the GN model accuracy were first pointed out, at the same time. They were simulatively investigated down to the level of span-by-span NLI accumulation (see Section 3.2).

Another group then independently found large discrepancies (several dBs) between the GN model prediction and the simulated NLI accumulation along the link, when assuming single-polarization lossless fibers or, but to a lesser extent, single-polarization lossy fiber with short spans.

These discrepancies were attributed in all papers to the signal-Gaussianity approximation (see Section 2.1.4). The first paper where a suitable procedure for removing it was outlined, limited to XPM. The full removal of this assumption from all NLI components has eventually led to the EGN model.

One question that needs to be addressed is: why were these GN model issues not detected in experiments? The likely answer is that experiments were mostly run on set-ups having typical realistic parameters, where the difference between the GN model predictions and the actual system performance is modest.

Note that in typical systems the inaccuracy of the GN model can be expected to be similar to what is shown in Figure 7.2, that is, quite small. Given the extent of the possible experimental uncertainty, such differences may well have gone undetected.

Another plausible explanation is the following. As it will be shown in detail in the next section, the GN model always overestimates non-linearity, leading to somewhat pessimistic maximum reach predictions.

In experiments, a number of small impairments often add up causing some penalty versus the expected performance, which may have brought the experimental results close to the GN model predictions.

Yet another aspect is that in most experimental papers the GN model predictions were made neglecting the non-linear effects produced by the co-propagating ASE noise, as well as neglecting channel power depletion.

These are reasonable approximations (see Sections 2.1.3 and 2.1.9) whose impact may however make up, in certain system configurations, for a substantial portion of the distance between the GN model predictions and the actual system performance.

### 3.2. Towards the EGN Model

Figure 7.2 indicates that the GN model returns a somewhat pessimistic maximum reach prediction, especially over lower-dispersion fibers. While the extent of the error has been found to be limited for typical system configurations at 28–32 GBaud (5–15%), it would obviously be desirable to obtain a model that would avoid such error and could therefore be used reliably in less-typical system configurations too. Such model would be useful not only in the analysis and design of current systems, but also in the exploration of possible innovative non-standard system solutions.

Research towards this goal started by analyzing not just the predicted system maximum reach, but the detailed prediction of the generation and accumulation of NLI, span by span, along the link. The first paper that performed such an analysis, by comparing accurate simulation results with the GN model prediction, as mentioned in the previous section.

Some of those results are shown in Figure 7.5. The plotted non-linearity parameter is:

\[\tag{7.4}\eta_\text{NLI}=P_\text{NLI}/P_\text{ch}^3\]

that has dimensions of (1/W\(^2\)).

The advantage of looking at the parameter \(\eta_\text{NLI}\) is that it is independent of the signal launch power used for testing, since \(P_\text{NLI}\propto{P}_\text{ch}^3\). Figure 7.5 shows that over the first few spans, where the signal is certainly farthest from Gaussian-distributed, the GN model (dashed-dotted line) substantially overestimates NLI noise power (black solid line), up to several dB’s. Such overestimation then abates considerably along the link and, in the examples of Figure 7.5, it drops to about 1.3 and 0.9 dB for PM-QPSK and PM-16QAM, respectively.

Note that these NLI errors translate into comparatively smaller maximum reach errors because the relationship between such quantities is approximately:

\[\tag{7.5}\Delta{L}^\text{max}[\text{dB}]\approx-\frac{1}{3}\Delta\eta_\text{NLI}[\text{dB}]\]

where \(L^\text{max}\) is the maximum reach (i.e., the system reach at the optimum launch power) and the symbol \(\Delta\) means the ratio of two values of the same quantity.

So a 1 dB overestimation of NLI power leads to only 1/3 dB (or 7%) maximum reach underestimation. Equation 7.5 is one of the main reasons why detailed NLI accumulation studies are necessary for in-depth model investigations: maximum reach alone is inadequate because it has too weak a sensitivity versus NLI estimation errors.

It was conjectured that NLI overestimation was due to the signal-Gaussianity approximation (see Section 2.1.4) used by the GN model. To test this hypothesis, system simulations were run where a considerable amount of pre-dispersion (200,000 ps/nm in figure, though similar results were obtained with 100,000 ps/nm) was applied, so that the signal started out very dispersed and did essentially behave as Gaussian noise.

In this case (Figure 7.5, light gray solid curves) there was an excellent coincidence between simulations and GN model. So it was surmised that it had to be the non-Gaussian nature of the signal that caused the actual NLI curve to depart from the GN model one, since a Gaussian-distributed signal behaved exactly as the GN model prediction.

Remarkable progress towards overcoming this GN model limitation was made, which succeeded in analytically removing the GN model signal Gaussianity approximation from one of the main contributions to NLI, the cross-phase-modulation one (XPM, see Section 2.1.10).

This result also confirmed the dependence of XPM generation on the fourth moment of the signal constellation, a result that other groups also found within different modeling approaches. Note that the GN model, due to the signal Gaussianity approximation, takes into account the signal constellation second moment only, both for the XPM contribution and for all other non-linearity contributions.

The approach was then extended and generalized, to derive a complete ‘enhanced’ GN model, the ‘EGN’ model, which addresses not just XPM (i.e., the X1 islands in Figure 7.1) but all NLI components (all the islands in Figure 7.1), including SCI.

Interestingly, SCI turned out to involve not just the fourth, but also the sixth moment of the signal constellation. Note that in this chapter the XPM model proposed, which includes the X1 regions exclusively, is called “FD-XPM” (frequency-domain XPM, see also Sect 2.1.10), to distinguish it from other models that were defined as XPM-based.

The accuracy of the EGN model has been since tested simulatively in various configurations. It has, so far, always been found excellent, with maximum reach errors on the order of few percent, at the limit of the accuracy of the computer simulations run for comparison.

In-depth tests carried out at the more error-sensitive level of NLI accumulation (the \(\eta_\text{NLI}\) parameter of Eq. 7.4) have shown the EGN model to supersede the limitations of both the GN model and the FD-XPM model (see Section 2.1.10)

In particular, very low dispersion (such as LS fibers) is handled successfully, as well as single or low span count, single-channel transmission, very low symbol rates (see Section 7.4.4), links with mixed fiber types (SMF and LS were tested), and special conditions found in dynamic reconfigurable networks (see Section 4.5).

A few examples of these NLI accumulation tests are shown in Figure 7.6. The EGN model curve is always much closer (or virtually superimposed) to the simulation results than either the GN model or the FD-XPM approximation.

On the other hand, the price to pay for such wide-scope predictive accuracy is increased complexity. This is not just measured in terms of computing time, but also in terms of the more complex management of the model.

This aspect will be made clearer in Section 4.5, where it will be shown that the EGN model, to be ideally capable of accurately predicting the NLI induced on any CUT propagating through a network, requires to be supplied with the detailed trace of the propagation history of all other INT channels.

This includes their format, where they started interacting with the CUT and where they stopped doing so, as well as how much dispersion they had accumulated when interacting with the CUT. Specifically, the latter is key information, which impacts NLI generation on the CUT in a substantial way.

We should also mention that, at present, the EGN model validation is mostly simulative. There certainly is the need for more targeted experiments, specifically designed for the purpose of confirming the GN–EGN model dichotomy.

The EGN model predictions regarding the existence of an optimum transmission symbol rate were experimentally confirmed. For more details on this topic, see Section 4.4.

At any rate, the field of non-linearity modeling is quite effervescent and though we tried to include the latest results in this section, we encourage the reader to check for possible further developments, which are likely to appear in the near future.

## 4. Model Selection Guide

In this section we provide practical guidance as to which model to use, within the GN–EGN model class, to best tackle certain specific analysis and design needs.

We make a first splitting distinction between PTP links and DRNs. The main difference is that in PTP links it is assumed that the channels travel together from source to destination, whereas in DRNs a certain CUT can change its INTs several times during propagation. This latter circumstance alters the non-linearity picture quite substantially.

Both for the PTP link and DRN cases we will provide a significant “case study.” In PTP links it consists of the determination of the symbol rate minimizing the generation of NLI (Section 4.4). In DRNs it is the analysis of the widely different NLI profile that spectrally identical WDM signals may generate depending on routing assumptions (Section 4.5.1).

### 4.1. From Model to System Performance

Before we discuss NLI modeling, it is necessary to clarify how system performance is calculated and what the model needs to deliver to make it possible to calculate it.

