1. Introduction

Space-division multiplexed (SDM) transmission based on multi-mode and multi-core fibers [1–3] offers the possibility to scale the capacity of fiber-optic systems by exploiting spatial diversity. In the following, for simplicity, we will refer generically to spatial modes, leaving the analogy with fiber cores implicit. While in principle the fiber modes do not couple during propagation, in practical fibers some coupling is unavoidable, owing to manufacturing imperfections and deployment-related issues, that are always present. Fiber design can be used to suppress inter-mode crosstalk, as is the case for uncoupled-core multi-core fibers, where the signal transmitted in the individual cores can be received with independent single-mode receivers [2]. In this work we focus on the opposite regime, where all the spatial modes undergo strong mixing during propagation, as is the case for coupled-core multi-core fibers [4]. Strong mode mixing has been shown to be beneficial in reducing the nonlinear signal distortion caused by the Kerr effect [5–8]. On the other hand, it entangles the spatially-multiplexed channels in a way that they must be treated as a spatial super-channel, thus requiring the use of multiple-input-multiple-output (MIMO) digital signal processing (DSP) to disentangle the transmitted signals at the receiver. The MIMO DSP complexity, that is, the number of the required filter taps, depends on the amount of spatial mode dispersion (SMD) experienced by the spatial modes during propagation. Spatial mode dispersion has also been shown to mitigate the accumulation of nonlinear distortions along the fiber [5,6], which makes the modeling of the interplay between the two effects key to the accurate assessment of system performance.

In this context, analytical perturbative models stand out thanks to their simplicity. In the majority of these models, the impairments due to the fiber nonlinearity are treated as additive noise, usually referred to as nonlinear interference (NLI). Of particular interest are the Gaussian noise (GN) model [9] and the nonlinear interference noise (NLIN) model [10], originally derived for single-mode fiber (SMF) transmissions, which provide an estimate of the NLI power as it accumulates during propagation. Contrary to the NLIN model, the GN model provides a worst-case prediction of the NLI variance, derived under the assumption of Gaussian distributed symbols. Modified versions of the GN model have been introduced in the literature to account for the dependence on the modulation format in SMF transmissions [11,12], and are generally called enhanced GN (EGN) models.

The GN and NLIN models have been extended to SDM transmissions in [13] and [14], respectively. However, both works account for the presence of mode dispersion under simplifying assumptions. In particular, the SMD impact on the cross-phase modulation (XPM) power was modeled in [14] for large SMD values while neglecting the effect of SMD within individual frequency channels. Arbitrary values of SMD were first accounted for in [15] for self-phase modulation (SPM) through a scaling of the SPM variance in the absence of SMD. Then, in [16] the dependency of the NLI variance on the link SDM level was characterized under the assumption of Gaussian distributed symbols, thus extending the GN model to the case of systems with strongly-coupled spatial modes in the presence of SMD. The resulting GN model was named the ergodic GN model, as it provides an estimate of the NLI variance, averaged with respect to the statistics of random mode coupling and SMD. In particular, the results in [16] showed the existence of a moderate-SMD region mitigating the XPM variance in strongly-coupled transmissions.

In this work, which is an extended version of [17], we investigate the dependence of the NLI variance on the modulation format in the same regime of operation of [16]. We first tackle the problem numerically to observe how the modulation format affects the variance of XPM for arbitrary levels of SMD. Then, exploiting the mathematical framework of the ergodic GN model [16], we provide a simple formula for a modulation-format-aware estimation of the XPM variance in the presence of mode dispersion, and thus of its impact on the signal-to-noise ratio (SNR). The proposed model can be used to analyze SDM systems with strongly coupled modes using arbitrary constellations with short computation times.

The paper is organized as follows. Section 2 presents the results of the preliminary numerical analysis. Section 3 focuses on the perturbative model and lays the ground for the computation of a modulation-format-dependent contribution to the XPM variance. The XPM formula, which is the key theoretical result of this work, and its numerical validation are discussed in Section 4 and 5, respectively. Finally, in Section 6 we draw our conclusions.

2. Preliminary numerical analysis

The simulations presented in this section are based on the spilt-step Fourier method (SSFM), where we included mode dispersion in the simulations through a fiber waveplate model [18] and collected results for various random waveplate realizations. In each waveplate of length $L_\mathrm {c}$, we used a deterministic differential group delay between polarizations of each mode equal to $\eta _{\mathrm {SMD}}\sqrt {L_\mathrm {c}}$ [19], with $\eta _{\mathrm {SMD}}$ the SMD coefficient. To avoid numerical artifacts, we adjusted the first SSFM step $h_1$ such that the worst-case walk-off with mode dispersion was $T/10$, with $T$ the symbol time, and set the waveplate length equal to $2h_1$. Smaller values of $h_1$ and waveplate length yielded a negligible difference in the results. The step was then updated with a constant local error criterion (CLE) [20].

