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A Closed-Form Approximation of the Gaussian Noise Model in the Presence of Inter-Channel Stimulated Raman Scattering

Daniel Semrau, Robert I. Killey, Polina Bayvel

arXiv:1808.07940v2eess.SP

TL;DR

Ultra-wideband systems require efficient NLI estimation because ISRS limits existing formulas and numerical ISRS GN evaluation is computationally costly. The paper derives a closed-form ISRS GN approximation, applicable up to 15 THz and incorporating dispersion slope and variable loading. Validation reports close agreement, including an average 0.2 dB NLI-power deviation for SMF spans across the C+L band.

  • Problem

    Existing analytical formulas do not apply beyond the 5 THz C-band with significant ISRS, while the ISRS GN model requires at least three-dimensional numerical integration.

  • Method

    The paper derives a closed-form approximation of the ISRS GN model for dispersion-unmanaged ultra-wideband systems, including dispersion slope and variably loaded fiber spans.

  • Results

    0.2 dB average deviation in nonlinear interference power was reported for SMF-based spans operating across the entire C+L band.

  • Takeaways & Limitations

    The result enables rapid evaluation of SNR, maximum reach, and optimum launch power for ultra-wideband transmission and optical-network optimization.

Abstract

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An accurate, closed-form expression evaluating the nonlinear interference (NLI) power in coherent optical transmission systems in the presence of inter-channel stimulated Raman scattering (ISRS) is derived. The analytical result enables a rapid estimate of the signal-to-noise ratio (SNR) and avoids the need for integral evaluations and split-step simulations. The formula also provides new insight into the underlying parameter dependence of ISRS on the NLI. The proposed result is applicable for dispersion unmanaged, ultra-wideband transmission systems that use optical bandwidths of up to 15 THz. The accuracy of the closed-form expression is compared to numerical integration of the ISRS Gaussian Noise model and split-step simulations in a point-to-point transmission, as well as in a mesh optical network scenario.

I. INTRODUCTION

The paper motivates closed-form NLI models for rapid optical-system and network estimation, addressing the lack of applicability and efficiency of existing approaches for ultra-wideband systems with ISRS.

  • Motivation: Analytical NLI models support rapid system design, achievable-rate estimation, and physical-layer-aware optical-network optimization.These applications support optical-network abstraction, virtualization, and capacity maximization.
  • Motivation: Numerical GN-model integration can require a few minutes per WDM channel and several hours for 200-channel ultra-wideband signals.Such runtimes are unsuitable for some performance-estimation applications.
  • Motivation: Closed-form approximations can reduce network-state performance estimation to a few microseconds while providing parameter-dependence and scaling-rule insight.The motivating applications include physical-layer-aware network optimization and network performance estimation.
  • Research gap: Existing formulas are not applicable beyond the 5 THz C-band because ISRS significantly changes the NLI distribution across the received spectrum.ISRS amplifies low-frequency components at the expense of high-frequency components within the same optical signal.
  • Research gap: The ISRS GN model rigorously accounts for ISRS but requires numerical integration of at least three dimensions, and no closed-form approximation had been reported.Comparable ISRS-aware models had also been published.
  • Contribution: This paper presents a closed-form ISRS GN approximation for dispersion-unmanaged ultra-wideband systems up to 15 THz and validates it extensively with split-step simulations.The formula also accounts for dispersion slope and variably loaded spans for mesh-network estimation.

II. THE ISRS GN MODEL IN CLOSED-FORM

This section presents the proposed closed-form approximation of the ISRS GN model and discusses its derivation and key assumptions.

  • II. THE ISRS GN MODEL IN CLOSED-FORM: The proposed closed-form approximation of the ISRS GN model is presented together with its main derivation steps and key assumptions.

A. The ISRS GN model

The ISRS GN model expresses SNR and NLI through channel launch power, ASE noise, frequency-dependent nonlinear interference, and ISRS-dependent power profiles.

