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An Efficient Correlation-based Evaluation of the GN-Model for General Multi-Span Optical Systems

Y. Jiang, Y. Gao, P. Poggiolini

arXiv:2610.11624v1eess.SP

TL;DR

Accurate, efficient NLI evaluation remains challenging for multi-span optical systems with heterogeneous spans and nonidentical channels. The paper introduces cGN, a correlation-based GN-model formulation, and reports that it reproduces benchmark results with high accuracy while running more than 2000× faster.

  • Problem

    Brute-force GN-model evaluation is computationally inefficient, while faster approximations may lose accuracy in systems without conventional simplifying conditions.

    Source: §I · p. 1

  • Method

    The cGN formulation decomposes NLI into single-span and cross-span contributions described by spatial auto- and cross-correlations or corresponding energy spectra.

    Source: §I · pp. 1–2, §VI · p. 13

  • Results

    Across tested configurations, cGN reproduced NLI spectra and coherent multi-span accumulation; in the heterogeneous 150-channel C+L+S case, maximum absolute difference was 0.098 dB and speedup exceeded 2000×.

    Source: §VI · p. 14

  • Takeaways & Limitations

    The formulation provides an efficient and physically transparent approach to NLI evaluation in general multi-band optical systems.

    Source: §VI · p. 14

Abstract

from arXiv · show

Accurate and computationally efficient evaluation of non-linear interference (NLI) remains challenging in multi-band WDM systems comprising heterogeneous spans and nonidentical channels. In such systems, channels may have distinct spatial power profiles and multi-subcarrier structures. Spans may use different fiber types with span-dependent dispersion and loss profiles. These systems may further employ general amplification schemes, including forward and backward Raman amplification. Inter-channel stimulated Raman scattering and multiple lumped gain or loss elements may also be present. In this encompassing framework, we provide an efficient decomposition of the GN-model NLI contributions. We show that the NLI power spectral density at any frequency across the WDM comb is fundamentally driven by the spatial auto- and cross-correlations of the channels' spatial power profiles or, equivalently, by the corresponding energy spectra. For this reason we call this formalism the `correlation GN-model', or cGN. Once these correlations or spectra have been computed, the remaining calculations reduces to a fast, well-behaved one-dimensional numerical integral. Moreover, cGN fully accounts for coherent interference of NLI in every pair of spans. Such coherence is often neglected or approximated in existing analytical and semi-analytical GN-model formulations, although it can increase the NLI by several dB. In short, the cGN framework applies to any general systems over any band and any bandwidth, with very high computational efficiency. It is capable of providing the frequency spectrum of NLI at multiple frequencies across each channel or subcarrier. Its accuracy is very high because virtually no approximations are invoked. The cGN framework is available as freely downloadable software from the European Union Zenodo website.

I. INTRODUCTION · II. GN MODEL AND SPAN-PAIR DECOMPOSITION · A. Integration domains and FWM efficiency

The cGN framework makes GN-model NLI evaluation efficient while retaining frequency-dependent integration domains and coherent span interactions. It supports heterogeneous channels and spans, including Raman and ISRS effects.

I. INTRODUCTION

  • The formulation combines correlation-based evaluation, arbitrarily accurate rectangular island covering, and arbitrary-frequency NLI evaluation for SCI, XCI, and MCI.It handles channels with arbitrary symbol rates, including multi-subcarrier channels.Source: §I · p. 1
  • Single-span NLI depends on spatial-profile autocorrelations, while cross-span NLI depends on span-pair cross-correlations and accumulated-dispersion phase shifts.The representation supports span-dependent fiber parameters and spatial power profiles shaped by attenuation, ISRS, distributed Raman amplification, and lumped gain or loss.Source: §I · pp. 1–2
  • The four nested GN-model integrals can be reduced either to a one-dimensional frequency-product integral or to a one-dimensional spatial-correlation integral.The first route performs space integrations before frequency integrations; the second evaluates frequency integrations analytically using a coherence-response function.Source: §I · p. 2
  • At better than 0.1 dB NLI-power accuracy versus the whole-link GN-model benchmark, cGN is about 2000x faster.Validation spans single- and multi-channel systems, frequency-resolved spectra, low- and conventional-dispersion regimes, nonuniform channels, Raman amplification, ISRS, and coherent multi-span accumulation.Source: §I · p. 2