We considered an optical fiber accommodating $N\!=\!2$ strongly-coupled spatial modes, each with two polarizations. The fiber had a length of $100$ km, attenuation coefficient $0.2$ dB/km, dispersion $D=17$ ps/$(\mathrm {nm\cdot km})$, and nonlinear coefficient $\gamma = 1.26/N$ $(\mathrm {W \cdot km})^{-1}$ [5], equal for all modes. Since we are interested on the characterization of XPM, which is the most relevant nonlinear effect and it is additive in the number of channels [9], we investigated a wavelength division multiplexing (WDM) signal consisting of two frequency channels spaced by $\Delta f = 100$ GHz on each polarization tributary. The data were modulated with either complex Gaussian distributed symbols, or 16 quadrature amplitude modulation (QAM), or quadrature phase-shift keying (QPSK), at a symbol rate of 49 Gbaud with sequences of $131072$ symbols. Nonlinear propagation was modelled by using the Manakov equation, with a Manakov coefficient of $\frac {4}{3}\frac {2N}{2N+1}$ [5]. To isolate the variance of XPM, we used different power levels on the two frequency channels. The frequency channel under test (CUT) had average power $P_{\mathrm {CUT}}=-30$ dBm, while the interfering channel had power $P_{\mathrm {INT}}=0$ dBm. The estimated XPM variance was then scaled to match the XPM variance of a flat power allocation with channel power $P=0$ dBm, hence it was normalized to $P_{\mathrm {CUT}}P_{\mathrm {INT}}^2$ and then multiplied by $P^3$ (in linear scale).

At first, we focused on single-span transmission, where the dependence of the NLI on the modulation format is expected to be stronger, as found for single-mode systems [11,12]. Figure 1(a) shows the XPM variance per polarization tributary as a function of the SMD coefficient. More details on the definition of the SMD coefficient are provided in Appendix C of [16]. The results are represented in the form of a 2D-histogram (also known as a density heat map) extracted from 500 independent waveplate realizations. The histogram of the XPM variance for Gaussian modulation is the same as in [16].

 

Fig. 1. XPM variance vs. SMD coefficient. $N\!=\!2$ strongly-coupled spatial modes, each carrying two frequency channels spaced 100 GHz. Gaussian distributed symbols, 16QAM, or QPSK signaling. Link composed of (a) single-span or (b) twenty spans of 100 km. Shaded regions: 2D-histograms of SSFM results for 500 waveplates realizations.

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The figure shows a strong dependence of the SMD-induced XPM reduction on the modulation format. This is more evident for the QPSK signaling, where the maximum XPM mitigation induced by the SMD is 4.5 dB, hence approximately 3 dB of extra reduction compared to the one observed in the case of Gaussian modulation. Moreover, for all the considered modulation formats, the minimum XPM variance occurs approximately at the same SMD value of $8$ ps/$\sqrt {\mathrm {km}}$, where the effect of mode dispersion within individual frequency channels starts to become relevant, as observed in [16]. It is worth noting that such a value represents a local minimum, as the XPM variance vanishes in the limiting case of SMD $\!\rightarrow \!+\infty$, when SMD induces a complete decorrelation among frequencies participating in the XPM process. For practical systems, the local minimum is of particular interest to alleviate the MIMO complexity.

We then considered the case of longer systems. Figure 1(b) shows that, after $20\times 100$ km, the modulation format still affects the dependence of the XPM variance on the SMD coefficient. Similarly to the single-span case, the difference among the XPM variance histograms still depends on the SMD although it is smaller in absolute values. In particular, the extra XPM mitigation observed at $8$ ps/$\sqrt {\mathrm {km}}$ for QPSK (compared to the Gaussian case) reduces from 3 dB to 0.5 dB.

It is worth noting that the 2D histograms in the longer link show a very small standard deviation, an indication of the fact that in this case the NLI variance only depends on the average interaction between mode dispersion and XPM, thus supporting the ergodic assumptions of [16].

The numerical analysis carried out in this section suggests that the interplay between the modulation format and the mode dispersion can have a significant impact on the XPM variance, motivating a theoretical investigation of the problem in the next sections.