  • Model formulation: After coherent detection and electronic dispersion compensation, the COI SNR is calculated while neglecting transceiver noise.
  • Model formulation: Pi denotes the launch power of channel i, while PASE denotes accumulated amplifier-generated ASE noise.
  • Model formulation: Without significant ISRS, total NLI follows PNLI = ηnP_i^3 because the NLI coefficient is independent of absolute launch power in first-order perturbation.
  • ISRS dependence: Beyond C-band, the NLI coefficient depends on total launch power and its absolute spectral distribution because ISRS changes effective channel attenuation.
  • ISRS dependence: Equation (2) is valid for optical bandwidths up to 15 THz because it assumes a linear Raman-gain spectrum.The model includes attenuation, effective length, Raman-gain slope, phase mismatch, GVD, and dispersion slope.
  • ISRS dependence: ISRS strength is assessed through net power transfer between the outer WDM channels, with transfer depending on total power, Raman-gain slope, effective length, and total bandwidth.The net transfer combines the lowest-frequency channel's gain and highest-frequency channel's loss.

B. The XPM assumption

The closed-form derivation decomposes NLI into SPM and XPM under stated accumulation, ISRS, and spectral-distribution assumptions, then evaluates interferer contributions analytically.

  • NLI decomposition: The XPM assumption sums each individual interferer's contribution while neglecting jointly generated two-interferer FWM/MCI terms.Those terms are typically small in highly dispersive links with high baud rates or channel spacings.
  • Coherent accumulation: Only SPM is assumed to accumulate coherently, XPM incoherently, and the coherence factor is redefined over the COI bandwidth.This makes coherent accumulation independent of the transmitted spectrum.
  • Coherent accumulation: Neglecting ISRS-induced coherence-factor changes yields estimated NLI errors of 0.1 dB after 10 spans and 0.2 dB after 50 spans.The neglected change was reported for 6.5 dB ISRS power transfer over 10 THz in SMF spans.
  • Single-interferer formulation: The derivation evaluates NLI from a single interferer on a COI using rectangular channel spectra, launch powers, bandwidths, and center frequencies.The COI–interferer frequency separation determines their relative placement.
  • NLI decomposition: Only two of six cross terms contribute to COI NLI: self-phase modulation (SPM) and cross-phase modulation (XPM).
  • ISRS treatment: Total-power and total-bandwidth quantities refer to the entire transmitted WDM signal, not only the COI–interferer pair.
  • ISRS treatment: ISRS changes the in-span signal-power profile as a function of total launch power and spectral distribution, motivating the approximation's power-profile assumptions.

C. The ISRS GN model in closed-form

The paper derives separate closed-form SPM and XPM approximations for the ISRS GN model, combining them into total NLI under three stated assumptions. Numerical comparisons show sub-0.1 dB accuracy without ISRS and small discrepancies under moderate ISRS.

  • Closed-form derivation: The total NLI is formed by separately solving closed-form SPM and XPM contributions and combining them with Eq. (5).The XPM expression sums contributions from individual interferers.
  • Key assumptions: The approximation assumes interferer separation much greater than half the channel bandwidth, weak ISRS, and signal attenuation dependent only on total launch power.The third assumption introduces no approximation error for uniform launch-power distributions.
  • Accuracy: < 0.1 dB on the NLI is achieved without ISRS, neglecting FWM, while ISRS causes a slightly higher discrepancy.The total-NLI accuracy is dominated by the XPM approximation.
  • Accuracy: 6.3 dB ISRS power transfer produces 2.3 dB net ISRS gain for the COI at 0 dBm launch power in a fully occupied 10 THz WDM signal.Under this condition, the SPM discrepancy is 0.3 dB and the discrepancy at 40 GHz separation is 0.1 dB.
  • Validity range: The weak-ISRS condition is not fully satisfied at 6.3 dB power transfer, so its assumption has a small impact on closed-form accuracy.The stated criterion gives 6.3 ≪ 26, which is not fully satisfied.
  • Launch-power dependence: Non-uniform launch-power distributions alter the ISRS signal power profile, whereas uniform distributions satisfy the launch-distribution assumption.The paper identifies mesh-network loading and sloped launch powers as relevant non-uniform cases.