II. GN MODEL AND SPAN-PAIR DECOMPOSITION

  • The GN-model reference formula gives NLI PSD at any frequency, with computational complexity driven chiefly by coherent FWM efficiency and integration-domain geometry.The formula is the starting point for the span-pair decomposition.Source: §II · p. 2

A. Integration domains and FWM efficiency

  • At an offset of −50 GHz, NLI PSD is approximately 5–6 dB below its channel-center value because both the integration domain and FWM efficiency change with frequency.Domains shift, clip, and become increasingly asymmetric away from channel center, so treating NLI PSD as flat across a channel can be inaccurate, particularly for low-dispersion and multi-subcarrier systems.Source: A. Integration domains and FWM… · pp. 2–3
  • For nonidentical channels, irregular integration islands can be covered by non-overlapping rectangles to any desired accuracy.Within each rectangle the interacting PSDs are treated as constant; refining the partition controls error, especially near spectral roll-off regions.Source: A. Integration domains and FWM… · p. 3

B. Span-pair decomposition

The section decomposes coherently accumulated NLI into same-span (SS) and cross-span (XS) contributions. It retains both in a unified framework because residual XS coherence can matter when accumulated dispersion does not rapidly decorrelate fields from different spans.

B. Span-pair decomposition

  • SS contributions arise within a span, while XS contributions arise from coherent interaction between distinct spans; SS terms are generally dominant and XS terms usually provide a correction.Existing formulations may approximate XS contributions, but residual XS coherence can remain significant in low-dispersion and/or low-baud-rate systems.Source: B. Span-pair decomposition · p. 4
  • The link’s NLI is expressed as a coherent sum of span-wise contributions, with each span’s NLI stimulus profile representing effective nonlinear-field generation along that span.The profile reflects both local optical power and the nonlinear coefficient, so regions with more signal power or a larger nonlinear coefficient contribute more strongly.Source: B. Span-pair decomposition · p. 5
  • Expanding the squared modulus yields a sum over span pairs: diagonal terms are SS contributions, and off-diagonal terms are XS contributions.The section identifies efficient evaluation of the core integral K(m,n) as the key step toward a closed-form or semi-analytical model.Source: B. Span-pair decomposition · p. 5

III. SEMI-ANALYTICAL EVALUATION OF SINGLE-SPAN CONTRIBUTIONS · A. Method 1: z-integral first

The section derives two equivalent semi-analytical representations of single-span contributions by changing the order of frequency and space integration. Method 1 evaluates space integrations first, reducing the calculation through spatial autocorrelation and energy spectra to a single integral.

III. SEMI-ANALYTICAL EVALUATION OF SINGLE-SPAN CONTRIBUTIONS

  • Two equivalent semi-analytical representations evaluate the core factor by integrating over space first or over the rectangular frequency domain first.Source: §III · p. 5

A. Method 1: z-integral first

  • A spatial change of variables maps the span’s square integration domain to a diamond, while extending the NSP by finite support makes the inner integral its autocorrelation.Source: A. Method 1: z-integral first · p. 5
  • For an exponential SPP, autocorrelation decreases monotonically with lag; a Raman-shaped profile from the 75th channel instead shows a secondary increase from overlap between high-power regions.Source: A. Method 1: z-integral first · p. 6
  • Because the energy spectrum is the Fourier transform of the autocorrelation, it peaks at perfect phase matching; increasing phase mismatch causes destructive cancellation and reduced NLI generation.Source: A. Method 1: z-integral first · p. 6
  • At |Ω| ≳ 2 km^-1, the energy spectrum is at least 30 dB below its maximum in both cases; the complicated profile exhibits interference peaks and nulls with period ∆Ω ∼ 0.074 km^-1 [30].Source: A. Method 1: z-integral first · p. 6
  • Evenness lets arbitrary rectangular domains be reconstructed from a first-quadrant primitive, yielding a single-integral expression whose possible logarithmic singularity at zero is integrable.Source: A. Method 1: z-integral first · pp. 6–7