For both PTP links and DRNs, we will accept the CIAGN approximation for NLI (see Section 2.1.5). Then, the system BER can be estimated by inserting the non-linear OSNR of Equation 7.2 into a suitable BER formula, which depends on the transmission format.

For instance, for PM-QPSK, the BER formula is:

\[\tag{7.6}\text{BER}_\text{PM-QPSK}=\frac{1}{2}\text{erfc}\left(\sqrt{\frac{\text{OSNR}_\text{NL}}{2}}\right)\]

The non-linear OSNR of Equation 7.2 must be computed as follows. Note that all PSDs are assumed to be unilateral. \(P_\text{ASE}\) in Equation 7.2 is:

\[\tag{7.7}P_\text{ASE}=G_\text{ASE}(f_\text{CUT})\cdot{R}_\text{CUT}\]

where \(G_\text{ASE}(f)\) is the PSD of ASE noise, \(f_\text{CUT}\) is the center frequency of the CUT and \(R_\text{CUT}\) is the CUT symbol rate. \(P_\text{NLI}\) in Equation 7.2 is given by:

\[\tag{7.8}P_\text{NLI}=\frac{R_\text{CUT}}{B_\text{H}}\int_{-\infty}^{\infty}G_\text{NLI}(f+f_\text{CUT})|H_\text{Rx}(f)|^2df\]

where \(B_\text{H}\) is a normalization factor, defined as:

\[\tag{7.9}B_\text{H}=\int_{-\infty}^{\infty}|H_\text{Rx}(f)|^2df\]

and \(H_\text{Rx}(f)\) is the receiver overall baseband transfer function, which is assumed to be matched to the transmitted signal baseband pulse. Note that, in coherent systems, the DSP adaptive equalizer tends to make \(H_\text{Rx}(f)\) matched, so this assumption appears reasonable. The above formulas also assume that inter-symbol interference (ISI) be absent. Otherwise, a penalty can be expected with respect to their predictions.

Equation 7.8 clearly shows that the key quantity that must be estimated through NLI models is the PSD of the NLI noise, that is, \(G_\text{NLI}(f)\), at least over the bandwidth of the CUT.

### 4.2. Point-to-Point Links

As shown in previous sections, the level of accuracy of the EGN model in characterizing the power of NLI interfering with a certain CUT has been found to be very good in all system situations tested so far, including some non-typical ones. The EGN model would therefore seem to be the natural first pick for the analysis and design of PTP links.

However, such good and reliable performance comes at the price of substantial model complexity. In addition, from its integral equations it is not possible to glean the actual dependence of NLI on many of the key system parameters, something that is of great interest from both a theoretical and a practical viewpoint.

So, we first introduce and discuss the full EGN model. Then, we provide a number of alternatives, found by gradually relaxing the accuracy constraints while gaining in either ease of use, speed of computation or parameter-dependence readability.

A comprehensive list of the symbols used in the following is provided here for convenience. The indicated units make them dimensionally consistent. All PSDs are assumed to be unilateral.

- \(z\) is the longitudinal spatial coordinate, along the link (km).
- \(\alpha\) is the fiber field loss coefficient (km\(^{-1}\)), such that the signal power is attenuated as \(\exp(-2\alpha{z})\).
- \(\beta_2\) is the dispersion coefficient (ps\(^2\)⋅km\(^{-1}\)). The relationship between \(\beta_2\) and the widely used dispersion parameter \(D\) in ps/(nm⋅km) is: \(D=-(2\pi{c}/\lambda^2)\beta_2\), with \(c\) the speed of light in km/s and \(\lambda\) the light wavelength in nm.
- \(\gamma\) is the fiber non-linearity coefficient (W\(^{-1}\)⋅km\(^{-1}\)).
- \(L_s\) is the span length (km).
- \(L_\text{eff}\) is the span effective length defined as: \([1-\exp(-2\alpha{L_s})]/2\alpha\) (km).
- \(N_s\) is the total number of spans in a link, sometimes written \(N_\text{span}\) when necessary for clarity.
- \(G_\text{WDM}(f)\) is the PSD of the overall WDM transmitted signal (W/Hz).
- \(G_\text{NLI}(f)\) is the PSD of NLI noise (W/Hz).
- \(N_\text{ch}\) is the total number of channels present in the WDM comb.
- \(P_n\) is the launch power of the nth channel in the WDM comb (W). The power of a single channel is also sometimes written \(P_\text{ch}\) when necessary for clarity.
- \(R_n\) is the symbol rate of the nth channel (TBaud). The symbol rate of a single channel is also written \(R\), or \(R_\text{ch}\) when necessary for clarity.
- \(T_n=R_n^{-1}\) is the symbol time of the \(n\)th channel (ps).
- \(\Delta{f}\) is the channel spacing, used for systems where it is uniform (THz).
- \(s_n(t)\) is the pulse used by the \(n\)th channel, in time domain. Its Fourier transform is \(s_n(f)\). The pulse is assumed to be normalized so that the integral of its absolute value squared is \(T_n\). If any pre-distortion or dispersion pre-compensation is applied at the transmitter, this should be taken into account in \(s_n(t)\) and \(s_n(f)\).
- \(B_n\) is the full bandwidth of the nth channel (THz). If the channel is Nyquist then \(B_n=R_n\).
- \(f_n\) is the center frequency of the \(n\)th channel (THz).
- \(a_{x,n}^k\), \(a_{y,n}^k\) are random variables corresponding to the symbols sent on the \(n\)th channel at the \(k\)th signaling time, on either the polarization \(\hat{x}\) or \(\hat{y}\); we will assume henceforth that they are all statistically independent of one another and that within each channel they are all equally distributed. They can have different distributions in different channels. Due to the units assumed for the pulses \(s_n(t)\), and the way the overall WDM signal is written in Equation 7.12, then \(|a_{x,n}^k|^2\), \(|a_{y,n}^k|^2\) must have dimensions of power (W). See also Equation 7.13.

### 4.3. The Complete EGN Model

In the following we provide the complete set of analytical formulas expressing theGN and EGN model.

According to the EGN model, the PSD of NLI generated at a certain frequency \(f\) in the optical spectrum is made up of two contributions:

\[\tag{7.10}G_\text{NLI}^\text{EGN}(f)=G_\text{NLI}^\text{GN}(f)-G_\text{NLI}^\text{corr}(f)\]

where the \(G_\text{NLI}^\text{GN}(f)\) contribution is calculated according to the GN model, that is, according to the signal Gaussianity assumption of Section 2.1.4, and the contribution \(G_\text{NLI}^\text{corr}(f)\) accounts for the actual non-Gaussian statistical features of the signal.

Note that for PM-QAM systems of any order, \(G_\text{NLI}^\text{corr}(f)\gt0\). This means that the correction term always detracts from the value of the PSD of NLI found through the GN model. This shows the GN model to be a guaranteed upper bound to NLI for all PM-QAM systems.

The GN model contribution is expressed by the GN model reference formula (GNRF):

\[\tag{7.11}G_\text{NLI}^\text{GN}(f)=\frac{16}{27}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}G_\text{WDM}(f_1)G_\text{WDM}(f_2)G_\text{WDM}(f_1+f_2-f)\cdot|\mu(f_1,f_2,f)|^2df_2df_1\]

where \(\mu(f_1,f_2,f)\) is the “link function” which depends only on the physical link parameters, that is, fiber parameters and amplification features, but not on the characteristics of the launched signal.

Equation 7.11 formally integrates over all frequencies in \(f_1\) and \(f_2\) but in practice its integration domain is shaped by the WDM signal PSD, \(G_\text{WDM}(f)\), whose presence in the integrand function induces a segmentation of the integration domain into many “islands.” Figure 7.1 shows an example of such islands for a 9-channel WDM system with rectangular spectra and quasi-Nyquist spacing.

Regarding the correction term \(G_\text{NLI}^\text{corr}(f)\) in Equation 7.10, again with reference to Figure 7.1, such term is zero over the white-filled islands (the M0 islands). It then takes different formal expressions over the eight other types of islands (SCI, X1-X4 and M1-M3) and therefore \(G_\text{NLI}^\text{corr}(f)\) cannot be expressed through a single simple formula as for the GN model contribution.