3. Background theory

We consider the propagation of a WDM signal in $N$ spatial modes with two polarization tributaries each. Following the notation of [16], we express the transmitted signal as:

According to perturbative theory [21], the received symbols after matched filtering, zero-forcing equalization of linear effects, and carrier-phase recovery, can be expressed as $u_\mathbf {i} = a_\mathbf {i} + n_\mathbf {i} + w_\mathbf {i}$, where $w_\mathbf {i}$ is the sampled amplified spontaneous emission (ASE) noise introduced by the lumped amplifiers, which are evenly spaced along the link, and $n_\mathbf {i}$ is the sampled NLI affecting the target symbol $\mathbf {i}$. In the most general case, such a sampled NLI can be expressed as follows [16]:

Our target is to assess the variance of the NLI in a generic space and polarization mode, defined as $\sigma ^2_\mathrm {NLI} = \mathbb {E}_\mathrm {a}\left [n_{\mathbf {i}}n_{\mathbf {i}}^{*}\right ]$, where the expected value is taken with respect to the random symbols (note that, in the regime of strong mode mixing, the NLI variance does not depend on the mode under test). Since the Kerr effect is cubic in the symbols, such expectation involves six-order moments of symbols. Some simplifications are possible since only combinations with an equal number of conjugate/non-conjugate pairs yield a non-zero average. The resulting expression of $\sigma ^2_\mathrm {NLI}$ is given by the master theorem in [22]:

In the particular case of XPM, only two frequency channels are involved in the interaction. It can be seen that the only combinations of frequency channel indexes that do not violate the principle of energy conservation are $\mathbf {kmni}_2\!=\!\left (\mathbf {kkii}_2,\mathbf {kiki}_2\right )$, where $\mathbf {kmni}_2$ is a shorthand notation for $(k_2,m_2,n_2,i_2)$. As a consequence, the HON contribution and the term $\left |{\cal X}_{\mathbf {nkki}}\right |^{2}$ in the FON contribution in Eq. (4) do not participate in XPM [11]. Therefore, the modulation format-dependent behavior observed in Sec. 2 is to be attributed to the following FON contribution to the XPM variance:

While the modeling of the XPM variance of the SON contribution in the presence of SMD was addressed in the ergodic GN model [16], in this work we focus on the XPM-FON term. To this aim, we start by manipulating the generic tensor expression in Eq. (3) in the Fourier domain. In particular, the Fourier transform of the shaping function at coordinate $z$ is given by:

Owing to the presence of the mode coupling matrix, the kernel is a random variable. It was shown in [16] that the link kernel expression can be significantly simplified by adopting an approximated model for mode dispersion, which neglects the effect of SMD within individual frequency channels. Such an approximation is equivalent to assuming that the random mode coupling is constant within the bandwidth of a frequency channel. According to jargon introduced in [16], we refer to such approximation as inter-channel SMD. The idea is illustrated in Fig. 2 for a single spatial mode and two frequency channels, for the sake of simplicity. Drawn in the figure by means of arrows is the state of polarization (SOP) at four generic frequencies. Figure 2 (a) represents the general SOP changes from frequency to frequency, while Fig. 2 (b) sketches the inter-channel SMD idea by imposing equal SOPs within a frequency channel. In the following, we adopt the inter-channel SMD approximation to derive an expression for the XPM-FON variance.

 

Fig. 2. Sketch of frequency-dependent SOPs in a simple scenario with a single spatial mode and two frequency channels. The SOPs depolarization is shown in (a) for the general case and (b) for the inter-channel SMD approximation.

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4. Simplified model with inter-channel SMD

According to the inter-channel SMD approximation, the transfer matrix $\mathbf {U}(z, \omega )$ in the bandwidth of a generic frequency channel is forced to coincide with its value at the channel carrier frequency. For instance, let us consider XPM between two frequency channels indexed $i_2$ and $k_2$ with spacing $\Delta \omega = 2 \pi \Delta f$, and take $\Omega _{i_2} = 0$ without losing generality. From Eq. (9) specialized to XPM, that is either $(k_{2}=m_{2}\neq i_{2},n_{2}=i_{2})$ or $(k_{2}=n_{2}\neq i_{2},m_{2}=i_{2})$, we infer that the sampling induced by the inter-channel SMD approximation is either $\left (\omega _{1},\omega _{2}, \omega \right )\approx \left (0,\Delta \omega,0\right )$ for the index combination $\mathbf {kmni}_{2}=\mathbf {kkii}_{2}$, or $\left (\omega _{1},\omega _{2},\omega \right )\approx \left (\Delta \omega,0,0\right )$ for $\mathbf {kmni}_{2}=\mathbf {kiki}_{2}$.