B. Method 2: f-integral first

The f-integral-first method expresses rectangular-island coherence through a response function of accumulated dispersion difference, enabling a semi-analytical self-span expression and characterizing coherence across SCI-, XCI-, and MCI-like islands.

B. Method 2: f-integral first

  • The regions z1 > z2 and z1 < z2 contribute complex conjugates, so their imaginary parts cancel and only the real part contributes to self-span NLI.Source: B. Method 2: f-integral first · p. 7
  • At B = 0, Frect(B) is maximal; as |B| increases, phase variation over the rectangle grows and coherent contribution decreases.B = 0 occurs at z1 = z2 or when β(m)_2,eff,rect = 0.Source: B. Method 2: f-integral first · p. 7
  • Island 1 has a broad positive, slowly decorrelating response; Island 2 shows zero crossings and negative side lobes; Island 3 oscillates fastest.Negative response values indicate destructive coherent addition, not negative NLI power.Source: Figure 8 · p. 8, B. Method 2: f-integral first · p. 8
  • Because Frect(B) has a closed form and B depends only on z1 − z2, the self-span contribution reduces to a semi-analytical expression.Equations (30) and (42) are equivalent representations; the preferred one depends on whether NSP energy spectrum or spatial autocorrelation is more convenient to compute and reuse, and both are extended to XS contributions next.Source: B. Method 2: f-integral first · p. 8

IV. SEMI-ANALYTICAL EVALUATION OF CROSS-SPAN CONTRIBUTIONS

Cross-span (XS) contributions describe coherent interference between NLI fields generated in distinct spans, with K(m,n) as the core factor. The evaluation accommodates differing span parameters and power evolutions, and can integrate in either space-first or frequency-first order.

IV. SEMI-ANALYTICAL EVALUATION OF CROSS-SPAN CONTRIBUTIONS

  • XS contributions describe coherent interference between NLI fields generated in distinct spans, represented by the core factor K(m,n).The passage introduces NLI,rect(f) in Eq. (12).Source: §IV · p. 8
  • Span parameters and power evolutions may differ, while the phase factor involving Bm − Bn accounts for relative accumulated dispersion between spans.Source: §IV · p. 8
  • Only pairs with m < n need evaluation, and the integral can be computed space-first or frequency-first.Source: §IV · p. 8

A. Method 1: z-integral first

The XS method recasts the inter-span phase in signed dispersion coordinates and evaluates contributions through correlations of dispersion-domain NSPs, retaining the generally asymmetric structure of cross-span correlations.

A. Method 1: z-integral first

  • Signed dispersion coordinates restore a difference-coordinate representation for the inter-span phase, which is not generally expressible using physical spatial separation; zero-dispersion cases are handled separately.The procedure applies the correlation-based approach used for SS contributions and assumes nonzero dispersion for this coordinate transformation.Source: A. Method 1: z-integral first · p. 8
  • The ordered cross-correlation of dispersion-domain NSPs is even for m = n but generally asymmetric for m ≠ n, reflecting different profile shapes and supports.It quantifies profile overlap versus dispersion-domain lag; exchanging the NSP order reverses the lag.Source: A. Method 1: z-integral first · pp. 8–9
  • The real part of the XS integrand is even and its imaginary part is odd, allowing quadrant mapping and the frequency-product transformation to be applied to the real part.Source: A. Method 1: z-integral first · p. 9

1) zero-dispersion cases:

Zero local dispersion prevents direct use of the dispersion-domain cross-correlation construction, but the original space integrals remain separable. The cGN contribution can still be evaluated through a single-integral expression, including when dispersion vanishes across the link.