Note that the acronyms used in Figure 7.1 refer to the type of non-linear interaction. Specifically:

- SCI: It is NLI caused by the CUT onto itself.
- XCI: It is NLI affecting the CUT caused by the non-linear interaction of the CUT with any single interfering (INT) channel.
- Multiple-channel interference (MCI): It is NLI affecting the CUT, caused either by the non-linear interaction of the CUT with two INTs or by the non-linear interaction of three INT channels.

Before we proceed, we have to introduce some notation, different from that used in Equation 7.11. The reason is that when dealing with \(G_\text{NLI}^\text{corr}(f)\) it is not possible to just look at the WDM signal PSD. Rather, the Fourier transforms of the individual channel pulses are called into play.

The overall WDM transmitted signal is written in time-domain as:

\[\tag{7.12}s_\text{WDM}(t)=\boldsymbol{\sum}_{n=1}^{N_\text{ch}}\boldsymbol{\sum}_k(a_{x,n}^k\hat{x}+a_{y,n}^k\hat{y})s_n(t-kT_n)e^{j2\pi{f_n}t}\]

where the index \(k\) identifies the signaling time-slot, and the index \(n\) the WDM channel. According to the symbol definitions established in Section 4.2, the power carried by each channel is given by:

\[\tag{7.13}P_n=\text{E}\left\{|a_{x,n}^k|^2+|a_{y,n}^k|^2\right\}\]

where E{⋅} is the statistical average (or “expectation”) operator.

We also define the following quantities related to the fourth and sixth moments of the channel constellations:

\[\tag{7.14}{\Phi}_n=2-\frac{\text{E}\{|a_n|^4\}}{\text{E}^2\{|a_n|^2\}},\quad\Psi_n=-\frac{\text{E}\{|a_n|^6\}}{\text{E}^3\{|a_n|^2\}}+9\frac{\text{E}\{|a_n|^4\}}{\text{E}^2\{|a_n|^2\}}-12\]

where \(a_n\) is any of the \(a_{x,n}^k\) or of the \(a_{y,n}^k\), which are assumed to be all identically distributed. The values of \(\Phi\) and \(\Psi\) for some relevant constellations are shown in Table 7.2.

The values of both \(\Phi\) and \(\Psi\) steadily decrease in absolute value when going from a simpler to a more complex PM-QAM constellation. We report their limit values for a QAM constellation made up of infinitely many signal points uniformly distributed within a square region whose center is the origin (the PM-\(\infty\)-QAM entry in Table 7.2).

Note that the values for PM-64QAM are very close to such limit. Also, if the distribution of the transmitted symbols \(a_{x,n}^k\) and \(a_{y,n}^k\) is assumed to be Gaussian and zero-mean, that is, a “Gaussian constellation” is used, then \(\Phi\) and \(\Psi\) vanish and \(G_\text{NLI}^\text{corr}(f)=0\).

We can now return to Equation 7.10. To compute the GN model contribution according to the signal notation introduced above, Equation 7.11 can still be used, with the substitution:

\[\tag{7.15}G_\text{WDM}(f)=\boldsymbol{\sum}_{n=1}^{N_\text{ch}}P_nR_n|s_n(f-f_n)|^2\]

Regarding the correction term \(G_\text{NLI}^\text{corr}(f)\) in Equation 7.10, in the following we show the equations related to the contributions of two specific types of islands, SCI and X1.

The complete set of formulas for all types of islands of Figure 7.1 is reported in Section A.3. Here we show them in a more general form, which can handle arbitrarily different WDM channels.

**The SCI Island**

We assume the CUT to be the \(m\)th channel, not necessarily the center channel in the WDM comb. The SCI island is the center integration region in Figure 7.1 accounting for the effect of non-linearity due to the CUT onto itself. Therefore, there is only one SCI island contributing to the NLI PSD correction term at a given frequency \(f\) within the CUT band, that is, for:

\[\tag{7.16}f\in[f_m-B_m/2,f_m+B_m/2]\]

The formulas for the SCI island contribution to \(G_\text{NLI}^\text{corr}(f)\) are as follows:

\[\tag{7.17}G_\text{SCI}^\text{corr}(f)=P_m^3[\Phi_m\kappa_\text{SCI}^m(f)+\Psi_m\varsigma_\text{SCI}^m(f)]\]

\[\tag{7.18}\begin{align}\kappa_\text{SCI}^m(f)&=\frac{80}{81}\int_{f_m-B_m/2}^{f_m+B_m/2}df_1\int_{f_m-B_m/2}^{f_m+B_m/2}df_2\int_{f_m-B_m/2}^{f_m+B_m/2}df_2'\cdot\\&\quad|s_m(f_1)|^2s_m(f_2)s_m^*(f_2')s_m^*(f_1+f_2-f)s_m(f_1+f_2'-f)\cdot\\&\quad\mu(f_1,f_2,f)\mu^*(f_1,f_2',f)\\&\quad+\frac{16}{81}R_m^2\int_{f_m-B_m/2}^{f_m+B_m/2}df_1\int_{f_m-B_m/2}^{f_m+B_m/2}df_2\int_{f_m-B_m/2}^{f_m+B_m/2}df_2'\cdot\\&\quad|s_m(f_1+f_2-f)|^2s_m(f_1)s_m(f_2)s_m^*(f_1+f_2-f_2')s_m^*(f_2')\cdot\\&\quad\mu(f_1,f_2,f)\mu^*(f_1+f_2-f_2',f_2',f)\end{align}\]

\[\tag{7.19}\begin{align}\varsigma_\text{SCI}^m(f)&=\frac{16}{81}\int_{f_m-B_m/2}^{f_m+B_m/2}df_1\int_{f_m-B_m/2}^{f_m+B_m/2}df_2\int_{f_m-B_m/2}^{f_m+B_m/2}df_1'\int_{f_m-B_m/2}^{f_m+B_m/2}df_2'\cdot\\&\quad{s_m}(f_1)s_m(f_2)s_m^*(f_1+f_2-f)s_m^*(f_1')s_m^*(f_2')s_m(f_1'+f_2'-f)\cdot\\&\quad\mu(f_1,f_2,f)\mu^*(f_1',f_2',f)\end{align}\]

**The X1 Islands**

Given a link with \(N_\text{ch}\) channels, there are \(2(N_\text{ch}-1)\) islands of type X1 contributing to \(G_\text{NLI}^\text{corr}(f)\), two for each INT channel. However, due to symmetries, the two islands related to a single INT produce exactly the same result, so that there are only \((N_\text{ch}-1)\) different contributions to be summed. Assuming again that the CUT is the \(m\)th channel, that is, that \(f\) obeys Equation 7.16, then:

\[\tag{7.20}G_\text{X1}^\text{corr}(f)=P_m\boldsymbol{\sum}_{\begin{split}n=1\\n\ne{m}\end{split}}^{N_\text{ch}}P_n^2\Phi_n\kappa_\text{X1}^n(f)\]

\[\tag{7.21}\begin{align}\kappa_\text{X1}^n(f)&=\frac{80}{81}R_mR_n\int_{f_m-B_m/2}^{f_m+B_m/2}df_1\int_{f_n-B_n/2}^{f_n+B_n/2}df_2\int_{f_n-B_n/2}^{f_n+B_n/2}df_2'\cdot\\&\quad|s_m(f_1)|^2s_n(f_2)s_n^*(f_2')s_n^*(f_1+f_2-f)s_n(f_1+f_2'-f)\cdot\\&\quad\mu(f_1,f_2,f)\mu^*(f_1,f_2',f)\end{align}\]

#### 4.3.1. Practical Remarks

The inspection of the above formulas for the SCI and X1 islands, together with those for the other six island types in Section A.3, clearly demonstrates the challenge posed by the EGN model. Although it can be shown that the triple and quadruple integrals in the formulas can always be reduced to double integrals, the number and diversity of all these contributions make the use of the full EGN model rather difficult.

Note that most of the complexity comes from the correction term. The GN model term, especially if cast in the form of Equation 7.11, is relatively easier to tackle, because it consists of a single comprehensive formula.

In addition, various analytical and semi-analytical results are available for the GN model, either to speed up its integration, or to obtain simpler and even closed-form approximate solutions.

So, if the full EGN model is too complex for a specific application, then it is mostly the correction term that should be targeted for substantial reduction of complexity, through suitable approximations. This will be done in the next sections.

However, it should be mentioned that any approximations to either the GN model or the correction term must be very carefully validated. The reason is that the EGN model PSD of Equation 7.10 is found by subtracting two terms which can be of comparable value. This circumstance greatly amplifies in the final result the possible errors incurred when approximating either one of them.