Such an inter-channel SMD approximation brings simplifications in the matrix products in Eq. (10) leveraging the unitary property of the coupling matrix and the assumption $\mathbf {U}(z,0) = \mathbf {I}$. The link kernel for the only two valid index combinations $\mathbf {kmni}$ of the XPM-FON in Eq. (6) takes the following expression:

We note that the dependence on mode dispersion disappears in Eq. (11) thanks to the applied inter-SMD approximation. In fact, it can be seen in Eq. (10) that the product of coupling matrices $\mathbf {U}$ reduces to the identity matrix for such a term, and thus the Kronecker’s delta $\delta _{i_3n_3}$ appears owing to the property $\langle{i_{3}|\mathbf{I}|n_{3}}\rangle = I_{i_3,n_3}$, with $I_{i_3,n_3}$ indicating the $(i_3,n_3)$ element of the matrix $\mathbf {I}$. Therefore, mode dispersion affects XPM-FON only through the configuration $\mathbf {knki}$, where only two coupling matrices remain after the inter-channel SMD approximation, thus relaxing the complexity of the problem.

The above kernel equations must be combined with the tensor expression in Eq. (8) specialized for the two valid index combinations. The resulting tensor can finally be used to evaluate the XPM-FON variance through Eq. (6). We thus proceed by applying the idea of the ergodic GN model [16] to the analysis of the XPM-FON variance, further motivated by the results observed in Fig. 1. Hence, we concentrate on the average value of Eq. (6) with respect to the random mode coupling. Let us analyze the expectation of each term in Eq. (6). The term $\sum _{\mathbf {k,n}}\left |{\cal X}_{\mathbf {kkni}}\right |^2$ is deterministic, being based on the link kernel in Eq. (11). Similarly, also the terms $\sum _{\mathbf {k,n}}{\cal X}_{\mathbf {knki}}{\cal X}^{*}_{\mathbf {kkni}}$ and $\sum _{\mathbf {k,n}}{\cal X}_{\mathbf {kkni}}{\cal X}^{*}_{\mathbf {knki}}$ are deterministic. This is visible by observing that the Kronecker delta in Eq. (11) forces $n_3=i_3$, hence, thanks to the unitary property of matrix $\mathbf {U}$: $\sum _{\mathbf {k,n}}{\cal X}_{\mathbf {knki}}{\cal X}^{*}_{\mathbf {kkni}}\propto \sum _{k_3} |U_{i_3,k_3}(z,\Delta \omega )|^2 = 1$. As a consequence, the only random term in Eq. (6), hence affected by mode dispersion, is $\sum _{\mathbf {k,n}}\left |{\cal X}_{\mathbf {knki}}\right |^2$, whose average value can be written as:

We recall that the sum over $(k_2,n_2)$ spans the frequency channels, while the sum over $(k_3, n_3)$ the hyper-polarization states. Note also that the summation with respect to the temporal indexes $(k_1, n_1)$ has been removed by using the Poisson formula, as well as two frequency integrals out of the overall six [22, Appendix B]. The expectation in the integral can be expressed as follows:

Regarding the remaining contributions in Eq. (6), i.e., those independent of mode dispersion, they can be evaluated using the corresponding index combination in Eq. (13), with the result:

4.1 Simplified formula of the XPM-FON variance

The ergodic XPM variance is the expectation of the XPM variance with respect to the random coupling statistics, and thus can be expressed as a combination of two main contributions:

For two frequency channels spaced by $\Delta \omega$, the obtained expression of the per-polarization XPM variance of the FON contribution can be expressed as:

Here $\eta _0$ is the link kernel in the absence of SMD whose expression is [9]:

Note that the ergodic XPM-FON variance expression in Eq. (20) has a structure similar to its GN counterpart in Eq. (22) of [16], with a first term being SMD-independent and a second term including the SMD as an extra attenuation. However, the terms have different weights.

In the two limit regimes of large and negligible SMD, Eq. (20) coincides with the FON expression derived asymptotically in [14], where Eq. (21) is related to the coefficient $\chi _2$ in Eqs. (18)–(19) of [14] as $\sigma ^2_{\mathrm {XPM,FON,1}} = |\kappa _2|\kappa _1 \frac {4}{5} \chi _2$.