1) zero-dispersion cases:

  • When either interacting span has zero local dispersion, its dispersion coordinate collapses and the cross-correlation construction cannot be used directly, but the Eq. (43) space integrals remain separable.Source: 1) zero-dispersion cases · p. 9
  • The Eq. (43) space integrals factorize as H_m(ξ)H_n*(ξ), allowing the Eq. (53) single-integral expression to remain applicable using integrated NSP.Source: 1) zero-dispersion cases · p. 9
  • When one local dispersion is zero, the expression becomes H_m(ξ)A_n; with both local dispersions zero, the contribution further reduces.An entirely dispersion-free link has B_m = B_n = 0; the accumulated-dispersion difference can nevertheless be nonzero when other spans are dispersive.Source: 1) zero-dispersion cases · p. 9
  • Each span requires only one independent space integration, after which its coherently accumulated contribution is obtained directly.Source: 1) zero-dispersion cases · p. 9

2) NLI PSD in homogeneous links:

For homogeneous links with identical spans, NLI contributions depend on span separation. The resulting expression combines self-span (SS) and cross-span (XS) contributions in one integral, with fully coherent accumulation when B2 = 0.

NLI PSD in homogeneous links

  • In a link of Ns identical spans, span-pair contributions depend only on their separation r, and there are Ns − r pairs at each separation.The r = 0 case is the SS contribution; r ≥ 1 covers XS contributions between distinct spans.Source: 2) NLI PSD in homogeneous links · pp. 9–10
  • The total NLI PSD combines SS and XS contributions into a single integral, with B2 = 0 corresponding to fully coherent NLI accumulation across all spans.The passage states that this limit applies throughout the integration domain when B2 = 0.Source: 2) NLI PSD in homogeneous links · p. 10

B. Method 2: f-integral first

Method 2 evaluates the rectangular-channel frequency integral before the two space integrals. Its analytic expression accounts for accumulated-dispersion differences between spans and remains valid when either local dispersion parameter is zero.

V. VALIDATION

The cGN framework is validated against full numerical integration of the GN-model, Eq. (1), across three systems of increasing complexity. The evaluation covers single-span cases and full 10-span links.

V. VALIDATION

  • Full numerical integration of the GN-model, Eq. (1), serves as the benchmark for evaluating the framework’s speed and accuracy.Validation proceeds through three systems of increasing complexity.Source: §V · p. 10
  • Config 1 is a single 140-GBaud channel with attenuation-only SPPs propagated over 10 identical 100-km spans.The study first examines a single span of this configuration.Source: §V · p. 10
  • Config 2 has five channels occupying approximately 500 GHz, with subcarriers in channels 2 and 4 and SPPs affected by attenuation, ISRS, and backward Raman amplification.The link has 10 identical 100-km spans; the study first examines a single span.Source: §V · p. 10
  • Config 3 is a 150-channel C+L+S system spanning 10 spans with span-dependent fiber parameters and per-channel SPPs accounting for ISRS and three backward Raman pumps.Its launch powers and pump powers were jointly optimized for throughput, producing a strongly nonuniform launch-power profile; the full 10-span link is examined.Source: §V · p. 10

A. Config 1 and Config 2, single-span · B. Multi-span validation · 1) Homogeneous links:

The cGN framework closely reproduces frequency-resolved single-span NLI and coherently accumulated multi-span GN-model predictions, including low-dispersion cases. In homogeneous links, cross-span contributions drive coherent enhancement, which decreases as dispersion increases.