If instead high accuracy is the target, especially in new untested system configurations, then of course the full EGN model should be used with no approximations.

#### 4.3.2. The EGN-SCI-X1 and EGN-X1 Approximate Models

One reasonable simplifying approximation consists of retaining only the SCI and X1 islands in the calculation of \(G_\text{NLI}^\text{corr}(f)\). This is suggested by the observation, substantiated below, that for typical systems these islands contribute the majority of the correction term.

The caveat is that this is a typical occurrence but not a general rule, and it should be kept in mind that especially for research-type applications where new non-standard systems are explored, the other islands contributing to \(G_\text{NLI}^\text{corr}(f)\) may be significant too. We call this approximate model EGN-SCI-X1.

As an even more drastic approximation, only the X1 islands can be considered for \(G_\text{NLI}^\text{corr}(f)\). This means that the signal non-Gaussianity correction related to SCI is neglected, that is, SCI is overestimated.

Clearly, this approximation is better suited for systems with a large number of channels, where the impact of SCI is smaller. Its inaccuracy also decreases as the channel spacing decreases, since the relative strength of SCI versus the other NLI contributions (XCI and MCI) tends to go down with the spacing. We call this model EGN-X1.

Both these approximations are conservative, in the sense that with PM-QAM systems they deliver a result that is guaranteed to be an upper bound to the NLI PSD delivered by the full EGN model, though a substantially tighter bound than the GN model.

In Fig 7.7 we show an example of the use of the EGN-SCI-X1 and EGN-X1 models. The system data are in the figure caption. The curves plot the NLI power on the center channel, normalized as shown in Equation 7.4, versus the number of spans. The picture shows that the EGN-SCI-X1 model incurs no practically relevant error. As expected, the EGN-X1 model is somewhat pessimistic.

Its NLI overestimation error is 0.5 dB at large span count. This is a very contained error, especially in view of Equation 7.5, which indicates that it would cause less than 4% max reach underestimation. Over NZDSF (not shown) a very similar results is found, with again virtually no appreciable error for the EGN-SCI-X1 model and a modest 0.65 dB error at large span count for the EGN-X1.

In passing, we remark that the simulation curve in Figure 7.7 matches very well the full EGN model, as it has been consistently found over a very wide range of system scenarios.

An intermediate-complexity approximation between the EGN-SCI-X1 and EGN-X1 models is also possible. It consists of the EGN-X1 model with a SCI correction including only the first term of \(\kappa_\text{SCI}^m(f)\) in Equation 7.18, namely the term whose leading factor is 80/81, while discarding the term whose leading factor is 16/81.

Also, the SCI correction contribution Equation 7.19 is neglected entirely. Preliminary results show the retained term to provide by far most of the SCI correction, for symbol rates of 28 GBaud or higher. In Figure 7.7, in particular, this approximation would be indistinguishable from the EGN-SCI-X1 curve.

In summary, when using the approximations proposed in this section, the overall model complexity decreases sharply, versus the full EGN model, and is no longer much larger than that of the GN model.

The practical usability of these models is hence much improved. Of course the EGN-SCI-X1 and EGN-X1 are approximate models and due caution must be exerted not to step out of their boundaries of validity.

A further substantial simplification that leads to a complexity which is comparable to that of the GN model is introduced in the next section. It consists of an approximate closed-form formula for \(G_\text{NLI}^\text{corr}(f)\).

#### 4.3.3. The Asymptotic EGN-X1 Approximate Model

Recently a closed-form approximate formula for the correction term \(G_\text{NLI}^\text{corr}(f)\) has been proposed. Its accuracy improves as the number of spans goes up, achieving an approximate asymptotic convergence versus \(N_s\).

The formula is:

\[\tag{7.22}G_\text{NLI}^\text{corr}(f)\approx\stackrel{G}{\rightarrow}_\text{NLI}^\text{corr}=\frac{40}{81}\frac{\gamma^2P_mN_s\bar{L}_\text{eff}^2}{R_m\pi\beta_2\bar{L}_s}\left(\boldsymbol{\sum}_{\begin{split}n=1\\n\ne{m}\end{split}}^{N_\text{ch}}\Phi_n\frac{P_n^2}{R_n|f_n-f_m|}+\Phi_m\frac{2P_m^2}{R_m^2}\right)\]

with the \(m\)th channel being the CUT.

The arrow underneath \(\stackrel{G}{\rightarrow}_\text{NLI}^\text{corr}\) is a reminder of the asymptotic behavior of the formula. If the channels are all identical and equally spaced, and the CUT is the center channel, then the formula can be re-written as:

\[\tag{7.23}\stackrel{G}{\rightarrow}_\text{NLI}^\text{corr}=\frac{80}{81}\Phi\frac{\gamma^2\bar{L}_\text{eff}^2P_\text{ch}^3N_s}{R^2\Delta{f}\pi\beta_2\bar{L}_s}\left[\text{HN}([N_\text{ch}-1]/2)+\frac{\Delta{f}}{R}\right]\]

where \(\text{HN}\) stands for harmonic number series, defined as: \(\text{HN}(N)=\sum_{n=1}^N(1/n)\).

Equation 7.22 can be used to correct the GN model term and therefore obtain an approximate EGN model, as follows:

\[\tag{7.24}G_\text{NLI}^\text{EGN}(f)\approx{G}_\text{NLI}^\text{GN}(f)-\stackrel{G}{\rightarrow}_\text{NLI}^\text{corr}\]

Because of the assumptions involved in the derivation of Equation 7.22, Equation 7.24 should be considered an asymptotic approximation, versus \(N_s\), of the EGN-SCI-X1 model discussed in Section 4.3.2.

Equation 7.22 has limitations, which impact Equation 7.24 as well. It assumes that the same type of fiber is used in all spans. Spans can be of different length, though: Equation 7.22 uses the average span length \(\bar{L}_s\) and the average span effective length \(\bar{L}_\text{eff}\). Accuracy is quite good for links having all individual span lengths within ±15% of the average. Caution should be used for larger deviations.

Equation 7.22 also assumes that lumped amplification is used, exactly compensating for span loss. Raman amplification can be present, provided that it contributes negligibly to NLI generation.

As a rule of thumb, this is the case if Raman is backward-pumped and the signal power at the end of the span is at least 6 dB lower than it is at the beginning of the span.

It is possible that for very low number of spans (1–3, i.e., completely outside of the range of validity of the formula), in conjunction with very low number of channels (1–3), Equation 7.24 returns negative values.

Equation 7.22 is derived assuming ideally rectangular channel spectra. If spectra have a significantly different shape (such as sinc-shaped), some error may be incurred. Finally, Equation 7.22 neglects the frequency-dependence of \(G_\text{NLI}^\text{corr}(f)\), that is, it assumes that the correction is flat over the CUT bandwidth.

In addition, one of the approximations used in deriving Equation 7.22 assumes that the symbol rate of the CUT channel (which we labeled as the \(m\)th) is not too low. For accuracy to be preserved, the following relation should be satisfied:

\[\tag{7.25}R_m\ge\left|\frac{1}{\pi\beta_2N_sL_s(f_n-R_n/2)}\right|,\quad{n=m\pm1}\]

Note that this constraint becomes less and less stringent for larger span count, span length and dispersion. At ideal Nyquist WDM, with identical channels, the constraint simplifies to: \(R\ge\sqrt{2/(\pi|\beta_2|L_sN_s)}\). Using SMF fiber parameters, 80 km spans and \(N_s=20\), the lower limit for the symbol rate \(R\) is about 4.4 GBaud. Going to NZDSF, it is 8.9 GBaud.

In practice, single-carrier type systems never pose any problems, whereas Equation 7.22 should not be used with either OFDM or multi-subcarrier channels with very low subcarrier symbol rate. Loss of accuracy is quite gradual when the symbol rate approaches the limit.

Incidentally, Equation 7.25, as well as Equation 7.22, have been found to have an important meaning in the context of the study of the dependence of NLI generation on the channel symbol rate.

The accuracy of Equation 7.22 in estimating \(G_\text{NLI}^\text{corr}(f)\) for large \(N_s\) was thoroughly tested. Despite all the above limitations and approximations, it was found to be excellent over a very broad range of system scenarios. Its effectiveness can be appreciated in Figure 7.8, plotted for a 31-channel, 32-GBaud PM-QPSK Nyquist WDM system, over three different fibers. Again, the NLI normalized power \(\eta_\text{NLI}\) defined in Equation 7.4 is represented, versus \(N_s\). The dashed curves are generated according to Equation 7.24. The residual error at large \(N_s\) is negligible.