A complete analytical expression of the FON correction to XPM can be derived by using a closed-form expression available for a single-mode system. In the case of systems with many spans ($N_s\gg 1$), the formula obtained in [24] gives the result:

5. Numerical results

We assess the accuracy of the proposed simplified formula for the XPM-FON variance by comparison with SSFM simulations. To this aim, we first extract the FON contribution from the QPSK and 16QAM simulation results shown in Fig. 1 by exploiting Eq. (19), where the GN contribution corresponds to the results obtained with the Gaussian modulation.

The XPM variance of the QPSK and 16QAM signaling, along with their GN and FON contributions, are reported in Figs. 3(a),(c) and Figs. 3(b),(d), respectively, for the single-span (top row) and the 20-span links (bottom row). All system parameters are the same as in Fig. 1. The markers identify the average values over the 500 different random realizations of the fiber waveplates, and the bars indicate the maximum and minimum XPM variance observed in the simulations. It can be seen from the numerical results that mode dispersion has a smaller impact on the FON variance compared to the GN variance, with a maximum excursion of 0.5 dB. The impact of the FON correction is maximum when it is closer to the GN variance, which occurs at SMD of $8$ ps/$\sqrt {\mathrm {km}}$, thus explaining the extra XPM variance reduction in the QPSK and 16QAM curves with respect to the GN. This phenomenon is less evident for the 20-span link, yet still present, due to the smaller relative importance of the FON correction, compared to the GN term.

 

Fig. 3. XPM variance vs. SMD coefficient after one (a)-(b) and twenty spans (c)-(d). Markers with error bars: SSFM results for 500 different random realizations of the waveplates. The triangles and the circles represent the GN and FON contribution, respectively, to the XPM variance for the case of QPSK (squares) and 16QAM (pentagrams) signaling. Solid lines: semi-analytical model, with the GN contribution evaluated by using Eqs. (17)–(21) of [16] and the FON contribution using Eqs. (20)–(21). Dotted lines: results obtained with the FON closed-form expression in Eqs. (20),(23).

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Figure 3 also shows by solid lines the semi-analytical estimations of each variance contribution. In particular, we computed the GN contribution through the ergodic GN model using Eqs. (17)–(21) of [16], and the FON with Eq. (20) of this work by using Monte Carlo integration [23,25] of Eq. (21). The second order cumulant in Eq. (21) is related to the per-polarization power through $\kappa _2/P_\mathrm {p}^2 = -1$ and $\kappa _2/P_\mathrm {p}^2 = -0.68$ for QPSK and 16QAM, respectively. The total variance is then the difference between the GN and FON terms according to Eq. (19). We note that the approximated inter-channel SMD model yielding the simplified formula in Eq. (20) captures well the impact of mode dispersion on the XPM-FON variance. As a result, the proposed XPM-FON formula provides an accurate estimation of the XPM variance for both QPSK and 16QAM signaling. The dotted curves in Figs. 3(c)-(d) are a plot of the results obtained with the closed-form expression of the XPM-FON variance, obtained by using the asymptotic formula Eq. (23) into Eq. (20). The excellent agreement with the semi-analytic result indicates the high accuracy of the closed-form approximation (the closed-form formula is not reported in Figs. 3(a),(c) since $N_\mathrm {s}=1$ is outside its range of validity).

To investigate the scaling properties of the maximum XPM reduction induced by SMD, we exploited the proposed semi-analytical model. We define the quantity $\Delta \mathrm {XPM}$ as the difference between the XPM variance, in dB scale, at $8$ ps/$\sqrt {\mathrm {km}}$ and its value in the absence of SMD. Figure 4(a) shows the dependence of $\Delta \mathrm {XPM}$ on the number of spans, for the settings of Fig. 3. The solid lines exploited Eq. (21), while the dotted lines the closed-form in Eq. (23). It can be seen that, for 16QAM and QPSK signaling, the reduction tends to decrease with the number of spans in the solid lines and eventually saturates. Nevertheless, the asymptotic result is different compared to the case of Gaussian modulation, thus suggesting the importance of the FON contribution even for long-haul links. As already commented in Section 4.1, the closed-form results are reliable only at large span count. Then, we considered a fixed link length of $20\times 100$ km and we increased the number of spatial modes. The results, which are reported in Fig. 4(b), show that the interplay between the modulation format and the mode dispersion is more evident for $N\!\gg \!1$, with a gap-to-Gaussian in the XPM reduction that achieves 1 dB at $N=16$. This is due to the opposite trend of the XPM-FON reduction (common to both modulation formats), which makes the FON impact more effective when $N\!\gg \!1$.