A. Config 1 and Config 2, single-span

B. Multi-span validation

1) Homogeneous links:

  • The SS contribution dominates Config 1’s ten-span SCI, while the nearest-neighbor XS contribution provides the most significant coherent correction.Other XS contributions generally decrease with span separation, and greater dispersion weakens them through faster phase variation.Source: 1) Homogeneous links · p. 11, Figure 12 · p. 11
  • 2.40 dB and 1.07 dB are the 10-span coherent SCI enhancements over the incoherent result at β2 = −1 and −10 ps2/km, respectively.Including only SS and nearest-neighbor XS terms captures 1.32 dB of the 2.40 dB enhancement at −1 ps2/km and 0.55 dB of the 1.07 dB enhancement at −10 ps2/km.Source: 1) Homogeneous links · p. 12, Figure 13 · p. 12
  • With SS terms alone, accumulated SCI power increases incoherently; progressively adding XS terms brings the cGN prediction into agreement with the coherent GN model.The analysis progressively includes span separations r = 1 to 9.Source: 1) Homogeneous links · p. 12, Figure 13 · p. 12
  • Below 0.065 dB is the maximum discrepancy between cGN and GN-model 10-span NLI PSDs in Config 2, while NLI power differs by at most 0.004 dB.The curves closely overlap across the entire frequency range for all three dispersion values, including the low-dispersion regime.Source: Figure 11 · p. 11, 1) Homogeneous links · p. 12

2) Heterogeneous Links: · C. Numerical convergence and computational efficiency · VI. CONCLUSION

The cGN framework evaluates NLI in general multi-span optical systems by decomposing GN-model contributions into spatial auto- and cross-correlation terms. The supplied results show benchmark agreement for heterogeneous links, while partition refinement controls accuracy and computational cost.

2) Heterogeneous Links:

  • Mean and maximum absolute differences were 0.062 and 0.098 dB at −21 ps2/km, versus 0.025 and 0.071 dB at −10 ps2/km; lower dispersion increased NLI across channels.Launch powers and all other link parameters were unchanged between the two dispersion settings.Source: Figure 15 · p. 12, 2) Heterogeneous Links · pp. 12–13
  • MCI rose from approximately 0.17% to 1.10%; at −10 ps2/km it stayed below 0.16% in L and C, reached approximately 3.30% at the S-band edge, and omission underestimated NLI by at most 0.146 dB.SCI and XCI dominated in both dispersion cases.Source: 2) Heterogeneous Links · p. 13

C. Numerical convergence and computational efficiency

  • As Nrect increased, MAE decreased and runtime generally increased; Config 3 converged more slowly than Configs 1 and 2, but partition refinement systematically controlled accuracy.The comparison kept kernel resolution and output-frequency quadrature fixed and measured MAE against the GN-model benchmark.Source: Figure 16 · p. 13, C. Numerical convergence and… · p. 13

VI. CONCLUSION

  • In the heterogeneous 150-channel C+L+S system, cGN’s maximum and mean absolute differences were 0.098 and 0.062 dB, and its 1.9 s runtime yielded a speedup above 2000× over 4689.2 s.The conclusion reports these results for refined numerical GN integration under the same computational conditions.Source: Table I · p. 13, C. Numerical convergence and… · p. 13, §VI · p. 14
  • The framework supports span-dependent fiber parameters, arbitrary channel arrangements and symbol rates, multi-subcarrier signals, and spatial power profiles shaped by attenuation, ISRS, Raman amplification, or lumped gain or loss.Rectangular contributions decompose into SS terms governed by spatial autocorrelations or energy spectra, and XS terms governed by cross-correlations or cross-spectra between spans.Source: §VI · p. 13

C. Numerical convergence and computational efficiency

  • Two equivalent semi-analytical methods reduce each rectangular contribution to a one-dimensional integral; the production method used reusable spatial Fourier-transform tables, and both methods remain defined at zero and near-zero dispersion.The alternative method supplies an independent numerical cross-check and a complementary correlation-domain interpretation.Source: §VI · p. 13, C. Numerical convergence and… · p. 13
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