In summary, the closed-form approximation for \(G_\text{NLI}^\text{corr}(f)\) provided by Equation 7.22 can be a quite effective tool in correcting for most of the GN model overestimation error in most cases of practical interest, adding essentially negligible complexity to the GN model itself. However, the GN model computational burden may still be excessive for certain applications and it may be desirable to reduce it further.

#### 4.3.4. Incoherent NLI Accumulation Models

A substantial problem in the numerical integration of both the GN and the EGN model equations is posed by the link function \(\mu(f_1,f_2,f)\), and in particular by its \(\nu(f_1,f_2,f)\) factor (see Eq. A.5). This latter factor is 1 for a single span but, as the number of spans grows, it displays increasingly sharp peaks which are difficult to integrate numerically.

These peaks may get smoothed somewhat if the spans are not all identical, but they are not eliminated. The question is then whether some suitable approximations may help in dealing with this problem.

In particular, if NLI could be estimated for any value of Ns based on calculations carried out for \(N_s=1\), then \(\nu(f_1,f_2,f)=1\) and such calculations would drastically simplify. We first discuss cross- and multi-channel effects (XCI and MCI), and then deal with SCI.

In identical-span systems, assuming transparency and conventional symbol rates (≥28 GBaud), XCI and MCI (called XMCI for brevity) turn out to accumulate very close to linearly as the number of spans.

In particular, the GN model term for XMCI, which we call \(G_\text{XMCI}^\text{GN}(f)\), grows as \(N_s\) already from the first span, whereas the correction term for XMCI does so asymptotically. In fact, removing the SCI correction from Equation 7.22, we get an asymptotic correction formula for XMCI:

\[\tag{7.26}\stackrel{G}{\rightarrow}_\text{XMCI}^\text{corr}=N_s\frac{40}{81}\frac{\gamma^2P_m\bar{L}_\text{eff}^2}{R_m\pi\beta_2\bar{L}_s}\boldsymbol{\sum}_{\begin{split}n=1\\n\ne{m}\end{split}}^{N_\text{ch}}\Phi_n\frac{P_n^2}{R_n|f_n-f_m|}\]

which grows as \(N_s\).

This circumstance suggests that, regarding XMCI, one could compute the GN model contribution \(G_\text{XMCI}^\text{GN}(f)\) in the first span and then scale it linearly versus \(N_s\). As for the correction \(G_\text{XMCI}^\text{corr}(f)\), its formula inherently scales as \(N_s\).

Regarding SCI, a possible coarse approximation is to calculate the GN model SCI contribution for \(N_s=1\), and then scale it linearly versus \(N_s\), while completely neglecting the SCI non-Gaussianity correction.

In reality, the GN model SCI contribution \(G_\text{SCI}^\text{GN}(f)\) scales super-linearly versus \(N_s\), so that scaling it just as \(N_s\) underestimates it. On the other hand, neglecting the SCI non-Gaussianity correction tends to lead to SCI overestimation, so that these two approximations tend to cancel each other out. At any rate, when the number of channels is large, the impact of these SCI approximations tends to be relatively minor (though non-negligible) because SCI tends to be a minority contribution to the overall NLI.

Pulling all these approximations together, we finally have:

\[\tag{7.27}G_\text{NLI}(f)\approx{N_s}\cdot{G}_\text{NLI}^\text{GN}(f)|_\text{1 span}-\stackrel{G}{\rightarrow}_\text{XMCI}^\text{corr}\]

This formula is well suited for a quick appraisal of identical span systems with large WDM bandwidths and large number of channels. Note that Equation 7.27, as opposed to the EGN-X1 approximation, is not guaranteed to either over- or underestimate the NLI PSD. As a rule of thumb, it can be expected to somewhat underestimate it for low channel count, whereas for high channel count nothing can be said.

One example of the use of Equation 7.27 is shown in Figure 7.9. A quasi-Nyquist-WDM system is considered, made up of 41 PM-QPSK rectangular channels at 32-GBaud, spaced either 33.6 or 50 GHz, over SMF. Once more, the NLI normalized power \(\eta_\text{NLI}\) defined in Equation 7.4 is represented, versus \(N_s\). The dashed line is Equation (7.27), which incurs a 0.5 and 0.65 dB error at \(N_s\) = 50, for \(\Delta{f}\) = 33.6 and 50 GHz, respectively. The error is towards underestimation, which confirms that now there is no guarantee of getting a conservative result. On the other hand the error is modest.

Equation 7.27 can be re-written as:

\[\tag{7.28}G_\text{NLI}(f)\approx{N_s}\cdot\left[G_\text{NLI}^\text{GN}(f)-\stackrel{G}{\rightarrow}_\text{XMCI}^\text{corr}\right]_\text{1 span}\]

where the quantities within square brackets are calculated for one span only.

The formula shows that this approximation inherently implies that the total PSD of NLI at the end of the system is given by the sum in power of the contribution of each single span.

This in turn suggests that such assumption could be generalized towards non-identical span links, using the incoherent accumulation approximation of Section 7.2.1.8, Equation 7.3, resulting into:

\[\tag{7.29}G_\text{NLI}(f)\approx\boldsymbol{\sum}_{n_s=1}^{N_s}\left[G_\text{NLI}^\text{GN}(f)-\stackrel{G}{\rightarrow}_\text{XMCI}^\text{corr}\right]_{n_s\text{th span}}\]

where the quantities within square brackets are calculated for the nth span only. that is, as if the nth span was the only one in the link.

An important caveat is that this heuristic formula is expected to be reasonably accurate if the spans are not very different in either fiber type or length. At present no extensive testing or error appraisal has been carried out for Equation 7.29 in more general cases where spans are substantially different.

Its use in such contexts should be subjected to proper pre-validation for the specific application of interest. Nonetheless, Equation 7.29 could potentially be a possible practical tool to be used in DRNs, as discussed in Section 7.4.5.

Finally, an even simpler and theoretically much coarser simplifying approach consists of entirely neglecting \(G_\text{NLI}^\text{corr}(f)\) in Equation 7.29, obtaining:

\[\tag{7.30}G_\text{NLI}(f)\approx\boldsymbol{\sum}_{n_s=1}^{N_s}\left[G_\text{NLI}^\text{GN}(f)\right]_{n_s\text{th span}}\]

This formula combines the incoherent accumulation approximation of Section 2.1.8 with the signal Gaussianity approximation of Section 2.1.4. The result is an extremely simple model that has been called the “incoherent GN model.” It was the first proposed version of the GN model.

Despite the drastic approximations, it actually delivers rather accurate maximum reach predictions for lumped amplification long-haul systems with moderate number of channels, operating in the 28–32 GBaud range, as extensive simulative tests showed. Good accuracy is also found at the NLI accumulation level for large span count, as shown in Figure 7.5, dashed-dotted line.

The reason for such good predictivity is that the signal Gaussianity approximation and the incoherent accumulation approximation used by Equation 7.30 tend to balance each other’s error out, when the number of channels is moderate.

For very large and very low channel count, this compensation is less accurate, leading to NLI overand underestimation, respectively. Also, some loss of accuracy can be expected when operating at lower symbol rates than 28–32 GBaud, a regime which appears to be of possible future interest according to the results discussed in Section 4.4.

In summary, both Equations 7.29 and 7.30 are rather coarse models, which may incur substantial errors. They cannot be recommended for highly accurate PTP link analysis and design. On the other hand, for certain applications they may be quite attractive, due to their simplicity and specific features. One such application can be DRNs, a topic which will be dealt with in Section 4.4.

#### 4.3.5. Closed-Form Analytical NLI Formulas

With reference to Equation 7.10, various approximate closed-form formulas for \(G_\text{NLI}^\text{GN}(f)\), or for quantities that can be traced back to it, have been proposed over the years. Regarding \(G_\text{NLI}^\text{corr}(f)\) instead, only Equations 7.22 and 7.23 are currently available due to the more recent introduction of the EGN model. It may be foreseen that more closed-form formulas will emerge in the near future for both \(G_\text{NLI}^\text{GN}(f)\) and \(G_\text{NLI}^\text{corr}(f)\).