III. NUMERICAL VALIDATION

The closed-form approximation is validated against SSFM and integral-form ISRS GN calculations in single- and multi-span settings. It closely matches the integral model, with accuracy decreasing as ISRS increases and with a larger gap for 64-QAM.

  • Validation setup: The validation covers point-to-point transmission using SSFM, numerical ISRS GN integration, and the proposed closed-form approximation.SSFM incorporates ISRS through frequency-dependent loss applied at every linear step.
  • Validation setup: Launch powers up to 3 dBm/ch. were tested, corresponding to ISRS power transfers up to 13 dB, to assess the weak-ISRS assumption.The simulations used logarithmic step-size distributions.
  • Single-span results: The integral-form ISRS GN model matches simulations with negligible error except at the outermost channels, while the closed form remains in good agreement.The outer-channel discrepancy is attributed to the local white-noise assumption.
  • Single-span results: 0.1 dB is the average closed-form gap without ISRS, increasing to 0.1 dB at 0 dBm/ch. and 0.2 dB at 2 dBm/ch. launch power.The increasing discrepancy with launch power is attributed to the weak-ISRS assumption.
  • Single-span results: The closed-form accuracy decreases with increasing ISRS power transfer because higher-order Taylor terms become significant.This behavior is shown through NLI-coefficient deviations across WDM channels.

A. A point-to-point transmission scenario

The six-span point-to-point study evaluates the closed-form approximation with and without ISRS, including coherent and incoherent NLI accumulation and 64-QAM comparisons.

  • Six identical 100 km SMF spans were evaluated at 0 dBm/ch., the optimum launch power for the central channel with 5 dB-noise-figure EDFAs.
  • The coherent closed-form approximation differed from the integral ISRS GN model by 0.1 dB without ISRS and 0.2 dB with ISRS.
  • Assuming incoherent accumulation caused an average accuracy loss of 0.2 dB in the studied system, with loss increasing with span count.
  • The Gaussian-based closed form exceeded SSFM results using 64-QAM by 1.6 dB in both the no-ISRS and ISRS cases.
  • A modulation-format heuristic reduced the average deviation from 1.6 dB to 0.8 dB for the studied 64-QAM comparison.
  • The approximation models ISRS effects accurately in fully occupied point-to-point scenarios and supports system design, optimization, and real-time performance estimation.

B. A mesh optical network scenario

The mesh-network study tests the approximation under variably loaded lightpaths where channels are added and dropped at ROADMs. Agreement with simulations remains close at 80% and 90% utilization.

  • Mesh transmission differs from point-to-point operation because ROADMs add and drop interfering channels, changing the NLI contributions along network edges.
  • Figure 7 compares every fifth channel after six spans using numerical simulations and the proposed closed-form approximation.
  • The ISRS-induced NLI change ranged from −1.6 dB to 1.5 dB at 80% utilization and from −1.8 dB to 1.6 dB at 90%.
  • Average discrepancies between the closed form and simulations were 0.1 dB at 80% utilization and 0.2 dB at 90%, while the 64-QAM gap was 1 dB.
  • The approximation enables performance evaluation of complicated lightpath configurations across an entire network topology within a few microseconds.

IV. ACCURACY IMPACT OF SLOPED LAUNCH POWER

This section examines the approximation’s accuracy under sloped launch-power distributions, extending the assessment beyond uniform loading while retaining the arbitrary-distribution extension as future work.

  • The derivation assumes effective channel attenuation depends on total launch power rather than its spectral distribution.
  • Sloped launch-power distributions may increase achievable information throughput or improve SNR margin.
  • A launch-power slope of ±2 dB introduces approximately ∓0.2 dB approximation error in predicted NLI.
  • Positive slopes increase measured NLI and reduce the closed-form-to-square-QAM gap through cancellation of approximation errors; negative slopes have the opposite effect.
  • Fully accounting for arbitrary launch-power distributions is left for future research.