 

Fig. 4. XPM variance reduction at $8$ ps/$\sqrt {\mathrm {km}}$ wrt the absence of SMD. Two frequency channels spaced 100 GHz, and variable modulation format. The left panel shows the dependence of XPM on the number of spans in the case of two spatial modes, while the right panel shows the scaling with the number of spatial modes along a 20-span link. Solid lines: results obtained from Eqs. (20)–(21). Dotted: results obtained with the FON closed-form expression in Eqs. (20),(23).

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To further test the proposed model, we varied the frequency spacing between the two WDM channels, and the results are shown in Fig. 5. In Fig. 5(a) we focused on QPSK signaling in a 10-span link and varied the SMD coefficient for selected channel spacing values. The curves confirm the existence of a local minimum in the XPM variance for an SMD coefficient of $\approx 8$ ps/$\sqrt {\mathrm {km}}$, regardless of channel spacing. Then, in Fig. 5(b) we fixed the SMD coefficient to such a relevant value of $8$ ps/$\sqrt {\mathrm {km}}$ and tested the impact of channel spacing at different link lengths, finding excellent agreement between the semi-analytical estimation and SSFM simulations, which exhibit infinitesimal error bars.

 

Fig. 5. QPSK transmission with two frequency channels and $N=2$ spatial modes. (a) XPM variance (normalized to its value without SMD) vs. SMD coefficient in a 10-span link at different frequency spacing. (b) XPM variance vs. channel spacing at SMD coefficient of $8$ ps/$\sqrt {\mathrm {km}}$ for 1 and 10 spans. Lines: theory in Eqs. (20)–(21). Markers: SSFM simulations.

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Finally, we investigated the transmission of $N_\mathrm {ch}$ WDM channels on each polarization of a spatial mode, spaced by 75 GHz and modulated at 64 Gbaud with QPSK symbols. In Fig. 6 we report with markers the NLI variance estimated through SSFM simulations as a function of the number of WDM channels. The link length was $20\times 100$ km. We considered an SMD coefficient of $3$ ps/$\sqrt {\mathrm {km}}$ and $8$ ps/$\sqrt {\mathrm {km}}$, and compared the results with those obtained in the absence of mode dispersion. Figure 6 shows that the impact of SMD is more beneficial in mitigating the NLI variance when the WDM comb is populated with more frequency channels.

 

Fig. 6. NLI variance vs. number of WDM channels at 75 GHz spacing modulated with 64 Gbaud QPSK signals. $N=2$ strongly-coupled spatial modes. $20\times 100$ km link. Markers: SSFM simulations average value. Lines: theory.

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Figure 6 also shows the theoretical estimation of the NLI variance through solid lines. Here we evaluated the GN contribution by including also SPM, FWM, and cross-channel interference (XCI) combinations [10,11,25] according to the complete model in [16]. We included the XPM-FON contribution according to Eqs. (20),(21), and the FON and HON parts of SPM through the heuristic scaling with SMD of [15,16]. The excellent agreement between theory and simulations is self-evident and it comes with computational times that are orders of magnitude smaller than SSFM simulations.

6. Conclusion

We investigated the impact of the interplay between mode dispersion and the modulation format on the NLI variance in a space-division multiplexed system supporting strongly-coupled spatial modes. We first performed a numerical analysis based on SSFM simulations which showed that the role of the modulation format in setting the NLI variance depends on the mode dispersion. In particular, the numerical results showed that the XPM variance reduction, brought by the presence of mode dispersion, is enhanced for lower cardinality constellations showing very limited randomness with respect to the random coupling realizations.

Consequently, we concentrated on its average value by extending the ergodic GN model in [16] to account for the effect of mode dispersion on the format-dependent contribution of the XPM variance, i.e., the FON contribution. A key result of this work is an expression of the average XPM-FON variance in Eq. (20). Such an expression can be computed numerically, for instance, by Monte Carlo integration in a few seconds or approximated by a closed-form as per Eq. (23) for a high span count.

Finally, we tested the accuracy of the proposed model against SSFM simulations reporting a good agreement between the estimated average values. Therefore, the formula derived in this work can be combined with the ergodic GN model in [16] and leveraged for quickly assessing the SNR of SDM strongly-coupled systems.