Due to the limited space available, only two closed-form EGN model approximate formulas will be presented here. The first one is significant as it addresses the limiting case of ideal Nyquist WDM. The second one allows to implement a generic-system quick approximate evaluator.

**• The Nyquist-WDM case with identical spans**

We remind the reader that by “Nyquist WDM” we mean a system where the WDM channels have a rectangular spectrum with bandwidth equal to the symbol rate \(R\), and the channel spacing is equal to \(R\) as well. In practice, the system spectrally looks like a compact rectangle with overall WDM bandwidth \(B_\text{WDM}=N_\text{ch}R\).

For the correction term Equation 7.23 is used. As a result, the following formula can be viewed as a fully closed-form approximation of the EGN-SCI-X1 model of Section 4.3.2, specific for Nyquist-WDM systems with identical spans:

\[\tag{7.31}\begin{align}G_\text{NLI}(f)&=\frac{8}{27}\frac{\gamma^2P_\text{ch}^3N_s}{\pi|\beta_2|R^3}\cdot\\&\left\{\frac{N_s^{\epsilon}}{2\alpha}\text{asinh}\left(\frac{|\beta_2|}{4\alpha}\pi^2B_\text{WDM}^2\right)-\Phi\frac{10}{3}\frac{L_\text{eff}^2}{L_s}\left[\text{HN}\left(\frac{N_\text{ch}-1}{2}\right)+1\right]\right\}\\&\epsilon\approx\frac{3}{10}\log_e\left(1+\frac{3}{\alpha{L_s}}/\text{asinh}\left(\frac{|\beta_2|}{4\alpha}\pi^2B_\text{WDM}^2\right)\right)\end{align}\]

The exponent \(\epsilon\gt0\) accounts for coherent NLI accumulation. Note that for large overall WDM bandwidths \(B_\text{WDM}\), then \(\epsilon\approx0\).

Equation 7.31 has the compounded limitations of Equation 7.23, listed in Section 4.3.3. The latter is essentially that the span loss \(L_s\) should not be too small. A value of 7 dB was indicated but a more conservative figure of 10 dB is advised. Within such constraints, the main source of inaccuracy of Equation 7.31 is the asymptotic convergence of Equation 7.23 versus \(N_s\). The inaccuracy due to the approximations involved in the reduction to closed-form is instead minor.

Despite its limitations, for large-enough number of spans, Equation 7.31 provides a very accurate estimate of the NLI PSD at the center of a Nyquist-WDM comb. In Figure 7.10, we show an example of a 31-channel Nyquist-WDM PM-QPSK system, at 32 GBaud, over SMF (\(L_s\) = 100 km). The asymptotic error for large-enough \(N_s\) is essentially negligible. Similar very good accuracy is found for NZDSF and LS.

**• The generic WDM comb case**

We adopt the incoherent accumulation approximation Equation 7.29 and then we use the closed-form expression for the GN model contribution. For the non-Gaussianity correction term we use Equation 7.26, as prescribed by Equation 7.29. We assume that the same type of fiber is used in all spans. The resulting formula provides \(G_\text{NLI}(f)\) at the center frequency of the \(m\)th channel in the comb:

\[\tag{7.32}\begin{align}G_\text{NLI}(f_m)&\approx\frac{8}{27}G_m\frac{N_s\gamma^2\bar{L}_\text{eff}}{\beta_2}\boldsymbol{\sum}_{n=1}^{N_\text{ch}}G_n^2\psi_{n,m}\\\psi_{n,m}&\approx\text{asinh}\left(\frac{\pi^2|\beta_2|}{2\alpha}[f_n-f_m+B_n/2]B_m\right)\\&\quad-\text{asinh}\left(\frac{\pi^2|\beta_2|}{2\alpha}[f_n-f_m-B_n/2]B_m\right)-\Phi_n\frac{R_n}{|f_n-f_m|}\frac{5}{3}\frac{\bar{L}_\text{eff}}{\bar{L}_s},\quad{n\ne{m}}\\\psi_{m,m}&\approx\text{asinh}\left(\frac{\pi^2|\beta_2|}{4\alpha}B_m^2\right)\end{align}\]

The limitations and constraints are similar to those of Equation 7.31, compounded by those induced by the use of the incoherent accumulation approximation, discussed in Section 4.3.4.

In Figure 7.9, we test Equation 7.32 on two 41-channel PM-QPSK 32-GBaud systems over SMF, with different channel spacing \(\Delta{f}\) = 33.6 or 50 GHz. The error versus the full EGN model at large span count is small: at 50 spans it is −0.2 and negligible for \(\Delta{f}\) = 33.6 and 50 GHz, respectively. Good results are found over NZDSF (not shown) too: −0.7 and negligible, respectively.

However, it must be understood that the high accuracy found in these cases is due in part to error cancellation. In general, possibly more substantial errors of either under- or overestimation could occur. On the other hand, the above results address typical and relevant scenarios. A small error in these cases is a significant result, especially considering that Equation 7.32 is a fully closed-form model.

Assuming that all channels are identical, a much more compact closed-form formula can be written, but the additional constraint of \(R\ge1/(16\sqrt{\beta_2})\) must be satisfied for sufficient accuracy.

### 4.4. Case Study: Determining the Optimum System Symbol Rate

A recent experiment has shown a rather strong maximum reach gain (20%) in long-haul transmission when a single serial-channel (SC) was broken up into either FDM quasi-Nyquist subcarriers. Simulative evidence of a dependence of performance on the per-subcarrier symbol rate has also been found. Investigating the behavior of NLI when changing the symbol-rate of WDM channels then appears to be an interesting case-study.

This case study is very well suited to show the different behavior of three of the modeling approaches mentioned in this tutorial: the GN model, the EGN model and the FD-XPM model (or FD-XPM approximation, see Section 2.1.10). It addresses a key question for PTP links, which is simply formulated as follows:

*given pre-determined total WDM bandwidth, spectral efficiency, spectrum roll-off and modulation format, what is the symbol rate which minimizes NLI generation?*

Note that the above assumptions make the total raw bit rate, conveyed by the overall WDM signal, a fixed constant, too.

We chose to address this case study using the following link parameters: SMF and NZDSF fibers (Table 7.1, att. 0.22 dB/km), span length 100 km, lumped amplification, roll-off 0.05, total WDM bandwidth 504 GHz, total raw bit rate 1920 Gb/s.

The channel spacing is 1.05 times the symbol rate, corresponding to a raw spectral efficiency of 3.81 b/(s Hz). The only free parameter is the number of channels \(N_\text{ch}\) that the overall WDM bandwidth is split into. We look at the accumulated NLI at 30 and 50 spans, for NZDSF and SMF, respectively.

The NLI-related quantity chosen for the study is \(\tilde{G}_\text{NLI}\), defined as the PSD of the non-linear noise NLI falling over the center channel and averaged over it. It is also normalized versus the transmission signal PSD cube, \(G_\text{ch}^3\). In math:

\[\tag{7.33}\tilde{G}_\text{NLI}=\frac{P_\text{NLI}}{R_\text{ch}\cdot{G}_\text{ch}^3}\]

where \(P_\text{NLI}\) is the total NLI power affecting the center channel, as defined in Equation 7.8. The convenient features of \(\tilde{G}_\text{NLI}\), are: it is independent of the power per channel launched into the link; the same value of \(\tilde{G}_\text{NLI}\) for different symbol rates means that the corresponding systems would achieve the same maximum reach.

The results are shown in Figure 7.11. The GN model line is essentially flat, that is, it predicts no change of performance versus the number of channels the total WDM bandwidth is split into.

The EGN model, on the contrary, shows a change, and in particular it shows a minimum, which for SMF and NZDSF is located at about 200 and 70 channels, that is, at about 2.4 and 6.8 GBaud, respectively. These results agree very well with the computer simulations (markers) shown in Figure 7.11. Interestingly, Figure 7.11 also shows that the GN and EGN model tend to come together both at very large and very small symbol rates.

The FD-XPM model does not include single-channel non-linearity. When plotted by itself it generates the curve marked “FD-XPM” in figure. We supplemented it with the SCI contribution calculated through the EGN model, so that a comparison could be carried out.

The plot shows that, as pointed out in Section 2.1.10, the FD-XPM (with SCI) model is accurate at large symbol rates. However, it departs from the EGN model when moving towards low symbol rates. At the optimum \(N_\text{ch}\), the FD-XPM (with SCI) model underestimates NLI by about 5 dB, both for SMF and NZDSF.