V. CONCLUSION

The paper concludes that its closed-form ISRS GN approximation is accurate across point-to-point and mesh scenarios, with small average NLI-power deviations and rapid performance evaluation.

  • The proposed closed-form approximation was verified against split-step simulations and integral-form ISRS GN numerical integrations.
  • The average deviation was 0.2 dB in NLI power for SMF spans operating across the C+L band.
  • The discrepancy primarily reflects neglected jointly generated FWM/MCI contributions and the first-order ISRS description, with little impact near optimum launch power in dispersion-unmanaged links.
  • The results support rapid evaluation of SNR, maximum reach, and optimum launch power for ultra-wideband transmission and network optimization.

APPENDIX A DERIVATION OF THE XPM CONTRIBUTION

The appendix derives closed-form XPM and SPM contributions by analytically approximating the integral NLI expression under channel-separation, weak-ISRS, and bandwidth-locality assumptions.

  • XPM contribution: The XPM derivation analytically approximates the integral describing interference on channel i from one interfering channel k, then sums individual contributions.The resulting total XPM contribution is expressed in the form of the paper’s closed-form equation (11).
  • XPM contribution: For XPM, the channel separation satisfies |∆f| ≫ Bk/2, allowing f2 + ∆f ≈ ∆f with small expected impact on total NLI.The approximation has small impact near the channel of interest and is expected to produce small total-NLI error because most ultra-wideband interferers satisfy the separation condition.
  • XPM contribution: The XPM phase mismatch uses φi,k = −4π^2(fk − fi)[β2 + πβ3(fi + fk)]ζ, with dispersion-slope impact treated as constant over Bi.The nonlinear-interference coefficient includes the suppressed prefactor 32/27 γ^2 before individual contributions are summed.
  • XPM contribution: The XPM ISRS term is expanded to first order under weak ISRS, while the signal power profile is treated as constant over the channel bandwidth Bi.The derivation also assumes e^-αL ≪ 1 and uses exact integral identities after simplifying the first-order expression.
  • SPM contribution: The SPM contribution is derived by analytically approximating the same integral expression for interference caused by channel i on itself.The derivation includes the SPM-specific factor 1/2, a suppressed prefactor, and a first-order ISRS expansion.
  • SPM contribution: The SPM approximation assumes power transfer is constant over Bi, negligible dispersion-slope variation over Bi, and 2α^2 ≫ φ1^2 f2^2.These assumptions enable the analytical approximation used to obtain the closed-form SPM coefficient.

APPENDIX C DERIVATION OF THE VALIDITY RANGE

The appendix derives a validity range for the weak-ISRS approximation by comparing first- and second-order Taylor terms and relating the resulting condition to span-end power transfer.

  • Weak-ISRS criterion: The weak-ISRS approximation is considered valid when the second-order Taylor term is negligible relative to the first-order term.The second Taylor coefficient is evaluated at the frequency component most affected by ISRS.
  • Weak-ISRS criterion: Requiring the second-order term to be negligible relative to the first-order approximation yields the validity condition.The condition is obtained directly from the ratio of the Taylor-series contributions.
  • Weak-ISRS criterion: The most ISRS-impacted channel is taken at center frequency fk = B/2 when evaluating the validity condition.This frequency is used as the representative worst-case point in the derivation.
  • Power-transfer interpretation: The derived condition is related to the end-of-span ISRS power transfer ∆ρ(L) [dB] using the power-transfer expression.Although ∆ρ(L) is reported in decibels, the relation compares numerical values linearly rather than comparing dB quantities directly.
  • Integral identities: The appendix also collects integral identities used in deriving the proposed closed-form expression and notes alternative approximations for one integral.A comparable asymptotic expansion based on the natural logarithm is cited as related work.
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