Note also that its prediction appears to decrease steadily for \(N_\text{ch}\rightarrow\infty\). The reason for this behavior is explained in Section 2.1.10 and has to do with the neglect by FD-XPM of all integration islands in Figure 7.1, except X1. This can be alternatively stated by saying that FD-XPM takes only XPM into account but neglects all of FWM. Depending chiefly on symbol rate and dispersion, this approximation may be accurate or not.

By using results from the derivation of the asymptotic closed-form correction formula of Section 4.3.3, a closed-form expression of the optimum symbol rate can be derived. Interestingly, it coincides with Equation 7.25. For quasi-Nyquist systems, with all identical spans, Equation 7.25 can be re-written as:

\[\tag{7.34}R_\text{opt}=\sqrt{2/(\pi|\beta_2|L_\text{span}N_\text{span})}\]

This formula indicates that the optimum rate is a function not only of the accumulated dispersion per span \(|\beta_2|\cdot{L}_\text{span}\) but also of the link length through \(N_\text{span}\).

Owing to the square root in Equation 7.34, the range of optimum rates is relatively narrow. It is difficult to push \(R_\text{opt}\) outside of the interval 2–10 GBaud, unless rather extreme scenarios are assumed.

Equation 7.34 was tested using the EGN model and found to be very accurate over a wide range of dispersions and span numbers (at 100 km span length). Another significant result has to do with the approximate asymptotic expression Equation 7.24.

It was indicated in Section 4.3.3 that it starts losing accuracy beyond the symbol rate given by Equation 7.25, which coincides with \(R_\text{opt}\). At that rate, Equation 7.24 is still rather accurate and can therefore be used to carry out approximate NLI calculations with reduced overall complexity.

Regarding the possible practical impact of the results of Figure 7.11, the NLI reduction between the optimum rates and the current industry-standard (32 GBaud, 15 channels in the plots) is 1.2 and 0.7 dB, respectively. According to Equation 7.5, this leads to about 0.4 and 0.27 dB (or 10% and 6%) max-reach increases for SMF and NZDSF, respectively. These numbers are significant but not disruptive.

Nonetheless, these results might still influence future trends. In particular, the general industry push towards higher symbol rates must be weighed versus the greater penalties that are incurred there.

In Figure 7.11, the NLI gap for SMF grows to 2 dB between \(R_\text{opt}\) and 100 GBaud. It is a full 3 dB between \(R_\text{opt}\) = 2.8 and 100 GBaud over 60-km-spans PSCF (plot not shown). This means that, apart from the obvious technological hurdles towards higher rates, there are also fundamental disadvantages to straightforward serial-rate increase, which are going to add up.

As for trying to move to lower rates, to take advantage of lower NLI, implementing this technique by increasing the number of optical carriers is clearly impractical. One solution could be subcarrier multiplexing over a single carrier by means of DSP-DAC enabled transmitters.

It should also be mentioned that for more complex formats than PM-QPSK, the potential NLI mitigation gets reduced. This is due to the correction term \(G_\text{NLI}^\text{corr}(f)\), which is responsible for the appearance of the NLI minimum, getting smaller because of the smaller \(\Phi\) and \(\psi\) coefficients (see Table 7.2).

Qualitatively similar plots to Figure 7.11 are found for instance for PM-16QAM. The general curve shapes are identical and the optimum symbol rates are the same, but the NLI mitigation is smaller: for SMF the drop from 32 to 2.4 GBaud is only 0.66 dB, resulting in about 5% potential max reach increase.

Further investigation is, however, in order, since PM-16QAM is affected more than PM-QPSK by long-correlated non-linear phase and polarization noise (see Section 2.1.5), whose thorough removal might improve the effectiveness of symbol-rate optimization.

Several other aspects of this topic need further investigation. One of them is the variation of the NLI mitigation versus the total WDM bandwidth. For NLI mitigation to be of interest, it must still be significant at C band and, in prospect, even at larger WDM bandwidths. The dependence of the NLI mitigation on the span number where NLI is assessed, as well as on span length and dispersion, also need to be carefully evaluated.

#### 4.4.1. Summary on PTP Links

In the subsections of Section 4 we have presented a range of alternative approaches that can be used to assess NLI in the context of PTP links. The ordering has roughly gone from greater accuracy and complexity towards simplicity and speed, at the cost of loss of accuracy.

For research purposes, the full EGN model is highly recommended. For routine accurate design applications, the EGN-SCI-X1 model is an efficient and still very accurate alternative. For fast preliminary assessment, the asymptotic EGN-SCI-X1 model is attractive, whereas all other presented solutions can be considered when even faster results are needed and accuracy can be relaxed.

### 4.5. NLI Modeling for Dynamically Reconfigurable Networks

As mentioned at the beginning of Section 4, in DRNs each optical transmission channel (or “lightpath”) can be re-routed at each network node and hence, contrary to PTP links, it can change its spectrally neighboring channels, possibly many times. Such neighboring channels can have a different symbol rate, format, and accumulated dispersion.

This complicates drastically the non-linearity modeling problem, since the final amount of NLI impacting any given channel (assumed as the CUT) depends on the detailed overall “propagation history” of the CUT itself and all of its INTs, from source to destination.

The EGN model can be extended to take such propagation history fully into account and deliver a very accurate end result. On the other hand, its complexity, already substantial for the PTP case, is further exacerbated.

In practice, in DRNs there is a need for a fast assessment of physical layer impairments, so that the control plane can enact “physical-layer aware” routing and traffic allocation decisions, essentially “real-time.”

Given this requirement, it is hard to picture the EGN model, made more complex by the need to take into account the propagation history of each CUT and INT, as a practical real-time solution for DRNs.

Besides practicality, there are other reasons why a full-fledged EGN model approach may not be the right option for DRNs. These reasons will be discussed after an initial set of examples is provided, in the next section.

#### 4.5.1. CUT Performance Dependence on INT History

We look at five different link scenarios, that have the following features in common:

- 50 spans of NZDSF (see Table 7.1), 100 km each. Transparency is assumed.
- 41 channels are transmitted, with symbol rate 32 GBaud and 33.6 GHz spacing; all spectra are raised-cosine with roll-off 0.05, all channels are launched with the same power, the total WDM bandwidth is 1.377 THz.
- The spectrum of the WDM signal launched is the same across the five scenarios at every point along the link.

Assuming that the CUT is the center channel in the WDM comb, the scenarios 1– 5 have the specific features:

- The CUT and the INTs are all PM-QPSK, and they propagate together from source to destination.
- The CUT and the INTs are all PM-16QAM, and they propagate together from source to destination.
- The CUT is PM-QPSK and the INTs are all PM-16QAM, and they propagate together from source to destination.
- The CUT and INTs are all PM-QPSK. The INTs are completely replaced every 10 spans with others with identical features but independent data. This mimicks a situation where the CUT is re-routed every 10 spans, changing all of its INTs. The new INTs are assumed not to originate at the CUT routing nodes. For simplicity, it is assumed that all of them have already travelled 10 spans before the CUT joins them.
- Same as #4 but all channels (CUT and INTs) are PM-16QAM.

The NLI accumulation curves for the five scenarios are shown in Figure 7.12. The GN model approximation is shown as a gray solid line. There is only one such line because the GN model prediction is the same for all these scenarios, since the GN model only looks at the PSD of the WDM signal, which is identical. Note also that the GN model curve is pessimistic, that is, it predicts more NLI, in all cases.

Regarding the EGN results, the lowest curve is that of scenario #1, that is, a PTP link with all PM-QPSK channels. The other curves are comprised between this curve and the GN model. In particular, scenario #4 shows that it is important to take the detailed INT history into account.

A comparison of scenarios #1 and #2 shows the impact of changing the format of the INTs. A comparison of scenarios #2 and #3 shows that the format of the INTs is more important in the generation of NLI than that of the CUT itself. Overall, Figure 7.12 shows that various scenarios whose spectrum is everywhere identical along the link may produce rather different NLI curves.

We also did the evaluation of scenarios #1–5 over SMF, and for other channel spacings (37.5 and 50 GHz). With SMF we also tested 60 and 80 km spans. The qualitative appearance of Figure 7.12 is maintained in all these links, including the incoherent GN model curve essentially merging into scenario #5 at large span count.

In an actual DRN, many more situations that are also spectrally identical to these could show up, where the INTs could change more or less frequently and could come into the link with any amount of accumulated dispersions. INTs and CUTs could have any mix of different formats.

Remarkably, it turns out that all the corresponding NLI curves would fall within the relatively narrow region, comprised between the curve of the PTP-like scenario using the lowest-cardinality format and the GN model curve, such as shown in Figure 7.12. The GN model curve is in fact an upper bound for all possible different situations that may present themselves.

Based on this circumstance, an approximate but conservative modeling approach could be that of adopting the GN model. This means that performance prediction would be pessimistic, to some variable degree.

On the other hand, the added complexity required to obtain the accurate EGN model curves shown in Figure 7.12 would be extremely large. Also, it would typically gain a relatively modest improvement in accuracy versus the GN model, considering what Equation 7.5 shows: an improvement of 1 dB in NLI noise estimation accuracy only improves the maximum reach estimation accuracy by about 1/3 dB.

As a result of the above remarks, the approach followed in the remainder of this section on DRNs will be that of taking as reference the GN model, on the basis that it appears to be a reasonable and conservative starting point, which represents a compromise between accuracy and complexity and possibly one of the few practicable alternatives to begin approaching real-time operation. However, not even the GN model itself is sufficiently lean to directly allow it.

#### 4.5.2. Real-Time DRN Physical-Layer Awareness

The GN model still requires to keep track of some of the propagation history of the CUT, namely the features of all the spans traversed by it, as well as the full WDM spectra present in such spans, though the format and propagation history of the INTs is no longer needed.

In addition, it still requires numerical integration, which may be particularly hard to perform due to the presence of the \(\nu\) factor in the link function, as discussed in Section 4.3.4. These requirements place the pure GN model approach still far away from handling real-time. Hence, further approximations are necessary.

An effective simplification strategy is that of combining the incoherent accumulation approximation of Section 2.1.8 with the GN model, resulting in the incoherent GN model of Equation 7.30. Per se, it does not remove the need for numerical integration, but it removes the problematic \(\nu\) factor from the link function.

If numerical integration is nonetheless too heavy, an approximate closed-form formula can be used. In particular, Equation 7.32, with \(\Phi_n\) set to zero, is an effective solution, though others are possible.

As for accuracy, it can be appreciated from Figure 7.12, where the incoherent GN model is shown as a light gray solid line. The interesting result is that the incoherent GN model appears to act as a tighter upper bound than the GN model to the bundle of curves of scenarios #1–5.

In any case, its prediction gets rather accurate, as the span number goes up towards values that are of practical interest for maximum reach. Considering the simplicity of the model and the extent of the approximations, the degree of accuracy is surprising.

Remarkably, the exact same behavior as shown in Figure 7.12 was found in a broader test encompassing 12 test sets (of which Figure 7.12 is one) generated by combining the following options in all possibleways: 15 or 41 WDM channels, NZDSF or SMF, 33.6 or 37.5 or 50 GHz channel spacing, all other parameters as in Figure 7.12.

Further test sets were obtained over SMF at 80 and 60 km spans. The qualitative appearance of Figure 7.12 is thoroughly maintained in all these sets, including the feature consisting of the incoherent GN model curve essentially merging into case #5 at large span count.

Subtler but perhaps crucial advantages of the incoherent accumulation approximation Equation 7.30 are the following:

- – it makes the non-linear effect of each span independent of that of any other span;
- – it makes the overall signal degradation due to ASE and NLI additive along the link.

The first of the above aspects means that, to evaluate each of the summation terms of Equation 7.30, one only needs link features “local” to each particular span. Specifically, one needs the WDM PSD of the signal launched into the span, the characteristics of the fiber used in the span and the span amplification features.

No “non-local” information is needed regarding the path previously travelled by the either the CUT or INTs, not even their respective formats. If NLI is evaluated through Equation 7.32, all that is needed regarding the WDM signal at each span is in fact the center frequencies of each channel \(f_n\), their respective bandwidths and symbol rates \(B_n\) and \(R_n\), and their flat-top PSDs \(G_n\). The latter can typically be approximated as \(P_n/R_n\).

Regarding the second aspect, we point out that under the incoherent accumulation approximation, not just NLI, but the total “signal degradation” due to both NLI and ASE together can be computed fully locally in each span, in the form of the reciprocal of a “span-local OSNR”.

Such signal degradations turn out to be additive, so that the total degradation for a given channel is the sum of its per-span degradations. Being local, the signal degradation can be minimized locally, which is equivalent to maximizing the span-local OSNR. This can be done by intervening locally on the launch power of each channel into that span.

The possibility of performing optimization for each span independently of all others has been called LOGO (for local-optimization, global-optimization). The specific optimization criteria can be varied, and will not be discussed here. We will only focus on one possible strategy, which has the merit of further drastically simplifying network management and ensuring that no lightpath disruption occurs under any circumstances, at the cost of some loss of efficiency. The goal is that of achieving the “optical ether” regime.

#### 4.5.3. The LOGON Approach and the “Optical Ether”

Any span-local OSNR maximization strategy for a given channel would require that the launch power of the channels present in that span be optimized, which in turn requires the detailed knowledge of the spectral loading of that span.

A drastic simplification occurs if such optimization can simply pre-suppose that all channels are always present in the span. In fact, an even more drastic approximation is to assume that the whole available WDM band is always fully and seamlessly saturated.

The details of this approach, called LOGON (for LOGO-Nyquist). One of its advantages is that the local optimization of each span relies on a simple pre-computation which results in a single number: the optimum launch PSD level \(G_n\), which is the same for all channels and which all channels must comply with.

Another substantial advantage of the LOGON strategy is the following. We assume that the CP has routed the channels in the DRN so that they are operational according to the performance predicted under the LOGON full-spectral-load assumption.

Then, the insertion of one or more channels in an already partially populated link cannot cause any disruption nor can it require any re-routing of the channels already present in the link.

This is because the NLI generated by the insertion of new channels was already factored in, to its worst (full-load) case, by the CP. The actual performance of the already present channels would of course degrade, but would be better, or at worst equal, to what the CP has already considered.

The LOGON strategy tends to achieve an operating regime which we would call “optical ether.” The term was used in the late 1980s and early 1990s. In essence, adapted to the current DRN scenario, the concept is that of reducing the complexities of the physical layer to a care-free transparent medium, where any turn-on, turn-off or re-routing of lightpaths is allowed and is without consequences for the already lit paths.

The strength of LOGON is also its main weakness: by always assuming full spectral loading, when a lightpath travels across a sparsely populated network, its degradation is substantially overestimated, possibly causing the CP to enact regeneration when it is not necessary. More sophisticated approaches than LOGON are currently being studied to attenuate this drawback.

The LOGON strategy has been recently used in experimental testbeds with good results in terms of predictivity and overall network optimization.

In general, the use of advanced NLI modeling in the control and management of next generation flexible DRNs is a very active field of research and new strategies and results are appearing at a fast pace.

There is no doubt however that the recent progress in NLI modeling, of which this chapter presents a partial overview, have already dramatically impacted this field and are bound to further significantly influence the future of next generation DRNs.

## 5. Conclusion

The field of non-linearity modeling has been very productive in recent years and will likely be so for the next few. Having been written in the midst of this very prolific and evolutionary phase, this tutorial is probably destined to age quickly and be superseded by a number of advancements.

Hopefully, some of the basic results and remarks reported here will endure and be useful for some extended period of time, at least as primer background material. One of the latest developments, the EGN model, appears to have now superseded the limitations of previous models (such as the GN and XPM models) which suggests that it might take some time for it to obsolesce, though there is no guarantee of this either.

Three very prominent topics, which were only touched on or briefly introduced here, are likely to provide very interesting results in the near future: one is that of the detailed study and the accurate assessment of the actual system impact of the phase-noise component of NLI; another is that of the mitigation of NLI by choosing the optimum per-carrier symbol rate, which appears to be a non-negligible effect which the EGN model allows to study in detail; yet another is that of flexible and reconfigurable-network physical-layer-aware control and management strategies.

Many other exciting topics exist and still more will certainly develop and come to the forefront in the coming years, perhaps having to do with spatial division multiplexed systems, or with non-conventional approaches which deal with fiber non-linear effects by actually exploiting them to increase capacity.

So, our conclusive advice is for the interested readers to keep monitoring this field closely for the new disruptive results which are certain to emerge.

The next tutorial introduces ** optical fibers and fiber optic communications**.